From c64dd138a06ca6b14bdd68ea51a5c9cd2e6274cb Mon Sep 17 00:00:00 2001 From: Christopher Koerber Date: Thu, 11 Feb 2021 10:47:20 +0100 Subject: [PATCH 1/4] Incorporated Jim's changes into main folder --- paper/main.bib | 979 +++++++++++++++++++++++++++++++++++++++++++++++++ paper/main.tex | 258 ++++++------- 2 files changed, 1110 insertions(+), 127 deletions(-) create mode 100644 paper/main.bib diff --git a/paper/main.bib b/paper/main.bib new file mode 100644 index 0000000..93aaf33 --- /dev/null +++ b/paper/main.bib @@ -0,0 +1,979 @@ +@ARTICLE{2020arXiv200713788H, + author = {{Hauke}, Philipp and {Mattiotti}, Giovanni and {Faccioli}, Pietro}, + title = "{Dominant Reaction Pathways by Quantum Computing}", + journal = {arXiv e-prints}, + keywords = {Quantum Physics, Condensed Matter - Disordered Systems and Neural Networks, Physics - Biological Physics}, + year = 2020, + month = jul, + eid = {arXiv:2007.13788}, + pages = {arXiv:2007.13788}, +archivePrefix = {arXiv}, + eprint = {2007.13788}, + primaryClass = {quant-ph}, + adsurl = {https://ui.adsabs.harvard.edu/abs/2020arXiv200713788H}, + adsnote = {Provided by the SAO/NASA Astrophysics Data System} +} + +@ARTICLE{2020RPPh...83e4401H, + author = {{Hauke}, Philipp and {Katzgraber}, Helmut G. and {Lechner}, Wolfgang and + {Nishimori}, Hidetoshi and {Oliver}, William D.}, + title = "{Perspectives of quantum annealing: methods and implementations}", + journal = {Reports on Progress in Physics}, + keywords = {review, quantum annealing, adiabatic quantum optimization, Quantum Physics}, + year = 2020, + month = may, + volume = {83}, + number = {5}, + eid = {054401}, + pages = {054401}, + doi = {10.1088/1361-6633/ab85b8}, +archivePrefix = {arXiv}, + eprint = {1903.06559}, + primaryClass = {quant-ph}, + adsurl = {https://ui.adsabs.harvard.edu/abs/2020RPPh...83e4401H}, + adsnote = {Provided by the SAO/NASA Astrophysics Data System} +} + +% tabu search +@article{GLOVER1986533, +title = {Future paths for integer programming and links to artificial intelligence}, +journal = {Computers \& Operations Research}, +volume = {13}, +number = {5}, +pages = {533 - 549}, +year = {1986}, +note = {Applications of Integer Programming}, +issn = {0305-0548}, +doi = {https://doi.org/10.1016/0305-0548(86)90048-1}, +url = {http://www.sciencedirect.com/science/article/pii/0305054886900481}, +author = {Glover, Fred}, +} + + +@article{doi:10.1287/ijoc.1.3.190, +author = {Glover, Fred}, +title = {Tabu Search—Part I}, +journal = {ORSA Journal on Computing}, +volume = {1}, +number = {3}, +pages = {190-206}, +year = {1989}, +doi = {10.1287/ijoc.1.3.190}, +URL = {https://doi.org/10.1287/ijoc.1.3.190}, +eprint = {https://doi.org/10.1287/ijoc.1.3.190} +} + +@article{doi:10.1287/ijoc.2.1.4, +author = {Glover, Fred}, +title = {Tabu Search—Part II}, +journal = {ORSA Journal on Computing}, +volume = {2}, +number = {1}, +pages = {4-32}, +year = {1990}, +doi = {10.1287/ijoc.2.1.4}, +URL = {https://doi.org/10.1287/ijoc.2.1.4}, +eprint = {https://doi.org/10.1287/ijoc.2.1.4} +} + +%floppy qubits +@ARTICLE{2016PhRvA..93e2320K, + author = {{King}, Andrew D. and {Hoskinson}, Emile and {Lanting}, Trevor and + {Andriyash}, Evgeny and {Amin}, Mohammad H.}, + title = "{Degeneracy, degree, and heavy tails in quantum annealing}", + journal = {\pra}, + keywords = {Quantum Physics}, + year = 2016, + month = may, + volume = {93}, + number = {5}, + eid = {052320}, + pages = {052320}, + doi = {10.1103/PhysRevA.93.052320}, +archivePrefix = {arXiv}, + eprint = {1512.07325}, + primaryClass = {quant-ph}, + adsurl = {https://ui.adsabs.harvard.edu/abs/2016PhRvA..93e2320K}, + adsnote = {Provided by the SAO/NASA Astrophysics Data System} +} + +@ARTICLE{2017PhRvL.118g0502M, + author = {{Mandr{\`a}}, Salvatore and {Zhu}, Zheng and {Katzgraber}, Helmut G.}, + title = "{Exponentially Biased Ground-State Sampling of Quantum Annealing Machines with Transverse-Field Driving Hamiltonians}", + journal = {\prl}, + keywords = {Quantum Physics, Condensed Matter - Disordered Systems and Neural Networks, Physics - Computational Physics}, + year = 2017, + month = feb, + volume = {118}, + number = {7}, + eid = {070502}, + pages = {070502}, + doi = {10.1103/PhysRevLett.118.070502}, +archivePrefix = {arXiv}, + eprint = {1606.07146}, + primaryClass = {quant-ph}, + adsurl = {https://ui.adsabs.harvard.edu/abs/2017PhRvL.118g0502M}, + adsnote = {Provided by the SAO/NASA Astrophysics Data System} +} + + +%% +@book{Goldenfeld:1992qy, + author = "Goldenfeld, N.", + title = "{Lectures on phase transitions and the renormalization group}", + year = "1992" +} + +@ARTICLE{2015PhRvB..91h1103L, + author = {{Luitz}, David J. and {Laflorencie}, Nicolas and {Alet}, Fabien}, + title = "{Many-body localization edge in the random-field Heisenberg chain}", + journal = {\prb}, + keywords = {75.10.Pq, 05.30.Rt, 72.15.Rn, Spin chain models, Localization effects, Condensed Matter - Disordered Systems and Neural Networks, Condensed Matter - Strongly Correlated Electrons}, + year = 2015, + month = feb, + volume = {91}, + number = {8}, + eid = {081103}, + pages = {081103}, + doi = {10.1103/PhysRevB.91.081103}, +archivePrefix = {arXiv}, + eprint = {1411.0660}, + primaryClass = {cond-mat.dis-nn}, + adsurl = {https://ui.adsabs.harvard.edu/abs/2015PhRvB..91h1103L}, + adsnote = {Provided by the SAO/NASA Astrophysics Data System} +} + +# flux qubit coherence time +@ARTICLE{2003Sci...299.1869C, + author = {{Chiorescu}, I. and {Nakamura}, Y. and {Harmans}, C.~J.~P.~M. and + {Mooij}, J.~E.}, + title = "{Coherent Quantum Dynamics of a Superconducting Flux Qubit}", + journal = {Science}, + keywords = {PHYSICS, Condensed Matter - Mesoscale and Nanoscale Physics, Condensed Matter - Superconductivity}, + year = 2003, + month = mar, + volume = {299}, + number = {5614}, + pages = {1869-1872}, + doi = {10.1126/science.1081045}, +archivePrefix = {arXiv}, + eprint = {cond-mat/0305461}, + primaryClass = {cond-mat.mes-hall}, + adsurl = {https://ui.adsabs.harvard.edu/abs/2003Sci...299.1869C}, + adsnote = {Provided by the SAO/NASA Astrophysics Data System} +} + +@misc{dwave_temp, +author = {D-Wave Systems}, +title = {The D-Wave 2000Q Quantum Computer Technology Overview}, +url = {https://www.dwavesys.com/sites/default/files/D-Wave%202000Q%20Tech%20Collateral_0117F_0.pdf}, +year = 2017, +} + +@misc{dwave_offset, +author = {D-Wave Systems}, +title = {Boosting integer factoring performance via quantum annealing offsets}, +url = {https://www.dwavesys.com/sites/default/files/14-1002A_B_tr_Boosting_integer_factorization_via_quantum_annealing_offsets.pdf}, +year = 2016, +} + +@misc{dwave_as, + author = {D-Wave Systems}, + title = {QPU-Specific Anneal Schedules}, + url = {https://support.dwavesys.com/hc/en-us/articles/360005267253-QPU-Specific-Anneal-Schedules}, + year = 2020, +} + +@misc{dwave_as_docu, + author = {D-Wave Systems}, + title = {Technical Description of the D-Wave Quantum Processing Unit}, + url = {https://docs.dwavesys.com/docs/latest/doc_qpu.html}, + month = August, + year = 2020, +} + +@misc{dwave_oceans, + author = {D-Wave Systems}, + title = {D-Wave Cloud Client}, + month = August, + year = 2020, + version = {2.5.0}, + url = {https://github.com/dwavesystems/dwave-ocean-sdk} + } + +@article{Chang:2018uoc, + author = "Chang, Chia Cheng and Gambhir, Arjun and Humble, Travis + S. and Sota, Shigetoshi", + title = "{Quantum annealing for systems of polynomial equations}", + journal = "Sci. Rep.", + volume = "9", + year = "2019", + number = "1", + pages = "10258", + doi = "10.1038/s41598-019-46729-0", + eprint = "1812.06917", + archivePrefix = "arXiv", + primaryClass = "quant-ph", + reportNumber = "RIKEN-iTHEMS-Report-19, RIKEN-iTHEMS-Report-18", + SLACcitation = "%%CITATION = ARXIV:1812.06917;%%" +} + +% Github code + + + +@article{Jiang2018, + title={Quantum Annealing for Prime Factorization}, + author={Shuxian Jiang and Keith A. Britt and Alexander J. McCaskey and Travis S. Humble and Sabre Kais }, + journal={Scientific Reports}, + volume={8}, + pages={17667}, + year={2086} +} + + +@article{denchev2016computational, + title={What is the computational value of finite-range tunneling?}, + author={Denchev, Vasil S and Boixo, Sergio and Isakov, Sergei V and Ding, Nan and Babbush, Ryan and Smelyanskiy, Vadim and Martinis, John and Neven, Hartmut}, + journal={Physical Review X}, + volume={6}, + number={3}, + pages={031015}, + year={2016}, + publisher={APS} +} + + +@article{katzgraber2014glassy, + title={Glassy chimeras could be blind to quantum speedup: Designing better benchmarks for quantum annealing machines}, + author={Katzgraber, Helmut G and Hamze, Firas and Andrist, Ruben S}, + journal={Physical Review X}, + volume={4}, + number={2}, + pages={021008}, + year={2014}, + publisher={APS} +} + + +@article{PhysRevA.94.022337, + title = {Strengths and weaknesses of weak-strong cluster problems: A detailed overview of state-of-the-art classical heuristics versus quantum approaches}, + author = {Mandr\`a, Salvatore and Zhu, Zheng and Wang, Wenlong and Perdomo-Ortiz, Alejandro and Katzgraber, Helmut G.}, + journal = {Phys. Rev. A}, + volume = {94}, + issue = {2}, + pages = {022337}, + numpages = {13}, + year = {2016}, + month = {Aug}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevA.94.022337}, + url = {https://link.aps.org/doi/10.1103/PhysRevA.94.022337} +} + + +@article{ronnow2014defining, + title={Defining and detecting quantum speedup}, + author={R{\o}nnow, Troels F and Wang, Zhihui and Job, Joshua and Boixo, Sergio and Isakov, Sergei V and Wecker, David and Martinis, John M and Lidar, Daniel A and Troyer, Matthias}, + journal={Science}, + volume={345}, + number={6195}, + pages={420--424}, + year={2014}, + publisher={American Association for the Advancement of Science} +} + + +@article{PhysRevX.8.031016, + title = {Demonstration of a Scaling Advantage for a Quantum Annealer over Simulated Annealing}, + author = {Albash, Tameem and Lidar, Daniel A.}, + journal = {Phys. Rev. X}, + volume = {8}, + issue = {3}, + pages = {031016}, + numpages = {26}, + year = {2018}, + month = {Jul}, + doi = {10.1103/PhysRevX.8.031016} +} + + + +@article{mandra2018, + author={Salvatore Mandrà and Helmut G Katzgraber}, + title={A deceptive step towards quantum speedup detection}, + journal={Quantum Science and Technology}, + volume={3}, + number={4}, + pages={04LT01}, + url={http://stacks.iop.org/2058-9565/3/i=4/a=04LT01}, + year={2018}, + abstract={There have been multiple attempts to design synthetic benchmark problems with the goal of detecting quantum speedup in current quantum annealing (QA) machines. To date, classical heuristics have consistently outperformed quantum annealing based approaches. Here we introduce a class of problems based on frustrated cluster loops—deceptive cluster loops—for which all currently known state of-the-art classical heuristics are outperformed by the DW2000Q QA machine. While there is a sizable constant speedup over all known classical heuristics, a noticeable improvement in the scaling remains elusive. These results represent the first steps towards a detection of potential quantum speedup, albeit without a scaling improvement and for synthetic benchmark problems.} +} + + +% Minor embedding +@ARTICLE{2008arXiv0804.4884C, + author = {{Choi}, V.}, + title = "{Minor-Embedding in Adiabatic Quantum Computation: I. The Parameter Setting Problem}", + journal = {ArXiv e-prints}, + archivePrefix = "arXiv", + eprint = {0804.4884}, + primaryClass = "quant-ph", + keywords = {Quantum Physics}, + year = 2008, + month = apr, + adsurl = {http://adsabs.harvard.edu/abs/2008arXiv0804.4884C}, + adsnote = {Provided by the SAO/NASA Astrophysics Data System} +} + + +@article{PhysRevB.39.11828, + title = {Sherrington-Kirkpatrick model in a transverse field: Absence of replica symmetry breaking due to quantum fluctuations}, + author = {Ray, P. and Chakrabarti, B. K. and Chakrabarti, Arunava}, + journal = {Phys. Rev. B}, + volume = {39}, + issue = {16}, + pages = {11828--11832}, + numpages = {0}, + year = {1989}, + month = {Jun}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevB.39.11828}, + url = {https://link.aps.org/doi/10.1103/PhysRevB.39.11828} +} + +@article{RevModPhys.80.1061, + title = {Colloquium: Quantum annealing and analog quantum computation}, + author = {Das, Arnab and Chakrabarti, Bikas K.}, + journal = {Rev. Mod. Phys.}, + volume = {80}, + issue = {3}, + pages = {1061--1081}, + numpages = {0}, + year = {2008}, + month = {Sep}, + publisher = {American Physical Society}, + doi = {10.1103/RevModPhys.80.1061}, + url = {https://link.aps.org/doi/10.1103/RevModPhys.80.1061} +} + + +%Nishimori +@ARTICLE{1998PhRvE..58.5355K, + author = {{Kadowaki}, T. and {Nishimori}, H.}, + title = "{Quantum annealing in the transverse Ising model}", + journal = {Phys. Rev. E}, + eprint = {cond-mat/9804280}, + keywords = {Quantum statistical mechanics, Spin-glass and other random models, Information theory and communication theory}, + year = 1998, + month = nov, + volume = 58, + pages = {5355-5363}, + doi = {10.1103/PhysRevE.58.5355}, + adsurl = {http://adsabs.harvard.edu/abs/1998PhRvE..58.5355K}, + adsnote = {Provided by the SAO/NASA Astrophysics Data System} +} + + +%Farhi +@ARTICLE{2000quant.ph..1106F, + author = {{Farhi}, E. and {Goldstone}, J. and {Gutmann}, S. and {Sipser}, M. + }, + title = "{Quantum Computation by Adiabatic Evolution}", + journal = {eprint arXiv:quant-ph/0001106}, + eprint = {quant-ph/0001106}, + keywords = {Quantum Physics}, + year = 2000, + month = jan, + adsurl = {http://adsabs.harvard.edu/abs/2000quant.ph..1106F}, + adsnote = {Provided by the SAO/NASA Astrophysics Data System} +} + +%chimera +@ARTICLE{2014ITAS...2418294B, + author = {{Bunyk}, P.~I. and {Hoskinson}, E.~M. and {Johnson}, M.~W. and + {Tolkacheva}, E. and {Altomare}, F. and {Berkley}, A.~J. and + {Harris}, R. and {Hilton}, J.~P. and {Lanting}, T. and {Przybysz}, A.~J. and + {Whittaker}, J.}, + title = "{Architectural Considerations in the Design of a Superconducting Quantum Annealing Processor}", + journal = {IEEE Transactions on Applied Superconductivity}, + archivePrefix = "arXiv", + eprint = {1401.5504}, + primaryClass = "quant-ph", + year = 2014, + month = aug, + volume = 24, + eid = {2318294}, + pages = {2318294}, + doi = {10.1109/TASC.2014.2318294}, + adsurl = {http://adsabs.harvard.edu/abs/2014ITAS...2418294B}, + adsnote = {Provided by the SAO/NASA Astrophysics Data System} +} + + +@ARTICLE{2017PhRvP...8f4025M, + author = {{Marshall}, J. and {Rieffel}, E.~G. and {Hen}, I.}, + title = "{Thermalization, Freeze-out, and Noise: Deciphering Experimental Quantum Annealers}", + journal = {Physical Review Applied}, + archivePrefix = "arXiv", + eprint = {1703.03902}, + primaryClass = "quant-ph", + year = 2017, + month = dec, + volume = 8, + number = 6, + eid = {064025}, + pages = {064025}, + doi = {10.1103/PhysRevApplied.8.064025}, + adsurl = {http://adsabs.harvard.edu/abs/2017PhRvP...8f4025M}, + adsnote = {Provided by the SAO/NASA Astrophysics Data System} +} + +@ARTICLE{2016PhRvE..94c2105N, + author = {{Nishimura}, K. and {Nishimori}, H. and {Ochoa}, A.~J. and {Katzgraber}, H.~G. + }, + title = "{Retrieving the ground state of spin glasses using thermal noise: Performance of quantum annealing at finite temperatures}", + journal = {Phys. Rev. E}, + archivePrefix = "arXiv", + eprint = {1605.03303}, + primaryClass = "cond-mat.dis-nn", + year = 2016, + month = sep, + volume = 94, + number = 3, + eid = {032105}, + pages = {032105}, + doi = {10.1103/PhysRevE.94.032105}, + adsurl = {http://adsabs.harvard.edu/abs/2016PhRvE..94c2105N}, + adsnote = {Provided by the SAO/NASA Astrophysics Data System} +} + +@ARTICLE{2016PhRvB..93v4414W, + author = {{Wang}, W. and {Machta}, J. and {Katzgraber}, H.~G.}, + title = "{Bond chaos in spin glasses revealed through thermal boundary conditions}", + journal = {Phys. Rev. B}, + archivePrefix = "arXiv", + eprint = {1603.00543}, + primaryClass = "cond-mat.dis-nn", + year = 2016, + month = jun, + volume = 93, + number = 22, + eid = {224414}, + pages = {224414}, + doi = {10.1103/PhysRevB.93.224414}, + adsurl = {http://adsabs.harvard.edu/abs/2016PhRvB..93v4414W}, + adsnote = {Provided by the SAO/NASA Astrophysics Data System} +} + +@ARTICLE{2015PhRvA..91f2320A, + author = {{Albash}, T. and {Lidar}, D.~A.}, + title = "{Decoherence in adiabatic quantum computation}", + journal = {Phys. Rev. A}, + archivePrefix = "arXiv", + eprint = {1503.08767}, + primaryClass = "quant-ph", + keywords = {Quantum computation, Decoherence, open systems, quantum statistical methods}, + year = 2015, + month = jun, + volume = 91, + number = 6, + eid = {062320}, + pages = {062320}, + doi = {10.1103/PhysRevA.91.062320}, + adsurl = {http://adsabs.harvard.edu/abs/2015PhRvA..91f2320A}, + adsnote = {Provided by the SAO/NASA Astrophysics Data System} +} + + +@ARTICLE{2009PhRvA..79b2107A, + author = {{Amin}, M.~H.~S. and {Averin}, D.~V. and {Nesteroff}, J.~A.}, + title = "{Decoherence in adiabatic quantum computation}", + journal = {Phys. Rev. A}, + archivePrefix = "arXiv", + eprint = {0708.0384}, + primaryClass = "cond-mat.mes-hall", + keywords = {Decoherence, open systems, quantum statistical methods, Quantum computation}, + year = 2009, + month = feb, + volume = 79, + number = 2, + eid = {022107}, + pages = {022107}, + doi = {10.1103/PhysRevA.79.022107}, + adsurl = {http://adsabs.harvard.edu/abs/2009PhRvA..79b2107A}, + adsnote = {Provided by the SAO/NASA Astrophysics Data System} +} + +@article{Dickson13, + title = "Thermally assisted quantum annealing of a 16-qubit problem", + abstract = "Efforts to develop useful quantum computers have been blocked primarily by environmental noise. Quantum annealing is a scheme of quantum computation that is predicted to be more robust against noise, because despite the thermal environment mixing the system's state in the energy basis, the system partially retains coherence in the computational basis, and hence is able to establish well-defined eigenstates. Here we examine the environment's effect on quantum annealing using 16 qubits of a superconducting quantum processor. For a problem instance with an isolated small-gap anticrossing between the lowest two energy levels, we experimentally demonstrate that, even with annealing times eight orders of magnitude longer than the predicted single-qubit decoherence time, the probabilities of performing a successful computation are similar to those expected for a fully coherent system. Moreover, for the problem studied, we show that quantum annealing can take advantage of a thermal environment to achieve a speedup factor of up to 1,000 over a closed system.", + author = "Dickson, {N. G.} and Johnson, {M. W.} and Amin, {M. H.} and R. Harris and F. Altomare and Berkley, {A. J.} and P. Bunyk and J. Cai and Chapple, {E. M.} and P. Chavez and F. Cioata and T. Cirip and P. Debuen and M. Drew-Brook and C. Enderud and S. Gildert and F. Hamze and Hilton, {J. P.} and E. Hoskinson and K. Karimi and E. Ladizinsky and N. Ladizinsky and T. Lanting and T. Mahon and R. Neufeld and T. Oh and I. Perminov and C. Petroff and A. Przybysz and C. Rich and P. Spear and A. Tcaciuc and Thom, {M. C.} and E. Tolkacheva and S. Uchaikin and J. Wang and Wilson, {A. B.} and Z. Merali and G. Rose", + year = "2013", + month = "6", + day = "12", + doi = "10.1038/ncomms2920", + language = "English", + volume = "4", + journal = "Nature Communications", + issn = "2041-1723", + publisher = "Nature Publishing Group", + +} + + + +@ARTICLE{2001Sci...292..472F, + author = {{Farhi}, E. and {Goldstone}, J. and {Gutmann}, S. and {Lapan}, J. and + {Lundgren}, A. and {Preda}, D.}, + title = "{A Quantum Adiabatic Evolution Algorithm Applied to Random Instances of an NP-Complete Problem}", + journal = {Science}, + eprint = {quant-ph/0104129}, + year = 2001, + month = apr, + volume = 292, + pages = {472-476}, + doi = {10.1126/science.1057726}, + adsurl = {http://adsabs.harvard.edu/abs/2001Sci...292..472F}, + adsnote = {Provided by the SAO/NASA Astrophysics Data System} +} + + +@article{aitken_1936, + title={IV.—On Least Squares and Linear Combination of Observations}, + volume={55}, + DOI={10.1017/S0370164600014346}, + journal={Proceedings of the Royal Society of Edinburgh}, + publisher={Royal Society of Edinburgh Scotland Foundation}, + author={Aitken, A. C.}, + year={1936}, + pages={42–48} +} + + +@ARTICLE{2004quant.ph..5098A, + author = {{Aharonov}, D. and {van Dam}, W. and {Kempe}, J. and {Landau}, Z. and + {Lloyd}, S. and {Regev}, O.}, + title = "{Adiabatic Quantum Computation is Equivalent to Standard Quantum Computation}", + journal = {eprint arXiv:quant-ph/0405098}, + eprint = {quant-ph/0405098}, + keywords = {Quantum Physics}, + year = 2004, + month = may, + adsurl = {http://adsabs.harvard.edu/abs/2004quant.ph..5098A}, + adsnote = {Provided by the SAO/NASA Astrophysics Data System} +} + + +@article{citeulike:9077321, + abstract = {{An iterative algorithm is given for solving a system Ax=k of n linear equations in n unknowns. The solution is given in n steps. It is shown that this method is a special case of a very general method which also includes Gaussian elimination. These general algorithms are essentilaly algorithms for finding an n dimensional ellipsoid. Connections are made with the theory of orthogonal polynomials and continued fractions.}}, + added-at = {2017-06-29T07:13:07.000+0200}, + author = {Hestenes, Magnus R. and Stiefel, Eduard}, + biburl = {https://www.bibsonomy.org/bibtex/2ad1ecd0242ef1e7c8a02b3f1ea173d28/gdmcbain}, + citeulike-article-id = {9077321}, + citeulike-attachment-1 = {V49N06A08.pdf; /pdf/user/gdmcbain/article/9077321/632277/V49N06A08.pdf; 466daddfb6340c28cb8da548007028c8cc5df687}, + citeulike-linkout-0 = {http://dx.doi.org/10.6028/jres.049.044}, + citeulike-linkout-1 = {http://www.ams.org/mathscinet-getitem?mr=0060307}, + citeulike-linkout-2 = {http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.129.6828}, + doi = {10.6028/jres.049.044}, + file = {V49N06A08.pdf}, + interhash = {bcb34a6f8b9fb2f2371e92430116a8ad}, + intrahash = {ad1ecd0242ef1e7c8a02b3f1ea173d28}, + journal = {Journal of Research of the National Bureau of Standards}, + keywords = {65f10-iterative-methods-for-linear-systems}, + month = dec, + number = 6, + pages = {409--436}, + posted-at = {2011-03-29 23:49:34}, + priority = {0}, + timestamp = {2017-06-29T07:13:07.000+0200}, + title = {{Methods of Conjugate Gradients for Solving Linear Systems}}, + url = {http://www.ams.org/mathscinet-getitem?mr=0060307}, + volume = 49, + year = 1952 +} + + +@inproceedings{Boros2012, + author = {E. Boros and A. Gruber}, + title = {On quadratization of pseudo-Boolean functions}, + booktitle = {ISAIM}, + year = {2012} +} + +@article{humble2014integrated, + title={An integrated programming and development environment for adiabatic quantum optimization}, + author={Humble, Travis S and McCaskey, Alex J and Bennink, Ryan S and Billings, Jay Jay and D'Azevedo, EF and Sullivan, Blair D and Klymko, Christine F and Seddiqi, Hadayat}, + journal={Computational Science and Discovery}, + volume={7}, + number={1}, + pages={015006}, + year={2014}, + publisher={IOP Publishing} +} + +@article{Klymko2014, + author="Christine F. Klymko and Blair D. Sullivan and Travis S. Humble", + title="Adiabatic quantum programming: minor embedding with hard faults", + year="2014", + journal={Quantum Inf. Process.}, + pages={709-729}, + volume=13 +} + +@article{Choi2008, + author="V. Choi", + title="Minor-embedding in adiabatic quantum computation: I. The parameter setting problem", + journal="Quantum Inf. Process", + volume="7", + year="2008", + pages="193--209" +} + +@article{Choi2011, + author="V. Choi", + title="Minor-embedding in adiabatic quantum computation: II. Minor-universal graph design", + journal="Quantum Inf. 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Fomin and Fabrizio Grandoni and Dieter Kratsch}, + title = {A measure {\&} conquer approach for the analysis of exact algorithms}, + journal = {Journal of the {ACM}} +} + +@incollection{vanRooij2009, + doi = {10.1007/978-3-642-04128-0_50}, + url = {https://doi.org/10.1007/978-3-642-04128-0_50}, + year = {2009}, + publisher = {Springer Berlin Heidelberg}, + pages = {554--565}, + author = {Johan M. M. van Rooij and Jesper Nederlof and Thomas C. van Dijk}, + title = {Inclusion/Exclusion Meets Measure and Conquer}, + booktitle = {Lecture Notes in Computer Science} +} + +@ARTICLE{2018Glover, + author = {{Glover}, Fred and {Kochenberger}, Gary and {Du}, Yu}, + title = "{A Tutorial on Formulating and Using QUBO Models}", + journal = {arXiv e-prints}, + keywords = {Computer Science - Data Structures and Algorithms, Computer Science - Discrete Mathematics, Mathematics - Optimization and Control, Quantum Physics, 90C27}, + year = 2018, + month = nov, + eid = {arXiv:1811.11538}, + pages = {arXiv:1811.11538}, + archivePrefix = {arXiv}, + eprint = {1811.11538}, + primaryClass = {cs.DS}, + adsurl = {https://ui.adsabs.harvard.edu/abs/2018arXiv181111538G}, + adsnote = {Provided by the SAO/NASA Astrophysics Data System} +} + + +@article{PhysRevE.90.022103, + title = {Lindblad-equation approach for the full counting statistics of work and heat in driven quantum systems}, + author = {Silaev, Mihail and Heikkil\"a, Tero T. and Virtanen, Pauli}, + journal = {Phys. Rev. E}, + volume = {90}, + issue = {2}, + pages = {022103}, + numpages = {8}, + year = {2014}, + month = {Aug}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevE.90.022103}, + url = {https://link.aps.org/doi/10.1103/PhysRevE.90.022103} +} + + +@article{PhysRevB.91.081103, + title = {Many-body localization edge in the random-field Heisenberg chain}, + author = {Luitz, David J. and Laflorencie, Nicolas and Alet, Fabien}, + journal = {Phys. Rev. B}, + volume = {91}, + issue = {8}, + pages = {081103}, + numpages = {5}, + year = {2015}, + month = {Feb}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevB.91.081103}, + url = {https://link.aps.org/doi/10.1103/PhysRevB.91.081103} +} + + +@article{RevModPhys.91.021001, + title = {Colloquium: Many-body localization, thermalization, and entanglement}, + author = {Abanin, Dmitry A. and Altman, Ehud and Bloch, Immanuel and Serbyn, Maksym}, + journal = {Rev. Mod. Phys.}, + volume = {91}, + issue = {2}, + pages = {021001}, + numpages = {26}, + year = {2019}, + month = {May}, + publisher = {American Physical Society}, + doi = {10.1103/RevModPhys.91.021001}, + url = {https://link.aps.org/doi/10.1103/RevModPhys.91.021001} +} + + + + + +@article{doi:10.1146/annurev-conmatphys-031214-014726, + author = {Nandkishore, Rahul and Huse, David A.}, + title = {Many-Body Localization and Thermalization in Quantum Statistical Mechanics}, + journal = {Annual Review of Condensed Matter Physics}, + volume = {6}, + number = {1}, + pages = {15-38}, + year = {2015}, + doi = {10.1146/annurev-conmatphys-031214-014726}, + + URL = { + https://doi.org/10.1146/annurev-conmatphys-031214-014726 + + }, + eprint = { + https://doi.org/10.1146/annurev-conmatphys-031214-014726 + + } +} + + + +@article{doi:10.7566/JPSJ.89.044001, + author = {Takada ,Kabuki and Yamashiro ,Yu and Nishimori ,Hidetoshi}, + title = {Mean-Field Solution of the Weak-Strong Cluster Problem for Quantum Annealing with Stoquastic and Non-Stoquastic Catalysts}, + journal = {Journal of the Physical Society of Japan}, + volume = {89}, + number = {4}, + pages = {044001}, + year = {2020}, + doi = {10.7566/JPSJ.89.044001}, + + URL = { + https://doi.org/10.7566/JPSJ.89.044001 + + }, + eprint = { + https://doi.org/10.7566/JPSJ.89.044001 + + } + +} + + +@article{PhysRevA.96.042322, + title = {Experimental demonstration of perturbative anticrossing mitigation using nonuniform driver Hamiltonians}, + author = {Lanting, Trevor and King, Andrew D. and Evert, Bram and Hoskinson, Emile}, + journal = {Phys. Rev. A}, + volume = {96}, + issue = {4}, + pages = {042322}, + numpages = {8}, + year = {2017}, + month = {Oct}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevA.96.042322}, + url = {https://link.aps.org/doi/10.1103/PhysRevA.96.042322} +} + +@misc{hsu2018quantum, + title={Quantum annealing with anneal path control: application to 2-SAT problems with known energy landscapes}, + author={Ting-Jui Hsu and Fengping Jin and Christian Seidel and Florian Neukart and Hans De Raedt and Kristel Michielsen}, + year={2018}, + eprint={1810.00194}, + archivePrefix={arXiv}, + primaryClass={quant-ph} +} + + +@InProceedings{10.1007/978-3-030-14082-3_14, + author="Yarkoni, Sheir + and Wang, Hao + and Plaat, Aske + and B{\"a}ck, Thomas", + editor="Feld, Sebastian + and Linnhoff-Popien, Claudia", + title="Boosting Quantum Annealing Performance Using Evolution Strategies for Annealing Offsets Tuning", + booktitle="Quantum Technology and Optimization Problems", + year="2019", + publisher="Springer International Publishing", + address="Cham", + pages="157--168", + abstract="In this paper we introduce a novel algorithm to iteratively tune annealing offsets for qubits in a D-Wave 2000Q quantum processing unit (QPU). Using a (1+1)-CMA-ES algorithm, we are able to improve the performance of the QPU by up to a factor of 12.4 in probability of obtaining ground states for small problems, and obtain previously inaccessible (i.e., better) solutions for larger problems. We also make efficient use of QPU samples as a resource, using 100 times less resources than existing tuning methods. The success of this approach demonstrates how quantum computing can benefit from classical algorithms, and opens the door to new hybrid methods of computing.", + isbn="978-3-030-14082-3" +} + + +@article{ALET2018498, + title = "Many-body localization: An introduction and selected topics", + journal = "Comptes Rendus Physique", + volume = "19", + number = "6", + pages = "498 - 525", + year = "2018", + note = "Quantum simulation / Simulation quantique", + issn = "1631-0705", + doi = "https://doi.org/10.1016/j.crhy.2018.03.003", + url = "http://www.sciencedirect.com/science/article/pii/S163107051830032X", + author = "Fabien Alet and Nicolas Laflorencie", + keywords = "Many-body localization, Thermalization, Simulations, Entanglement, Localisation à corps, Thermalisation, Simulations, Intrication" +} + + +@article{PhysRevB.82.174411, + title = {Many-body localization phase transition}, + author = {Pal, Arijeet and Huse, David A.}, + journal = {Phys. Rev. B}, + volume = {82}, + issue = {17}, + pages = {174411}, + numpages = {7}, + year = {2010}, + month = {Nov}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevB.82.174411}, + url = {https://link.aps.org/doi/10.1103/PhysRevB.82.174411} +} + + +@article{PhysRevLett.109.017202, + title = {Unbounded Growth of Entanglement in Models of Many-Body Localization}, + author = {Bardarson, Jens H. and Pollmann, Frank and Moore, Joel E.}, + journal = {Phys. Rev. Lett.}, + volume = {109}, + issue = {1}, + pages = {017202}, + numpages = {5}, + year = {2012}, + month = {Jul}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevLett.109.017202}, + url = {https://link.aps.org/doi/10.1103/PhysRevLett.109.017202} +} + + +@book{10.5555/1972505, + author = {Nielsen, Michael A. and Chuang, Isaac L.}, + title = {Quantum Computation and Quantum Information: 10th Anniversary Edition}, + year = {2011}, + isbn = {1107002176}, + publisher = {Cambridge University Press}, + address = {USA}, + edition = {10th} +} + +@article{preskill1998lecture, + title={Lecture notes for physics 229: Quantum information and computation}, + author={Preskill, John}, + journal={California Institute of Technology}, + volume={16}, + year={1998} +} + +@article{RevModPhys.81.1665, + title = {Nonequilibrium fluctuations, fluctuation theorems, and counting statistics in quantum systems}, + author = {Esposito, Massimiliano and Harbola, Upendra and Mukamel, Shaul}, + journal = {Rev. Mod. Phys.}, + volume = {81}, + issue = {4}, + pages = {1665--1702}, + numpages = {0}, + year = {2009}, + month = {Dec}, + publisher = {American Physical Society}, + doi = {10.1103/RevModPhys.81.1665}, + url = {https://link.aps.org/doi/10.1103/RevModPhys.81.1665} +} + + +@misc{github:cchang5/quantum_linear_programming, + author = {Chang, Chia Cheng and Chen, Chih-Chieh and K\"o{}rber, Christopher}, + title = {Quantum Linear Programming}, + year = {2020}, + publisher = {GitHub}, + journal = {GitHub repository}, + howpublished = {\url{https://github.com/cchang5/quantum_linear_programming} \texttt{tag: arXiv}}, + tag = {arXiv}, +} + +@article{Chang:2019khk, + author = {Chang, Chia Cheng and K\"orber, Christopher and Walker-Loud, Andr\'e}, + title = "{EspressoDB: A scientific database for managing high-performance computing workflows}", + eprint = "1912.03580", + archivePrefix = "arXiv", + primaryClass = "hep-lat", + reportNumber = "RIKEN-iTHEMS-Report-19", + doi = "10.21105/joss.02007", + journal = "J. Open Source Softw.", + volume = "5", + number = "46", + pages = "2007", + year = "2020" +} diff --git a/paper/main.tex b/paper/main.tex index 17a2463..443b68e 100644 --- a/paper/main.tex +++ b/paper/main.tex @@ -1,5 +1,5 @@ -%! suppress = LineBreak -\documentclass[prd,twocolumn,tightenlines,preprintnumbers,showpacs,superscriptaddress,notitlepage,nofootinbib,eqsecnum,floatfix,longbibliography,aps,10pt]{revtex4-2} +\documentclass[prd,twocolumn,tightenlines,preprintnumbers,showpacs,superscriptaddress,notitlepage,nofootinbib,eqsecnum,floatfix,longbibliography,aps,10pt]{revtex4-1} +\pdfoutput=1 \usepackage{CJK} \usepackage[utf8]{inputenc} \usepackage[T1]{fontenc} @@ -10,7 +10,7 @@ \usepackage{graphicx} \usepackage{xcolor} \usepackage{enumitem} - +\usepackage{soul} \usepackage{hyperref} \hypersetup{ colorlinks=true, % false: boxed links; true: colored links @@ -27,7 +27,7 @@ \begin{document} -\title{Integer Programming from Quantum Annealing and Open Quantum Systems} +\title{Integer Programming with Quantum Annealing from an Open Quantum System} \author{Chia~Cheng~Chang} \affiliation{RIKEN iTHEMS, Wako, Saitama 351-0198, Japan} @@ -35,12 +35,12 @@ \affiliation{Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA} \author{Chih-Chieh~Chen } \affiliation{R\&D Group, Grid Inc., Tokyo 107-0061, Japan} -\affiliation{Department of Physics, University of California, Berkeley, California 94720, USA} +\affiliation{Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA} \author{Christopher K\"orber} \affiliation{Institut f\"ur Theoretische Physik II, Ruhr-Universit\"at Bochum, D-44780 Bochum, Germany} \affiliation{Department of Physics, University of California, Berkeley, California 94720, USA} \affiliation{Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA} -\author{Travis~Humble} +\author{Travis~S.~Humble} \affiliation{Computational Sciences and Engineering, Oak Ridge National Laboratory, Oak Ridge, Tennessee, 37831, USA} \author{Jim~Ostrowski} \affiliation{Industrial and Systems Engineering, University of Tennessee, Knoxville, Tennessee 37996, USA} @@ -61,8 +61,8 @@ While quantum computing proposes promising solutions to computational problems not accessible with classical approaches, due to current hardware constraints, most quantum algorithms are not yet capable of computing systems of practical relevance, and classical counterparts outperform them. To practically benefit from quantum architecture, one has to identify problems and algorithms with favorable scaling and improve on corresponding limitations depending on available hardware. For this reason, we developed an algorithm that solves integer linear programming problems, a classically NP-hard problem, on a quantum annealer, and investigated problem and hardware-specific limitations. - This work presents the formalism of how to map ILP problems to the annealing architectures, how to systematically improve computations utilizing optimized anneal schedules and models the anneal process through a simulation. - It illustrates the effects of decoherence and Many-Body Localization, for the Minimum Dominating Set problem, and compares annealing results against numerical simulations of the quantum architecture. + This work presents the formalism of how to map ILP problems to the annealing architectures, how to systematically improve computations utilizing optimized anneal schedules, and models the anneal process through a simulation. + It illustrates the effects of decoherence and many body localization for the minimum dominating set problem, and compares annealing results against numerical simulations of the quantum architecture. We find that the algorithm outperforms random guessing but is limited to small problems and that annealing schedules can be adjusted to reduce the effects of decoherence. Simulations qualitatively reproduce algorithmic improvements of the modified annealing schedule, suggesting the improvements have origins from quantum effects. \end{abstract} @@ -82,7 +82,6 @@ \section{INTRODUCTION} \label{sec:introduction} %======================================================================================== - Integer Linear Programming (ILP) is an integer optimization problem subject to inequality constraints \begin{align} \label{eq:initial-ip-def} @@ -96,23 +95,26 @@ \section{INTRODUCTION} \end{align} where $i=1, \cdots, N$ is the number of dependent variables and $a=1, \cdots, M$ the number of constraint equations. -ILP is a commonly tackled problem applicable to situations such as scheduling, network optimization, and graph optimization such as the Minimum Dominating Set Problem (MDS). -In general, ILP is classically NP-complete, and as a result, heuristic methods are employed~\cite{GLOVER1986533, doi:10.1287/ijoc.1.3.190, doi:10.1287/ijoc.2.1.4}. -Standard classical heuristic algorithms employ a greedy scheme which iteratively approximates optimal solutions--starting from a random initial guess, these algorithms apply locally optimal choices at each step. -The NP-hardness of ILP can be understood by realizing that while the solution to an $n$-dimensional linear problem can be obtained in polynomial time, the optimal integer solution in general must be found in the $2^n$ integer solutions which surround the real number solution. -While greedy algorithms do not guarantee optimal global solutions, they find approximate solutions in polynomial time, which can be utilized in further computations. +While ILP is NP-Hard, extremely large instances of ILP are rutinely solved on a daily basis in a variety of industries such as transportation, energy, online dating, network optimization, graph optimization and has recently been explored in the context of quantum annealing~\cite{2020arXiv200713788H}. +The fact that instances with millions of integer variables can be solved is due to the incredible improvement in classical ILP methods. +One estimate has that the speed of ILP algorithms, combined with hardware improvements, have lead to a speedup of around 200 billion\cite{bertsimas2014statistics}. +Despite this fact, there are still very small problems that still require several months, or even years, of computational effort to solve. For example, the Steiner Triple Covering problems can be notoriously difficult; an instance with only 135 binary variables, \texttt{STS135}, required over 120 days of computations~\cite{ostrowski2011solving}. Another example is the football pool problem, where an instance with only 729 binary variables still remains unsolved after years of computational effort~\cite{linderoth2009improving}. +In both these cases, the underlying problem is a covering problem. + +In general, heuristic methods are employed~\cite{GLOVER1986533, doi:10.1287/ijoc.1.3.190, doi:10.1287/ijoc.2.1.4} to solve ILP problems. +Standard classical heuristic algorithms follow a greedy scheme which iteratively approximates optimal solutions--starting from a random initial guess, these algorithms apply locally optimal choices at each step. +While greedy algorithms do not guarantee optimal global solutions, they find feasible solutions in polynomial time, which can be utilized in further computations. -An important ILP application is the Minimum Dominating Set problem, which is representatively considered in this work. -For a given a graph $G(E,V)$, defined by the set of $V$ vertices and $E$ edges, a dominating set $D$ is a specific subset of vertices $D \subseteq V$. +In this paper we will consider instances of the minimum dominating set (MDS) problem. For a given a graph $G(E,V)$, defined by the set of $V$ vertices and $E$ edges, a dominating set $D$ is a specific subset of vertices $D \subseteq V$. In particular, $D$ is a dominating set if all vertices in $V$ but not in $D$ are adjacent to at least one vertex in $D$. This is equivalent to requiring the set of nearest-neighbor vertices of $D$ (exclusive) and $D$ cover all vertices $N(D) \cup D = V$ (an example is given by Fig.~\ref{fig:dominating_sets}a). The set $D$ is a minimal dominating set if there is no proper subset of $D$ that is a dominating set, {\it{i.e.}}, the removal of any vertex in $D$ results in $N(D) \cup D \neq V$. An example is given by Fig.~\ref{fig:dominating_sets}b. The domination number of $D$ is given by the cardinality of $|D| \equiv \overline{\overline{D}}$. -The Minimum Dominating Set (MDS) is defined by $D$ with the smallest domination number. -Fig.~\ref{fig:dominating_sets}c shows an example of the minimum dominating set of $G(V, E)$ and is different from the minimal dominating set. -We emphasize that while the maximum independent set is always a minimal dominating set as exemplified by Fig.~\ref{fig:dominating_sets}b, the minimum dominating set, in general, can have a smaller domination number. -As a result, the solution to the dominating set problem can not be obtained by solving for the maximum independent set, a well-studied problem for quantum annealers. +The MDS is defined by $D$ with the smallest domination number. +Fig.~\ref{fig:dominating_sets}c shows an example of the MDS of $G(V, E)$ and is different from the minimal dominating set. +We emphasize that while the maximum independent set is always a minimal dominating set as exemplified by Fig.~\ref{fig:dominating_sets}b, the MDS, in general, can have a smaller domination number. +As a result, the solution to the dominating set problem can not be obtained by solving the maximum independent set problem, a well-studied problem for quantum annealers. \begin{figure*} \centering @@ -124,16 +126,16 @@ \section{INTRODUCTION} \includegraphics[width=0.2\textwidth]{./new_figures/MDS_mds2.pdf}\\ \centering\textbf{a} && \centering\textbf{b} && \centering\textbf{c} \end{tabular} - \caption{Example of different dominating sets for $G(V, E)$. Vertices in the dominating set $D$ are highlighted in blue. {\textbf{a)}} A dominating set of $G$ with domination number $\overline{\overline{D}} = 2$. {\textbf{b)}} A minimal dominating set of $G$ with domination number of $\overline{\overline{D}} = 2$. {\textbf{c)}} The minimum dominating set of $G$ with domination number of $\overline{\overline{D}} = 1$.} + \caption{Example of different dominating sets for $G(V, E)$. Vertices in the dominating set $D$ are highlighted in blue. {\textbf{a)}} A dominating set of $G$ with domination number $\overline{\overline{D}} = 2$. {\textbf{b)}} A minimal dominating set of $G$ with domination number of $\overline{\overline{D}} = 2$. {\textbf{c)}} The MDS of $G$ with domination number of $\overline{\overline{D}} = 1$.} \label{fig:dominating_sets} \end{figure*} For general graphs, existing algorithms on classical computers find MDS solutions in exponential time $\sim O( 1.5^n)$ \cite{Fomin2009, vanRooij2009} or approximate solutions in polynomial time. For example, greedy algorithms locally optimize decisions about which nodes to add to the dominating set. Thus one is guaranteed to find a dominating set but not necessarily an MDS. -We present a method to obtain optimal solutions to ILP by employing quantum annealing methods. +In this work, we present a method to obtain optimal solutions to ILP by employing quantum annealing methods. -Current implementations of quantum annealing solves the quadratic binary optimization problem (QUBO) by slowly varying a time-dependent Hamiltonian~\cite{1998PhRvE..58.5355K, 2000quant.ph..1106F, RevModPhys.80.1061}. +Current implementations of quantum annealing solve the quadratic binary optimization problem (QUBO) by slowly varying a time-dependent Hamiltonian~\cite{1998PhRvE..58.5355K, 2000quant.ph..1106F, RevModPhys.80.1061}. Through the adiabatic theorem of quantum mechanics, the annealer is initially prepared in a trivial ground state while the final Hamiltonian encodes the solution to the ILP. Due to the explosion in research efforts towards hardware implementations of quantum annealers and future improvements to the annealing schedule~\cite{doi:10.7566/JPSJ.89.044001}, mapping ILP to QUBO provides a path forward towards obtaining optimal solutions to the class of integer optimization problems~\cite{2018Glover}. @@ -141,41 +143,40 @@ \section{INTRODUCTION} \begin{equation} H(t) = A(t) H^{\textrm{init}} + B(t) H^{\textrm{problem}}, \label{eq:tdhamiltonian} \end{equation} -and $H^\textrm{init}=-\sum_i\sigma^x_i$ on the DWave, while $H^\textrm{problem}$ encodes the problem to be solved. In practice, and as discussed in Sec.~\ref{sec:methods:lindblad}, modeling dynamics that arise during quantum annealing requires a more robust description of the thermally populated mixed quantum states and the open dynamical processes that govern population of the sought-after eigenstate. +and $H^\textrm{init}=-\sum_i\sigma^x_i$ (on the D-Wave), while $H^\textrm{problem}$ encodes the problem to be solved. In practice, and as discussed in Sec.~\ref{sec:methods:lindblad}, modeling dynamics that arise during quantum annealing requires a more robust description of the thermally populated mixed quantum states and the open dynamical processes that govern population of the sought-after eigenstate. -In this paper, we propose an algorithm that maps ILP problems to QUBO. -This is achieved by the introduction of slack variables $s_a$ which turns the inequalities Eq.~\eqref{eq:ilp-constraints} to equalities +The mapping of ILPs to QUBOs we propose is realized by introducing slack variables $s_a$ which turn the inequalities Eq.~\eqref{eq:ilp-constraints} to equalities \begin{align} \label{eq:ilp:slack} & \sum_i A_{a i}x_i + s_a + b_a = 0, \\ & x_i, s_a \in \mathbb{Z} \geq 0. \end{align} -While the coefficients of the inequality constraints are not required to be integer valued, the inequalities can be trivially rescaled such that $s_i \in \mathbbm{Z}$ given fixed precision coefficients $A_{ij}$ and $b_i$. +While, in general, the coefficients of the inequality constraints are not required to be integer valued, this real valued inequalities can be rescaled such that $s_i \in \mathbbm{Z}$ given fixed precision coefficients $A_{ij}$ and $b_i$. Current hardware however, offers approximately 10 bits of precision over the control of value of the coupling and biases, limiting the usefulness of rescaling. + Furthermore, this formalism can be generalized to constrained quadratic optimizations \ref{sec:methods:ilp:quadratic}. -We improve the quantum annealer's performance by utilizing annealing offsets, which effectively delays the annealing schedule on a per-qubit basis~\cite{PhysRevA.96.042322,hsu2018quantum,10.1007/978-3-030-14082-3_14}. +We improve the quantum annealer's performance by utilizing annealing offsets, which effectively delay the annealing schedule on a per-qubit basis~\cite{PhysRevA.96.042322,hsu2018quantum,10.1007/978-3-030-14082-3_14,2020RPPh...83e4401H}. Converting to the Ising representation of the the problem Hamiltonian, \begin{equation} \label{eq:HIsing} - H^{\textrm{Ising}} = \sum_{ij} J_{ij} \sigma^z_i \sigma^z_j + \sum_i h_i \sigma^z_i , + H^{\textrm{problem}} \leftrightarrow H^{\textrm{Ising}} = \sum_{ij} J_{ij} \sigma^z_i \sigma^z_j + \sum_i h_i \sigma^z_i , \end{equation} we recognize that Eq.~(\ref{eq:HIsing}) exhibits spin-glass properties. More specifically, if the $h_i$ coefficients are randomly drawn from a Bernoulli distribution, one expects spin-localization behavior to influence the outcome of the anneal. In the case of quantum annealing, when an algorithm is mapped to its Ising representation, the values of $h_i$ will frequently take on various values, mimicking spin-glass like behavior. -More explicitly, the spin-glass enters a glassy state when more disorder is introduced ($|h_i|$ becomes large), and as a consequence, the wavefunction experiences Many-Body Localization (MBL) effects, the many-body analog of Anderson Localization~\cite{doi:10.1146/annurev-conmatphys-031214-014726,PhysRevE.90.022103,RevModPhys.91.021001,ALET2018498,PhysRevB.82.174411,PhysRevLett.109.017202}. +More explicitly, the spin-glass enters a glassy state when more disorder is introduced ($|h_i|$ becomes large), and as a consequence, the wavefunction experiences many body localization (MBL) effects, the many-body analog of Anderson localization~\cite{doi:10.1146/annurev-conmatphys-031214-014726,PhysRevE.90.022103,RevModPhys.91.021001,ALET2018498,PhysRevB.82.174411,PhysRevLett.109.017202}. Our improvement strategy is motivated by the MBL hypothesis. -As a result, we employ a modified annealing schedule which relies on partitioning the Hamiltonian into regions of relatively weak and strong external magnetic fields. - +As a result, we employ a modified annealing schedule that relies on partitioning the Hamiltonian into regions of relatively weak and strong external magnetic fields. -To understand which phenomena are relevant for describing the proposed offset study's scaling behavior, whether they are rooted in the quantum nature or related to hardware constraints, we simulate the anneal for a small MDS problem. This simulation solves the von Neumann equations accounting for different models of quantum decoherence, and explores whether algorithmic improvements on hardware are present in idealizes systems. +To understand which phenomena are relevant for describing the proposed offset study's scaling behavior, whether they are rooted in the quantum nature or related to hardware constraints, we simulate the anneal for a small MDS problem. This simulation solves the von Neumann equations accounting for different quantum decoherence models and explores whether algorithmic improvements on hardware are present in idealized systems. %======================================================================================== \section{RESULTS} \label{sec:results} %======================================================================================== -We first present the QUBO mapping for ILP (\ref{sec:results:qa1}), and demonstrate the methodology on an example implementation in case of the Minimal Dominating Set problem (\ref{sec:results:qa}) on the DWave quantum annealer. Results from quantum annealing are compared and discussed in contrast to simulations (\ref{sec:results:simulation}). +We first present the QUBO mapping for ILP (\ref{sec:results:qa1}), and demonstrate the methodology on an example implementation in case of the Minimal Dominating Set problem (\ref{sec:results:qa}) on the D-Wave quantum annealer. Results from quantum annealing are compared and discussed in contrast to simulations (\ref{sec:results:simulation}). %---------------------------------------------------------------------------------------- \subsection{QUBO Formulation of ILP} @@ -220,6 +221,7 @@ \subsection{QUBO Formulation of ILP} As a result, the $\vec x$ and $\vec s$ map to the binary vector $\vec \Psi$ under the transformation \begin{equation} + \label{eq:bit-vector-mapping} \begin{pmatrix} \vec x \\ \vec s \end{pmatrix} @@ -240,11 +242,10 @@ \subsection{QUBO Formulation of ILP} \label{eq:ilp-slack-bit-energy} \chi^2(\vec \Psi) &= - \vec c^T T_x \vec \Psi_x + p \left\| A T_s \vec \Psi_x + T_s \vec \Psi_s + \vec b \right\|^2 + \vec c^T T_x \vec \Psi_x + p \left\| A T_x \vec \Psi_x + T_s \vec \Psi_s + \vec b \right\|^2 \end{align} where $p$ is an external parameter representing the strength of the penalty when violating constraints. -This parameter needs to be sufficiently large, e.g., $p \geq \vec c \cdot \vec x$, to ensure the constraints are indeed fulfilled\footnote{Depending on the problem, $p$ can be smaller as well. -For example, in the case of the MDS, $p\geq 1$ suffices.}. +This parameter needs to be sufficiently large, e.g., $p \geq \vec c \cdot \vec x$, to ensure the constraints are indeed fulfilled\footnote{Depending on the problem, $p$ can be smaller as well. For example, in the case of the MDS, $p\geq 1$ suffices.}. The exact value of a sufficiently large penalty coefficient changes the excited-state spectrum of the time-dependent Hamiltonian, and while leaving the ground state unchanged, will impact the performance on quantum annealers due to the limited control over couplings and biases. Therefore, choosen the smallest valid penalty coefficient is observed to be optimal for quantum annealers today. The objective function $\chi^2$ can be represented as a QUBO Hamiltonian \begin{align} \chi^2(\vec \Psi) = & @@ -268,10 +269,10 @@ \subsection{QUBO Formulation of ILP} The function $\mathrm{diag}(\vec v)$ transforms a vector $\vec v$ into a diagonal matrix, and absorbs the linear contributions of the QUBO into the diagonal elements of the quadratic representation. The integer solution to the original ILP is computed by $\vec x^{(0)} = T_x \vec \Psi_x^{(0)}$ and the original solution to the problem is computed by shifting the annealer extracted energy $E \equiv \chi^2(\vec \Psi^{(0)}) $ by $p \left \| \vec b \right\|^2$. -The slack components of this vector $\vec \Psi_s$ can be utilized to check if the contraints are indeed fulfilled. +The slack components of this vector $\vec \Psi_s$ can be utilized to check if the constraints are indeed fulfilled. %---------------------------------------------------------------------------------------- -\subsection{Application to the Dominating Set} +\subsection{Annealer Results for the Dominating Set} \label{sec:results:qa} %---------------------------------------------------------------------------------------- \begin{figure}[b] @@ -284,7 +285,7 @@ \subsection{Application to the Dominating Set} \label{fig:linear} \end{figure} -We demonstrate the proposed algorithm in order to obtain the MDS on a series of linear graphs $G(n)$ as shown in Fig.~\ref{fig:linear}. This type of graph is chosen because the small number of nearest-neighbor connections is more efficiently embedded into the chimera graph allowing for scaling plots to be generated when using a DWave quantum annealer. In particular, the number of qubits required to solve MDS scales at worse as $n_V \log_2 n_V$ where $n_V$ is the number of vertices for a generic graph before minor embedding. Details of the mapping of ILP to MDS is given in Sec.~\ref{sec:methods:mds-qubo}. +We demonstrate the proposed algorithm in order to obtain the MDS on a series of linear graphs $G(n)$, as shown in Fig.~\ref{fig:linear}. This type of graph is chosen because the small number of nearest-neighbor connections is more efficiently embedded into the chimera graph, allowing scaling plots to be generated when using a D-Wave quantum annealer. In particular, the number of qubits required to solve MDS scales at worse as $n_V \log_2 n_V$ where $n_V$ is the number of vertices for a generic graph before minor embedding. Details of the mapping of ILP to MDS is given in Sec.~\ref{sec:methods:mds-qubo}. For the graph $G(n)$ the MDS solution is known analytically, and contains both unique and degenerate solutions. In particular, the domination number for $G(n)$ is $\lceil n/3 \rceil$ while the number of MDS solutions for $n$ vertices is \begin{align} @@ -294,83 +295,89 @@ \subsection{Application to the Dominating Set} \end{align} and gives the probability of randomly guessing the MDS of $G(n)$. -For even the smallest graph $G(2)$, the MDS problem is not native to the chimera graph and must be embedded. Following the hypothesis of MBL, we therefore, must look at the values of $h_i$ after embedding. The qubits then split into two groups depending on the value of $h_i$ relative to $(\textrm{max}|\{h\}| + \textrm{min}|\{h\}|) / 2$ given the set of external magnetic fields $\{h\}$ defined by a specific embedding. Further detail is given in Sec.~\ref{sec:methods:minor_embedding}. We study the effects of delaying the anneal schedule of one group of qubits over the other and present the results of this study is shown in Fig.~\ref{fig:baseline}. +For even the smallest graph $G(2)$, the MDS problem is not native to the chimera graph and must be embedded. Following the hypothesis of MBL, we, therefore, must look at the values of $h_i$ after embedding. The qubits split into two groups depending on the value of $h_i$ relative to $(\textrm{max}|\{h\}| + \textrm{min}|\{h\}|) / 2$ given the set of external magnetic fields $\{h\}$ defined by a specific embedding. Further detail is given in Sec.~\ref{sec:methods:minor_embedding}. We study the effects of delaying the anneal schedule of one group of qubits over the other and present the results of this study is shown in Fig.~\ref{fig:baseline}. -For the following studies, we perform experiments on the \texttt{DW\_2000Q\_6} solver. The annealing time is set to 500$\mu$s after performing a study on various annealing times the $G(6)$ graph. The black line (offset$=0.0$) in Fig.~\ref{fig:baseline} shows results from the baseline experiment, without modification to the DWave annealing schedule, and observe improvement over random guessing (blue). +Due to near-term limitations, hardware realizations of quantum annealing are unique, and possess for example, different lattice layouts (due to faulty qubits), annealing schedules, and qubit fidelity. For the following studies, we perform experiments specifically on the \texttt{DW\_2000Q\_6} solver. The annealing time is set to 500$\mu$s after performing a study on various annealing times the $G(6)$ graph. The black line (offset$=0.0$) in Fig.~\ref{fig:baseline} shows results from the baseline experiment, without modification to the D-Wave annealing schedule, and observe improvement over random guessing (dashed green). \begin{figure} \centering \includegraphics[width=\columnwidth]{./new_figures/DWave_scaling.pdf} - \caption{Baseline result of DWave (black) compared to random guessing (dashed green). The jagged nature of random guessing reflects the degeneracy of the ground state. Negative offsets with a `s' tag (blue) are results from delays of large values of $|h_i|$, while negative offsets with the `w' label (red) delay the schedule of qubits with small values of $h_i$.} + \caption{Baseline result of D-Wave (black) compared to random guessing (dashed green). The jagged nature of random guessing reflects the degeneracy of the ground state. Negative offsets with a `s' tag (blue) are results from delays of large values of $|h_i|$, while negative offsets with the `w' label (red) delay the schedule of qubits with small values of $h_i$.} \label{fig:baseline} \end{figure} We explore one avenue towards improving the experiment results by introducing per-qubit annealing offsets into the time evolution. -The results in yellow (green) delays the annealing of qubits subject to stronger (weaker) external fields. We observe improvement (diminishment) in the experimental results when qubits subject to stronger (weaker) final external fields are delayed in the anneal schedule, in agreement with the MBL hypothesis. The phenomena is observed across different problem sizes and hints at the possibility of a generic improvement strategy. +The blue (red) results delay qubits' annealing subject to stronger (weaker) external fields. We observe improvement (diminishment) in the experimental results when qubits subject to stronger (weaker) final external fields are delayed in the annealing schedule, in agreement with the MBL hypothesis. The phenomena is observed across different problem sizes and hints at the possibility of a generic improvement strategy. %---------------------------------------------------------------------------------------- \subsection{Simulation Results} %---------------------------------------------------------------------------------------- \label{sec:results:simulation} -In order to understand the effects of annealing offsets, we simulate the annealing process for the $G(2)$ graph embedded in chimera topography. -In order to shorten the simulation time, we repeat the same anneal process with a total annealing time of 1 $\mu s$ and count the number of correct ground state occurances. +To understand the effects of annealing offsets, we simulate the annealing process for the $G(2)$ graph embedded in chimera topography. +We repeat the same anneal process with a shortend total annealing time of 1 $\mu s$, reducing the computational demands of the simulation, and count the number of correct ground state occurrences. The resulting ground state probability as a function of offset measured over 100,000 observations is shown in Fig.~\ref{fig:dwave1us}. \begin{figure}[b] \centering \includegraphics[width=\columnwidth]{./new_figures/NN2_offset_scaling.pdf} - \caption{The probability of finding the MDS for $G(2)$ from DWave (black) and simulation (dashed yellow) at annealing times of 1 $\mu s$.} + \caption{The probability of finding the MDS for $G(2)$ from D-Wave (black) and simulation (dashed yellow) at annealing times of 1 $\mu s$.} \label{fig:dwave1us} \end{figure} To solve for time evolution dynamics of quantum annealing including thermal and the decoherence effects, we solve for the master equation in Lindblad form \begin{align} + \label{eq:sim:linblad-eq} \partial_t \rho (t) = \frac{-i}{\hbar} [H(t) , \rho(t)] + \mathcal{L}(\rho(t), H(t)) \end{align} -where $\rho (t)$ is the density matrix at time $t$, $H(t)$ is the time-dependent Hamiltonian and $\mathcal{L}$ is the Linblad operator implementing the decoherence models. +where $\hbar$ is the reduced Plank's constant, $\rho (t)$ is the density matrix at time $t$, $H(t)$ is the time-dependent Hamiltonian and $\mathcal{L}$ is the Linblad operator implementing the decoherence models. In this work, we consider two types of decoherence models: full-counting statistics~\cite{PhysRevE.90.022103,RevModPhys.81.1665} and single-qubit amplitude damping (local damping)~\cite{10.5555/1972505,preskill1998lecture}. The full-counting statistics term models the global decoherence to all the qubits due to the classical reservoir. The local damping term models the decoherence of each qubit independently. Details of the master equation are given in Sec.~\ref{sec:methods:lindblad}. -Implementation of the anneal schedule and annealing offsets for the simulation are discussed in Sec.~\ref{sec:methods:annealing-schedule}. +Implementation of the annealing schedule and annealing offsets for the simulation are discussed in Sec.~\ref{sec:methods:annealing-schedule}. -The graph $G(2)$ has the interesting feature of having a degenerate ground state depending on whether qubit 0 or 1 is chosen to be the MDS solution. +The graph $G(2)$ has degenerate ground states depending on whether qubit 0 or 1 is chosen to be the MDS solution. This degeneracy is reflected in the experimental result and provides a non-trivial benchmark for our simulation. -Fig.~\ref{fig:final_state_distribution} show the final state distribution of the three lowest lying state. -The states $(0, 1, 0, 0, 0)$ and $(1, 0, 0, 1, 0)$ are the two degenerate ground states of the embedded Hamiltonian, while $(1, 1, 1, 1, 1)$ is the first excited state which yields an incorrect solution. +Fig.~\ref{fig:final_state_distribution} show the final state distribution of the three lowest-lying state. +The states $(0, 1, 0, 0, 0)$ and $(1, 0, 0, 1, 0)$ are the two degenerate ground states of the embedded Hamiltonian, while $(1, 1, 1, 1, 1)$ is the first excited state which yields an incorrect solution\footnote{According to Eq.~\eqref{eq:bit-vector-mapping}, the first two vector components represent the nodes of the graph while the following components represent the slack variables after embedding.}. All other states receive negligible probability at the end of annealing. -The simulation (black) captures main features of the experimental result: 1) significant probability to populate both ground states (rather than populating only one), 2) asymmetry in ground state population due to systematic effects of offset lifting the final state degeneracy spanning approximately the correct range, 3) population of first excited state with systematically lower probability when the strong field is delayed. The asymmetry in the ground state distribution at non-zero offset is a result of annealing offsets lifting the ground state degeneracy. -Within the broken symmetry, the non-degenerate ground state switches between the two states depending on which qubit group is delayed. At zero offset, a slight asymmetry exists in the simulation because the Hamiltonian is only degenerate at the last moment, while the DWave is also subject to sampling bias~\cite{2016PhRvA..93e2320K, 2017PhRvL.118g0502M}. - -This result can be obtained by tuning three free parameters: the simulation temperature to the order of 10 milliKelvin, and the two coefficients of the two decoherence models at the order of 1 to 10 $ns$. These values are consistent with the reported DWave operating temperature~\cite{dwave_temp} and coherence times for flux qubits~\cite{2003Sci...299.1869C}. +The simulation (black) captures the main features of the experimental result: +\begin{enumerate} + \item Significant probability of populating both ground states (rather than populating only one) + \item Asymmetry in ground state population due to offsets and spanning approximately the correct range + \item Population of first excited state with systematically lower probability when the strong field is delayed +\end{enumerate} +The asymmetry in the ground state distribution at non-zero offset results from annealing offsets lifting the ground state degeneracy. +The non-degenerate ground state switches between the two states within the broken symmetry, depending on which qubit group is delayed. At zero offset, a slight asymmetry exists in the simulation because the Hamiltonian is only degenerate at the last moment, while the D-Wave is also subject to sampling bias~\cite{2016PhRvA..93e2320K, 2017PhRvL.118g0502M}. + +This result can be obtained by tuning three free parameters: the simulation temperature to the order of 10 milliKelvin, and the two coefficients of the two decoherence models at the order of 1 to 10 $ns$. These values are consistent with the reported D-Wave operating temperature~\cite{dwave_temp} and coherence times for flux qubits~\cite{2003Sci...299.1869C}. Additional insights of the simulation are given in Sec.~\ref{sec:discussion:time_evolution}. The resulting probability of recovering the correct solution as a function of annealing offset is given in Fig.~\ref{fig:dwave1us}. -We confirm that scaling with respect to offset from the simulation matches what is observed from experiment, and provides evidence suggesting that the improvements are related to quantum mechanics. -An additional study where an extended annealing schedule is employed in the simulation which removes systematic errors introduced by annealing offsets and effects of local damping are presented in Sec.~\ref{sec:discussion:idealqa}, and further supports this observation. +We confirm that the simulation's offset-scaling follows the experiment's scaling, which suggests that the improvements are related to quantum mechanics. +An additional study where an extended annealing schedule is employed in the simulation, which removes systematic errors introduced by annealing offsets and effects of local damping are presented in Sec.~\ref{sec:discussion:idealqa}, and further supports this observation. \begin{figure} \centering \includegraphics[width=\columnwidth]{./new_figures/final_state_distribution.pdf} - \caption{Final state distribution from DWave (solid bars) and simulation (black outline). The colors label the type of offset The $(0, 1, 0, 0, 0)$ state is the first solution where vertex 1 is in the dominating set. The $(1, 0, 0, 1, 0)$ state is the second solution where vertex 0 is in the dominating set. The first excited state is the $(1, 1, 1, 1, 1)$ state where both vertices are in the dominating set.} + \caption{Final state distribution from D-Wave (solid bars) and simulation (black outline). The colors label the type of offset The $(0, 1, 0, 0, 0)$ state is the first solution where vertex 1 is in the dominating set. The $(1, 0, 0, 1, 0)$ state is the second solution where vertex 0 is in the dominating set. The first excited state is the $(1, 1, 1, 1, 1)$ state where both vertices are in the dominating set.} \label{fig:final_state_distribution} \end{figure} \section{Discussion} - \subsection{Dynamics of Time Evolution} \label{sec:discussion:time_evolution} - The time-dependent overlap with the exact Ising ground state is shown in Fig.~\ref{fig:td_prob} from applying the simulator to $G(2)$. We observe for all cases that the system undergoes what is analogous to a magnetic phase transition around $s\sim 0.4$. -After the phase transition, we are able to confirm that the system collapses to effectively a classical state in the sense that the density matrix becomes a diagonal matrix. +The time-dependent overlap with the exact Ising ground state is shown in Fig.~\ref{fig:td_prob} from applying the simulator to $G(2)$. We observe for all cases that the system undergoes what is analogous to a magnetic phase transition around $s\sim 0.4$. +After the phase transition, we can confirm that the system collapses to effectively a classical state in the sense that the density matrix becomes a diagonal matrix. The steady increase in probability after the phase transition is a sensitive balance between the competing effects between full-counting statistics and local damping. -In our example, full-counting statistics is tuned to be slightly stronger, resulting in what amounts to a final thermal annealing step. -If local damping is relatively stronger, then the probability after the phase transition will slowly decrease as the system decoheres into its local ground state. -While we believe both effects are important in DWave, the experimental results are not precise enough for us to conclude which is the dominant effect. -We emphasize however, that the simulation suggests that the ground state is recovered predominantly due to the quantum phase transition. +In our example, the full-counting statistics decoherance rate is tuned to be slightly stronger compared to the local dampening decoherance rate, effectively resulting in a final thermal annealing process after the phase transition. +If local damping were relatively larger, then the probability after the phase transition will slowly decrease as the system decoheres into its local ground state. +While we believe both effects are essential to simulate D-Wave, due to the competing effects of both decoherence models, we emphasize that a fully quantitative comparison of both decoherence models cannot be made just considering the $G(2)$ graph. +However, we emphasize that the simulation suggests that the ground state is recovered predominantly due to the quantum phase transition happing around $s\sim 0.4$. \begin{figure} \centering @@ -379,9 +386,10 @@ \subsection{Dynamics of Time Evolution} \label{fig:td_prob} \end{figure} -Finally, we comment that in order for the simulation to obtain the final state distribution shown in Fig.~\ref{fig:final_state_distribution}, effects of full-counting statistics are required. -Absent some dynamical thermalization effect, the $G(2)$ problem is too small such that at a total annealing time of 1$\mu s$, the annealing offsets lift the ground state degeneracy by an amount such that diabatic transitions are absent, and the simulation is populated predominantly by the unique ground state. -The final state distribution gives us a very sensitive observable to tune the simulation temperature, and agrees well with the experimental operating temperature. +Finally, we comment that effects of full-counting statistics are required for the simulation to obtain the final state distribution shown in Fig.~\ref{fig:final_state_distribution}. +Because of the smallness of the $G(2)$ problem (and the utilized total annealing time of 1$\mu s$), the annealing offsets lift the ground state degeneracy in a manor that diabatic transitions do not occur--setting a discrete, fixed groundstate probability for given offsets. +Thus, if dynamical thermalization effects were absent, the simulation would populate predominantly one of the two unique ground states depending on the offsets. +Once dynamical thermalization effects are present, the final state distribution continously depends on offsets and becomes a very sensitive observable to tune the simulation temperature, which, if properly tuned, agrees well with the experimental operating temperature. \subsection{Idealized Quantum Annealing} @@ -401,19 +409,16 @@ \subsection{Idealized Quantum Annealing} \label{fig:anneal_schedule_ext} \end{figure*} -In the simulation, we also reserve the ability to remove the systematic errors by extending the annealing schedule as shown by the red data points in Fig.~\ref{fig:anneal_schedule_ext}. With the extended schedule, all qubits start and end with the same values of $A$ and $B$ and faithfully preserves the initial and final Hamiltonians. The dependence on offset for $G(2)$ (Fig.~\ref{fig:dwave1us}) remains the same under the extended anneal schedule (Fig.~\ref{fig:anneal_schedule_ext}), confirming that the behavior is not a systematic artifact. +In the simulation, we also reserve the ability to remove systematic errors associated with finite annealing schedules by extending them, as shown by the red data points in Fig.~\ref{fig:anneal_schedule_ext}. With the extended schedule, all qubits start and end with the same values of $A$ and $B$ and faithfully preserves the initial and final Hamiltonians. The dependence on offset for $G(2)$ (Fig.~\ref{fig:dwave1us}) remains the same under the extended anneal schedule (Fig.~\ref{fig:anneal_schedule_ext}), confirming that the behavior is not a systematic artifact. -The quantum system can be further idealized by including only the full-counting statistics model (blue), or local decoherence (green). We observe in both cases that the overall scaling follows the story of the MBL hypothesis. Perhaps more importantly, we observe that a simulation without local decoherence, where the relaxation is dependent on precisely the instantaneous value of $|h_i|$, exhibits the same scaling as experiment. This result suggests that our strategy for setting annealing offsets is improving the algorithm beyond the simple explanation of qubits freezing due to single-particle amplitude damping. In fact, the simulation with only amplitude damping (green) does not fully capture the results of the experiment. We observe hints of a phase transition with respect to offset (which may be thought of as some measure of disorder) when dynamical thermalization effects are removed. This is a tantalizing observation, and also is strong evidence for the inclusion of the full-counting statistics model. - -It unfortunately does not make much sense to remove all decoherence models for the $G(2)$ graph for an anneal time of $1\mu s$ since all probabilities go to unity. +The quantum system can be further idealized by including either solely the full-counting statistics model (blue) or solely the local decoherence model (green)\footnote{For problems without diabatic transitions, like the MDS problem for a $G(2)$ graph and an anneal time of $1\mu s$, removing all decoherence models results in ground state probabilities equal to unity.}. We observe in both cases that the overall scaling follows the story of the MBL hypothesis. Perhaps more importantly, we observe that a simulation without local decoherence, where the relaxation is dependent on precisely the instantaneous value of $|h_i|$, exhibits the same scaling as the experiment. This result suggests that our strategy for setting annealing offsets is improving the algorithm beyond the simple explanation of qubits freezing due to single-particle amplitude damping. The simulation with only amplitude damping (green) does not fully capture the results of the experiment. We observe hints of a phase transition depending on the offset (which may be considered some measure of disorder) when dynamical thermalization effects are removed. This observation is a strong evidence for the inclusion of the full-counting statistics model. \subsection{Final Remarks} \label{sec:results:final} -We would like to emphasize that while the annealing offsets are motivated by the MBL hypothesis, and the results also follow those expectations, we do not have definitive proof that MBL plays a crucial role. -The reason is because observations of MBL inevitably require the study of finite-size scaling~\cite{2015PhRvB..91h1103L}, and our current simulation while being extremely thorough and explicit, is exponentially slow to evaluate making evaluations of even $G(3)$ untenable. +We want to emphasize that while the annealing offsets are motivated by the MBL hypothesis, and the results also follow those expectations, we do not have definitive proof that MBL plays a crucial role. +Observations of MBL inevitably require the study of finite-size scaling~\cite{2015PhRvB..91h1103L}, and our current simulation, while being extremely thorough and explicit, is exponentially slow to evaluate, making evaluations of even $G(3)$ unfeasible. However, the intersection of time-dependent quantum mechanics and emergent phenomena~\cite{Goldenfeld:1992qy} is an exciting direction that is pertinent to adiabatic quantum computing. - %======================================================================================== \section{METHODS} \label{sec:methods} @@ -427,7 +432,7 @@ \subsubsection{Minimum Dominating Set QUBO} \label{sec:methods:mds-qubo} %---------------------------------------------------------------------------------------- -The solution to the Minimum Dominating Set problem can be expressed as an integer optimization problem given by, +The solution to the MDS problem can be expressed as an integer optimization problem given by, \begin{align} f(\vec x) = & \min\left(\sum_{v \in V} x_v\right), \\ \end{align} @@ -442,7 +447,8 @@ \subsubsection{Minimum Dominating Set QUBO} \begin{equation} \vec s \in \left\{ \mathbb{Z}^{n_V} \, \middle| \, 0 \leq s_{v} \leq |\mathcal{N}(v)| \quad \forall v\in V \right\} \, , \end{equation} -which is related to the qubit vector $\vec \Psi_s$ by $\vec s = T_s \vec \Psi_s$, to encode the inequality constraint such that +which is related to the qubit vector $\vec \Psi_s$ by $\vec s = T_s \vec \Psi_s$. +The inequality constraint is encoded by \begin{align} f(\vec \Psi_x) = @@ -451,31 +457,30 @@ \subsubsection{Minimum Dominating Set QUBO} subject to \begin{align} & - \vec \Psi_x + J \vec \Psi_x - T_s \Psi_s - \vec 1 = 0\,, + (\mathbbm{1} + J)\vec \Psi_x - T_s \Psi_s - \vec 1 = 0\,, \\ & (\vec \Psi_s)_\nu \in \{ 0, 1\} \end{align} -where the nearest-neighbor sum is expressed by $J$ (zero diagonal and symmetric for non-directional graphs), the adjacency matrix for $G$. +where the nearest-neighbor sum is expressed by the adjacency matrix $J$ (zero diagonal and symmetric for non-directional graphs). The algorithm uses \begin{equation} N_q = \overline{\overline{V}} + \sum_{v \in V} \log_2 \mathcal{N}(v) \end{equation} -qubits to encode the vertices and to embed the slack variables. -Therefore, the (logical) qubit vector $\vec \Psi$ at worst scales with $n_V \log_2 n_V$ qubits for fully connected graphs. +qubits to encode the vertices and slack variables before embedding. +Therefore, the (logical) binary vector $\vec \Psi$ at worst scales with $n_V \log_2 n_V$ qubits for fully connected graphs. The target QUBO in the notation of Eq.~\eqref{eq:matrix_form} reads {\small \begin{align} - Q_{xx} & = \mathbbm{1} + p\left[J^T J + J^T + J - 2 \mathrm{diag}(|J|) - \mathbbm{1} \right] \,, \\ - Q_{sx} & = - p(\mathbbm{1}+J^T)T_s\,, \\ - Q_{ss} & = p\left[{T_s}^T T_s + 2\mathrm{diag}(|T_s|)\right]\,, \\ - C & = p \overline{\overline{V}}\,, + Q_{xx} & = \mathbbm{1} + p\left[J^T J + J^T + J - \mathrm{diag}\left(J^T \vec 1 + \vec 1^T J\right) - \mathbbm{1} \right] \,, \\ + Q_{sx} & = - p(\mathbbm{1}+J)^TT_s\,, \\ + Q_{ss} & = p\left[{T_s}^T T_s + \mathrm{diag}\left(T_s^T \vec 1 + \vec 1^T T_s\right)\right]\,, \\ + C & = p \overline{\overline{V}}\,. \end{align}} -where $ |M| \equiv \sum_{\nu} M_{\nu \mu}$. %---------------------------------------------------------------------------------------- -\subsubsection{Integer quadratic optimization} +\subsubsection{Integer Quadratic Optimization} \label{sec:methods:ilp:quadratic} %---------------------------------------------------------------------------------------- @@ -491,18 +496,18 @@ \subsubsection{Integer quadratic optimization} \end{equation} %---------------------------------------------------------------------------------------- -\subsection{ILP on the DWave} -\label{sec:methods:ILP-on-DWave} +\subsection{ILP on the D-Wave} +\label{sec:methods:ILP-on-D-Wave} %---------------------------------------------------------------------------------------- %---------------------------------------------------------------------------------------- -\subsubsection{Comment on ILP QUBO penalty term} +\subsubsection{Comment on ILP QUBO Penalty Term} \label{sec:methods:ilp-qubo-comments} %---------------------------------------------------------------------------------------- -The minimal energy solution to Eq.~\eqref{eq:initial-ip-def} and Eq.~\eqref{eq:ilp-slack-bit-energy} are exactly the same if the penalty term is larger than the energy gap of the first excited solution: $p > E_1 - E_0$. +The minimal energy solution to Eq.~\eqref{eq:initial-ip-def} and Eq.~\eqref{eq:ilp-slack-bit-energy} are exactly the same if the penalty term is greater or equal to the energy gap of the first excited solution: $p \geq E_1 - E_0$. Thus some knowledge of the problem is required. -In principle, it is possible to set the penalty term arbitrarily large, at the cost of problem resolution: large values of For $p$ increase the highest available energy of the system by multiples of $p$. +In principle, it is possible to set the penalty term arbitrarily large, at the cost of problem resolution: large values for $p$ increase the highest available energy of the system by multiples of $p$. After normalization of the QUBO, this corresponds to decreasing the energy gap between the ground state and the first excited state. Thus, if solvers have finite precision, one must estimate reasonable values for $p$: for large $p$ more solutions fulfill the constraints, while for small $p$, more solutions minimize the objective function. @@ -511,13 +516,13 @@ \subsubsection{Minor Embedding} \label{sec:methods:minor_embedding} %---------------------------------------------------------------------------------------- Our proposed offset strategy is motivated by the structure of the Hamiltonian being evaluated by the annealer. -As a result, details of the embedding are important. We obtain an embedding for $G(n)$ with the \texttt{embed\_qubo} function provided by the DWave Oceans Python package~\cite{dwave_oceans} under the \texttt{dwave.embedding} module~\cite{2008arXiv0804.4884C}. -The same embedding is used for all DWave solves of the same graph (independent of offset), and consequently the simulation solves the resulting embedded Hamiltonian for $G(2)$. +As a result, details of the embedding are important. We obtain an embedding for $G(n)$ with the \texttt{embed\_qubo} function provided by the D-Wave Ocean Python package~\cite{dwave_oceans} under the \texttt{dwave.embedding} module~\cite{2008arXiv0804.4884C}. +The same embedding is used for all D-Wave solves of the same graph (independent of offset), and consequently the simulation solves the resulting embedded Hamiltonian for $G(2)$. Additionally, solving the same graph as a function of offset on the exact same qubits removes (or at least keeps consistent) the systematic effects due to solving a problem on different physical qubits. We note that comparisons between different graphs in Fig.~\ref{fig:baseline} are subject to this uncontrolled systematic. After embedding the QUBO for $G(2)$ requires 5 qubits (an increase from 4), where by construction, qubits 0 and 3 form the qubit chain. -We confirm through brute force evaluation of the eigenvalue decomposition of the 5 qubit Hamiltonian, that the embedding provided by DWave solves the expected ILP problem for $G(2)$, with degenerate ground states at $(0, 1, 0, 0, 0)$ and $(1, 0, 0, 1, 0)$ corresponding to whether vertex 0 or 1 is chosen for the MDS solution, and $(1, 1, 1, 1, 1)$ as the first (non-degenerate) excited state where both vertex 1 and 0 are in the set yielding a valid dominating set but not the minimum dominating set. +We confirm through brute force evaluation of the eigenvalue decomposition of the 5 qubit Hamiltonian, that the embedding provided by D-Wave solves the expected ILP problem for $G(2)$, with degenerate ground states at $(0, 1, 0, 0, 0)$ and $(1, 0, 0, 1, 0)$ corresponding to whether vertex 0 or 1 is chosen for the MDS solution, and $(1, 1, 1, 1, 1)$ as the first (non-degenerate) excited state where both vertex 1 and 0 are in the set yielding a valid dominating set but not the MDS. The resulting Ising Hamiltonian has external field equal to $h = (2.75, 1.5, -1.0, -1.25, -1.0)$. Following the offset strategy described in Sec.~\ref{sec:results:qa}, qubit(s) 0 (1, 2, 3, 4) are placed in the set with relatively stronger (weaker) final external fields. @@ -535,37 +540,32 @@ \subsubsection{The Lindblad Equation} \label{sec:methods:lindblad} %---------------------------------------------------------------------------------------- -To solve for time evolution dynamics of quantum annealing including thermal and the decoherence effects, we evaluate the master equation in Lindblad form -\begin{align} - \partial_t \rho (t) = & \frac{-i}{\hbar} \left[H(t) , \rho(t)\right] \nonumber \\ - & + \sum_j \left(2L_j \rho(t) L_j^\dagger - \{ L^\dagger_j L_j, \rho(t) \}\right) , -\end{align} -where $\rho (t)$ is the density matrix at time $t$. -$H(t)$ is the time-dependent Hamiltonian +To solve for time evolution dynamics of quantum annealing including thermal and the decoherence effects, we evaluate the master equation Eq.~\eqref{eq:sim:linblad-eq} in Lindblad form. +The explicit time-dependence of the Hamiltonian is given by \begin{align} \label{eq:annealH} - H_{anneal}(t) = & - \sum_i A_i(t)\sigma^x_i +\sum_i B_{i}(t) h_i \sigma^z_i \notag \\ + H(t) = & - \sum_i A_i(t)\sigma^x_i +\sum_i B_{i}(t) h_i \sigma^z_i \notag \\ & + \sum_{i>j} \sqrt{B_{i}(t)B_{j}(t)} J_{ij} \sigma^z_i \sigma^z_j , \end{align} where $A_i(t)$ and $B_{i}(t)$ are site-dependent annealing schedule functions. -The site dependency takes into account of the annealing offset. -$[,]$ denotes the Lie bracket. -$L_j$ are the Lindblad operators for decoherence. -$\{, \}$ denotes the anti-commutator. +The site dependency takes into account of the annealing offsets. +The simulation takes two decoherance models into account. +Full-counting statistics and local decoherence. For full-counting statistics, the Lindblad operator is \begin{align} -\mathcal{L}[\rho (t)] = & \Gamma_{\textrm{fc}} \sum_j \left(2 S_j \rho(t) S_j^\dagger - \{ S^\dagger_j S_j, \rho(t) \}\right) \notag \\ -& + e^{-\beta \Delta E_j} \left(2 S_j^\dagger \rho(t) S_j - \{ S_j S^\dagger_j, \rho(t) \} \right) +\mathcal{L}_{\textrm{fc}}(\rho (t), H(t)) = & \Gamma_{\textrm{fc}} \sum_{j, i0$. $S_j=|E_\nu \rangle \langle E_\mu|$ denotes the many-body lowering operator. -$\{, \}$ denotes the anti-commutator. $\Gamma = 1/T_c$ is the decoherence rate for coherence time $T_c$. +where $\{, \}$ denotes the anti-commutator, $(ij)$ is the index for the inter-level spacing $\Delta E_{ij}=E_j-E_i>0$, $S_{ij}=|E_i \rangle \langle E_j|$ denotes the many-body lowering operator and $\Gamma_{fc} = 1/T_{fc}$ is the full-counting decoherence rate for coherence time $T_{fc}$. That is, due to the interaction with the classical thermal bath, there is a probability that the system hops from each higher-energy many-body state to a lower-energy many-body state. The probability of the inverse process is given by a Boltzmann factor. + To model the local decoherence of each qubits in the non-interacting limit, we also consider the amplitude damping for non-interacting qubits. -The Lindblad operator is +For the local decoherance model, the Lindblad operator is \begin{align} -\mathcal{L}[\rho (t)] = & \Gamma_{\textrm{loc}} \sum_j \left(2L_j \rho(t) L_j^\dagger - \{ L^\dagger_j L_j, \rho(t) \} \right)\notag \\ +\mathcal{L}_{\textrm{loc}}(\rho (t), H(t)) = & \Gamma_{\textrm{loc}} \sum_j \left(2L_j \rho(t) L_j^\dagger - \{ L^\dagger_j L_j, \rho(t) \} \right)\notag \\ & + e^{- 2 \beta |h_j| } \left(2L_j^\dagger \rho(t) L_j - \{ L_j L_j^\dagger, \rho(t) \} \right) , \end{align} where $j$ is the index for qubit. @@ -588,13 +588,13 @@ \subsubsection{The Lindblad Equation} The coherence time for local decoherence is set to 15$ns$, and 1$ns$ for full-counting statistics. %---------------------------------------------------------------------------------------- -\subsubsection{Annealing schedule and offsets} +\subsubsection{Annealing Schedule and Offsets} \label{sec:methods:annealing-schedule} %---------------------------------------------------------------------------------------- In this section, we discuss the details of the annealing schedule with respect to the dimensionless normalized time $s$. -On DWave solvers, annealing offsets effectively advance or delay the annealing schedule of individual qubits ($B(s) \to B_i(s)$). -In Fig.~\ref{fig:anneal_schedule}, the default DWave annealing schedule is shown in black, in addition to the effects of applying negative offsets (effective time delay) to $A(s)$ and $B(s)$ in blue. -Further documentation is provided by DWave in Ref.~\cite{dwave_as, dwave_as_docu}. +On D-Wave solvers, annealing offsets effectively advance or delay the annealing schedule of individual qubits (see E.~\eqref{eq:annealH}). +In Fig.~\ref{fig:anneal_schedule}, the default D-Wave annealing schedule is shown in black, in addition to the effects of applying negative offsets (effective time delay) to $A(s)$ and $B(s)$ in blue. +Further documentation is provided by D-Wave in Ref.~\cite{dwave_as, dwave_as_docu}. \begin{figure}[htb] \centering @@ -607,13 +607,13 @@ \subsubsection{Annealing schedule and offsets} \label{fig:anneal_schedule} \end{figure} -The coefficients are nonlinear in time because the underlying control variable $c(s)$ which is designed to grow the persistent current $I_p(s)$ linearly in time. +The coefficients $A(s)$ and $B(s)$ follow the underlying control variable $c(s)$, which is designed to grow the persistent current $I_p(s)$ linearly in time. The effective time delay is implemented by introducing an offset as $c(s) + \delta$. -As a result, systematic errors are introduced because the final values of $A(s)$ and $B(s)$ will differ for qubits with different offsets. +Because annealing schedules are finite, systematic errors are introduced because the final values of $A(s)$ and $B(s)$ will differ for qubits with different offsets. Additionally, the values of the coefficients are unknown outside of $s\in [0, 1]$. -We extrapolate their values by a linear extrapolation, and only employ negative offsets such that this effect only enters at the beginning of the annealing process rather than the end. +We only employ negative offsets such that this unkonwn coefficient range, approximated by a linear extrapolation, only enters at the beginning of the annealing process. -We verify that our implementation of annealing offsets on the simulator is consistent with DWave by solving the following three qubit Hamiltonian +We verify that our implementation of annealing offsets on the simulator is consistent with D-Wave by solving the following three qubit Hamiltonian \begin{align} \label{eq:offset_test_hamiltonian} H^{\textrm{problem}} = @@ -623,9 +623,9 @@ \subsubsection{Annealing schedule and offsets} 0 & 0 & -0.25 \end{pmatrix} \end{align} -which has a doubly-degenerate ground state of $(0, 1, 1)$ and $(1, 0, 1)$. An annealing offset is then applied to either qubit 0 or 1, and breaks the ground state degeneracy of the system. Because of the systematic error introduced when assigning an offset to a qubit, the final Hamiltonian given by Eq.~(\ref{eq:annealH}) will have small deviations from the input problem Hamiltonian. For example, we expect a negative offset to qubit 0 to Eq.~(\ref{eq:offset_test_hamiltonian}) will yield $(1, 0, 1)$ as the unique ground state given Eq.~(\ref{eq:annealH}). +which has a doubly-degenerate ground state of $(0, 1, 1)$ and $(1, 0, 1)$. An annealing offset is then applied to either qubit 0 or 1, and thus breaks the ground state degeneracy of the system. Because of the systematic error introduced when assigning an offset to a qubit, the final Hamiltonian given by Eq.~(\ref{eq:annealH}) will have small deviations from the input problem Hamiltonian. For example, we expect a negative offset to qubit 0 to Eq.~(\ref{eq:offset_test_hamiltonian}) will yield $(1, 0, 1)$ as the unique ground state given Eq.~(\ref{eq:annealH}). -We confirm that with different combinations of annealing offsets, the degeneracy is lifted on the DWave results as expected by solving for the modified problem Hamiltonian spectrum, as well as dialing in annealing offset in the simulation of this three qubit test case. +We confirm that with different combinations of annealing offsets, the degeneracy is lifted on the D-Wave results as expected by solving for the modified problem Hamiltonian spectrum, as well as dialing in annealing offset in the simulation of this three qubit test case. %---------------------------------------------------------------------------------------- @@ -642,7 +642,7 @@ \subsubsection{Pure Transverse Field Simulation} \begin{figure} \centering \includegraphics[width=\columnwidth]{./new_figures/vacuum_probability.pdf} - \caption{Time dependent vacuum probability of 5 qubits system under pure transverse field.} + \caption{Time-dependent vacuum probability of 5 qubits system under pure transverse field.} \label{figcheck} \end{figure} @@ -690,6 +690,11 @@ \section{AUTHOR CONTRIBUTIONS} Chang, Chen, K\"orber, and Humble interpreted the results. All authors contributed to writing and editing of the final manuscript. +%======================================================================================== +\section{CORRESPONDING AUTHOR} +%======================================================================================== +Correspondence to chiachang@berkeley.edu. + %======================================================================================== \section{ADDITIONAL INFORMATION} %======================================================================================== @@ -697,7 +702,6 @@ \section{ADDITIONAL INFORMATION} \textbf{Competing Interests:} The authors declare no competing interests. \bibliographystyle{apsrev4-1} -\bibliography{qilp} +\bibliography{main.bib} %======================================================================================== - \end{document} From 6d9e26a0a4ff8353e1cf8794a9b0fc512afc1dec Mon Sep 17 00:00:00 2001 From: Christopher Koerber Date: Thu, 11 Feb 2021 10:56:19 +0100 Subject: [PATCH 2/4] Added reference version of v2 folder --- paper/v2/main.tex | 37 +++++++++++++++++++------------------ 1 file changed, 19 insertions(+), 18 deletions(-) diff --git a/paper/v2/main.tex b/paper/v2/main.tex index 508b610..443b68e 100644 --- a/paper/v2/main.tex +++ b/paper/v2/main.tex @@ -95,14 +95,15 @@ \section{INTRODUCTION} \end{align} where $i=1, \cdots, N$ is the number of dependent variables and $a=1, \cdots, M$ the number of constraint equations. -While ILP is NP-Hard, extremely large instances of ILP are solved are rutinely solved on a daily basis in a variety of industries such as transportation, energy, and online dating. The fact that instances with millions of integer variables can be solved is due to the incredible improvement in classical ILP methods. One estimate has that the speed of ILP algorithms, combined with hardware improvements, have lead to a speedup of around 200 billion\cite{bertsimas2014statistics}. Despite this fact, there are still very small problems that still require several months, or even years of computational effort to solve. For example, the Steiner Triple Covering problems can be notoriously difficult, an instance with only 135 binary variables, \mathtt{STS135}, required over 120 days of computations~\cite{ostrowski2011solving}. Another example is the football pool problem, where an instance with only 729 binary variables still remains unsolved after years of computational effort~\cite{linderoth2009improving}. In both these cases, the underlying problem is a covering problem. +While ILP is NP-Hard, extremely large instances of ILP are rutinely solved on a daily basis in a variety of industries such as transportation, energy, online dating, network optimization, graph optimization and has recently been explored in the context of quantum annealing~\cite{2020arXiv200713788H}. +The fact that instances with millions of integer variables can be solved is due to the incredible improvement in classical ILP methods. +One estimate has that the speed of ILP algorithms, combined with hardware improvements, have lead to a speedup of around 200 billion\cite{bertsimas2014statistics}. +Despite this fact, there are still very small problems that still require several months, or even years, of computational effort to solve. For example, the Steiner Triple Covering problems can be notoriously difficult; an instance with only 135 binary variables, \texttt{STS135}, required over 120 days of computations~\cite{ostrowski2011solving}. Another example is the football pool problem, where an instance with only 729 binary variables still remains unsolved after years of computational effort~\cite{linderoth2009improving}. +In both these cases, the underlying problem is a covering problem. - - -{\color{red} In general, ILP is classically NP-hard, and as a result, heuristic methods are employed~\cite{GLOVER1986533, doi:10.1287/ijoc.1.3.190, doi:10.1287/ijoc.2.1.4}. +In general, heuristic methods are employed~\cite{GLOVER1986533, doi:10.1287/ijoc.1.3.190, doi:10.1287/ijoc.2.1.4} to solve ILP problems. Standard classical heuristic algorithms follow a greedy scheme which iteratively approximates optimal solutions--starting from a random initial guess, these algorithms apply locally optimal choices at each step. -The NP-hardness of ILP can be understood by realizing that while the solution to an $n$-dimensional linear problem must lie on the vertices of the feasibility region, while the optimal integer solution may in general be at any integer solution inside the feasibility region. -While greedy algorithms do not guarantee optimal global solutions, they find feasible solutions in polynomial time, which can be utilized in further computations. ILP is a commonly tackled problem applicable to situations such as scheduling, network optimization, and graph optimization such as the minimum dominating set problem (MDS). ILP can also be used to identify reaction pathways of biological system, and has recently been explored in the context of quantum annealing~\cite{2020arXiv200713788H}.} +While greedy algorithms do not guarantee optimal global solutions, they find feasible solutions in polynomial time, which can be utilized in further computations. In this paper we will consider instances of the minimum dominating set (MDS) problem. For a given a graph $G(E,V)$, defined by the set of $V$ vertices and $E$ edges, a dominating set $D$ is a specific subset of vertices $D \subseteq V$. In particular, $D$ is a dominating set if all vertices in $V$ but not in $D$ are adjacent to at least one vertex in $D$. @@ -118,11 +119,11 @@ \section{INTRODUCTION} \begin{figure*} \centering \begin{tabular}{p{0.2\textwidth}p{0.1\textwidth}p{0.2\textwidth}p{0.1\textwidth}p{0.2\textwidth}} - \includegraphics[width=0.2\textwidth]{./MDS_mds0.pdf} + \includegraphics[width=0.2\textwidth]{./new_figures/MDS_mds0.pdf} && - \includegraphics[width=0.2\textwidth]{./MDS_mds1.pdf} + \includegraphics[width=0.2\textwidth]{./new_figures/MDS_mds1.pdf} && - \includegraphics[width=0.2\textwidth]{./MDS_mds2.pdf}\\ + \includegraphics[width=0.2\textwidth]{./new_figures/MDS_mds2.pdf}\\ \centering\textbf{a} && \centering\textbf{b} && \centering\textbf{c} \end{tabular} \caption{Example of different dominating sets for $G(V, E)$. Vertices in the dominating set $D$ are highlighted in blue. {\textbf{a)}} A dominating set of $G$ with domination number $\overline{\overline{D}} = 2$. {\textbf{b)}} A minimal dominating set of $G$ with domination number of $\overline{\overline{D}} = 2$. {\textbf{c)}} The MDS of $G$ with domination number of $\overline{\overline{D}} = 1$.} @@ -278,7 +279,7 @@ \subsection{Annealer Results for the Dominating Set} \centering \begin{tabular}{cc} $G(n):$ & - \raisebox{-.4\height}{\includegraphics[width=0.8\columnwidth]{./linear_graph.pdf}} + \raisebox{-.4\height}{\includegraphics[width=0.8\columnwidth]{./new_figures/linear_graph.pdf}} \end{tabular} \caption{Linear graphs $G(n)$ used in this study. Nodes denote vertices of the graphs and lines are undirected edges.} \label{fig:linear} @@ -300,7 +301,7 @@ \subsection{Annealer Results for the Dominating Set} \begin{figure} \centering - \includegraphics[width=\columnwidth]{./DWave_scaling.pdf} + \includegraphics[width=\columnwidth]{./new_figures/DWave_scaling.pdf} \caption{Baseline result of D-Wave (black) compared to random guessing (dashed green). The jagged nature of random guessing reflects the degeneracy of the ground state. Negative offsets with a `s' tag (blue) are results from delays of large values of $|h_i|$, while negative offsets with the `w' label (red) delay the schedule of qubits with small values of $h_i$.} \label{fig:baseline} \end{figure} @@ -319,7 +320,7 @@ \subsection{Simulation Results} \begin{figure}[b] \centering - \includegraphics[width=\columnwidth]{./NN2_offset_scaling.pdf} + \includegraphics[width=\columnwidth]{./new_figures/NN2_offset_scaling.pdf} \caption{The probability of finding the MDS for $G(2)$ from D-Wave (black) and simulation (dashed yellow) at annealing times of 1 $\mu s$.} \label{fig:dwave1us} \end{figure} @@ -360,7 +361,7 @@ \subsection{Simulation Results} \begin{figure} \centering - \includegraphics[width=\columnwidth]{./final_state_distribution.pdf} + \includegraphics[width=\columnwidth]{./new_figures/final_state_distribution.pdf} \caption{Final state distribution from D-Wave (solid bars) and simulation (black outline). The colors label the type of offset The $(0, 1, 0, 0, 0)$ state is the first solution where vertex 1 is in the dominating set. The $(1, 0, 0, 1, 0)$ state is the second solution where vertex 0 is in the dominating set. The first excited state is the $(1, 1, 1, 1, 1)$ state where both vertices are in the dominating set.} \label{fig:final_state_distribution} \end{figure} @@ -380,7 +381,7 @@ \subsection{Dynamics of Time Evolution} \begin{figure} \centering - \includegraphics[width=\columnwidth]{./time_dependent_probability.pdf} + \includegraphics[width=\columnwidth]{./new_figures/time_dependent_probability.pdf} \caption{Time-dependent probability for resolving the ground state for simulation results presented in Sec.~\ref{sec:results:simulation}. Different offsets are labeled in the same way as Fig.~\ref{fig:baseline}.} \label{fig:td_prob} \end{figure} @@ -397,9 +398,9 @@ \subsection{Idealized Quantum Annealing} \begin{figure*} \centering \begin{tabular}{p{0.5\textwidth}p{0.5\textwidth}} - \includegraphics[width=0.5\textwidth]{./anneal_schedule_extended.pdf} + \includegraphics[width=0.5\textwidth]{./new_figures/anneal_schedule_extended.pdf} & - \includegraphics[width=0.5\textwidth]{./NN2_offset_scaling_extended.pdf}\\ + \includegraphics[width=0.5\textwidth]{./new_figures/NN2_offset_scaling_extended.pdf}\\ \centering \textbf{a} & \centering \textbf{b} \end{tabular} \centering @@ -597,7 +598,7 @@ \subsubsection{Annealing Schedule and Offsets} \begin{figure}[htb] \centering - \includegraphics[width=\columnwidth]{./anneal_schedule.pdf} + \includegraphics[width=\columnwidth]{./new_figures/anneal_schedule.pdf} \caption{ Anneal schedules for amplitudes of initial Hamiltonian (dashed) and final Hamiltonian (solid). Offset 0.0 (black) and -0.05 (purple) are displayed. @@ -640,7 +641,7 @@ \subsubsection{Pure Transverse Field Simulation} \begin{figure} \centering - \includegraphics[width=\columnwidth]{./vacuum_probability.pdf} + \includegraphics[width=\columnwidth]{./new_figures/vacuum_probability.pdf} \caption{Time-dependent vacuum probability of 5 qubits system under pure transverse field.} \label{figcheck} \end{figure} From 12a76289c92f3416cf9a07ba065222c12f3cf80b Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Christopher=20K=C3=B6rber?= <35035055+ckoerber@users.noreply.github.com> Date: Thu, 11 Feb 2021 11:37:28 +0100 Subject: [PATCH 3/4] Update compile-tex.yaml Remove slack notification from latex action --- .github/workflows/compile-tex.yaml | 7 ------- 1 file changed, 7 deletions(-) diff --git a/.github/workflows/compile-tex.yaml b/.github/workflows/compile-tex.yaml index 58ff8e4..28a1090 100644 --- a/.github/workflows/compile-tex.yaml +++ b/.github/workflows/compile-tex.yaml @@ -16,10 +16,3 @@ jobs: with: name: main.pdf path: paper/main.pdf - - name: Send slack notification - uses: rtCamp/action-slack-notify@v2.0.2 - env: - SLACK_WEBHOOK: https://hooks.slack.com/services/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 - SLACK_USERNAME: "CompileBot" - SLACK_TITLE: "PDF compiled!" - SLACK_MESSAGE: "I was able to compile the most recent pdf. 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Lett.}\ }\textbf {\bibinfo {volume} {109}},\ + \bibinfo {pages} {017202} (\bibinfo {year} {2012})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Chang}\ \emph {et~al.}(2019)\citenamefont {Chang}, + \citenamefont {Gambhir}, \citenamefont {Humble},\ and\ \citenamefont + {Sota}}]{Chang:2018uoc}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {C.~C.}\ \bibnamefont + {Chang}}, \bibinfo {author} {\bibfnamefont {A.}~\bibnamefont {Gambhir}}, + \bibinfo {author} {\bibfnamefont {T.~S.}\ \bibnamefont {Humble}}, \ and\ + \bibinfo {author} {\bibfnamefont {S.}~\bibnamefont {Sota}},\ }\href {\doibase + 10.1038/s41598-019-46729-0} {\bibfield {journal} {\bibinfo {journal} {Sci. + Rep.}\ }\textbf {\bibinfo {volume} {9}},\ \bibinfo {pages} {10258} (\bibinfo + {year} {2019})},\ \Eprint {http://arxiv.org/abs/1812.06917} {arXiv:1812.06917 + [quant-ph]} \BibitemShut {NoStop}% +%%CITATION = ARXIV:1812.06917;%% +\bibitem [{\citenamefont {Esposito}\ \emph {et~al.}(2009)\citenamefont + {Esposito}, \citenamefont {Harbola},\ and\ \citenamefont + {Mukamel}}]{RevModPhys.81.1665}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {M.}~\bibnamefont + {Esposito}}, \bibinfo {author} {\bibfnamefont {U.}~\bibnamefont {Harbola}}, \ + and\ \bibinfo {author} {\bibfnamefont {S.}~\bibnamefont {Mukamel}},\ }\href + {\doibase 10.1103/RevModPhys.81.1665} {\bibfield {journal} {\bibinfo + {journal} {Rev. Mod. Phys.}\ }\textbf {\bibinfo {volume} {81}},\ \bibinfo + {pages} {1665} (\bibinfo {year} {2009})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Nielsen}\ and\ \citenamefont + {Chuang}(2011)}]{10.5555/1972505}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {M.~A.}\ \bibnamefont + {Nielsen}}\ and\ \bibinfo {author} {\bibfnamefont {I.~L.}\ \bibnamefont + {Chuang}},\ }\href@noop {} {\emph {\bibinfo {title} {Quantum Computation and + Quantum Information: 10th Anniversary Edition}}},\ \bibinfo {edition} {10th}\ + ed.\ (\bibinfo {publisher} {Cambridge University Press},\ \bibinfo {address} + {USA},\ \bibinfo {year} {2011})\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Preskill}(1998)}]{preskill1998lecture}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {J.}~\bibnamefont + {Preskill}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} + {California Institute of Technology}\ }\textbf {\bibinfo {volume} {16}} + (\bibinfo {year} {1998})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {{King}}\ \emph {et~al.}(2016)\citenamefont {{King}}, + \citenamefont {{Hoskinson}}, \citenamefont {{Lanting}}, \citenamefont + {{Andriyash}},\ and\ \citenamefont {{Amin}}}]{2016PhRvA..93e2320K}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {A.~D.}\ \bibnamefont + {{King}}}, \bibinfo {author} {\bibfnamefont {E.}~\bibnamefont {{Hoskinson}}}, + \bibinfo {author} {\bibfnamefont {T.}~\bibnamefont {{Lanting}}}, \bibinfo + {author} {\bibfnamefont {E.}~\bibnamefont {{Andriyash}}}, \ and\ \bibinfo + {author} {\bibfnamefont {M.~H.}\ \bibnamefont {{Amin}}},\ }\href {\doibase + 10.1103/PhysRevA.93.052320} {\bibfield {journal} {\bibinfo {journal} + {\pra}\ }\textbf {\bibinfo {volume} {93}},\ \bibinfo {eid} {052320} (\bibinfo + {year} {2016})},\ \Eprint {http://arxiv.org/abs/1512.07325} {arXiv:1512.07325 + [quant-ph]} \BibitemShut {NoStop}% +\bibitem [{\citenamefont {{Mandr{\`a}}}\ \emph {et~al.}(2017)\citenamefont + {{Mandr{\`a}}}, \citenamefont {{Zhu}},\ and\ \citenamefont + {{Katzgraber}}}]{2017PhRvL.118g0502M}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {S.}~\bibnamefont + {{Mandr{\`a}}}}, \bibinfo {author} {\bibfnamefont {Z.}~\bibnamefont {{Zhu}}}, + \ and\ \bibinfo {author} {\bibfnamefont {H.~G.}\ \bibnamefont + {{Katzgraber}}},\ }\href {\doibase 10.1103/PhysRevLett.118.070502} {\bibfield + {journal} {\bibinfo {journal} {\prl}\ }\textbf {\bibinfo {volume} {118}},\ + \bibinfo {eid} {070502} (\bibinfo {year} {2017})},\ \Eprint + {http://arxiv.org/abs/1606.07146} {arXiv:1606.07146 [quant-ph]} \BibitemShut + {NoStop}% +\bibitem [{\citenamefont {Systems}(2017)}]{dwave_temp}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {D.-W.}\ \bibnamefont + {Systems}},\ }\href + {https://www.dwavesys.com/sites/default/files/D-Wave%202000Q%20Tech%20Collateral_0117F_0.pdf} + {\enquote {\bibinfo {title} {The d-wave 2000q quantum computer technology + overview},}\ } (\bibinfo {year} {2017})\BibitemShut {NoStop}% +\bibitem [{\citenamefont {{Chiorescu}}\ \emph {et~al.}(2003)\citenamefont + {{Chiorescu}}, \citenamefont {{Nakamura}}, \citenamefont {{Harmans}},\ and\ + \citenamefont {{Mooij}}}]{2003Sci...299.1869C}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {I.}~\bibnamefont + {{Chiorescu}}}, \bibinfo {author} {\bibfnamefont {Y.}~\bibnamefont + {{Nakamura}}}, \bibinfo {author} {\bibfnamefont {C.~J.~P.~M.}\ \bibnamefont + {{Harmans}}}, \ and\ \bibinfo {author} {\bibfnamefont {J.~E.}\ \bibnamefont + {{Mooij}}},\ }\href {\doibase 10.1126/science.1081045} {\bibfield {journal} + {\bibinfo {journal} {Science}\ }\textbf {\bibinfo {volume} {299}},\ \bibinfo + {pages} {1869} (\bibinfo {year} {2003})},\ \Eprint + {http://arxiv.org/abs/cond-mat/0305461} {arXiv:cond-mat/0305461 + [cond-mat.mes-hall]} \BibitemShut {NoStop}% +\bibitem [{\citenamefont {{Luitz}}\ \emph {et~al.}(2015)\citenamefont + {{Luitz}}, \citenamefont {{Laflorencie}},\ and\ \citenamefont + {{Alet}}}]{2015PhRvB..91h1103L}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {D.~J.}\ \bibnamefont + {{Luitz}}}, \bibinfo {author} {\bibfnamefont {N.}~\bibnamefont + {{Laflorencie}}}, \ and\ \bibinfo {author} {\bibfnamefont {F.}~\bibnamefont + {{Alet}}},\ }\href {\doibase 10.1103/PhysRevB.91.081103} {\bibfield + {journal} {\bibinfo {journal} {\prb}\ }\textbf {\bibinfo {volume} {91}},\ + \bibinfo {eid} {081103} (\bibinfo {year} {2015})},\ \Eprint + {http://arxiv.org/abs/1411.0660} {arXiv:1411.0660 [cond-mat.dis-nn]} + \BibitemShut {NoStop}% +\bibitem [{\citenamefont {Goldenfeld}(1992)}]{Goldenfeld:1992qy}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {N.}~\bibnamefont + {Goldenfeld}},\ }\href@noop {} {\emph {\bibinfo {title} {{Lectures on phase + transitions and the renormalization group}}}}\ (\bibinfo {year} + {1992})\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Systems}(2020{\natexlab{a}})}]{dwave_oceans}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {D.-W.}\ \bibnamefont + {Systems}},\ }\href {https://github.com/dwavesystems/dwave-ocean-sdk} + {\enquote {\bibinfo {title} {D-wave cloud client},}\ } (\bibinfo {year} + {2020}{\natexlab{a}})\BibitemShut {NoStop}% +\bibitem [{\citenamefont {{Choi}}(2008)}]{2008arXiv0804.4884C}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {V.}~\bibnamefont + {{Choi}}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {ArXiv + e-prints}\ } (\bibinfo {year} {2008})},\ \Eprint + {http://arxiv.org/abs/0804.4884} {arXiv:0804.4884 [quant-ph]} \BibitemShut + {NoStop}% +\bibitem [{\citenamefont {Systems}(2020{\natexlab{b}})}]{dwave_as}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {D.-W.}\ \bibnamefont + {Systems}},\ }\href + {https://support.dwavesys.com/hc/en-us/articles/360005267253-QPU-Specific-Anneal-Schedules} + {\enquote {\bibinfo {title} {Qpu-specific anneal schedules},}\ } (\bibinfo + {year} {2020}{\natexlab{b}})\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Systems}(2020{\natexlab{c}})}]{dwave_as_docu}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {D.-W.}\ \bibnamefont + {Systems}},\ }\href {https://docs.dwavesys.com/docs/latest/doc_qpu.html} + {\enquote {\bibinfo {title} {Technical description of the d-wave quantum + processing unit},}\ } (\bibinfo {year} {2020}{\natexlab{c}})\BibitemShut + {NoStop}% +\bibitem [{\citenamefont {Chang}\ \emph + {et~al.}(2020{\natexlab{a}})\citenamefont {Chang}, \citenamefont {Chen},\ + and\ \citenamefont {K\"o{}rber}}]{github:cchang5/quantum_linear_programming}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {C.~C.}\ \bibnamefont + {Chang}}, \bibinfo {author} {\bibfnamefont {C.-C.}\ \bibnamefont {Chen}}, \ + and\ \bibinfo {author} {\bibfnamefont {C.}~\bibnamefont {K\"o{}rber}},\ + }\href@noop {} {\enquote {\bibinfo {title} {Quantum linear programming},}\ + }\bibinfo {howpublished} + {\url{https://github.com/cchang5/quantum_linear_programming} \texttt{tag: + arXiv}} (\bibinfo {year} {2020}{\natexlab{a}})\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Chang}\ \emph + {et~al.}(2020{\natexlab{b}})\citenamefont {Chang}, \citenamefont {K\"orber},\ + and\ \citenamefont {Walker-Loud}}]{Chang:2019khk}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {C.~C.}\ \bibnamefont + {Chang}}, \bibinfo {author} {\bibfnamefont {C.}~\bibnamefont {K\"orber}}, \ + and\ \bibinfo {author} {\bibfnamefont {A.}~\bibnamefont {Walker-Loud}},\ + }\href {\doibase 10.21105/joss.02007} {\bibfield {journal} {\bibinfo + {journal} {J. Open Source Softw.}\ }\textbf {\bibinfo {volume} {5}},\ + \bibinfo {pages} {2007} (\bibinfo {year} {2020}{\natexlab{b}})},\ \Eprint + {http://arxiv.org/abs/1912.03580} {arXiv:1912.03580 [hep-lat]} \BibitemShut + {NoStop}% +\end{thebibliography}% diff --git a/paper/v2/main.bib b/paper/v2/main.bib new file mode 100644 index 0000000..2a1aa6e --- /dev/null +++ b/paper/v2/main.bib @@ -0,0 +1,1010 @@ +@inproceedings{bertsimas2014statistics, + title={Statistics and machine learning via a modern optimization lens}, + author={Bertsimas, Dimitris}, + booktitle={INFORMS Annual Meeting}, + year={2014} +} + + +@article{ostrowski2011solving, + title={Solving large Steiner triple covering problems}, + author={Ostrowski, James and Linderoth, Jeff and Rossi, Fabrizio and Smriglio, Stefano}, + journal={Operations Research Letters}, + volume={39}, + number={2}, + pages={127--131}, + year={2011}, + publisher={Elsevier} +} + +@article{linderoth2009improving, + title={Improving bounds on the football pool problem by integer programming and high-throughput computing}, + author={Linderoth, Jeff and Margot, Fran{\c{c}}ois and Thain, Greg}, + journal={INFORMS Journal on Computing}, + volume={21}, + number={3}, + pages={445--457}, + year={2009}, + publisher={INFORMS} +} + + +@ARTICLE{2020arXiv200713788H, + author = {{Hauke}, Philipp and {Mattiotti}, Giovanni and {Faccioli}, Pietro}, + title = "{Dominant Reaction Pathways by Quantum Computing}", + journal = {arXiv e-prints}, + keywords = {Quantum Physics, Condensed Matter - Disordered Systems and Neural Networks, Physics - Biological Physics}, + year = 2020, + month = jul, + eid = {arXiv:2007.13788}, + pages = {arXiv:2007.13788}, +archivePrefix = {arXiv}, + eprint = {2007.13788}, + primaryClass = {quant-ph}, + adsurl = {https://ui.adsabs.harvard.edu/abs/2020arXiv200713788H}, + adsnote = {Provided by the SAO/NASA Astrophysics Data System} +} + +@ARTICLE{2020RPPh...83e4401H, + author = {{Hauke}, Philipp and {Katzgraber}, Helmut G. and {Lechner}, Wolfgang and + {Nishimori}, Hidetoshi and {Oliver}, William D.}, + title = "{Perspectives of quantum annealing: methods and implementations}", + journal = {Reports on Progress in Physics}, + keywords = {review, quantum annealing, adiabatic quantum optimization, Quantum Physics}, + year = 2020, + month = may, + volume = {83}, + number = {5}, + eid = {054401}, + pages = {054401}, + doi = {10.1088/1361-6633/ab85b8}, +archivePrefix = {arXiv}, + eprint = {1903.06559}, + primaryClass = {quant-ph}, + adsurl = {https://ui.adsabs.harvard.edu/abs/2020RPPh...83e4401H}, + adsnote = {Provided by the SAO/NASA Astrophysics Data System} +} + +% tabu search +@article{GLOVER1986533, +title = {Future paths for integer programming and links to artificial intelligence}, +journal = {Computers \& Operations Research}, +volume = {13}, +number = {5}, +pages = {533 - 549}, +year = {1986}, +note = {Applications of Integer Programming}, +issn = {0305-0548}, +doi = {https://doi.org/10.1016/0305-0548(86)90048-1}, +url = {http://www.sciencedirect.com/science/article/pii/0305054886900481}, +author = {Glover, Fred}, +} + + +@article{doi:10.1287/ijoc.1.3.190, +author = {Glover, Fred}, +title = {Tabu Search—Part I}, +journal = {ORSA Journal on Computing}, +volume = {1}, +number = {3}, +pages = {190-206}, +year = {1989}, +doi = {10.1287/ijoc.1.3.190}, +URL = {https://doi.org/10.1287/ijoc.1.3.190}, +eprint = {https://doi.org/10.1287/ijoc.1.3.190} +} + +@article{doi:10.1287/ijoc.2.1.4, +author = {Glover, Fred}, +title = {Tabu Search—Part II}, +journal = {ORSA Journal on Computing}, +volume = {2}, +number = {1}, +pages = {4-32}, +year = {1990}, +doi = {10.1287/ijoc.2.1.4}, +URL = {https://doi.org/10.1287/ijoc.2.1.4}, +eprint = {https://doi.org/10.1287/ijoc.2.1.4} +} + +%floppy qubits +@ARTICLE{2016PhRvA..93e2320K, + author = {{King}, Andrew D. and {Hoskinson}, Emile and {Lanting}, Trevor and + {Andriyash}, Evgeny and {Amin}, Mohammad H.}, + title = "{Degeneracy, degree, and heavy tails in quantum annealing}", + journal = {\pra}, + keywords = {Quantum Physics}, + year = 2016, + month = may, + volume = {93}, + number = {5}, + eid = {052320}, + pages = {052320}, + doi = {10.1103/PhysRevA.93.052320}, +archivePrefix = {arXiv}, + eprint = {1512.07325}, + primaryClass = {quant-ph}, + adsurl = {https://ui.adsabs.harvard.edu/abs/2016PhRvA..93e2320K}, + adsnote = {Provided by the SAO/NASA Astrophysics Data System} +} + +@ARTICLE{2017PhRvL.118g0502M, + author = {{Mandr{\`a}}, Salvatore and {Zhu}, Zheng and {Katzgraber}, Helmut G.}, + title = "{Exponentially Biased Ground-State Sampling of Quantum Annealing Machines with Transverse-Field Driving Hamiltonians}", + journal = {\prl}, + keywords = {Quantum Physics, Condensed Matter - Disordered Systems and Neural Networks, Physics - Computational Physics}, + year = 2017, + month = feb, + volume = {118}, + number = {7}, + eid = {070502}, + pages = {070502}, + doi = {10.1103/PhysRevLett.118.070502}, +archivePrefix = {arXiv}, + eprint = {1606.07146}, + primaryClass = {quant-ph}, + adsurl = {https://ui.adsabs.harvard.edu/abs/2017PhRvL.118g0502M}, + adsnote = {Provided by the SAO/NASA Astrophysics Data System} +} + + +%% +@book{Goldenfeld:1992qy, + author = "Goldenfeld, N.", + title = "{Lectures on phase transitions and the renormalization group}", + year = "1992" +} + +@ARTICLE{2015PhRvB..91h1103L, + author = {{Luitz}, David J. and {Laflorencie}, Nicolas and {Alet}, Fabien}, + title = "{Many-body localization edge in the random-field Heisenberg chain}", + journal = {\prb}, + keywords = {75.10.Pq, 05.30.Rt, 72.15.Rn, Spin chain models, Localization effects, Condensed Matter - Disordered Systems and Neural Networks, Condensed Matter - Strongly Correlated Electrons}, + year = 2015, + month = feb, + volume = {91}, + number = {8}, + eid = {081103}, + pages = {081103}, + doi = {10.1103/PhysRevB.91.081103}, +archivePrefix = {arXiv}, + eprint = {1411.0660}, + primaryClass = {cond-mat.dis-nn}, + adsurl = {https://ui.adsabs.harvard.edu/abs/2015PhRvB..91h1103L}, + adsnote = {Provided by the SAO/NASA Astrophysics Data System} +} + +# flux qubit coherence time +@ARTICLE{2003Sci...299.1869C, + author = {{Chiorescu}, I. and {Nakamura}, Y. and {Harmans}, C.~J.~P.~M. and + {Mooij}, J.~E.}, + title = "{Coherent Quantum Dynamics of a Superconducting Flux Qubit}", + journal = {Science}, + keywords = {PHYSICS, Condensed Matter - Mesoscale and Nanoscale Physics, Condensed Matter - Superconductivity}, + year = 2003, + month = mar, + volume = {299}, + number = {5614}, + pages = {1869-1872}, + doi = {10.1126/science.1081045}, +archivePrefix = {arXiv}, + eprint = {cond-mat/0305461}, + primaryClass = {cond-mat.mes-hall}, + adsurl = {https://ui.adsabs.harvard.edu/abs/2003Sci...299.1869C}, + adsnote = {Provided by the SAO/NASA Astrophysics Data System} +} + +@misc{dwave_temp, +author = {D-Wave Systems}, +title = {The D-Wave 2000Q Quantum Computer Technology Overview}, +url = {https://www.dwavesys.com/sites/default/files/D-Wave%202000Q%20Tech%20Collateral_0117F_0.pdf}, +year = 2017, +} + +@misc{dwave_offset, +author = {D-Wave Systems}, +title = {Boosting integer factoring performance via quantum annealing offsets}, +url = {https://www.dwavesys.com/sites/default/files/14-1002A_B_tr_Boosting_integer_factorization_via_quantum_annealing_offsets.pdf}, +year = 2016, +} + +@misc{dwave_as, + author = {D-Wave Systems}, + title = {QPU-Specific Anneal Schedules}, + url = {https://support.dwavesys.com/hc/en-us/articles/360005267253-QPU-Specific-Anneal-Schedules}, + year = 2020, +} + +@misc{dwave_as_docu, + author = {D-Wave Systems}, + title = {Technical Description of the D-Wave Quantum Processing Unit}, + url = {https://docs.dwavesys.com/docs/latest/doc_qpu.html}, + month = August, + year = 2020, +} + +@misc{dwave_oceans, + author = {D-Wave Systems}, + title = {D-Wave Cloud Client}, + month = August, + year = 2020, + version = {2.5.0}, + url = {https://github.com/dwavesystems/dwave-ocean-sdk} + } + +@article{Chang:2018uoc, + author = "Chang, Chia Cheng and Gambhir, Arjun and Humble, Travis + S. and Sota, Shigetoshi", + title = "{Quantum annealing for systems of polynomial equations}", + journal = "Sci. Rep.", + volume = "9", + year = "2019", + number = "1", + pages = "10258", + doi = "10.1038/s41598-019-46729-0", + eprint = "1812.06917", + archivePrefix = "arXiv", + primaryClass = "quant-ph", + reportNumber = "RIKEN-iTHEMS-Report-19, RIKEN-iTHEMS-Report-18", + SLACcitation = "%%CITATION = ARXIV:1812.06917;%%" +} + +% Github code + + + +@article{Jiang2018, + title={Quantum Annealing for Prime Factorization}, + author={Shuxian Jiang and Keith A. Britt and Alexander J. McCaskey and Travis S. Humble and Sabre Kais }, + journal={Scientific Reports}, + volume={8}, + pages={17667}, + year={2086} +} + + +@article{denchev2016computational, + title={What is the computational value of finite-range tunneling?}, + author={Denchev, Vasil S and Boixo, Sergio and Isakov, Sergei V and Ding, Nan and Babbush, Ryan and Smelyanskiy, Vadim and Martinis, John and Neven, Hartmut}, + journal={Physical Review X}, + volume={6}, + number={3}, + pages={031015}, + year={2016}, + publisher={APS} +} + + +@article{katzgraber2014glassy, + title={Glassy chimeras could be blind to quantum speedup: Designing better benchmarks for quantum annealing machines}, + author={Katzgraber, Helmut G and Hamze, Firas and Andrist, Ruben S}, + journal={Physical Review X}, + volume={4}, + number={2}, + pages={021008}, + year={2014}, + publisher={APS} +} + + +@article{PhysRevA.94.022337, + title = {Strengths and weaknesses of weak-strong cluster problems: A detailed overview of state-of-the-art classical heuristics versus quantum approaches}, + author = {Mandr\`a, Salvatore and Zhu, Zheng and Wang, Wenlong and Perdomo-Ortiz, Alejandro and Katzgraber, Helmut G.}, + journal = {Phys. Rev. A}, + volume = {94}, + issue = {2}, + pages = {022337}, + numpages = {13}, + year = {2016}, + month = {Aug}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevA.94.022337}, + url = {https://link.aps.org/doi/10.1103/PhysRevA.94.022337} +} + + +@article{ronnow2014defining, + title={Defining and detecting quantum speedup}, + author={R{\o}nnow, Troels F and Wang, Zhihui and Job, Joshua and Boixo, Sergio and Isakov, Sergei V and Wecker, David and Martinis, John M and Lidar, Daniel A and Troyer, Matthias}, + journal={Science}, + volume={345}, + number={6195}, + pages={420--424}, + year={2014}, + publisher={American Association for the Advancement of Science} +} + + +@article{PhysRevX.8.031016, + title = {Demonstration of a Scaling Advantage for a Quantum Annealer over Simulated Annealing}, + author = {Albash, Tameem and Lidar, Daniel A.}, + journal = {Phys. Rev. X}, + volume = {8}, + issue = {3}, + pages = {031016}, + numpages = {26}, + year = {2018}, + month = {Jul}, + doi = {10.1103/PhysRevX.8.031016} +} + + + +@article{mandra2018, + author={Salvatore Mandrà and Helmut G Katzgraber}, + title={A deceptive step towards quantum speedup detection}, + journal={Quantum Science and Technology}, + volume={3}, + number={4}, + pages={04LT01}, + url={http://stacks.iop.org/2058-9565/3/i=4/a=04LT01}, + year={2018}, + abstract={There have been multiple attempts to design synthetic benchmark problems with the goal of detecting quantum speedup in current quantum annealing (QA) machines. To date, classical heuristics have consistently outperformed quantum annealing based approaches. Here we introduce a class of problems based on frustrated cluster loops—deceptive cluster loops—for which all currently known state of-the-art classical heuristics are outperformed by the DW2000Q QA machine. While there is a sizable constant speedup over all known classical heuristics, a noticeable improvement in the scaling remains elusive. These results represent the first steps towards a detection of potential quantum speedup, albeit without a scaling improvement and for synthetic benchmark problems.} +} + + +% Minor embedding +@ARTICLE{2008arXiv0804.4884C, + author = {{Choi}, V.}, + title = "{Minor-Embedding in Adiabatic Quantum Computation: I. The Parameter Setting Problem}", + journal = {ArXiv e-prints}, + archivePrefix = "arXiv", + eprint = {0804.4884}, + primaryClass = "quant-ph", + keywords = {Quantum Physics}, + year = 2008, + month = apr, + adsurl = {http://adsabs.harvard.edu/abs/2008arXiv0804.4884C}, + adsnote = {Provided by the SAO/NASA Astrophysics Data System} +} + + +@article{PhysRevB.39.11828, + title = {Sherrington-Kirkpatrick model in a transverse field: Absence of replica symmetry breaking due to quantum fluctuations}, + author = {Ray, P. and Chakrabarti, B. K. and Chakrabarti, Arunava}, + journal = {Phys. Rev. B}, + volume = {39}, + issue = {16}, + pages = {11828--11832}, + numpages = {0}, + year = {1989}, + month = {Jun}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevB.39.11828}, + url = {https://link.aps.org/doi/10.1103/PhysRevB.39.11828} +} + +@article{RevModPhys.80.1061, + title = {Colloquium: Quantum annealing and analog quantum computation}, + author = {Das, Arnab and Chakrabarti, Bikas K.}, + journal = {Rev. Mod. Phys.}, + volume = {80}, + issue = {3}, + pages = {1061--1081}, + numpages = {0}, + year = {2008}, + month = {Sep}, + publisher = {American Physical Society}, + doi = {10.1103/RevModPhys.80.1061}, + url = {https://link.aps.org/doi/10.1103/RevModPhys.80.1061} +} + + +%Nishimori +@ARTICLE{1998PhRvE..58.5355K, + author = {{Kadowaki}, T. and {Nishimori}, H.}, + title = "{Quantum annealing in the transverse Ising model}", + journal = {Phys. Rev. E}, + eprint = {cond-mat/9804280}, + keywords = {Quantum statistical mechanics, Spin-glass and other random models, Information theory and communication theory}, + year = 1998, + month = nov, + volume = 58, + pages = {5355-5363}, + doi = {10.1103/PhysRevE.58.5355}, + adsurl = {http://adsabs.harvard.edu/abs/1998PhRvE..58.5355K}, + adsnote = {Provided by the SAO/NASA Astrophysics Data System} +} + + +%Farhi +@ARTICLE{2000quant.ph..1106F, + author = {{Farhi}, E. and {Goldstone}, J. and {Gutmann}, S. and {Sipser}, M. + }, + title = "{Quantum Computation by Adiabatic Evolution}", + journal = {eprint arXiv:quant-ph/0001106}, + eprint = {quant-ph/0001106}, + keywords = {Quantum Physics}, + year = 2000, + month = jan, + adsurl = {http://adsabs.harvard.edu/abs/2000quant.ph..1106F}, + adsnote = {Provided by the SAO/NASA Astrophysics Data System} +} + +%chimera +@ARTICLE{2014ITAS...2418294B, + author = {{Bunyk}, P.~I. and {Hoskinson}, E.~M. and {Johnson}, M.~W. and + {Tolkacheva}, E. and {Altomare}, F. and {Berkley}, A.~J. and + {Harris}, R. and {Hilton}, J.~P. and {Lanting}, T. and {Przybysz}, A.~J. and + {Whittaker}, J.}, + title = "{Architectural Considerations in the Design of a Superconducting Quantum Annealing Processor}", + journal = {IEEE Transactions on Applied Superconductivity}, + archivePrefix = "arXiv", + eprint = {1401.5504}, + primaryClass = "quant-ph", + year = 2014, + month = aug, + volume = 24, + eid = {2318294}, + pages = {2318294}, + doi = {10.1109/TASC.2014.2318294}, + adsurl = {http://adsabs.harvard.edu/abs/2014ITAS...2418294B}, + adsnote = {Provided by the SAO/NASA Astrophysics Data System} +} + + +@ARTICLE{2017PhRvP...8f4025M, + author = {{Marshall}, J. and {Rieffel}, E.~G. and {Hen}, I.}, + title = "{Thermalization, Freeze-out, and Noise: Deciphering Experimental Quantum Annealers}", + journal = {Physical Review Applied}, + archivePrefix = "arXiv", + eprint = {1703.03902}, + primaryClass = "quant-ph", + year = 2017, + month = dec, + volume = 8, + number = 6, + eid = {064025}, + pages = {064025}, + doi = {10.1103/PhysRevApplied.8.064025}, + adsurl = {http://adsabs.harvard.edu/abs/2017PhRvP...8f4025M}, + adsnote = {Provided by the SAO/NASA Astrophysics Data System} +} + +@ARTICLE{2016PhRvE..94c2105N, + author = {{Nishimura}, K. and {Nishimori}, H. and {Ochoa}, A.~J. and {Katzgraber}, H.~G. + }, + title = "{Retrieving the ground state of spin glasses using thermal noise: Performance of quantum annealing at finite temperatures}", + journal = {Phys. Rev. E}, + archivePrefix = "arXiv", + eprint = {1605.03303}, + primaryClass = "cond-mat.dis-nn", + year = 2016, + month = sep, + volume = 94, + number = 3, + eid = {032105}, + pages = {032105}, + doi = {10.1103/PhysRevE.94.032105}, + adsurl = {http://adsabs.harvard.edu/abs/2016PhRvE..94c2105N}, + adsnote = {Provided by the SAO/NASA Astrophysics Data System} +} + +@ARTICLE{2016PhRvB..93v4414W, + author = {{Wang}, W. and {Machta}, J. and {Katzgraber}, H.~G.}, + title = "{Bond chaos in spin glasses revealed through thermal boundary conditions}", + journal = {Phys. Rev. B}, + archivePrefix = "arXiv", + eprint = {1603.00543}, + primaryClass = "cond-mat.dis-nn", + year = 2016, + month = jun, + volume = 93, + number = 22, + eid = {224414}, + pages = {224414}, + doi = {10.1103/PhysRevB.93.224414}, + adsurl = {http://adsabs.harvard.edu/abs/2016PhRvB..93v4414W}, + adsnote = {Provided by the SAO/NASA Astrophysics Data System} +} + +@ARTICLE{2015PhRvA..91f2320A, + author = {{Albash}, T. and {Lidar}, D.~A.}, + title = "{Decoherence in adiabatic quantum computation}", + journal = {Phys. Rev. A}, + archivePrefix = "arXiv", + eprint = {1503.08767}, + primaryClass = "quant-ph", + keywords = {Quantum computation, Decoherence, open systems, quantum statistical methods}, + year = 2015, + month = jun, + volume = 91, + number = 6, + eid = {062320}, + pages = {062320}, + doi = {10.1103/PhysRevA.91.062320}, + adsurl = {http://adsabs.harvard.edu/abs/2015PhRvA..91f2320A}, + adsnote = {Provided by the SAO/NASA Astrophysics Data System} +} + + +@ARTICLE{2009PhRvA..79b2107A, + author = {{Amin}, M.~H.~S. and {Averin}, D.~V. and {Nesteroff}, J.~A.}, + title = "{Decoherence in adiabatic quantum computation}", + journal = {Phys. Rev. A}, + archivePrefix = "arXiv", + eprint = {0708.0384}, + primaryClass = "cond-mat.mes-hall", + keywords = {Decoherence, open systems, quantum statistical methods, Quantum computation}, + year = 2009, + month = feb, + volume = 79, + number = 2, + eid = {022107}, + pages = {022107}, + doi = {10.1103/PhysRevA.79.022107}, + adsurl = {http://adsabs.harvard.edu/abs/2009PhRvA..79b2107A}, + adsnote = {Provided by the SAO/NASA Astrophysics Data System} +} + +@article{Dickson13, + title = "Thermally assisted quantum annealing of a 16-qubit problem", + abstract = "Efforts to develop useful quantum computers have been blocked primarily by environmental noise. Quantum annealing is a scheme of quantum computation that is predicted to be more robust against noise, because despite the thermal environment mixing the system's state in the energy basis, the system partially retains coherence in the computational basis, and hence is able to establish well-defined eigenstates. Here we examine the environment's effect on quantum annealing using 16 qubits of a superconducting quantum processor. For a problem instance with an isolated small-gap anticrossing between the lowest two energy levels, we experimentally demonstrate that, even with annealing times eight orders of magnitude longer than the predicted single-qubit decoherence time, the probabilities of performing a successful computation are similar to those expected for a fully coherent system. Moreover, for the problem studied, we show that quantum annealing can take advantage of a thermal environment to achieve a speedup factor of up to 1,000 over a closed system.", + author = "Dickson, {N. G.} and Johnson, {M. W.} and Amin, {M. H.} and R. Harris and F. Altomare and Berkley, {A. J.} and P. Bunyk and J. Cai and Chapple, {E. M.} and P. Chavez and F. Cioata and T. Cirip and P. Debuen and M. Drew-Brook and C. Enderud and S. Gildert and F. Hamze and Hilton, {J. P.} and E. Hoskinson and K. Karimi and E. Ladizinsky and N. Ladizinsky and T. Lanting and T. Mahon and R. Neufeld and T. Oh and I. Perminov and C. Petroff and A. Przybysz and C. Rich and P. Spear and A. Tcaciuc and Thom, {M. C.} and E. Tolkacheva and S. Uchaikin and J. Wang and Wilson, {A. B.} and Z. Merali and G. Rose", + year = "2013", + month = "6", + day = "12", + doi = "10.1038/ncomms2920", + language = "English", + volume = "4", + journal = "Nature Communications", + issn = "2041-1723", + publisher = "Nature Publishing Group", + +} + + + +@ARTICLE{2001Sci...292..472F, + author = {{Farhi}, E. and {Goldstone}, J. and {Gutmann}, S. and {Lapan}, J. and + {Lundgren}, A. and {Preda}, D.}, + title = "{A Quantum Adiabatic Evolution Algorithm Applied to Random Instances of an NP-Complete Problem}", + journal = {Science}, + eprint = {quant-ph/0104129}, + year = 2001, + month = apr, + volume = 292, + pages = {472-476}, + doi = {10.1126/science.1057726}, + adsurl = {http://adsabs.harvard.edu/abs/2001Sci...292..472F}, + adsnote = {Provided by the SAO/NASA Astrophysics Data System} +} + + +@article{aitken_1936, + title={IV.—On Least Squares and Linear Combination of Observations}, + volume={55}, + DOI={10.1017/S0370164600014346}, + journal={Proceedings of the Royal Society of Edinburgh}, + publisher={Royal Society of Edinburgh Scotland Foundation}, + author={Aitken, A. C.}, + year={1936}, + pages={42–48} +} + + +@ARTICLE{2004quant.ph..5098A, + author = {{Aharonov}, D. and {van Dam}, W. and {Kempe}, J. and {Landau}, Z. and + {Lloyd}, S. and {Regev}, O.}, + title = "{Adiabatic Quantum Computation is Equivalent to Standard Quantum Computation}", + journal = {eprint arXiv:quant-ph/0405098}, + eprint = {quant-ph/0405098}, + keywords = {Quantum Physics}, + year = 2004, + month = may, + adsurl = {http://adsabs.harvard.edu/abs/2004quant.ph..5098A}, + adsnote = {Provided by the SAO/NASA Astrophysics Data System} +} + + +@article{citeulike:9077321, + abstract = {{An iterative algorithm is given for solving a system Ax=k of n linear equations in n unknowns. The solution is given in n steps. It is shown that this method is a special case of a very general method which also includes Gaussian elimination. These general algorithms are essentilaly algorithms for finding an n dimensional ellipsoid. Connections are made with the theory of orthogonal polynomials and continued fractions.}}, + added-at = {2017-06-29T07:13:07.000+0200}, + author = {Hestenes, Magnus R. and Stiefel, Eduard}, + biburl = {https://www.bibsonomy.org/bibtex/2ad1ecd0242ef1e7c8a02b3f1ea173d28/gdmcbain}, + citeulike-article-id = {9077321}, + citeulike-attachment-1 = {V49N06A08.pdf; /pdf/user/gdmcbain/article/9077321/632277/V49N06A08.pdf; 466daddfb6340c28cb8da548007028c8cc5df687}, + citeulike-linkout-0 = {http://dx.doi.org/10.6028/jres.049.044}, + citeulike-linkout-1 = {http://www.ams.org/mathscinet-getitem?mr=0060307}, + citeulike-linkout-2 = {http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.129.6828}, + doi = {10.6028/jres.049.044}, + file = {V49N06A08.pdf}, + interhash = {bcb34a6f8b9fb2f2371e92430116a8ad}, + intrahash = {ad1ecd0242ef1e7c8a02b3f1ea173d28}, + journal = {Journal of Research of the National Bureau of Standards}, + keywords = {65f10-iterative-methods-for-linear-systems}, + month = dec, + number = 6, + pages = {409--436}, + posted-at = {2011-03-29 23:49:34}, + priority = {0}, + timestamp = {2017-06-29T07:13:07.000+0200}, + title = {{Methods of Conjugate Gradients for Solving Linear Systems}}, + url = {http://www.ams.org/mathscinet-getitem?mr=0060307}, + volume = 49, + year = 1952 +} + + +@inproceedings{Boros2012, + author = {E. Boros and A. Gruber}, + title = {On quadratization of pseudo-Boolean functions}, + booktitle = {ISAIM}, + year = {2012} +} + +@article{humble2014integrated, + title={An integrated programming and development environment for adiabatic quantum optimization}, + author={Humble, Travis S and McCaskey, Alex J and Bennink, Ryan S and Billings, Jay Jay and D'Azevedo, EF and Sullivan, Blair D and Klymko, Christine F and Seddiqi, Hadayat}, + journal={Computational Science and Discovery}, + volume={7}, + number={1}, + pages={015006}, + year={2014}, + publisher={IOP Publishing} +} + +@article{Klymko2014, + author="Christine F. Klymko and Blair D. Sullivan and Travis S. Humble", + title="Adiabatic quantum programming: minor embedding with hard faults", + year="2014", + journal={Quantum Inf. Process.}, + pages={709-729}, + volume=13 +} + +@article{Choi2008, + author="V. Choi", + title="Minor-embedding in adiabatic quantum computation: I. The parameter setting problem", + journal="Quantum Inf. Process", + volume="7", + year="2008", + pages="193--209" +} + +@article{Choi2011, + author="V. Choi", + title="Minor-embedding in adiabatic quantum computation: II. Minor-universal graph design", + journal="Quantum Inf. Process", + volume="10", + year="2011", + pages="343--353" +} + +@article{zheng2017solving, + title={Solving systems of linear equations with a superconducting quantum processor}, + author={Zheng, Yarui and Song, Chao and Chen, Ming-Cheng and Xia, Benxiang and Liu, Wuxin and Guo, Qiujiang and Zhang, Libo and Xu, Da and Deng, Hui and Huang, Keqiang and others}, + journal={Physical review letters}, + volume={118}, + number={21}, + pages={210504}, + year={2017}, + publisher={APS} +} + +@article{subasi2018quantum, + title={Quantum algorithms for linear systems of equations inspired by adiabatic quantum computing}, + author={Subasi, Yigit and Somma, Rolando D and Orsucci, Davide}, + journal={arXiv preprint arXiv:1805.10549}, + year={2018} +} + +@article{xin2018quantum, + title={A Quantum Algorithm for Solving Linear Differential Equations: Theory and Experiment}, + author={Xin, Tao and Wei, Shijie and Cui, Jianlian and Xiao, Junxiang and Arrazola, I{\~n}igo and Lamata, Lucas and Kong, Xiangyu and Lu, Dawei and Solano, Enrique and Long, Guilu}, + journal={arXiv preprint arXiv:1807.04553}, + year={2018} +} + +@article{wen2018experimental, + title={Experimental realization of quantum algorithms for linear system inspired by adiabatic quantum computing}, + author={Wen, Jingwei and Kong, Xiangyu and Wei, Shijie and Xin, Tao and Wang, Bixue and Li, Keren and Zhu, Yuanye and Long, Guilu}, + journal={arXiv preprint arXiv:1806.03295}, + year={2018} +} + +@ARTICLE{2013arXiv1303.1377F, + author = {{Frommer}, Andreas and {Kahl}, Karsten and {Krieg}, Stefan and + {Leder}, Bj{\"o}rn and {Rottmann}, Matthias}, + title = "{Adaptive Aggregation Based Domain Decomposition Multigrid for the Lattice Wilson Dirac Operator}", + journal = {arXiv e-prints}, + keywords = {High Energy Physics - Lattice}, + year = "2013", + month = "Mar", + eid = {arXiv:1303.1377}, + pages = {arXiv:1303.1377}, + archivePrefix = {arXiv}, + eprint = {1303.1377}, + primaryClass = {hep-lat}, + adsurl = {https://ui.adsabs.harvard.edu/abs/2013arXiv1303.1377F}, + adsnote = {Provided by the SAO/NASA Astrophysics Data System} +} + + + +@article{Fomin2009, + doi = {10.1145/1552285.1552286}, + url = {https://doi.org/10.1145/1552285.1552286}, + year = {2009}, + month = aug, + publisher = {Association for Computing Machinery ({ACM})}, + volume = {56}, + number = {5}, + pages = {1--32}, + author = {Fedor V. Fomin and Fabrizio Grandoni and Dieter Kratsch}, + title = {A measure {\&} conquer approach for the analysis of exact algorithms}, + journal = {Journal of the {ACM}} +} + +@incollection{vanRooij2009, + doi = {10.1007/978-3-642-04128-0_50}, + url = {https://doi.org/10.1007/978-3-642-04128-0_50}, + year = {2009}, + publisher = {Springer Berlin Heidelberg}, + pages = {554--565}, + author = {Johan M. M. van Rooij and Jesper Nederlof and Thomas C. van Dijk}, + title = {Inclusion/Exclusion Meets Measure and Conquer}, + booktitle = {Lecture Notes in Computer Science} +} + +@ARTICLE{2018Glover, + author = {{Glover}, Fred and {Kochenberger}, Gary and {Du}, Yu}, + title = "{A Tutorial on Formulating and Using QUBO Models}", + journal = {arXiv e-prints}, + keywords = {Computer Science - Data Structures and Algorithms, Computer Science - Discrete Mathematics, Mathematics - Optimization and Control, Quantum Physics, 90C27}, + year = 2018, + month = nov, + eid = {arXiv:1811.11538}, + pages = {arXiv:1811.11538}, + archivePrefix = {arXiv}, + eprint = {1811.11538}, + primaryClass = {cs.DS}, + adsurl = {https://ui.adsabs.harvard.edu/abs/2018arXiv181111538G}, + adsnote = {Provided by the SAO/NASA Astrophysics Data System} +} + + +@article{PhysRevE.90.022103, + title = {Lindblad-equation approach for the full counting statistics of work and heat in driven quantum systems}, + author = {Silaev, Mihail and Heikkil\"a, Tero T. and Virtanen, Pauli}, + journal = {Phys. Rev. E}, + volume = {90}, + issue = {2}, + pages = {022103}, + numpages = {8}, + year = {2014}, + month = {Aug}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevE.90.022103}, + url = {https://link.aps.org/doi/10.1103/PhysRevE.90.022103} +} + + +@article{PhysRevB.91.081103, + title = {Many-body localization edge in the random-field Heisenberg chain}, + author = {Luitz, David J. and Laflorencie, Nicolas and Alet, Fabien}, + journal = {Phys. Rev. B}, + volume = {91}, + issue = {8}, + pages = {081103}, + numpages = {5}, + year = {2015}, + month = {Feb}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevB.91.081103}, + url = {https://link.aps.org/doi/10.1103/PhysRevB.91.081103} +} + + +@article{RevModPhys.91.021001, + title = {Colloquium: Many-body localization, thermalization, and entanglement}, + author = {Abanin, Dmitry A. and Altman, Ehud and Bloch, Immanuel and Serbyn, Maksym}, + journal = {Rev. Mod. Phys.}, + volume = {91}, + issue = {2}, + pages = {021001}, + numpages = {26}, + year = {2019}, + month = {May}, + publisher = {American Physical Society}, + doi = {10.1103/RevModPhys.91.021001}, + url = {https://link.aps.org/doi/10.1103/RevModPhys.91.021001} +} + + + + + +@article{doi:10.1146/annurev-conmatphys-031214-014726, + author = {Nandkishore, Rahul and Huse, David A.}, + title = {Many-Body Localization and Thermalization in Quantum Statistical Mechanics}, + journal = {Annual Review of Condensed Matter Physics}, + volume = {6}, + number = {1}, + pages = {15-38}, + year = {2015}, + doi = {10.1146/annurev-conmatphys-031214-014726}, + + URL = { + https://doi.org/10.1146/annurev-conmatphys-031214-014726 + + }, + eprint = { + https://doi.org/10.1146/annurev-conmatphys-031214-014726 + + } +} + + + +@article{doi:10.7566/JPSJ.89.044001, + author = {Takada ,Kabuki and Yamashiro ,Yu and Nishimori ,Hidetoshi}, + title = {Mean-Field Solution of the Weak-Strong Cluster Problem for Quantum Annealing with Stoquastic and Non-Stoquastic Catalysts}, + journal = {Journal of the Physical Society of Japan}, + volume = {89}, + number = {4}, + pages = {044001}, + year = {2020}, + doi = {10.7566/JPSJ.89.044001}, + + URL = { + https://doi.org/10.7566/JPSJ.89.044001 + + }, + eprint = { + https://doi.org/10.7566/JPSJ.89.044001 + + } + +} + + +@article{PhysRevA.96.042322, + title = {Experimental demonstration of perturbative anticrossing mitigation using nonuniform driver Hamiltonians}, + author = {Lanting, Trevor and King, Andrew D. and Evert, Bram and Hoskinson, Emile}, + journal = {Phys. Rev. A}, + volume = {96}, + issue = {4}, + pages = {042322}, + numpages = {8}, + year = {2017}, + month = {Oct}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevA.96.042322}, + url = {https://link.aps.org/doi/10.1103/PhysRevA.96.042322} +} + +@misc{hsu2018quantum, + title={Quantum annealing with anneal path control: application to 2-SAT problems with known energy landscapes}, + author={Ting-Jui Hsu and Fengping Jin and Christian Seidel and Florian Neukart and Hans De Raedt and Kristel Michielsen}, + year={2018}, + eprint={1810.00194}, + archivePrefix={arXiv}, + primaryClass={quant-ph} +} + + +@InProceedings{10.1007/978-3-030-14082-3_14, + author="Yarkoni, Sheir + and Wang, Hao + and Plaat, Aske + and B{\"a}ck, Thomas", + editor="Feld, Sebastian + and Linnhoff-Popien, Claudia", + title="Boosting Quantum Annealing Performance Using Evolution Strategies for Annealing Offsets Tuning", + booktitle="Quantum Technology and Optimization Problems", + year="2019", + publisher="Springer International Publishing", + address="Cham", + pages="157--168", + abstract="In this paper we introduce a novel algorithm to iteratively tune annealing offsets for qubits in a D-Wave 2000Q quantum processing unit (QPU). Using a (1+1)-CMA-ES algorithm, we are able to improve the performance of the QPU by up to a factor of 12.4 in probability of obtaining ground states for small problems, and obtain previously inaccessible (i.e., better) solutions for larger problems. We also make efficient use of QPU samples as a resource, using 100 times less resources than existing tuning methods. The success of this approach demonstrates how quantum computing can benefit from classical algorithms, and opens the door to new hybrid methods of computing.", + isbn="978-3-030-14082-3" +} + + +@article{ALET2018498, + title = "Many-body localization: An introduction and selected topics", + journal = "Comptes Rendus Physique", + volume = "19", + number = "6", + pages = "498 - 525", + year = "2018", + note = "Quantum simulation / Simulation quantique", + issn = "1631-0705", + doi = "https://doi.org/10.1016/j.crhy.2018.03.003", + url = "http://www.sciencedirect.com/science/article/pii/S163107051830032X", + author = "Fabien Alet and Nicolas Laflorencie", + keywords = "Many-body localization, Thermalization, Simulations, Entanglement, Localisation à corps, Thermalisation, Simulations, Intrication" +} + + +@article{PhysRevB.82.174411, + title = {Many-body localization phase transition}, + author = {Pal, Arijeet and Huse, David A.}, + journal = {Phys. Rev. B}, + volume = {82}, + issue = {17}, + pages = {174411}, + numpages = {7}, + year = {2010}, + month = {Nov}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevB.82.174411}, + url = {https://link.aps.org/doi/10.1103/PhysRevB.82.174411} +} + + +@article{PhysRevLett.109.017202, + title = {Unbounded Growth of Entanglement in Models of Many-Body Localization}, + author = {Bardarson, Jens H. and Pollmann, Frank and Moore, Joel E.}, + journal = {Phys. Rev. Lett.}, + volume = {109}, + issue = {1}, + pages = {017202}, + numpages = {5}, + year = {2012}, + month = {Jul}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevLett.109.017202}, + url = {https://link.aps.org/doi/10.1103/PhysRevLett.109.017202} +} + + +@book{10.5555/1972505, + author = {Nielsen, Michael A. and Chuang, Isaac L.}, + title = {Quantum Computation and Quantum Information: 10th Anniversary Edition}, + year = {2011}, + isbn = {1107002176}, + publisher = {Cambridge University Press}, + address = {USA}, + edition = {10th} +} + +@article{preskill1998lecture, + title={Lecture notes for physics 229: Quantum information and computation}, + author={Preskill, John}, + journal={California Institute of Technology}, + volume={16}, + year={1998} +} + +@article{RevModPhys.81.1665, + title = {Nonequilibrium fluctuations, fluctuation theorems, and counting statistics in quantum systems}, + author = {Esposito, Massimiliano and Harbola, Upendra and Mukamel, Shaul}, + journal = {Rev. Mod. Phys.}, + volume = {81}, + issue = {4}, + pages = {1665--1702}, + numpages = {0}, + year = {2009}, + month = {Dec}, + publisher = {American Physical Society}, + doi = {10.1103/RevModPhys.81.1665}, + url = {https://link.aps.org/doi/10.1103/RevModPhys.81.1665} +} + + +@misc{github:cchang5/quantum_linear_programming, + author = {Chang, Chia Cheng and Chen, Chih-Chieh and K\"o{}rber, Christopher}, + title = {Quantum Linear Programming}, + year = {2020}, + publisher = {GitHub}, + journal = {GitHub repository}, + howpublished = {\url{https://github.com/cchang5/quantum_linear_programming} \texttt{tag: arXiv}}, + tag = {arXiv}, +} + +@article{Chang:2019khk, + author = {Chang, Chia Cheng and K\"orber, Christopher and Walker-Loud, Andr\'e}, + title = "{EspressoDB: A scientific database for managing high-performance computing workflows}", + eprint = "1912.03580", + archivePrefix = "arXiv", + primaryClass = "hep-lat", + reportNumber = "RIKEN-iTHEMS-Report-19", + doi = "10.21105/joss.02007", + journal = "J. 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Open Quantum System}{}% 2 +\BOOKMARK [1][-]{section*.1}{Abstract}{section*.2}% 1 +\BOOKMARK [1][-]{section*.3}{INTRODUCTION}{section*.2}% 3 +\BOOKMARK [1][-]{section*.4}{RESULTS}{section*.2}% 4 +\BOOKMARK [2][-]{section*.5}{QUBO Formulation of ILP}{section*.4}% 5 +\BOOKMARK [2][-]{section*.6}{Annealer Results for the Dominating Set}{section*.4}% 6 +\BOOKMARK [2][-]{section*.7}{Simulation Results}{section*.4}% 7 +\BOOKMARK [1][-]{section*.8}{Discussion}{section*.2}% 8 +\BOOKMARK [2][-]{section*.9}{Dynamics of Time Evolution}{section*.8}% 9 +\BOOKMARK [2][-]{section*.10}{Idealized Quantum Annealing}{section*.8}% 10 +\BOOKMARK [2][-]{section*.11}{Final Remarks}{section*.8}% 11 +\BOOKMARK [1][-]{section*.12}{METHODS}{section*.2}% 12 +\BOOKMARK [2][-]{section*.13}{Additional Mappings to QUBO}{section*.12}% 13 +\BOOKMARK [3][-]{section*.14}{Minimum Dominating Set QUBO}{section*.13}% 14 +\BOOKMARK [3][-]{section*.15}{Integer Quadratic Optimization}{section*.13}% 15 +\BOOKMARK [2][-]{section*.16}{ILP on the D-Wave}{section*.12}% 16 +\BOOKMARK [3][-]{section*.17}{Comment on ILP QUBO Penalty Term}{section*.16}% 17 +\BOOKMARK [3][-]{section*.18}{Minor Embedding}{section*.16}% 18 +\BOOKMARK [2][-]{section*.19}{Simulation of a Quantum Annealer}{section*.12}% 19 +\BOOKMARK [3][-]{section*.20}{The Lindblad Equation}{section*.19}% 20 +\BOOKMARK [3][-]{section*.21}{Annealing Schedule and Offsets}{section*.19}% 21 +\BOOKMARK [3][-]{section*.22}{Pure Transverse Field Simulation}{section*.19}% 22 +\BOOKMARK [1][-]{section*.23}{DATA AVAILABILITY}{section*.2}% 23 +\BOOKMARK [1][-]{section*.24}{ACKNOWLEDGEMENTS}{section*.2}% 24 +\BOOKMARK [1][-]{section*.25}{AUTHOR CONTRIBUTIONS}{section*.2}% 25 +\BOOKMARK [1][-]{section*.26}{CORRESPONDING AUTHOR}{section*.2}% 26 +\BOOKMARK [1][-]{section*.27}{ADDITIONAL INFORMATION}{section*.2}% 27 +\BOOKMARK [1][-]{section*.28}{References}{section*.2}% 28 diff --git a/paper/v2/main.pdf b/paper/v2/main.pdf new file mode 100644 index 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