210:, one may consider the variables in the problem to be classical degrees of freedom, and the cost functions to be the potential energy function (classical Hamiltonian). Then a suitable term consisting of non-commuting variable(s) (i.e. variables that have non-zero commutator with the variables of the original mathematical problem) has to be introduced artificially in the Hamiltonian to play the role of the tunneling field (kinetic part). Then one may carry out the simulation with the quantum Hamiltonian thus constructed (the original function + non-commuting part) just as described above. Here, there is a choice in selecting the non-commuting term and the efficiency of annealing may depend on that.
182:, whose "temperature" parameter plays a similar role to QA's tunneling field strength. In simulated annealing, the temperature determines the probability of moving to a state of higher "energy" from a single current state. In quantum annealing, the strength of transverse field determines the quantum-mechanical probability to change the amplitudes of all states in parallel. Analytical and numerical evidence suggests that quantum annealing outperforms simulated annealing under certain conditions (see for a careful analysis, and, for a fully solvable model of quantum annealing to arbitrary target Hamiltonian and comparison of different computation approaches).
191:
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918:) problems, the general structure of quantum annealing-based algorithms and two examples of this kind of algorithms for solving instances of the max-SAT and Minimum Multicut problems, together with an overview of the quantum annealing systems manufactured by D-Wave Systems. Hybrid quantum-classic algorithms for large-scale discrete-continuous optimization problems were reported to illustrate the quantum advantage.
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868:(1QBit) and cancer research group DNA-SEQ to focus on solving real-world problems with quantum hardware. As the first company dedicated to producing software applications for commercially available quantum computers, 1QBit's research and development arm has focused on D-Wave's quantum annealing processors and has successfully demonstrated that these processors are suitable for solving real-world applications.
880:, found "no quantum speedup" across the entire range of their tests, and only inconclusive results when looking at subsets of the tests. Their work illustrated "the subtle nature of the quantum speedup question". Further work has advanced understanding of these test metrics and their reliance on equilibrated systems, thereby missing any signatures of advantage due to quantum dynamics.
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43:
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because Shor's algorithm is not a hillclimbing process. Shor's algorithm requires a universal quantum computer. During the Qubits 2021 conference held by D-Wave, it was announced that the company is developing their first universal quantum computers, capable of running Shor's algorithm in addition to
883:
There are many open questions regarding quantum speedup. The ETH reference in the previous section is just for one class of benchmark problems. Potentially there may be other classes of problems where quantum speedup might occur. Researchers at Google, LANL, USC, Texas A&M, and D-Wave are working
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in June 2014, described as "likely the most thorough and precise study that has been done on the performance of the D-Wave machine" and "the fairest comparison yet", attempted to define and measure quantum speedup. Several definitions were put forward as some may be unverifiable by empirical tests,
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It has been demonstrated experimentally as well as theoretically, that quantum annealing can indeed outperform thermal annealing (simulated annealing) in certain cases, especially where the potential energy (cost) landscape consists of very high but thin barriers surrounding shallow local minima.
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of the barriers, for very high barriers, it is extremely difficult for thermal fluctuations to get the system out from such local minima. However, as argued earlier in 1989 by Ray, Chakrabarti & Chakrabarti, the quantum tunneling probability through the same barrier (considered in isolation)
194:
Quantum
Annealing (blue line) efficiently traverses energy landscapes by leveraging quantum tunneling to find the global minimum. Quantum annealing offers a significant performance advantage over Simulated Annealing (magenta line), unlocking the potential to solve massive optimization problems
165:
that corresponds to the solution to the original optimization problem. An experimental demonstration of the success of quantum annealing for random magnets was reported immediately after the initial theoretical proposal. Quantum annealing has also been proven to provide a fast
766:, such simulations would be much more efficient and exact than that done in a classical computer, because it can perform the tunneling directly, rather than needing to add it by hand. Moreover, it may be able to do this without the tight error controls needed to harness the
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while others, though falsified, would nonetheless allow for the existence of performance advantages. The study found that the D-Wave chip "produced no quantum speedup" and did not rule out the possibility in future tests. The researchers, led by
Matthias Troyer at the
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Smelyanskiy, Vadim N.; Rieffel, Eleanor G.; Knysh, Sergey I.; Williams, Colin P.; Johnson, Mark W.; Thom, Murray C.; Macready, William G.; Pudenz, Kristen L. (2012). "A Near-Term
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144:, a natural quantum-mechanical evolution of physical systems. The amplitudes of all candidate states keep changing, realizing a quantum parallelism, according to the time-dependent strength of the transverse field, which causes
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133:. The term "quantum annealing" was first proposed in 1988 by B. Apolloni, N. Cesa Bianchi and D. De Falco as a quantum-inspired classical algorithm. It was formulated in its present form by T. Kadowaki and H. Nishimori (
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1989 Idea was presented that quantum fluctuations could help explore rugged energy landscapes of the classical Ising spin glasses by escaping from local minima (having tall but thin barriers) using tunneling;
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purchased an adiabatic quantum computer from D-Wave
Systems with 512 qubits. An extensive study of its performance as quantum annealer, compared to some classical annealing algorithms, is already available.
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between states or essentially tunneling through peaks. If the rate of change of the transverse field is slow enough, the system stays close to the ground state of the instantaneous
Hamiltonian (also see
153:). If the rate of change of the transverse field is accelerated, the system may leave the ground state temporarily but produce a higher likelihood of concluding in the ground state of the final problem
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With demonstrations of entanglement published, the question of whether or not the D-Wave machine can demonstrate quantum speedup over all classical computers remains unanswered. A study published in
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The tunneling field is basically a kinetic energy term that does not commute with the classical potential energy part of the original glass. The whole process can be simulated in a computer using
261:
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Bapst, V.; Foini, L.; Krzakala, F.; Semerjian, G.; Zamponi, F. (2013). "The quantum adiabatic algorithm applied to random optimization problems: The quantum spin glass perspective".
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Quantum annealing starts from a quantum-mechanical superposition of all possible states (candidate states) with equal weights. Then the system evolves following the time-dependent
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announced the first commercial quantum annealer on the market by the name D-Wave One and published a paper in Nature on its performance. The company claims this system uses a 128
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Ray, P.; Chakrabarti, B. K.; Chakrabarti, A. (1989). "Sherrington-Kirkpatrick model in a transverse field: Absence of replica symmetry breaking due to quantum fluctuations".
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137:) in 1998 though an imaginary-time variant without quantum coherence had been discussed by A. B. Finnila, M. A. Gomez, C. Sebenik and J. D. Doll in 1994.
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quantum computation. The transverse field is finally switched off, and the system is expected to have reached the ground state of the classical
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Lanting, T.; Przybysz, A. J.; Smirnov, A. Yu.; Spedalieri, F. M.; et al. (2014-05-29). "Entanglement in a quantum annealing processor".
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914:"A cross-disciplinary introduction to quantum annealing-based algorithms" presents an introduction to combinatorial optimization (
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Li, F.; Chernyak, V. Y. & Sinitsyn, N. A. (2018). "Quantum annealing and thermalization: insights from integrability".
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Electro-Optical and
Infrared Systems: Technology and Applications XII; and Quantum Information Science and Technology
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D-Wave's architecture differs from traditional quantum computers. It is not known to be polynomially equivalent to a
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Mukherjee, S. & Chakrabarti, B. K. (2015). "Multivariable Optimization: Quantum Annealing & Computation".
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2265:"D-Wave Systems Building Quantum Application Ecosystem, Announces Partnerships with DNA-SEQ Alliance and 1QBit"
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2641:"Quantum computing based hybrid solution strategies for large-scale discrete-continuous optimization problems"
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1998 Formulation of quantum annealing and numerical test demonstrating its advantages in Ising glass systems;
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used in more traditional quantum algorithms. Some confirmation of this is found in exactly solvable models.
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Quantum Phase Transitions in Transverse Field Spin Models: From Statistical Physics to Quantum Information
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Dutta, A.; Aeppli, G.; Chakrabarti, B. K.; Divakaran, U.; Rosenbaum, T.F. & Sen, D. (2015).
461:, in presence of quantum tunneling, can be of major help: If the barriers are thin enough (i.e.
1157:"A Quantum adiabatic evolution algorithm applied to random instances of an NP-Complete problem"
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1547:"Analytical solution for nonadiabatic quantum annealing to arbitrary Ising spin Hamiltonian"
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2011 Superconducting-circuit quantum annealing machine built and marketed by D-Wave Systems.
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491:), quantum fluctuations can surely bring the system out of the shallow local minima. For an
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989:. Stochastic Processes, Physics and Geometry, Proceedings of the Ascona-Locarno Conference.
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Corporation entered into an agreement to purchase a D-Wave One system. On October 28, 2011
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1999 First experimental demonstration of quantum annealing in LiHoYF Ising glass magnets;
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2120:"D-Wave Systems sells its first Quantum Computing System to Lockheed Martin Corporation"
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1439:"Optimization using quantum mechanics: quantum annealing through adiabatic evolution"
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logic elements that exhibit controllable and tunable coupling to perform operations.
113:. Quantum annealing is used mainly for problems where the search space is discrete (
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2561:"D-Wave's Next-Generation Roadmap: Bringing Clarity to Practical Quantum Computing"
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Farhi, E.; Goldstone, J.; Gutmann, S.; Lapan, J.; Ludgren, A.; Preda, D. (2001).
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over a given set of candidate solutions (candidate states), by a process using
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by up to a factor of 100,000,000 on a set of hard optimization problems.
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oracle for the square-root speedup in solving many NP-complete problems.
2694:"Community detection in brain connectomes with hybrid quantum computing"
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Timeline of ideas related to quantum annealing in Ising spin glasses:
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2944:"Quantum Annealing & Computation: Challenges & Perspectives"
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Apolloni, Bruno; Cesa-Bianchi, Nicolo; De Falco, Diego (July 1988).
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2207:"Evidence for quantum annealing with more than one hundred qubits"
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Quantum Ising Phases & Transitions in Transverse Ising Models
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Johnson, M. W.; Amin, M. H. S.; Gildert, S.; et al. (2011).
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is the tunneling field. This additional handle through the width
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2835:. Lecture Note in Physics. Vol. 679. Heidelberg: Springer.
2816:. Lecture Note in Physics. Vol. 802. Heidelberg: Springer.
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Das, A.; Chakrabarti, B. K. & Stinchcombe, R. B. (2005).
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Apolloni, Bruno; Carvalho, Maria C.; De Falco, Diego (1989).
27:
Quantum physics-based metaheuristic for optimization problems
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2639:
Ajagekar, Akshay; Humble, Travis; You, Fengqi (2020-01-04).
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Brooke, J.; Bitko, D.; Rosenbaum, T. F.; Aeppli, G. (1999).
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Heim, B.; Rønnow, T. F.; Isakov, S. V.; Troyer, M. (2015).
1437:
Santoro, Giuseppe E. & Tosatti, Erio (18 August 2006).
2394:"Quantum or not, controversial computer yields no speedup"
1621:"Local Maxima and Minima, and, Absolute Maxima and Minima"
1488:"Quantum versus classical annealing of Ising spin glasses"
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processor chipset. On May 25, 2011, D-Wave announced that
414:{\displaystyle e^{-{\frac {{\sqrt {\Delta }}w}{\Gamma }}}}
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Tanaka, S.; Tamura, R. & Chakrabarti, B. K. (2017).
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Since thermal transition probabilities (proportional to
1640:"Quantum annealing in a kinetically constrained system"
57:
2928:. Cambridge & Delhi: Cambridge University Press.
2856:. Cambridge & Delhi: Cambridge University Press.
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Das, Arnab & Chakrabarti, Bikas K., eds. (2005).
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2692:Wierzbiński, M.; Falo-Roget, J.; Crimi, A. (2023).
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may be too technical for most readers to understand
2833:Quantum Annealing and Related Optimization Methods
911:other gate-model algorithms such as QAOA and VQE.
848:In May 2013 it was announced that a consortium of
813:, mounted and wire-bonded in a sample holder. The
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2925:Quantum Spin Glasses, Annealing & Computation
1039:"Quantum annealing in the transverse Ising model"
987:"A numerical implementation of quantum annealing"
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206:In the case of annealing a purely mathematical
256:{\displaystyle e^{-{\frac {\Delta }{k_{B}T}}}}
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101:) is an optimization process for finding the
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2814:Quantum Quenching, Annealing and Computation
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887:In December 2015, Google announced that the
2036:"Quantum annealing with manufactured spins"
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2150:"Google and NASA snap up quantum computer"
1968:"Quantum Annealing of a Disordered Magnet"
1266:"Quantum annealing of a disordered magnet"
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80:Learn how and when to remove this message
64:, without removing the technical details.
3215:Optimization computes maxima and minima.
845:took delivery of Lockheed's D-Wave One.
186:Quantum mechanics: analogy and advantage
4231:Continuous-variable quantum information
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858:Universities Space Research Association
2954:. Royal Society, London. January 2023.
2907:(2nd ed.). Heidelberg: Springer.
797:D-Wave Systems § Computer systems
484:{\displaystyle w\ll {\sqrt {\Delta }}}
355:of the barrier, but also on its width
3411:Principal pivoting algorithm of Lemke
878:Swiss Federal Institute of Technology
178:Quantum annealing can be compared to
62:make it understandable to non-experts
7:
2808:Chandra, Anjan K.; Das, Arnab &
2646:Computers & Chemical Engineering
2090:"Learning to program the D-Wave One"
1037:Kadowaki, T.; Nishimori, H. (1998).
817:'s processor is designed to use 128
809:Photograph of a chip constructed by
195:previously thought to be impossible.
4591:Optimization algorithms and methods
2379:
906:and, in particular, cannot execute
620:for the annealing time (instead of
3055:Successive parabolic interpolation
2499:. Vol. 9648. p. 964816.
518:
476:
428:
404:
395:
342:
321:
231:
25:
3375:Projective algorithm of Karmarkar
2670:10.1016/j.compchemeng.2019.106630
2535:"When can Quantum Annealing win?"
1891:Yan, B.; Sinitsyn, N. A. (2022).
1545:Yan, B.; Sinitsyn, N. A. (2022).
1002:"Quantum stochastic optimization"
174:Comparison to Simulated Annealing
4560:
4559:
4550:
4549:
3370:Ellipsoid algorithm of Khachiyan
3273:Sequential quadratic programming
3110:Broyden–Fletcher–Goldfarb–Shanno
511:-spin glass, the barrier height
41:
1385:Journal of Mathematical Physics
335:depends not only on the height
3328:Reduced gradient (Frank–Wolfe)
2378:Helmut Katzgraber, quoted in (
1862:10.1103/PhysRevLett.121.190601
884:to find such problem classes.
843:Information Sciences Institute
667:for thermal annealing), while
375:and is approximately given by
1:
4226:Adiabatic quantum computation
3658:Spiral optimization algorithm
3278:Successive linear programming
2793:10.1016/j.physrep.2012.10.002
2618:10.1080/00107514.2018.1450720
2559:D-Wave Systems (2021-10-05).
2419:10.1126/science.344.6190.1330
752:{\displaystyle 1/{\sqrt {N}}}
707:-independent for cases where
613:{\displaystyle e^{\sqrt {N}}}
151:adiabatic quantum computation
4277:Topological quantum computer
3396:Simplex algorithm of Dantzig
3268:Augmented Lagrangian methods
2948:Philosophical Transactions A
2392:Cho, Adrian (20 June 2014).
2126:. 2011-05-25. Archived from
2002:10.1126/science.284.5415.779
1300:10.1126/science.284.5415.779
1134:10.1016/0009-2614(94)00117-0
1020:10.1016/0304-4149(89)90040-9
866:1QB Information Technologies
314:) depend only on the height
4555:Quantum information science
3722:Quantum information science
1743:10.1140/epjst/e2015-02339-y
1465:10.1088/0305-4470/39/36/R01
1361:10.1103/PhysRevA.108.022412
762:It is speculated that in a
4612:
3950:quantum gate teleportation
2718:10.1038/s41598-023-30579-y
2473:10.1103/PhysRevA.92.052323
1927:10.1038/s41467-022-29887-0
1807:10.1103/RevModPhys.80.1061
1676:10.1103/PhysRevE.72.026701
1581:10.1038/s41467-022-29887-0
904:universal quantum computer
794:
131:traveling salesman problem
115:combinatorial optimization
32:Annealing (disambiguation)
29:
4545:
4079:Quantum Fourier transform
3975:Post-quantum cryptography
3918:Entanglement distillation
3675:
3628:
3615:
3599:Push–relabel maximum flow
3444:
3431:
3401:Revised simplex algorithm
3307:
3294:
3235:
3222:
3208:
3021:
3008:
2358:10.1103/PhysRevX.4.021041
2162:10.1038/nature.2013.12999
964:10.1103/PhysRevB.39.11828
4565:Quantum mechanics topics
4260:Quantum machine learning
4236:One-way quantum computer
4089:Quantum phase estimation
3990:Quantum key distribution
3923:Monogamy of entanglement
3124:Symmetric rank-one (SR1)
3105:Berndt–Hall–Hall–Hausman
1103:Chemical Physics Letters
1075:10.1103/PhysRevE.58.5355
551:. For constant value of
4586:Stochastic optimization
4172:Randomized benchmarking
4034:Amplitude amplification
3648:Parallel metaheuristics
3456:Approximation algorithm
3167:Powell's dog leg method
3119:Davidon–Fletcher–Powell
3015:Unconstrained nonlinear
1831:Physical Review Letters
1523:10.1126/science.aaa4170
1193:10.1126/science.1057726
524:{\displaystyle \Delta }
434:{\displaystyle \Gamma }
348:{\displaystyle \Delta }
327:{\displaystyle \Delta }
4272:Quantum Turing machine
4265:quantum neural network
4012:Quantum secret sharing
3633:Evolutionary algorithm
3216:
822:
791:D-Wave implementations
753:
721:
701:
681:
661:
634:
614:
585:
565:
545:
525:
505:
485:
455:
435:
415:
369:
349:
328:
304:
277:
257:
196:
121:; such as finding the
4344:Entanglement-assisted
4305:quantum convolutional
3980:Quantum coin flipping
3945:Quantum teleportation
3906:entanglement-assisted
3736:DiVincenzo's criteria
3406:Criss-cross algorithm
3229:Constrained nonlinear
3214:
3035:Golden-section search
2810:Chakrabarti, Bikas K.
1897:Nature Communications
1551:Nature Communications
808:
795:Further information:
754:
722:
702:
682:
680:{\displaystyle \tau }
662:
660:{\displaystyle e^{N}}
635:
633:{\displaystyle \tau }
615:
586:
584:{\displaystyle \tau }
566:
546:
526:
506:
486:
456:
436:
416:
370:
350:
329:
305:
303:{\displaystyle k_{B}}
278:
258:
193:
4155:processor benchmarks
4084:Quantum optimization
3967:Quantum cryptography
3778:physical vs. logical
3323:Cutting-plane method
2587:Contemporary Physics
1444:Journal of Physics A
768:quantum entanglement
731:
711:
691:
671:
644:
624:
595:
575:
555:
535:
515:
495:
465:
445:
425:
379:
359:
339:
318:
287:
283:the temperature and
267:
218:
142:Schrödinger equation
117:problems) with many
111:quantum fluctuations
30:For other uses, see
3868:Quantum speed limit
3763:Quantum programming
3758:Quantum information
3653:Simulated annealing
3471:Integer programming
3461:Dynamic programming
3301:Convex optimization
3162:Levenberg–Marquardt
2895:2013arXiv1310.1339G
2873:Science and Culture
2785:2013PhR...523..127B
2710:2023NatSR..13.3446W
2610:2018ConPh..59..174V
2505:2015SPIE.9648E..16S
2465:2015PhRvA..92e2323A
2410:2014Sci...344.1330C
2404:(6190): 1330–1331.
2350:2014PhRvX...4b1041L
2275:on 31 December 2019
2235:2014NatPh..10..218B
2094:D-Wave Systems blog
2060:10.1038/nature10012
2052:2011Natur.473..194J
1994:1999Sci...284..779B
1919:2022NatCo..13.2212Y
1854:2018arXiv180400371L
1789:2008RvMP...80.1061D
1735:2015EPJST.224...17M
1668:2005PhRvE..72b6701D
1573:2022NatCo..13.2212Y
1514:2015Sci...348..215H
1457:2006JPhA...39R.393S
1408:2008JMP....49l5210M
1353:2023PhRvA.108b2412S
1292:1999Sci...284..779B
1185:2001Sci...292..472F
1126:1994CPL...219..343F
1067:1998PhRvE..58.5355K
956:1989PhRvB..3911828R
950:(16): 11828–11832.
897:Quantum Monte Carlo
893:simulated annealing
856:and the non-profit
201:quantum Monte Carlo
180:simulated annealing
4596:Quantum algorithms
4517:Forest/Rigetti QCS
4253:quantum logic gate
4039:Bernstein–Vazirani
4026:Quantum algorithms
3901:Classical capacity
3785:Quantum processors
3768:Quantum simulation
3333:Subgradient method
3217:
3142:Conjugate gradient
3050:Nelder–Mead method
2698:Scientific Reports
2513:10.1117/12.2202661
2148:Jones, N. (2013).
823:
749:
717:
697:
677:
657:
630:
610:
581:
561:
541:
521:
501:
481:
451:
431:
411:
365:
345:
324:
312:Boltzmann constant
300:
273:
253:
208:objective function
197:
107:objective function
4573:
4572:
4484:
4483:
4381:Linear optical QC
4162:Quantum supremacy
4116:complexity theory
4069:Quantum annealing
4020:
4019:
3957:Superdense coding
3746:Quantum computing
3688:
3687:
3671:
3670:
3611:
3610:
3607:
3606:
3570:
3569:
3531:
3530:
3427:
3426:
3423:
3422:
3419:
3418:
3290:
3289:
3286:
3285:
3206:
3205:
3202:
3201:
3180:
3179:
2935:978-1-10711-319-0
2914:978-3-64233-038-4
2863:978-1-10706-879-7
2842:978-3-54027-987-7
2823:978-3-64211-469-4
2541:. 8 December 2015
2443:Physical Review A
2327:Physical Review X
2243:10.1038/nphys2900
1978:(5415): 779–781.
1498:(6231): 215–217.
1451:(36): R393–R431.
1416:10.1063/1.2995837
1331:Physical Review A
1007:Stoc. Proc. Appl.
943:Physical Review B
891:outperforms both
747:
720:{\displaystyle w}
700:{\displaystyle N}
607:
564:{\displaystyle w}
544:{\displaystyle N}
531:becomes of order
504:{\displaystyle N}
479:
454:{\displaystyle w}
407:
398:
368:{\displaystyle w}
276:{\displaystyle T}
249:
146:quantum tunneling
95:Quantum annealing
90:
89:
82:
16:(Redirected from
4603:
4563:
4562:
4553:
4552:
4359:
4289:error correction
4218:computing models
4184:Relaxation times
4074:Quantum counting
3963:
3911:quantum capacity
3858:No-teleportation
3843:No-communication
3715:
3708:
3701:
3692:
3617:
3533:
3499:
3476:Branch and bound
3466:Greedy algorithm
3446:
3433:
3353:
3309:
3296:
3237:
3224:
3172:Truncated Newton
3087:Wolfe conditions
3070:
3023:
3010:
2983:
2976:
2969:
2960:
2955:
2939:
2918:
2898:
2888:
2867:
2846:
2827:
2804:
2778:
2748:
2747:
2729:
2689:
2683:
2682:
2672:
2662:
2636:
2630:
2629:
2603:
2581:
2575:
2574:
2572:
2571:
2556:
2550:
2549:
2547:
2546:
2531:
2525:
2524:
2491:
2485:
2484:
2458:
2438:
2432:
2431:
2421:
2389:
2383:
2376:
2370:
2369:
2343:
2321:
2315:
2314:
2312:
2310:
2301:. Archived from
2295:"1QBit Research"
2291:
2285:
2284:
2282:
2280:
2271:. Archived from
2261:
2255:
2254:
2228:
2202:
2196:
2195:
2193:
2180:
2174:
2173:
2145:
2139:
2138:
2136:
2135:
2130:on July 23, 2011
2116:
2110:
2109:
2107:
2105:
2100:on July 23, 2011
2096:. Archived from
2086:
2080:
2079:
2031:
2022:
2021:
1987:
1985:cond-mat/0105238
1963:
1957:
1956:
1938:
1912:
1888:
1882:
1881:
1847:
1825:
1819:
1818:
1800:
1782:
1773:(3): 1061–1081.
1761:
1755:
1754:
1728:
1706:
1700:
1699:
1694:. Archived from
1661:
1659:cond-mat/0502167
1635:
1629:
1628:
1617:
1611:
1610:
1592:
1566:
1542:
1536:
1535:
1525:
1507:
1483:
1477:
1476:
1434:
1428:
1427:
1401:
1379:
1373:
1372:
1346:
1326:
1320:
1319:
1285:
1283:cond-mat/0105238
1276:(5415): 779–81.
1261:
1255:
1254:
1252:
1240:
1234:
1233:
1231:
1219:
1213:
1212:
1178:
1176:quant-ph/0104129
1152:
1146:
1145:
1119:
1110:(5–6): 343–348.
1097:
1091:
1090:
1085:. Archived from
1060:
1058:cond-mat/9804280
1034:
1025:
1024:
1022:
997:
991:
990:
982:
976:
975:
937:
908:Shor's algorithm
764:quantum computer
758:
756:
755:
750:
748:
743:
741:
726:
724:
723:
718:
706:
704:
703:
698:
687:can even become
686:
684:
683:
678:
666:
664:
663:
658:
656:
655:
640:proportional to
639:
637:
636:
631:
619:
617:
616:
611:
609:
608:
603:
591:proportional to
590:
588:
587:
582:
570:
568:
567:
562:
550:
548:
547:
542:
530:
528:
527:
522:
510:
508:
507:
502:
490:
488:
487:
482:
480:
475:
460:
458:
457:
452:
440:
438:
437:
432:
420:
418:
417:
412:
410:
409:
408:
403:
399:
394:
391:
374:
372:
371:
366:
354:
352:
351:
346:
333:
331:
330:
325:
309:
307:
306:
301:
299:
298:
282:
280:
279:
274:
262:
260:
259:
254:
252:
251:
250:
248:
244:
243:
230:
85:
78:
74:
71:
65:
45:
44:
37:
21:
18:Quantum annealer
4611:
4610:
4606:
4605:
4604:
4602:
4601:
4600:
4576:
4575:
4574:
4569:
4541:
4491:
4480:
4453:Superconducting
4447:
4413:
4404:Neutral atom QC
4396:Ultracold atoms
4390:
4355:implementations
4354:
4348:
4288:
4281:
4248:Quantum circuit
4216:
4210:
4204:
4194:
4154:
4148:
4115:
4108:
4064:Hidden subgroup
4016:
4005:other protocols
3961:
3938:quantum network
3933:Quantum channel
3893:
3887:
3833:No-broadcasting
3823:Gottesman–Knill
3796:
3724:
3719:
3689:
3684:
3667:
3624:
3603:
3566:
3527:
3504:
3493:
3486:
3440:
3415:
3379:
3346:
3337:
3314:
3303:
3282:
3256:
3252:Penalty methods
3247:Barrier methods
3231:
3218:
3198:
3194:Newton's method
3176:
3128:
3091:
3059:
3040:Powell's method
3017:
3004:
2987:
2942:
2936:
2921:
2915:
2902:
2870:
2864:
2849:
2843:
2830:
2824:
2812:, eds. (2010).
2807:
2763:Physics Reports
2760:
2757:
2755:Further reading
2752:
2751:
2691:
2690:
2686:
2638:
2637:
2633:
2583:
2582:
2578:
2569:
2567:
2558:
2557:
2553:
2544:
2542:
2533:
2532:
2528:
2493:
2492:
2488:
2440:
2439:
2435:
2391:
2390:
2386:
2377:
2373:
2323:
2322:
2318:
2308:
2306:
2305:on 19 June 2014
2293:
2292:
2288:
2278:
2276:
2263:
2262:
2258:
2204:
2203:
2199:
2182:
2181:
2177:
2147:
2146:
2142:
2133:
2131:
2118:
2117:
2113:
2103:
2101:
2088:
2087:
2083:
2046:(7346): 194–8.
2033:
2032:
2025:
1965:
1964:
1960:
1890:
1889:
1885:
1827:
1826:
1822:
1798:10.1.1.563.9990
1767:Rev. Mod. Phys.
1763:
1762:
1758:
1709:
1707:
1703:
1637:
1636:
1632:
1619:
1618:
1614:
1544:
1543:
1539:
1485:
1484:
1480:
1436:
1435:
1431:
1381:
1380:
1376:
1328:
1327:
1323:
1263:
1262:
1258:
1242:
1241:
1237:
1221:
1220:
1216:
1169:(5516): 472–5.
1154:
1153:
1149:
1117:chem-ph/9404003
1099:
1098:
1094:
1036:
1035:
1028:
999:
998:
994:
984:
983:
979:
939:
938:
929:
924:
835:Lockheed Martin
819:superconducting
803:
793:
729:
728:
709:
708:
689:
688:
669:
668:
647:
642:
641:
622:
621:
598:
593:
592:
573:
572:
553:
552:
533:
532:
513:
512:
493:
492:
463:
462:
443:
442:
423:
422:
392:
382:
377:
376:
357:
356:
337:
336:
316:
315:
290:
285:
284:
265:
264:
235:
234:
221:
216:
215:
188:
176:
86:
75:
69:
66:
58:help improve it
55:
46:
42:
35:
28:
23:
22:
15:
12:
11:
5:
4609:
4607:
4599:
4598:
4593:
4588:
4578:
4577:
4571:
4570:
4568:
4567:
4557:
4546:
4543:
4542:
4540:
4539:
4537:many others...
4534:
4529:
4524:
4519:
4510:
4496:
4494:
4486:
4485:
4482:
4481:
4479:
4478:
4473:
4468:
4463:
4457:
4455:
4449:
4448:
4446:
4445:
4440:
4435:
4430:
4424:
4422:
4415:
4414:
4412:
4411:
4409:Trapped-ion QC
4406:
4400:
4398:
4392:
4391:
4389:
4388:
4383:
4378:
4373:
4367:
4365:
4363:Quantum optics
4356:
4350:
4349:
4347:
4346:
4341:
4340:
4339:
4332:
4327:
4322:
4317:
4312:
4307:
4302:
4293:
4291:
4283:
4282:
4280:
4279:
4274:
4269:
4268:
4267:
4257:
4256:
4255:
4245:
4244:
4243:
4233:
4228:
4222:
4220:
4212:
4211:
4209:
4208:
4207:
4206:
4202:
4196:
4192:
4181:
4180:
4179:
4169:
4167:Quantum volume
4164:
4158:
4156:
4150:
4149:
4147:
4146:
4141:
4136:
4131:
4126:
4120:
4118:
4110:
4109:
4107:
4106:
4101:
4096:
4091:
4086:
4081:
4076:
4071:
4066:
4061:
4056:
4051:
4046:
4044:Boson sampling
4041:
4036:
4030:
4028:
4022:
4021:
4018:
4017:
4015:
4014:
4009:
4008:
4007:
4002:
3997:
3987:
3982:
3977:
3971:
3969:
3960:
3959:
3954:
3953:
3952:
3942:
3941:
3940:
3930:
3925:
3920:
3915:
3914:
3913:
3908:
3897:
3895:
3889:
3888:
3886:
3885:
3880:
3878:Solovay–Kitaev
3875:
3870:
3865:
3860:
3855:
3850:
3845:
3840:
3835:
3830:
3825:
3820:
3815:
3810:
3804:
3802:
3798:
3797:
3795:
3794:
3793:
3792:
3782:
3781:
3780:
3770:
3765:
3760:
3755:
3754:
3753:
3743:
3738:
3732:
3730:
3726:
3725:
3720:
3718:
3717:
3710:
3703:
3695:
3686:
3685:
3683:
3682:
3676:
3673:
3672:
3669:
3668:
3666:
3665:
3660:
3655:
3650:
3645:
3640:
3635:
3629:
3626:
3625:
3622:Metaheuristics
3620:
3613:
3612:
3609:
3608:
3605:
3604:
3602:
3601:
3596:
3594:Ford–Fulkerson
3591:
3586:
3580:
3578:
3572:
3571:
3568:
3567:
3565:
3564:
3562:Floyd–Warshall
3559:
3554:
3553:
3552:
3541:
3539:
3529:
3528:
3526:
3525:
3520:
3515:
3509:
3507:
3496:
3488:
3487:
3485:
3484:
3483:
3482:
3468:
3463:
3458:
3452:
3450:
3442:
3441:
3436:
3429:
3428:
3425:
3424:
3421:
3420:
3417:
3416:
3414:
3413:
3408:
3403:
3398:
3392:
3390:
3381:
3380:
3378:
3377:
3372:
3367:
3365:Affine scaling
3361:
3359:
3357:Interior point
3350:
3339:
3338:
3336:
3335:
3330:
3325:
3319:
3317:
3305:
3304:
3299:
3292:
3291:
3288:
3287:
3284:
3283:
3281:
3280:
3275:
3270:
3264:
3262:
3261:Differentiable
3258:
3257:
3255:
3254:
3249:
3243:
3241:
3233:
3232:
3227:
3220:
3219:
3209:
3207:
3204:
3203:
3200:
3199:
3197:
3196:
3190:
3188:
3182:
3181:
3178:
3177:
3175:
3174:
3169:
3164:
3159:
3154:
3149:
3144:
3138:
3136:
3130:
3129:
3127:
3126:
3121:
3116:
3107:
3101:
3099:
3093:
3092:
3090:
3089:
3084:
3078:
3076:
3067:
3061:
3060:
3058:
3057:
3052:
3047:
3042:
3037:
3031:
3029:
3019:
3018:
3013:
3006:
3005:
2988:
2986:
2985:
2978:
2971:
2963:
2957:
2956:
2940:
2934:
2919:
2913:
2900:
2868:
2862:
2847:
2841:
2828:
2822:
2805:
2769:(3): 127–205.
2756:
2753:
2750:
2749:
2684:
2631:
2594:(2): 174–196.
2576:
2551:
2526:
2486:
2433:
2384:
2371:
2316:
2286:
2269:D-Wave Systems
2256:
2219:(3): 218–224.
2212:Nature Physics
2197:
2175:
2140:
2111:
2081:
2023:
1958:
1883:
1838:(19): 190601.
1820:
1756:
1701:
1698:on 2014-01-13.
1630:
1612:
1537:
1478:
1429:
1392:(12): 125210.
1374:
1321:
1256:
1235:
1214:
1147:
1092:
1089:on 2013-08-11.
1026:
1013:(2): 233–244.
992:
977:
926:
925:
923:
920:
827:D-Wave Systems
811:D-Wave Systems
792:
789:
788:
787:
784:
781:
778:
746:
740:
736:
716:
696:
676:
654:
650:
629:
606:
601:
580:
560:
540:
520:
500:
478:
473:
470:
450:
430:
406:
402:
397:
389:
385:
364:
344:
323:
297:
293:
272:
247:
242:
238:
233:
228:
224:
187:
184:
175:
172:
103:global minimum
88:
87:
49:
47:
40:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4608:
4597:
4594:
4592:
4589:
4587:
4584:
4583:
4581:
4566:
4558:
4556:
4548:
4547:
4544:
4538:
4535:
4533:
4530:
4528:
4525:
4523:
4520:
4518:
4514:
4511:
4509:
4505:
4501:
4498:
4497:
4495:
4493:
4487:
4477:
4474:
4472:
4469:
4467:
4464:
4462:
4459:
4458:
4456:
4454:
4450:
4444:
4441:
4439:
4436:
4434:
4433:Spin qubit QC
4431:
4429:
4426:
4425:
4423:
4420:
4416:
4410:
4407:
4405:
4402:
4401:
4399:
4397:
4393:
4387:
4384:
4382:
4379:
4377:
4374:
4372:
4369:
4368:
4366:
4364:
4360:
4357:
4351:
4345:
4342:
4338:
4337:
4333:
4331:
4328:
4326:
4323:
4321:
4318:
4316:
4313:
4311:
4308:
4306:
4303:
4301:
4298:
4297:
4295:
4294:
4292:
4290:
4284:
4278:
4275:
4273:
4270:
4266:
4263:
4262:
4261:
4258:
4254:
4251:
4250:
4249:
4246:
4242:
4241:cluster state
4239:
4238:
4237:
4234:
4232:
4229:
4227:
4224:
4223:
4221:
4219:
4213:
4205:
4201:
4197:
4195:
4191:
4187:
4186:
4185:
4182:
4178:
4175:
4174:
4173:
4170:
4168:
4165:
4163:
4160:
4159:
4157:
4151:
4145:
4142:
4140:
4137:
4135:
4132:
4130:
4127:
4125:
4122:
4121:
4119:
4117:
4111:
4105:
4102:
4100:
4097:
4095:
4092:
4090:
4087:
4085:
4082:
4080:
4077:
4075:
4072:
4070:
4067:
4065:
4062:
4060:
4057:
4055:
4052:
4050:
4049:Deutsch–Jozsa
4047:
4045:
4042:
4040:
4037:
4035:
4032:
4031:
4029:
4027:
4023:
4013:
4010:
4006:
4003:
4001:
3998:
3996:
3993:
3992:
3991:
3988:
3986:
3985:Quantum money
3983:
3981:
3978:
3976:
3973:
3972:
3970:
3968:
3964:
3958:
3955:
3951:
3948:
3947:
3946:
3943:
3939:
3936:
3935:
3934:
3931:
3929:
3926:
3924:
3921:
3919:
3916:
3912:
3909:
3907:
3904:
3903:
3902:
3899:
3898:
3896:
3894:communication
3890:
3884:
3881:
3879:
3876:
3874:
3871:
3869:
3866:
3864:
3861:
3859:
3856:
3854:
3851:
3849:
3846:
3844:
3841:
3839:
3836:
3834:
3831:
3829:
3826:
3824:
3821:
3819:
3816:
3814:
3811:
3809:
3806:
3805:
3803:
3799:
3791:
3788:
3787:
3786:
3783:
3779:
3776:
3775:
3774:
3771:
3769:
3766:
3764:
3761:
3759:
3756:
3752:
3749:
3748:
3747:
3744:
3742:
3739:
3737:
3734:
3733:
3731:
3727:
3723:
3716:
3711:
3709:
3704:
3702:
3697:
3696:
3693:
3681:
3678:
3677:
3674:
3664:
3661:
3659:
3656:
3654:
3651:
3649:
3646:
3644:
3641:
3639:
3638:Hill climbing
3636:
3634:
3631:
3630:
3627:
3623:
3618:
3614:
3600:
3597:
3595:
3592:
3590:
3587:
3585:
3582:
3581:
3579:
3577:
3576:Network flows
3573:
3563:
3560:
3558:
3555:
3551:
3548:
3547:
3546:
3543:
3542:
3540:
3538:
3537:Shortest path
3534:
3524:
3521:
3519:
3516:
3514:
3511:
3510:
3508:
3506:
3505:spanning tree
3500:
3497:
3495:
3489:
3481:
3477:
3474:
3473:
3472:
3469:
3467:
3464:
3462:
3459:
3457:
3454:
3453:
3451:
3447:
3443:
3439:
3438:Combinatorial
3434:
3430:
3412:
3409:
3407:
3404:
3402:
3399:
3397:
3394:
3393:
3391:
3389:
3386:
3382:
3376:
3373:
3371:
3368:
3366:
3363:
3362:
3360:
3358:
3354:
3351:
3349:
3344:
3340:
3334:
3331:
3329:
3326:
3324:
3321:
3320:
3318:
3316:
3310:
3306:
3302:
3297:
3293:
3279:
3276:
3274:
3271:
3269:
3266:
3265:
3263:
3259:
3253:
3250:
3248:
3245:
3244:
3242:
3238:
3234:
3230:
3225:
3221:
3213:
3195:
3192:
3191:
3189:
3187:
3183:
3173:
3170:
3168:
3165:
3163:
3160:
3158:
3155:
3153:
3150:
3148:
3145:
3143:
3140:
3139:
3137:
3135:
3134:Other methods
3131:
3125:
3122:
3120:
3117:
3115:
3111:
3108:
3106:
3103:
3102:
3100:
3098:
3094:
3088:
3085:
3083:
3080:
3079:
3077:
3075:
3071:
3068:
3066:
3062:
3056:
3053:
3051:
3048:
3046:
3043:
3041:
3038:
3036:
3033:
3032:
3030:
3028:
3024:
3020:
3016:
3011:
3007:
3003:
2999:
2995:
2991:
2984:
2979:
2977:
2972:
2970:
2965:
2964:
2961:
2953:
2949:
2945:
2941:
2937:
2931:
2927:
2926:
2920:
2916:
2910:
2906:
2901:
2896:
2892:
2887:
2882:
2878:
2874:
2869:
2865:
2859:
2855:
2854:
2848:
2844:
2838:
2834:
2829:
2825:
2819:
2815:
2811:
2806:
2802:
2798:
2794:
2790:
2786:
2782:
2777:
2772:
2768:
2764:
2759:
2758:
2754:
2745:
2741:
2737:
2733:
2728:
2723:
2719:
2715:
2711:
2707:
2703:
2699:
2695:
2688:
2685:
2680:
2676:
2671:
2666:
2661:
2656:
2652:
2648:
2647:
2642:
2635:
2632:
2627:
2623:
2619:
2615:
2611:
2607:
2602:
2597:
2593:
2589:
2588:
2580:
2577:
2566:
2562:
2555:
2552:
2540:
2539:Research Blog
2536:
2530:
2527:
2522:
2518:
2514:
2510:
2506:
2502:
2498:
2490:
2487:
2482:
2478:
2474:
2470:
2466:
2462:
2457:
2452:
2449:(5): 052323.
2448:
2444:
2437:
2434:
2429:
2425:
2420:
2415:
2411:
2407:
2403:
2399:
2395:
2388:
2385:
2381:
2375:
2372:
2367:
2363:
2359:
2355:
2351:
2347:
2342:
2337:
2334:(2): 021041.
2333:
2329:
2328:
2320:
2317:
2304:
2300:
2296:
2290:
2287:
2274:
2270:
2266:
2260:
2257:
2252:
2248:
2244:
2240:
2236:
2232:
2227:
2222:
2218:
2214:
2213:
2208:
2201:
2198:
2192:
2187:
2179:
2176:
2171:
2167:
2163:
2159:
2155:
2151:
2144:
2141:
2129:
2125:
2121:
2115:
2112:
2099:
2095:
2091:
2085:
2082:
2077:
2073:
2069:
2065:
2061:
2057:
2053:
2049:
2045:
2041:
2037:
2030:
2028:
2024:
2019:
2015:
2011:
2007:
2003:
1999:
1995:
1991:
1986:
1981:
1977:
1973:
1969:
1962:
1959:
1954:
1950:
1946:
1942:
1937:
1932:
1928:
1924:
1920:
1916:
1911:
1906:
1902:
1898:
1894:
1887:
1884:
1879:
1875:
1871:
1867:
1863:
1859:
1855:
1851:
1846:
1841:
1837:
1833:
1832:
1824:
1821:
1816:
1812:
1808:
1804:
1799:
1794:
1790:
1786:
1781:
1776:
1772:
1769:
1768:
1760:
1757:
1752:
1748:
1744:
1740:
1736:
1732:
1727:
1722:
1718:
1715:
1714:
1713:Eur. Phys. J.
1705:
1702:
1697:
1693:
1689:
1685:
1681:
1677:
1673:
1669:
1665:
1660:
1655:
1652:(2): 026701.
1651:
1647:
1646:
1641:
1634:
1631:
1626:
1622:
1616:
1613:
1608:
1604:
1600:
1596:
1591:
1586:
1582:
1578:
1574:
1570:
1565:
1560:
1556:
1552:
1548:
1541:
1538:
1533:
1529:
1524:
1519:
1515:
1511:
1506:
1501:
1497:
1493:
1489:
1482:
1479:
1474:
1470:
1466:
1462:
1458:
1454:
1450:
1446:
1445:
1440:
1433:
1430:
1425:
1421:
1417:
1413:
1409:
1405:
1400:
1395:
1391:
1387:
1386:
1378:
1375:
1370:
1366:
1362:
1358:
1354:
1350:
1345:
1340:
1337:(2): 022412.
1336:
1332:
1325:
1322:
1317:
1313:
1309:
1305:
1301:
1297:
1293:
1289:
1284:
1279:
1275:
1271:
1267:
1260:
1257:
1251:
1246:
1239:
1236:
1230:
1225:
1218:
1215:
1210:
1206:
1202:
1198:
1194:
1190:
1186:
1182:
1177:
1172:
1168:
1164:
1163:
1158:
1151:
1148:
1143:
1139:
1135:
1131:
1127:
1123:
1118:
1113:
1109:
1105:
1104:
1096:
1093:
1088:
1084:
1080:
1076:
1072:
1068:
1064:
1059:
1054:
1050:
1046:
1045:
1040:
1033:
1031:
1027:
1021:
1016:
1012:
1009:
1008:
1003:
996:
993:
988:
981:
978:
973:
969:
965:
961:
957:
953:
949:
945:
944:
936:
934:
932:
928:
921:
919:
917:
912:
909:
905:
900:
898:
894:
890:
885:
881:
879:
874:
869:
867:
862:
859:
855:
851:
846:
844:
840:
836:
832:
828:
820:
816:
812:
807:
802:
798:
790:
785:
782:
779:
776:
775:
774:
771:
769:
765:
760:
744:
738:
734:
727:decreases as
714:
694:
674:
652:
648:
627:
604:
599:
578:
558:
538:
498:
471:
468:
448:
400:
387:
383:
362:
313:
295:
291:
270:
245:
240:
236:
226:
222:
211:
209:
204:
202:
192:
185:
183:
181:
173:
171:
169:
164:
160:
156:
152:
147:
143:
138:
136:
132:
128:
124:
120:
116:
112:
108:
104:
100:
96:
92:
84:
81:
73:
63:
59:
53:
50:This article
48:
39:
38:
33:
19:
4461:Charge qubit
4386:KLM protocol
4335:
4199:
4189:
4068:
3883:Purification
3813:Eastin–Knill
3643:Local search
3589:Edmonds–Karp
3545:Bellman–Ford
3315:minimization
3147:Gauss–Newton
3097:Quasi–Newton
3082:Trust region
2990:Optimization
2951:
2947:
2924:
2904:
2876:
2872:
2852:
2832:
2813:
2766:
2762:
2701:
2697:
2687:
2650:
2644:
2634:
2591:
2585:
2579:
2568:. Retrieved
2564:
2554:
2543:. Retrieved
2538:
2529:
2496:
2489:
2446:
2442:
2436:
2401:
2397:
2387:
2374:
2331:
2325:
2319:
2307:. Retrieved
2303:the original
2298:
2289:
2277:. Retrieved
2273:the original
2268:
2259:
2216:
2210:
2200:
2178:
2153:
2143:
2132:. Retrieved
2128:the original
2123:
2114:
2102:. Retrieved
2098:the original
2093:
2084:
2043:
2039:
1975:
1971:
1961:
1900:
1896:
1886:
1835:
1829:
1823:
1770:
1765:
1759:
1719:(1): 17–24.
1716:
1711:
1704:
1696:the original
1649:
1645:Phys. Rev. E
1643:
1633:
1624:
1615:
1554:
1550:
1540:
1495:
1491:
1481:
1448:
1442:
1432:
1389:
1383:
1377:
1334:
1330:
1324:
1273:
1269:
1259:
1238:
1217:
1166:
1160:
1150:
1107:
1101:
1095:
1087:the original
1048:
1044:Phys. Rev. E
1042:
1010:
1005:
995:
980:
947:
941:
913:
901:
886:
882:
870:
863:
847:
824:
772:
761:
212:
207:
205:
198:
177:
139:
123:ground state
119:local minima
98:
94:
93:
91:
76:
70:January 2022
67:
51:
4492:programming
4471:Phase qubit
4376:Circuit QED
3848:No-deleting
3790:cloud-based
3663:Tabu search
3074:Convergence
3045:Line search
2879:: 485–500.
2704:(1): 3446.
2154:Nature News
1903:(1): 2212.
1557:(1): 2212.
1051:(5): 5355.
163:Ising model
155:Hamiltonian
105:of a given
4580:Categories
4532:libquantum
4466:Flux qubit
4371:Cavity QED
4320:Bacon–Shor
4310:stabilizer
3838:No-cloning
3494:algorithms
3002:heuristics
2994:Algorithms
2660:1910.13045
2653:: 106630.
2601:1803.03372
2570:2021-11-12
2545:2016-01-21
2456:1503.04216
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