199:, 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.
171:, 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).
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907:) 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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857:(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.
869:, 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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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
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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
864:
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)
183:
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
154:
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
755:, 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
Quantum Computing Approach for Hard Computational Problems in Space Exploration".
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Steiger, Damian; Heim, Bettina; Rønnow, Troels; Troyer, Matthias (October 22, 2015). "Performance of quantum annealing hardware". In
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133:, 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
2912:
122:. 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.
137:
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
142:). 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
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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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Venegas-Andraca, Salvador E.; Cruz-Santos, William; McGeoch, Catherine; Lanzagorta, Marco (2018). "A cross-disciplinary introduction to quantum annealing-based algorithms".
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126:) 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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903:"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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2254:"D-Wave Systems Building Quantum Application Ecosystem, Announces Partnerships with DNA-SEQ Alliance and 1QBit"
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2630:"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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450:, in presence of quantum tunneling, can be of major help: If the barriers are thin enough (i.e.
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2011 Superconducting-circuit quantum annealing machine built and marketed by D-Wave Systems.
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480:), quantum fluctuations can surely bring the system out of the shallow local minima. For an
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978:. 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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2109:"D-Wave Systems sells its first Quantum Computing System to Lockheed Martin Corporation"
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logic elements that exhibit controllable and tunable coupling to perform operations.
102:. Quantum annealing is used mainly for problems where the search space is discrete (
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2550:"D-Wave's Next-Generation Roadmap: Bringing Clarity to Practical Quantum Computing"
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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.
2683:"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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2933:"Quantum Annealing & Computation: Challenges & Perspectives"
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Apolloni, Bruno; Cesa-Bianchi, Nicolo; De Falco, Diego (July 1988).
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2196:"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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2805:. 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).
16:
Quantum physics-based metaheuristic for optimization problems
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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).
1426:
Santoro, Giuseppe E. & Tosatti, Erio (18 August 2006).
2383:"Quantum or not, controversial computer yields no speedup"
1610:"Local Maxima and Minima, and, Absolute Maxima and Minima"
1477:"Quantum versus classical annealing of Ising spin glasses"
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processor chipset. On May 25, 2011, D-Wave announced that
403:{\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
1629:"Quantum annealing in a kinetically constrained system"
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2917:. Cambridge & Delhi: Cambridge University Press.
2845:. Cambridge & Delhi: Cambridge University Press.
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Das, Arnab & Chakrabarti, Bikas K., eds. (2005).
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2681:Wierzbiński, M.; Falo-Roget, J.; Crimi, A. (2023).
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may be too technical for most readers to understand
2822:Quantum Annealing and Related Optimization Methods
900:other gate-model algorithms such as QAOA and VQE.
837:In May 2013 it was announced that a consortium of
802:, mounted and wire-bonded in a sample holder. The
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2914:Quantum Spin Glasses, Annealing & Computation
1028:"Quantum annealing in the transverse Ising model"
976:"A numerical implementation of quantum annealing"
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195:In the case of annealing a purely mathematical
245:{\displaystyle e^{-{\frac {\Delta }{k_{B}T}}}}
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90:) is an optimization process for finding the
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2803:Quantum Quenching, Annealing and Computation
2018:
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876:In December 2015, Google announced that the
2025:"Quantum annealing with manufactured spins"
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2139:"Google and NASA snap up quantum computer"
1957:"Quantum Annealing of a Disordered Magnet"
1255:"Quantum annealing of a disordered magnet"
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69:Learn how and when to remove this message
53:, without removing the technical details.
3204:Optimization computes maxima and minima.
834:took delivery of Lockheed's D-Wave One.
175:Quantum mechanics: analogy and advantage
4220:Continuous-variable quantum information
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847:Universities Space Research Association
2943:. Royal Society, London. January 2023.
2896:(2nd ed.). Heidelberg: Springer.
786:D-Wave Systems § Computer systems
473:{\displaystyle w\ll {\sqrt {\Delta }}}
344:of the barrier, but also on its width
3400:Principal pivoting algorithm of Lemke
867:Swiss Federal Institute of Technology
167:Quantum annealing can be compared to
51:make it understandable to non-experts
7:
2797:Chandra, Anjan K.; Das, Arnab &
2635:Computers & Chemical Engineering
2079:"Learning to program the D-Wave One"
1026:Kadowaki, T.; Nishimori, H. (1998).
806:'s processor is designed to use 128
798:Photograph of a chip constructed by
184:previously thought to be impossible.
4580:Optimization algorithms and methods
2368:
895:and, in particular, cannot execute
609:for the annealing time (instead of
3044:Successive parabolic interpolation
2488:. Vol. 9648. p. 964816.
507:
465:
417:
393:
384:
331:
310:
220:
14:
3364:Projective algorithm of Karmarkar
2659:10.1016/j.compchemeng.2019.106630
2524:"When can Quantum Annealing win?"
1880:Yan, B.; Sinitsyn, N. A. (2022).
1534:Yan, B.; Sinitsyn, N. A. (2022).
991:"Quantum stochastic optimization"
163:Comparison to Simulated Annealing
4549:
4548:
4539:
4538:
3359:Ellipsoid algorithm of Khachiyan
3262:Sequential quadratic programming
3099:Broyden–Fletcher–Goldfarb–Shanno
500:-spin glass, the barrier height
30:
1374:Journal of Mathematical Physics
324:depends not only on the height
3317:Reduced gradient (Frank–Wolfe)
2367:Helmut Katzgraber, quoted in (
1851:10.1103/PhysRevLett.121.190601
873:to find such problem classes.
832:Information Sciences Institute
656:for thermal annealing), while
364:and is approximately given by
1:
4215:Adiabatic quantum computation
3647:Spiral optimization algorithm
3267:Successive linear programming
2782:10.1016/j.physrep.2012.10.002
2607:10.1080/00107514.2018.1450720
2548:D-Wave Systems (2021-10-05).
2408:10.1126/science.344.6190.1330
741:{\displaystyle 1/{\sqrt {N}}}
696:-independent for cases where
602:{\displaystyle e^{\sqrt {N}}}
140:adiabatic quantum computation
4266:Topological quantum computer
3385:Simplex algorithm of Dantzig
3257:Augmented Lagrangian methods
2937:Philosophical Transactions A
2381:Cho, Adrian (20 June 2014).
2115:. 2011-05-25. Archived from
1991:10.1126/science.284.5415.779
1289:10.1126/science.284.5415.779
1123:10.1016/0009-2614(94)00117-0
1009:10.1016/0304-4149(89)90040-9
855:1QB Information Technologies
303:) depend only on the height
4544:Quantum information science
3711:Quantum information science
1732:10.1140/epjst/e2015-02339-y
1454:10.1088/0305-4470/39/36/R01
1350:10.1103/PhysRevA.108.022412
751:It is speculated that in a
4601:
3939:quantum gate teleportation
2707:10.1038/s41598-023-30579-y
2462:10.1103/PhysRevA.92.052323
1916:10.1038/s41467-022-29887-0
1796:10.1103/RevModPhys.80.1061
1665:10.1103/PhysRevE.72.026701
1570:10.1038/s41467-022-29887-0
893:universal quantum computer
783:
120:traveling salesman problem
104:combinatorial optimization
21:Annealing (disambiguation)
18:
4534:
4068:Quantum Fourier transform
3964:Post-quantum cryptography
3907:Entanglement distillation
3664:
3617:
3604:
3588:Push–relabel maximum flow
3433:
3420:
3390:Revised simplex algorithm
3296:
3283:
3224:
3211:
3197:
3010:
2997:
2347:10.1103/PhysRevX.4.021041
2151:10.1038/nature.2013.12999
953:10.1103/PhysRevB.39.11828
4554:Quantum mechanics topics
4249:Quantum machine learning
4225:One-way quantum computer
4078:Quantum phase estimation
3979:Quantum key distribution
3912:Monogamy of entanglement
3113:Symmetric rank-one (SR1)
3094:Berndt–Hall–Hall–Hausman
1092:Chemical Physics Letters
1064:10.1103/PhysRevE.58.5355
540:. For constant value of
4575:Stochastic optimization
4161:Randomized benchmarking
4023:Amplitude amplification
3637:Parallel metaheuristics
3445:Approximation algorithm
3156:Powell's dog leg method
3108:Davidon–Fletcher–Powell
3004:Unconstrained nonlinear
1820:Physical Review Letters
1512:10.1126/science.aaa4170
1182:10.1126/science.1057726
513:{\displaystyle \Delta }
423:{\displaystyle \Gamma }
337:{\displaystyle \Delta }
316:{\displaystyle \Delta }
4261:Quantum Turing machine
4254:quantum neural network
4001:Quantum secret sharing
3622:Evolutionary algorithm
3205:
811:
780:D-Wave implementations
742:
710:
690:
670:
650:
623:
603:
574:
554:
534:
514:
494:
474:
444:
424:
404:
358:
338:
317:
293:
266:
246:
185:
110:; such as finding the
4333:Entanglement-assisted
4294:quantum convolutional
3969:Quantum coin flipping
3934:Quantum teleportation
3895:entanglement-assisted
3725:DiVincenzo's criteria
3395:Criss-cross algorithm
3218:Constrained nonlinear
3203:
3024:Golden-section search
2799:Chakrabarti, Bikas K.
1886:Nature Communications
1540:Nature Communications
797:
784:Further information:
743:
711:
691:
671:
669:{\displaystyle \tau }
651:
649:{\displaystyle e^{N}}
624:
622:{\displaystyle \tau }
604:
575:
573:{\displaystyle \tau }
555:
535:
515:
495:
475:
445:
425:
405:
359:
339:
318:
294:
292:{\displaystyle k_{B}}
267:
247:
182:
4144:processor benchmarks
4073:Quantum optimization
3956:Quantum cryptography
3767:physical vs. logical
3312:Cutting-plane method
2576:Contemporary Physics
1433:Journal of Physics A
757:quantum entanglement
720:
700:
680:
660:
633:
613:
584:
564:
544:
524:
504:
484:
454:
434:
414:
368:
348:
328:
307:
276:
272:the temperature and
256:
207:
131:Schrödinger equation
106:problems) with many
100:quantum fluctuations
19:For other uses, see
3857:Quantum speed limit
3752:Quantum programming
3747:Quantum information
3642:Simulated annealing
3460:Integer programming
3450:Dynamic programming
3290:Convex optimization
3151:Levenberg–Marquardt
2884:2013arXiv1310.1339G
2862:Science and Culture
2774:2013PhR...523..127B
2699:2023NatSR..13.3446W
2599:2018ConPh..59..174V
2494:2015SPIE.9648E..16S
2454:2015PhRvA..92e2323A
2399:2014Sci...344.1330C
2393:(6190): 1330–1331.
2339:2014PhRvX...4b1041L
2264:on 31 December 2019
2224:2014NatPh..10..218B
2083:D-Wave Systems blog
2049:10.1038/nature10012
2041:2011Natur.473..194J
1983:1999Sci...284..779B
1908:2022NatCo..13.2212Y
1843:2018arXiv180400371L
1778:2008RvMP...80.1061D
1724:2015EPJST.224...17M
1657:2005PhRvE..72b6701D
1562:2022NatCo..13.2212Y
1503:2015Sci...348..215H
1446:2006JPhA...39R.393S
1397:2008JMP....49l5210M
1342:2023PhRvA.108b2412S
1281:1999Sci...284..779B
1174:2001Sci...292..472F
1115:1994CPL...219..343F
1056:1998PhRvE..58.5355K
945:1989PhRvB..3911828R
939:(16): 11828–11832.
886:Quantum Monte Carlo
882:simulated annealing
845:and the non-profit
190:quantum Monte Carlo
169:simulated annealing
4585:Quantum algorithms
4506:Forest/Rigetti QCS
4242:quantum logic gate
4028:Bernstein–Vazirani
4015:Quantum algorithms
3890:Classical capacity
3774:Quantum processors
3757:Quantum simulation
3322:Subgradient method
3206:
3131:Conjugate gradient
3039:Nelder–Mead method
2687:Scientific Reports
2502:10.1117/12.2202661
2137:Jones, N. (2013).
812:
738:
706:
686:
666:
646:
619:
599:
570:
550:
530:
510:
490:
470:
440:
420:
400:
354:
334:
313:
301:Boltzmann constant
289:
262:
242:
197:objective function
186:
96:objective function
4562:
4561:
4473:
4472:
4370:Linear optical QC
4151:Quantum supremacy
4105:complexity theory
4058:Quantum annealing
4009:
4008:
3946:Superdense coding
3735:Quantum computing
3677:
3676:
3660:
3659:
3600:
3599:
3596:
3595:
3559:
3558:
3520:
3519:
3416:
3415:
3412:
3411:
3408:
3407:
3279:
3278:
3275:
3274:
3195:
3194:
3191:
3190:
3169:
3168:
2924:978-1-10711-319-0
2903:978-3-64233-038-4
2852:978-1-10706-879-7
2831:978-3-54027-987-7
2812:978-3-64211-469-4
2530:. 8 December 2015
2432:Physical Review A
2316:Physical Review X
2232:10.1038/nphys2900
1967:(5415): 779–781.
1487:(6231): 215–217.
1440:(36): R393–R431.
1405:10.1063/1.2995837
1320:Physical Review A
996:Stoc. Proc. Appl.
932:Physical Review B
880:outperforms both
736:
709:{\displaystyle w}
689:{\displaystyle N}
596:
553:{\displaystyle w}
533:{\displaystyle N}
520:becomes of order
493:{\displaystyle N}
468:
443:{\displaystyle w}
396:
387:
357:{\displaystyle w}
265:{\displaystyle T}
238:
135:quantum tunneling
84:Quantum annealing
79:
78:
71:
4592:
4552:
4551:
4542:
4541:
4348:
4278:error correction
4207:computing models
4173:Relaxation times
4063:Quantum counting
3952:
3900:quantum capacity
3847:No-teleportation
3832:No-communication
3704:
3697:
3690:
3681:
3606:
3522:
3488:
3465:Branch and bound
3455:Greedy algorithm
3435:
3422:
3342:
3298:
3285:
3226:
3213:
3161:Truncated Newton
3076:Wolfe conditions
3059:
3012:
2999:
2972:
2965:
2958:
2949:
2944:
2928:
2907:
2887:
2877:
2856:
2835:
2816:
2793:
2767:
2737:
2736:
2718:
2678:
2672:
2671:
2661:
2651:
2625:
2619:
2618:
2592:
2570:
2564:
2563:
2561:
2560:
2545:
2539:
2538:
2536:
2535:
2520:
2514:
2513:
2480:
2474:
2473:
2447:
2427:
2421:
2420:
2410:
2378:
2372:
2365:
2359:
2358:
2332:
2310:
2304:
2303:
2301:
2299:
2290:. Archived from
2284:"1QBit Research"
2280:
2274:
2273:
2271:
2269:
2260:. Archived from
2250:
2244:
2243:
2217:
2191:
2185:
2184:
2182:
2169:
2163:
2162:
2134:
2128:
2127:
2125:
2124:
2119:on July 23, 2011
2105:
2099:
2098:
2096:
2094:
2089:on July 23, 2011
2085:. Archived from
2075:
2069:
2068:
2020:
2011:
2010:
1976:
1974:cond-mat/0105238
1952:
1946:
1945:
1927:
1901:
1877:
1871:
1870:
1836:
1814:
1808:
1807:
1789:
1771:
1762:(3): 1061–1081.
1750:
1744:
1743:
1717:
1695:
1689:
1688:
1683:. Archived from
1650:
1648:cond-mat/0502167
1624:
1618:
1617:
1606:
1600:
1599:
1581:
1555:
1531:
1525:
1524:
1514:
1496:
1472:
1466:
1465:
1423:
1417:
1416:
1390:
1368:
1362:
1361:
1335:
1315:
1309:
1308:
1274:
1272:cond-mat/0105238
1265:(5415): 779–81.
1250:
1244:
1243:
1241:
1229:
1223:
1222:
1220:
1208:
1202:
1201:
1167:
1165:quant-ph/0104129
1141:
1135:
1134:
1108:
1099:(5–6): 343–348.
1086:
1080:
1079:
1074:. Archived from
1049:
1047:cond-mat/9804280
1023:
1014:
1013:
1011:
986:
980:
979:
971:
965:
964:
926:
897:Shor's algorithm
753:quantum computer
747:
745:
744:
739:
737:
732:
730:
715:
713:
712:
707:
695:
693:
692:
687:
676:can even become
675:
673:
672:
667:
655:
653:
652:
647:
645:
644:
629:proportional to
628:
626:
625:
620:
608:
606:
605:
600:
598:
597:
592:
580:proportional to
579:
577:
576:
571:
559:
557:
556:
551:
539:
537:
536:
531:
519:
517:
516:
511:
499:
497:
496:
491:
479:
477:
476:
471:
469:
464:
449:
447:
446:
441:
429:
427:
426:
421:
409:
407:
406:
401:
399:
398:
397:
392:
388:
383:
380:
363:
361:
360:
355:
343:
341:
340:
335:
322:
320:
319:
314:
298:
296:
295:
290:
288:
287:
271:
269:
268:
263:
251:
249:
248:
243:
241:
240:
239:
237:
233:
232:
219:
74:
67:
63:
60:
54:
34:
33:
26:
4600:
4599:
4595:
4594:
4593:
4591:
4590:
4589:
4565:
4564:
4563:
4558:
4530:
4480:
4469:
4442:Superconducting
4436:
4402:
4393:Neutral atom QC
4385:Ultracold atoms
4379:
4344:implementations
4343:
4337:
4277:
4270:
4237:Quantum circuit
4205:
4199:
4193:
4183:
4143:
4137:
4104:
4097:
4053:Hidden subgroup
4005:
3994:other protocols
3950:
3927:quantum network
3922:Quantum channel
3882:
3876:
3822:No-broadcasting
3812:Gottesman–Knill
3785:
3713:
3708:
3678:
3673:
3656:
3613:
3592:
3555:
3516:
3493:
3482:
3475:
3429:
3404:
3368:
3335:
3326:
3303:
3292:
3271:
3245:
3241:Penalty methods
3236:Barrier methods
3220:
3207:
3187:
3183:Newton's method
3165:
3117:
3080:
3048:
3029:Powell's method
3006:
2993:
2976:
2931:
2925:
2910:
2904:
2891:
2859:
2853:
2838:
2832:
2819:
2813:
2801:, eds. (2010).
2796:
2752:Physics Reports
2749:
2746:
2744:Further reading
2741:
2740:
2680:
2679:
2675:
2627:
2626:
2622:
2572:
2571:
2567:
2558:
2556:
2547:
2546:
2542:
2533:
2531:
2522:
2521:
2517:
2482:
2481:
2477:
2429:
2428:
2424:
2380:
2379:
2375:
2366:
2362:
2312:
2311:
2307:
2297:
2295:
2294:on 19 June 2014
2282:
2281:
2277:
2267:
2265:
2252:
2251:
2247:
2193:
2192:
2188:
2171:
2170:
2166:
2136:
2135:
2131:
2122:
2120:
2107:
2106:
2102:
2092:
2090:
2077:
2076:
2072:
2035:(7346): 194–8.
2022:
2021:
2014:
1954:
1953:
1949:
1879:
1878:
1874:
1816:
1815:
1811:
1787:10.1.1.563.9990
1756:Rev. Mod. Phys.
1752:
1751:
1747:
1698:
1696:
1692:
1626:
1625:
1621:
1608:
1607:
1603:
1533:
1532:
1528:
1474:
1473:
1469:
1425:
1424:
1420:
1370:
1369:
1365:
1317:
1316:
1312:
1252:
1251:
1247:
1231:
1230:
1226:
1210:
1209:
1205:
1158:(5516): 472–5.
1143:
1142:
1138:
1106:chem-ph/9404003
1088:
1087:
1083:
1025:
1024:
1017:
988:
987:
983:
973:
972:
968:
928:
927:
918:
913:
824:Lockheed Martin
808:superconducting
792:
782:
718:
717:
698:
697:
678:
677:
658:
657:
636:
631:
630:
611:
610:
587:
582:
581:
562:
561:
542:
541:
522:
521:
502:
501:
482:
481:
452:
451:
432:
431:
412:
411:
381:
371:
366:
365:
346:
345:
326:
325:
305:
304:
279:
274:
273:
254:
253:
224:
223:
210:
205:
204:
177:
165:
75:
64:
58:
55:
47:help improve it
44:
35:
31:
24:
17:
12:
11:
5:
4598:
4596:
4588:
4587:
4582:
4577:
4567:
4566:
4560:
4559:
4557:
4556:
4546:
4535:
4532:
4531:
4529:
4528:
4526:many others...
4523:
4518:
4513:
4508:
4499:
4485:
4483:
4475:
4474:
4471:
4470:
4468:
4467:
4462:
4457:
4452:
4446:
4444:
4438:
4437:
4435:
4434:
4429:
4424:
4419:
4413:
4411:
4404:
4403:
4401:
4400:
4398:Trapped-ion QC
4395:
4389:
4387:
4381:
4380:
4378:
4377:
4372:
4367:
4362:
4356:
4354:
4352:Quantum optics
4345:
4339:
4338:
4336:
4335:
4330:
4329:
4328:
4321:
4316:
4311:
4306:
4301:
4296:
4291:
4282:
4280:
4272:
4271:
4269:
4268:
4263:
4258:
4257:
4256:
4246:
4245:
4244:
4234:
4233:
4232:
4222:
4217:
4211:
4209:
4201:
4200:
4198:
4197:
4196:
4195:
4191:
4185:
4181:
4170:
4169:
4168:
4158:
4156:Quantum volume
4153:
4147:
4145:
4139:
4138:
4136:
4135:
4130:
4125:
4120:
4115:
4109:
4107:
4099:
4098:
4096:
4095:
4090:
4085:
4080:
4075:
4070:
4065:
4060:
4055:
4050:
4045:
4040:
4035:
4033:Boson sampling
4030:
4025:
4019:
4017:
4011:
4010:
4007:
4006:
4004:
4003:
3998:
3997:
3996:
3991:
3986:
3976:
3971:
3966:
3960:
3958:
3949:
3948:
3943:
3942:
3941:
3931:
3930:
3929:
3919:
3914:
3909:
3904:
3903:
3902:
3897:
3886:
3884:
3878:
3877:
3875:
3874:
3869:
3867:Solovay–Kitaev
3864:
3859:
3854:
3849:
3844:
3839:
3834:
3829:
3824:
3819:
3814:
3809:
3804:
3799:
3793:
3791:
3787:
3786:
3784:
3783:
3782:
3781:
3771:
3770:
3769:
3759:
3754:
3749:
3744:
3743:
3742:
3732:
3727:
3721:
3719:
3715:
3714:
3709:
3707:
3706:
3699:
3692:
3684:
3675:
3674:
3672:
3671:
3665:
3662:
3661:
3658:
3657:
3655:
3654:
3649:
3644:
3639:
3634:
3629:
3624:
3618:
3615:
3614:
3611:Metaheuristics
3609:
3602:
3601:
3598:
3597:
3594:
3593:
3591:
3590:
3585:
3583:Ford–Fulkerson
3580:
3575:
3569:
3567:
3561:
3560:
3557:
3556:
3554:
3553:
3551:Floyd–Warshall
3548:
3543:
3542:
3541:
3530:
3528:
3518:
3517:
3515:
3514:
3509:
3504:
3498:
3496:
3485:
3477:
3476:
3474:
3473:
3472:
3471:
3457:
3452:
3447:
3441:
3439:
3431:
3430:
3425:
3418:
3417:
3414:
3413:
3410:
3409:
3406:
3405:
3403:
3402:
3397:
3392:
3387:
3381:
3379:
3370:
3369:
3367:
3366:
3361:
3356:
3354:Affine scaling
3350:
3348:
3346:Interior point
3339:
3328:
3327:
3325:
3324:
3319:
3314:
3308:
3306:
3294:
3293:
3288:
3281:
3280:
3277:
3276:
3273:
3272:
3270:
3269:
3264:
3259:
3253:
3251:
3250:Differentiable
3247:
3246:
3244:
3243:
3238:
3232:
3230:
3222:
3221:
3216:
3209:
3208:
3198:
3196:
3193:
3192:
3189:
3188:
3186:
3185:
3179:
3177:
3171:
3170:
3167:
3166:
3164:
3163:
3158:
3153:
3148:
3143:
3138:
3133:
3127:
3125:
3119:
3118:
3116:
3115:
3110:
3105:
3096:
3090:
3088:
3082:
3081:
3079:
3078:
3073:
3067:
3065:
3056:
3050:
3049:
3047:
3046:
3041:
3036:
3031:
3026:
3020:
3018:
3008:
3007:
3002:
2995:
2994:
2977:
2975:
2974:
2967:
2960:
2952:
2946:
2945:
2929:
2923:
2908:
2902:
2889:
2857:
2851:
2836:
2830:
2817:
2811:
2794:
2758:(3): 127–205.
2745:
2742:
2739:
2738:
2673:
2620:
2583:(2): 174–196.
2565:
2540:
2515:
2475:
2422:
2373:
2360:
2305:
2275:
2258:D-Wave Systems
2245:
2208:(3): 218–224.
2201:Nature Physics
2186:
2164:
2129:
2100:
2070:
2012:
1947:
1872:
1827:(19): 190601.
1809:
1745:
1690:
1687:on 2014-01-13.
1619:
1601:
1526:
1467:
1418:
1381:(12): 125210.
1363:
1310:
1245:
1224:
1203:
1136:
1081:
1078:on 2013-08-11.
1015:
1002:(2): 233–244.
981:
966:
915:
914:
912:
909:
816:D-Wave Systems
800:D-Wave Systems
781:
778:
777:
776:
773:
770:
767:
735:
729:
725:
705:
685:
665:
643:
639:
618:
595:
590:
569:
549:
529:
509:
489:
467:
462:
459:
439:
419:
395:
391:
386:
378:
374:
353:
333:
312:
286:
282:
261:
236:
231:
227:
222:
217:
213:
176:
173:
164:
161:
92:global minimum
77:
76:
38:
36:
29:
15:
13:
10:
9:
6:
4:
3:
2:
4597:
4586:
4583:
4581:
4578:
4576:
4573:
4572:
4570:
4555:
4547:
4545:
4537:
4536:
4533:
4527:
4524:
4522:
4519:
4517:
4514:
4512:
4509:
4507:
4503:
4500:
4498:
4494:
4490:
4487:
4486:
4484:
4482:
4476:
4466:
4463:
4461:
4458:
4456:
4453:
4451:
4448:
4447:
4445:
4443:
4439:
4433:
4430:
4428:
4425:
4423:
4422:Spin qubit QC
4420:
4418:
4415:
4414:
4412:
4409:
4405:
4399:
4396:
4394:
4391:
4390:
4388:
4386:
4382:
4376:
4373:
4371:
4368:
4366:
4363:
4361:
4358:
4357:
4355:
4353:
4349:
4346:
4340:
4334:
4331:
4327:
4326:
4322:
4320:
4317:
4315:
4312:
4310:
4307:
4305:
4302:
4300:
4297:
4295:
4292:
4290:
4287:
4286:
4284:
4283:
4281:
4279:
4273:
4267:
4264:
4262:
4259:
4255:
4252:
4251:
4250:
4247:
4243:
4240:
4239:
4238:
4235:
4231:
4230:cluster state
4228:
4227:
4226:
4223:
4221:
4218:
4216:
4213:
4212:
4210:
4208:
4202:
4194:
4190:
4186:
4184:
4180:
4176:
4175:
4174:
4171:
4167:
4164:
4163:
4162:
4159:
4157:
4154:
4152:
4149:
4148:
4146:
4140:
4134:
4131:
4129:
4126:
4124:
4121:
4119:
4116:
4114:
4111:
4110:
4108:
4106:
4100:
4094:
4091:
4089:
4086:
4084:
4081:
4079:
4076:
4074:
4071:
4069:
4066:
4064:
4061:
4059:
4056:
4054:
4051:
4049:
4046:
4044:
4041:
4039:
4038:Deutsch–Jozsa
4036:
4034:
4031:
4029:
4026:
4024:
4021:
4020:
4018:
4016:
4012:
4002:
3999:
3995:
3992:
3990:
3987:
3985:
3982:
3981:
3980:
3977:
3975:
3974:Quantum money
3972:
3970:
3967:
3965:
3962:
3961:
3959:
3957:
3953:
3947:
3944:
3940:
3937:
3936:
3935:
3932:
3928:
3925:
3924:
3923:
3920:
3918:
3915:
3913:
3910:
3908:
3905:
3901:
3898:
3896:
3893:
3892:
3891:
3888:
3887:
3885:
3883:communication
3879:
3873:
3870:
3868:
3865:
3863:
3860:
3858:
3855:
3853:
3850:
3848:
3845:
3843:
3840:
3838:
3835:
3833:
3830:
3828:
3825:
3823:
3820:
3818:
3815:
3813:
3810:
3808:
3805:
3803:
3800:
3798:
3795:
3794:
3792:
3788:
3780:
3777:
3776:
3775:
3772:
3768:
3765:
3764:
3763:
3760:
3758:
3755:
3753:
3750:
3748:
3745:
3741:
3738:
3737:
3736:
3733:
3731:
3728:
3726:
3723:
3722:
3720:
3716:
3712:
3705:
3700:
3698:
3693:
3691:
3686:
3685:
3682:
3670:
3667:
3666:
3663:
3653:
3650:
3648:
3645:
3643:
3640:
3638:
3635:
3633:
3630:
3628:
3627:Hill climbing
3625:
3623:
3620:
3619:
3616:
3612:
3607:
3603:
3589:
3586:
3584:
3581:
3579:
3576:
3574:
3571:
3570:
3568:
3566:
3565:Network flows
3562:
3552:
3549:
3547:
3544:
3540:
3537:
3536:
3535:
3532:
3531:
3529:
3527:
3526:Shortest path
3523:
3513:
3510:
3508:
3505:
3503:
3500:
3499:
3497:
3495:
3494:spanning tree
3489:
3486:
3484:
3478:
3470:
3466:
3463:
3462:
3461:
3458:
3456:
3453:
3451:
3448:
3446:
3443:
3442:
3440:
3436:
3432:
3428:
3427:Combinatorial
3423:
3419:
3401:
3398:
3396:
3393:
3391:
3388:
3386:
3383:
3382:
3380:
3378:
3375:
3371:
3365:
3362:
3360:
3357:
3355:
3352:
3351:
3349:
3347:
3343:
3340:
3338:
3333:
3329:
3323:
3320:
3318:
3315:
3313:
3310:
3309:
3307:
3305:
3299:
3295:
3291:
3286:
3282:
3268:
3265:
3263:
3260:
3258:
3255:
3254:
3252:
3248:
3242:
3239:
3237:
3234:
3233:
3231:
3227:
3223:
3219:
3214:
3210:
3202:
3184:
3181:
3180:
3178:
3176:
3172:
3162:
3159:
3157:
3154:
3152:
3149:
3147:
3144:
3142:
3139:
3137:
3134:
3132:
3129:
3128:
3126:
3124:
3123:Other methods
3120:
3114:
3111:
3109:
3106:
3104:
3100:
3097:
3095:
3092:
3091:
3089:
3087:
3083:
3077:
3074:
3072:
3069:
3068:
3066:
3064:
3060:
3057:
3055:
3051:
3045:
3042:
3040:
3037:
3035:
3032:
3030:
3027:
3025:
3022:
3021:
3019:
3017:
3013:
3009:
3005:
3000:
2996:
2992:
2988:
2984:
2980:
2973:
2968:
2966:
2961:
2959:
2954:
2953:
2950:
2942:
2938:
2934:
2930:
2926:
2920:
2916:
2915:
2909:
2905:
2899:
2895:
2890:
2885:
2881:
2876:
2871:
2867:
2863:
2858:
2854:
2848:
2844:
2843:
2837:
2833:
2827:
2823:
2818:
2814:
2808:
2804:
2800:
2795:
2791:
2787:
2783:
2779:
2775:
2771:
2766:
2761:
2757:
2753:
2748:
2747:
2743:
2734:
2730:
2726:
2722:
2717:
2712:
2708:
2704:
2700:
2696:
2692:
2688:
2684:
2677:
2674:
2669:
2665:
2660:
2655:
2650:
2645:
2641:
2637:
2636:
2631:
2624:
2621:
2616:
2612:
2608:
2604:
2600:
2596:
2591:
2586:
2582:
2578:
2577:
2569:
2566:
2555:
2551:
2544:
2541:
2529:
2528:Research Blog
2525:
2519:
2516:
2511:
2507:
2503:
2499:
2495:
2491:
2487:
2479:
2476:
2471:
2467:
2463:
2459:
2455:
2451:
2446:
2441:
2438:(5): 052323.
2437:
2433:
2426:
2423:
2418:
2414:
2409:
2404:
2400:
2396:
2392:
2388:
2384:
2377:
2374:
2370:
2364:
2361:
2356:
2352:
2348:
2344:
2340:
2336:
2331:
2326:
2323:(2): 021041.
2322:
2318:
2317:
2309:
2306:
2293:
2289:
2285:
2279:
2276:
2263:
2259:
2255:
2249:
2246:
2241:
2237:
2233:
2229:
2225:
2221:
2216:
2211:
2207:
2203:
2202:
2197:
2190:
2187:
2181:
2176:
2168:
2165:
2160:
2156:
2152:
2148:
2144:
2140:
2133:
2130:
2118:
2114:
2110:
2104:
2101:
2088:
2084:
2080:
2074:
2071:
2066:
2062:
2058:
2054:
2050:
2046:
2042:
2038:
2034:
2030:
2026:
2019:
2017:
2013:
2008:
2004:
2000:
1996:
1992:
1988:
1984:
1980:
1975:
1970:
1966:
1962:
1958:
1951:
1948:
1943:
1939:
1935:
1931:
1926:
1921:
1917:
1913:
1909:
1905:
1900:
1895:
1891:
1887:
1883:
1876:
1873:
1868:
1864:
1860:
1856:
1852:
1848:
1844:
1840:
1835:
1830:
1826:
1822:
1821:
1813:
1810:
1805:
1801:
1797:
1793:
1788:
1783:
1779:
1775:
1770:
1765:
1761:
1758:
1757:
1749:
1746:
1741:
1737:
1733:
1729:
1725:
1721:
1716:
1711:
1707:
1704:
1703:
1702:Eur. Phys. J.
1694:
1691:
1686:
1682:
1678:
1674:
1670:
1666:
1662:
1658:
1654:
1649:
1644:
1641:(2): 026701.
1640:
1636:
1635:
1630:
1623:
1620:
1615:
1611:
1605:
1602:
1597:
1593:
1589:
1585:
1580:
1575:
1571:
1567:
1563:
1559:
1554:
1549:
1545:
1541:
1537:
1530:
1527:
1522:
1518:
1513:
1508:
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1500:
1495:
1490:
1486:
1482:
1478:
1471:
1468:
1463:
1459:
1455:
1451:
1447:
1443:
1439:
1435:
1434:
1429:
1422:
1419:
1414:
1410:
1406:
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1398:
1394:
1389:
1384:
1380:
1376:
1375:
1367:
1364:
1359:
1355:
1351:
1347:
1343:
1339:
1334:
1329:
1326:(2): 022412.
1325:
1321:
1314:
1311:
1306:
1302:
1298:
1294:
1290:
1286:
1282:
1278:
1273:
1268:
1264:
1260:
1256:
1249:
1246:
1240:
1235:
1228:
1225:
1219:
1214:
1207:
1204:
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1195:
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1183:
1179:
1175:
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1157:
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1147:
1140:
1137:
1132:
1128:
1124:
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1116:
1112:
1107:
1102:
1098:
1094:
1093:
1085:
1082:
1077:
1073:
1069:
1065:
1061:
1057:
1053:
1048:
1043:
1039:
1035:
1034:
1029:
1022:
1020:
1016:
1010:
1005:
1001:
998:
997:
992:
985:
982:
977:
970:
967:
962:
958:
954:
950:
946:
942:
938:
934:
933:
925:
923:
921:
917:
910:
908:
906:
901:
898:
894:
889:
887:
883:
879:
874:
870:
868:
863:
858:
856:
851:
848:
844:
840:
835:
833:
829:
825:
821:
817:
809:
805:
801:
796:
791:
787:
779:
774:
771:
768:
765:
764:
763:
760:
758:
754:
749:
733:
727:
723:
716:decreases as
703:
683:
663:
641:
637:
616:
593:
588:
567:
547:
527:
487:
460:
457:
437:
389:
376:
372:
351:
302:
284:
280:
259:
234:
229:
225:
215:
211:
200:
198:
193:
191:
181:
174:
172:
170:
162:
160:
158:
153:
149:
145:
141:
136:
132:
127:
125:
121:
117:
113:
109:
105:
101:
97:
93:
89:
85:
81:
73:
70:
62:
52:
48:
42:
39:This article
37:
28:
27:
22:
4450:Charge qubit
4375:KLM protocol
4324:
4188:
4178:
4057:
3872:Purification
3802:Eastin–Knill
3632:Local search
3578:Edmonds–Karp
3534:Bellman–Ford
3304:minimization
3136:Gauss–Newton
3086:Quasi–Newton
3071:Trust region
2979:Optimization
2940:
2936:
2913:
2893:
2865:
2861:
2841:
2821:
2802:
2755:
2751:
2690:
2686:
2676:
2639:
2633:
2623:
2580:
2574:
2568:
2557:. Retrieved
2553:
2543:
2532:. Retrieved
2527:
2518:
2485:
2478:
2435:
2431:
2425:
2390:
2386:
2376:
2363:
2320:
2314:
2308:
2296:. Retrieved
2292:the original
2287:
2278:
2266:. Retrieved
2262:the original
2257:
2248:
2205:
2199:
2189:
2167:
2142:
2132:
2121:. Retrieved
2117:the original
2112:
2103:
2091:. Retrieved
2087:the original
2082:
2073:
2032:
2028:
1964:
1960:
1950:
1889:
1885:
1875:
1824:
1818:
1812:
1759:
1754:
1748:
1708:(1): 17–24.
1705:
1700:
1693:
1685:the original
1638:
1634:Phys. Rev. E
1632:
1622:
1613:
1604:
1543:
1539:
1529:
1484:
1480:
1470:
1437:
1431:
1421:
1378:
1372:
1366:
1323:
1319:
1313:
1262:
1258:
1248:
1227:
1206:
1155:
1149:
1139:
1096:
1090:
1084:
1076:the original
1037:
1033:Phys. Rev. E
1031:
999:
994:
984:
969:
936:
930:
902:
890:
875:
871:
859:
852:
836:
813:
761:
750:
201:
196:
194:
187:
166:
128:
112:ground state
108:local minima
87:
83:
82:
80:
65:
59:January 2022
56:
40:
4481:programming
4460:Phase qubit
4365:Circuit QED
3837:No-deleting
3779:cloud-based
3652:Tabu search
3063:Convergence
3034:Line search
2868:: 485–500.
2693:(1): 3446.
2143:Nature News
1892:(1): 2212.
1546:(1): 2212.
1040:(5): 5355.
152:Ising model
144:Hamiltonian
94:of a given
4569:Categories
4521:libquantum
4455:Flux qubit
4360:Cavity QED
4309:Bacon–Shor
4299:stabilizer
3827:No-cloning
3483:algorithms
2991:heuristics
2983:Algorithms
2649:1910.13045
2642:: 106630.
2590:1803.03372
2559:2021-11-12
2534:2016-01-21
2445:1503.04216
2123:2011-05-30
1899:2110.12354
1834:1804.00371
1697:See e.g.,
1614:Mathonline
1553:2110.12354
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