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so in order to obtain the expectation value of the energy, one can measure the expectation value of each Pauli string (number of counts for a given value over the total number of counts). This step corresponds to measuring each qubit in the axis provided by the Pauli string. For example, for the
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The objective of the VQE is to find a set of quantum operations that prepares the lowest energy state (or minima) of a close approximation to some target quantity or observable. While the only strict requirement for the representation of an observable is that it is efficient to estimate its
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Another variant of the ansatz circuit is the hardware efficient ansatz, which consists of sequence of 1 qubit rotational gates and 2 qubit entangling gates. The number of repetitions of 1-qubit rotational gates and 2-qubit entangling gates is called the depth of the circuit.
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2384:{\displaystyle \nabla _{\theta }f(\theta )={\frac {\partial }{\partial \theta }}\langle \phi |U^{\dagger }AU|\phi \rangle =\langle \phi |\left({\frac {i}{2}}P\right)U^{\dagger }AU|\phi \rangle +\langle \phi |U^{\dagger }A\left(-{\frac {i}{2}}P\right)U|\phi \rangle }
3475:
Tilly, Jules; Chen, Hongxiang; Cao, Shuxiang; Picozzi, Dario; Setia, Kanav; Li, Ying; Grant, Edward; Wossnig, Leonard; Rungger, Ivan; Booth, George H.; Tennyson, Jonathan (2022-06-12). "The
Variational Quantum Eigensolver: A review of methods and best practices".
3277:
Tilly, Jules; Chen, Hongxiang; Cao, Shuxiang; Picozzi, Dario; Setia, Kanav; Li, Ying; Grant, Edward; Wossnig, Leonard; Rungger, Ivan; Booth, George H.; Tennyson, Jonathan (2022-06-12). "The
Variational Quantum Eigensolver: A review of methods and best practices".
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By running the circuit many times and constantly updating the parameters to find the global minima of the expectation value of the desired observable, one can approach the ground state of the given system and store it in a quantum processor as a series of
3409:
Moll, Nikolaj; Barkoutsos, Panagiotis; Bishop, Lev S; Chow, Jerry M; Cross, Andrew; Egger, Daniel J; Filipp, Stefan; Fuhrer, Andreas; Gambetta, Jay M; Ganzhorn, Marc; Kandala, Abhinav; Mezzacapo, Antonio; MĂĽller, Peter; Riess, Walter; Salis, Gian (2018).
3130:
Bharti, Kishor; Cervera-Lierta, Alba; Kyaw, Thi Ha; Haug, Tobias; Alperin-Lea, Sumner; Anand, Abhinav; Degroote, Matthias; Heimonen, Hermanni; Kottmann, Jakob S.; Menke, Tim; Mok, Wai-Keong; Sim, Sukin; Kwek, Leong-Chuan; Aspuru-Guzik, Alán (2022-02-15).
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1851:{\displaystyle \Delta f({\vec {\theta }}^{({\text{old}})})=\left({\frac {\partial f({\vec {\theta }}^{({\text{old}})})}{\partial \theta _{1}}},{\frac {\partial f({\vec {\theta }}^{({\text{old}})})}{\partial \theta _{2}}},\ldots \right)^{\top }}
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2594:{\displaystyle ={\frac {1}{2}}\langle \phi |U^{\dagger }(\theta +{\frac {\pi }{2}})AU(\theta +{\frac {\pi }{2}})|\phi \rangle -{\frac {1}{2}}\langle \phi |U^{\dagger }(\theta -{\frac {\pi }{2}})AU(\theta -{\frac {\pi }{2}})|\phi \rangle }
353:, whose parameters can be updated after each run. The ansatz has to be adaptable enough to not miss the desired state. A common method to obtain a valid ansatz is given by the unitary coupled cluster (UCC) framework and its extensions.
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where the number of parameters chosen are enough to lend the algorithm expressive power to compute the ground state of the system, but not too big to increase the computational cost of the optimization step.
703:{\displaystyle E(\theta _{1},\cdots ,\theta _{n})=\langle {\hat {H}}\rangle =\sum _{i}\alpha _{i}\langle \psi (\theta _{1},\cdots ,\theta _{N})|{\hat {P}}_{i}|\psi (\theta _{1},\cdots ,\theta _{N})\rangle }
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If the ansatz is not chosen adequately the procedure may halt at suboptimal parameters that do not correspond to a minima. In this situation, the algorithm is said to have reached a 'barren plateau'.
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can be used to transform between axes. If two Pauli strings commute, then they can be both measured simultaneously using the same circuit and interpreting the result according to the Pauli algebra.
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Given a parametrized ansatz for the ground state eigenstate, with parameters that can be modified, one is sure to find the parametrized state that is closest to the ground state based on the
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Its a heuristic method and thus does not guarantee convergence to the ground state value. The method is highly influenced by the choice of ansatz circuit and the optimization methods.
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Full authors: Alberto
Peruzzo, Jarrod McClean, Peter Shadbolt, Man-Hong Yung, Xiao-Qi Zhou, Peter J. Love, Alan Aspuru-Guzik and Jeremy L. O’Brien. All equally contributing.
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Number of measurements required to conclude the value of ground state is higher compared to the QPE and scales approximately with the number of terms in the
Hamiltonian.
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expectation values, it is often simplest if that operator has a compact or simple expression in terms of Pauli operators or tensor products of Pauli operators.
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is written in terms of Pauli operators and irrelevant states are discarded (finite-dimensional space), it would consist of a linear combination of Pauli strings
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2082:{\displaystyle \nabla _{\theta }U={\frac {\partial U}{\partial \theta }}=-{\frac {i}{2}}Pe^{-i{\frac {\theta }{2}}P}=-{\frac {i}{2}}PU=-{\frac {i}{2}}UP}
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Peruzzo, Alberto; McClean, Jarrod; Shadbolt, Peter; Yung, Man-Hong; Zhou, Xiao-Qi; Love, Peter J.; Aspuru-Guzik, Alán; O’Brien, Jeremy L. (2014).
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molecule. Larger molecules can be simulated by taking into account symmetry considerations. In 2020, a 12-qubit simulation of a hydrogen chain (H
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are numerical coefficients. Based on the coefficients, the number of Pauli strings can be reduced in order to optimize the calculation.
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1636:{\displaystyle {\vec {\theta }}^{({\text{new}})}={\vec {\theta }}^{({\text{old}})}-r\nabla f({\vec {\theta }}^{({\text{old}})})}
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The ansatz can be set to an initial trial function to start the algorithm. For example, for a molecular system, one can use the
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to only a subset of possible n-qubit states which is based on conventional physics, chemistry and quantum mechanics knowledge.
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dimensional, is impractical. Thus to restrict the search space to a more practical size (eg. poly(n)), we need to restrict the
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and has been further substantiated by general hybrid algorithms between quantum and classical computers. It is an example of a
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Kandala, Abhinav; Mezzacapo, Antonio; Temme, Kristan; Takita, Maika; Brink, Markus; Chow, Jerry M.; Gambetta, Jay M. (2017).
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would be a ground state (assuming no degeneracy). But the above minimization problem over all possible states
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1362:{\displaystyle {\vec {\theta }}={\begin{pmatrix}\theta _{1}\\\theta _{2}\\\vdots \\\theta _{p}\end{pmatrix}}}
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The VQE can be adapted to other optimization problems by adapting the
Hamiltonian to be a cost function.
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2690:{\displaystyle ={\frac {1}{2}}\left(f(\theta +{\frac {\pi }{2}})-f(\theta -{\frac {\pi }{2}})\right)}
1514:{\displaystyle f({\vec {\theta }})=\langle \psi ({\vec {\theta }})|H|\psi ({\vec {\theta }})\rangle }
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For a fermionic system, it is often most convenient to qubitize: that is to write the many-body
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VQE is highly versatile, as problems (apart from chemistry) can be expressed as
Hamiltonians.
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3531:"Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets"
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As of 2022, the variational quantum eigensolver can only simulate small molecules like the
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2959:(QPE), it is more robust to errors and lends itself well to error mitigation strategies.
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McClean, Jarrod R; Romero, Jonathan; Babbush, Ryan; Aspuru-Guzik, Alán (2016-02-04).
1072:{\displaystyle \operatorname {argmin} _{|\psi \rangle }\langle \psi |H|\psi \rangle }
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3333:"The Bravyi-Kitaev transformation for quantum computation of electronic structure"
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3412:"Quantum optimization using variational algorithms on near-term quantum devices"
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It was originally proposed in 2014, with corresponding authors
Alberto Peruzzo,
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controls the subset of possible states that can be created, and the parameter
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The adjoining figure illustrates the high level steps in the VQE algorithm.
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Methods to simulate fermions on quantum computers with hardware limitations
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In case of gradient descent, its required to minimize a cost function
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2155:{\displaystyle f(\theta )=\langle \phi |U^{\dagger }AU|\phi \rangle }
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In order to compute the gradients, the parameter shift rule is used.
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to provide a starting state that is close to the real ground state.
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Seeley, Jacob T.; Richard, Martin J.; Love, Peter J. (2012-12-12).
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1009:{\displaystyle \min _{|\psi \rangle }\langle \psi |H|\psi \rangle }
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3057:"A variational eigenvalue solver on a photonic quantum processor"
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The choice of ansatz state depends on the system of interest. In
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in terms of Pauli operators. Common schemes for fermions include
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3192:"The theory of variational hybrid quantum-classical algorithms"
439:{\displaystyle |\psi (\theta _{1},\cdots ,\theta _{N})\rangle }
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Arute, Frank; Arya, Kunal; Babbush, Ryan; et al. (2020).
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needs to be evaluated 2 times to arrive at the gradient value
296:{\displaystyle {\hat {H}}=\sum _{i}\alpha _{i}{\hat {P}}_{i}}
119:, Bravyi-Kitaev transformation, and parity transformation.
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is used to improve the guess. The algorithm is based on the
3604:"Hartree-Fock on a superconducting qubit quantum computer"
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The VQE circuit does not require many gates compared with
1197:
High Level illustration of
Variational Quantum Algorithm
1926:{\displaystyle U(\theta )=e^{-i{\frac {\theta }{2}}P},}
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750:-axis, while the last two are to be measured in the
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2700:The above result has interesting properties as:
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3133:"Noisy intermediate-scale quantum algorithms"
51:of a given physical system. Given a guess or
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222:{\displaystyle X\otimes I\otimes Z\otimes X}
2878:. Unsourced material may be challenged and
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1869:Considering a single Pauli gate example:
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4211:Continuous-variable quantum information
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776:variational method of quantum mechanics
383:The expectation value of a given state
363:Example of a hardware efficient ansatz.
1650:is the learning rate (step size) and
111:, and then use a mapping to write the
2828:{\displaystyle \pm {\frac {\pi }{2}}}
1276:contains the variational parameters,
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2876:adding citations to reliable sources
1016:will be the ground state energy and
822:adding citations to reliable sources
3263:(PhD Thesis). University of Leiden.
1412:{\displaystyle f({\vec {\theta }})}
1240:{\displaystyle U({\vec {\theta }})}
770:Variational method and optimization
739:{\displaystyle X\otimes Y\otimes Y}
2957:quantum phase estimation algorithm
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2891:"Variational quantum eigensolver"
837:"Variational quantum eigensolver"
341:Ansatz and initial trial function
187:consisting of tensor products of
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1269:{\displaystyle {\vec {\theta }}}
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113:creation-annihiliation operators
84:noisy intermediate-scale quantum
3337:The Journal of Chemical Physics
3014:Quantum optimization algorithms
25:variational quantum eigensolver
3416:Quantum Science and Technology
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180:{\displaystyle {\hat {P}}_{i}}
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4206:Adiabatic quantum computation
3508:10.1016/j.physrep.2022.08.003
3310:10.1016/j.physrep.2022.08.003
3227:10.1088/1367-2630/18/2/023023
2968:VQE can run on NISQ hardware.
762:-axis is only possible, then
4257:Topological quantum computer
3167:10.1103/RevModPhys.94.015004
2840:Advantages and disadvantages
347:gate-based quantum computing
117:Jordan–Wigner transformation
4535:Quantum information science
3702:Quantum information science
327:{\displaystyle \alpha _{i}}
4582:
3930:quantum gate teleportation
3002:Sycamore quantum processor
2726:{\displaystyle f(\theta )}
144:{\displaystyle {\hat {H}}}
4525:
4059:Quantum Fourier transform
3955:Post-quantum cryptography
3898:Entanglement distillation
3137:Reviews of Modern Physics
2997:) was demonstrated using
2795:{\displaystyle f(\cdot )}
4545:Quantum mechanics topics
4240:Quantum machine learning
4216:One-way quantum computer
4069:Quantum phase estimation
3970:Quantum key distribution
3903:Monogamy of entanglement
3446:10.1088/2058-9565/aab822
758:. If measurement in the
80:quantum machine learning
4152:Randomized benchmarking
4014:Amplitude amplification
3638:10.1126/science.abb9811
3438:2018QS&T....3c0503M
2805:As the angle precision
1521:. The update rule is:
1419:where for the VQE case
4252:Quantum Turing machine
4245:quantum neural network
3992:Quantum secret sharing
3196:New Journal of Physics
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67:of quantum mechanics.
4324:Entanglement-assisted
4285:quantum convolutional
3960:Quantum coin flipping
3925:Quantum teleportation
3886:entanglement-assisted
3716:DiVincenzo's criteria
3257:Steudtner, M (2019).
3061:Nature Communications
2830:
2797:
2767:
2728:
2692:
2596:
2386:
2157:
2084:
1928:
1853:
1638:
1516:
1414:
1364:
1271:
1242:
1196:
1185:
1157:
1155:{\displaystyle 2^{n}}
1130:
1102:
1074:
1011:
951:
923:
741:
705:
492:
441:
362:
329:
298:
224:
182:
146:
122:Once the Hamiltonian
45:optimization problems
4135:processor benchmarks
4064:Quantum optimization
3947:Quantum cryptography
3758:physical vs. logical
2872:improve this section
2809:
2777:
2737:
2708:
2608:
2398:
2169:
2096:
1951:
1876:
1657:
1528:
1423:
1385:
1280:
1251:
1213:
1166:
1139:
1111:
1083:
1020:
960:
932:
904:
818:improve this section
718:
504:
450:
387:
311:
236:
195:
155:
126:
107:of the system using
3848:Quantum speed limit
3743:Quantum programming
3738:Quantum information
3630:2020Sci...369.1084.
3614:(6507): 1084–1089.
3565:10.1038/nature23879
3557:2017Natur.549..242K
3500:2022PhR...986....1T
3359:2012JChPh.137v4109S
3302:2022PhR...986....1T
3218:2016NJPh...18b3023M
3159:2022RvMP...94a5004B
3083:2014NatCo...5.4213P
486:
369:Hartree–Fock method
109:second quantization
41:quantum simulations
4566:Quantum algorithms
4497:Forest/Rigetti QCS
4233:quantum logic gate
4019:Bernstein–Vazirani
4006:Quantum algorithms
3881:Classical capacity
3765:Quantum processors
3748:Quantum simulation
3091:10.1038/ncomms5213
2987:helium hydride ion
2825:
2792:
2762:
2723:
2687:
2591:
2381:
2152:
2079:
1923:
1848:
1633:
1511:
1409:
1359:
1353:
1266:
1237:
1199:
1180:
1152:
1125:
1097:
1069:
1006:
980:
946:
918:
736:
700:
578:
487:
466:
436:
365:
324:
293:
263:
219:
177:
141:
86:(NISQ) algorithm.
65:variational method
4553:
4552:
4464:
4463:
4361:Linear optical QC
4142:Quantum supremacy
4096:complexity theory
4049:Quantum annealing
4000:
3999:
3937:Superdense coding
3726:Quantum computing
3541:(7671): 242–246.
3367:10.1063/1.4768229
2991:beryllium hydride
2952:
2951:
2944:
2926:
2823:
2677:
2649:
2622:
2575:
2547:
2507:
2480:
2452:
2412:
2357:
2277:
2210:
2071:
2049:
2028:
2004:
1988:
1913:
1829:
1803:
1793:
1766:
1740:
1730:
1689:
1679:
1623:
1613:
1581:
1571:
1551:
1541:
1502:
1468:
1441:
1403:
1292:
1263:
1231:
963:
956:arbitrarily then
898:
897:
890:
872:
645:
569:
560:
284:
254:
248:
168:
138:
72:Alán Aspuru-Guzik
37:quantum chemistry
33:quantum algorithm
21:quantum computing
16:Quantum algorithm
4573:
4543:
4542:
4533:
4532:
4339:
4269:error correction
4198:computing models
4164:Relaxation times
4054:Quantum counting
3943:
3891:quantum capacity
3838:No-teleportation
3823:No-communication
3695:
3688:
3681:
3672:
3666:
3665:
3623:
3599:
3593:
3592:
3550:
3526:
3520:
3519:
3493:
3472:
3466:
3465:
3431:
3406:
3395:
3394:
3352:
3328:
3322:
3321:
3295:
3274:
3265:
3264:
3254:
3248:
3247:
3229:
3211:
3187:
3181:
3180:
3178:
3152:
3127:
3121:
3120:
3110:
3076:
3052:
3033:
3030:
2947:
2940:
2936:
2933:
2927:
2925:
2884:
2852:
2844:
2834:
2832:
2831:
2826:
2824:
2816:
2801:
2799:
2798:
2793:
2771:
2769:
2768:
2763:
2749:
2748:
2732:
2730:
2729:
2724:
2696:
2694:
2693:
2688:
2686:
2682:
2678:
2670:
2650:
2642:
2623:
2615:
2600:
2598:
2597:
2592:
2584:
2576:
2568:
2548:
2540:
2529:
2528:
2519:
2508:
2500:
2489:
2481:
2473:
2453:
2445:
2434:
2433:
2424:
2413:
2405:
2390:
2388:
2387:
2382:
2374:
2366:
2362:
2358:
2350:
2337:
2336:
2327:
2307:
2296:
2295:
2286:
2282:
2278:
2270:
2263:
2243:
2232:
2231:
2222:
2211:
2209:
2198:
2181:
2180:
2161:
2159:
2158:
2153:
2145:
2134:
2133:
2124:
2088:
2086:
2085:
2080:
2072:
2064:
2050:
2042:
2034:
2033:
2029:
2021:
2005:
1997:
1989:
1987:
1979:
1971:
1963:
1962:
1932:
1930:
1929:
1924:
1919:
1918:
1914:
1906:
1857:
1855:
1854:
1849:
1847:
1846:
1841:
1837:
1830:
1828:
1827:
1826:
1813:
1809:
1808:
1804:
1801:
1795:
1794:
1786:
1772:
1767:
1765:
1764:
1763:
1750:
1746:
1745:
1741:
1738:
1732:
1731:
1723:
1709:
1695:
1694:
1690:
1687:
1681:
1680:
1672:
1642:
1640:
1639:
1634:
1629:
1628:
1624:
1621:
1615:
1614:
1606:
1587:
1586:
1582:
1579:
1573:
1572:
1564:
1557:
1556:
1552:
1549:
1543:
1542:
1534:
1520:
1518:
1517:
1512:
1504:
1503:
1495:
1486:
1478:
1470:
1469:
1461:
1443:
1442:
1434:
1418:
1416:
1415:
1410:
1405:
1404:
1396:
1368:
1366:
1365:
1360:
1358:
1357:
1350:
1349:
1329:
1328:
1315:
1314:
1294:
1293:
1285:
1275:
1273:
1272:
1267:
1265:
1264:
1256:
1246:
1244:
1243:
1238:
1233:
1232:
1224:
1189:
1187:
1186:
1181:
1173:
1161:
1159:
1158:
1153:
1151:
1150:
1134:
1132:
1131:
1126:
1118:
1106:
1104:
1103:
1098:
1090:
1078:
1076:
1075:
1070:
1062:
1054:
1040:
1039:
1032:
1015:
1013:
1012:
1007:
999:
991:
979:
972:
955:
953:
952:
947:
939:
927:
925:
924:
919:
911:
893:
886:
882:
879:
873:
871:
830:
798:
790:
780:gradient descent
745:
743:
742:
737:
709:
707:
706:
701:
693:
692:
674:
673:
658:
653:
652:
647:
646:
638:
634:
626:
625:
607:
606:
588:
587:
577:
562:
561:
553:
541:
540:
522:
521:
496:
494:
493:
488:
485:
480:
465:
464:
446:with parameters
445:
443:
442:
437:
429:
428:
410:
409:
394:
333:
331:
330:
325:
323:
322:
302:
300:
299:
294:
292:
291:
286:
285:
277:
273:
272:
262:
250:
249:
241:
228:
226:
225:
220:
186:
184:
183:
178:
176:
175:
170:
169:
161:
150:
148:
147:
142:
140:
139:
131:
4581:
4580:
4576:
4575:
4574:
4572:
4571:
4570:
4556:
4555:
4554:
4549:
4521:
4471:
4460:
4433:Superconducting
4427:
4393:
4384:Neutral atom QC
4376:Ultracold atoms
4370:
4335:implementations
4334:
4328:
4268:
4261:
4228:Quantum circuit
4196:
4190:
4184:
4174:
4134:
4128:
4095:
4088:
4044:Hidden subgroup
3996:
3985:other protocols
3941:
3918:quantum network
3913:Quantum channel
3873:
3867:
3813:No-broadcasting
3803:Gottesman–Knill
3776:
3704:
3699:
3669:
3601:
3600:
3596:
3528:
3527:
3523:
3478:Physics Reports
3474:
3473:
3469:
3408:
3407:
3398:
3330:
3329:
3325:
3280:Physics Reports
3276:
3275:
3268:
3256:
3255:
3251:
3189:
3188:
3184:
3129:
3128:
3124:
3054:
3053:
3046:
3042:
3037:
3036:
3031:
3027:
3022:
3010:
2996:
2983:
2978:
2948:
2937:
2931:
2928:
2885:
2883:
2869:
2853:
2842:
2807:
2806:
2775:
2774:
2740:
2735:
2734:
2706:
2705:
2628:
2624:
2606:
2605:
2520:
2425:
2396:
2395:
2345:
2341:
2328:
2287:
2268:
2264:
2223:
2202:
2172:
2167:
2166:
2125:
2094:
2093:
2009:
1980:
1972:
1954:
1949:
1948:
1894:
1874:
1873:
1867:
1818:
1814:
1783:
1773:
1755:
1751:
1720:
1710:
1707:
1703:
1702:
1669:
1655:
1654:
1603:
1561:
1531:
1526:
1525:
1421:
1420:
1383:
1382:
1352:
1351:
1341:
1338:
1337:
1331:
1330:
1320:
1317:
1316:
1306:
1299:
1278:
1277:
1249:
1248:
1211:
1210:
1204:
1164:
1163:
1142:
1137:
1136:
1109:
1108:
1081:
1080:
1023:
1018:
1017:
958:
957:
930:
929:
928:if we can vary
902:
901:
894:
883:
877:
874:
831:
829:
815:
799:
788:
772:
716:
715:
684:
665:
635:
617:
598:
579:
532:
513:
502:
501:
456:
448:
447:
420:
401:
385:
384:
381:
351:quantum circuit
343:
314:
309:
308:
274:
264:
234:
233:
193:
192:
189:Pauli operators
158:
153:
152:
124:
123:
97:
92:
17:
12:
11:
5:
4579:
4577:
4569:
4568:
4558:
4557:
4551:
4550:
4548:
4547:
4537:
4526:
4523:
4522:
4520:
4519:
4517:many others...
4514:
4509:
4504:
4499:
4490:
4476:
4474:
4466:
4465:
4462:
4461:
4459:
4458:
4453:
4448:
4443:
4437:
4435:
4429:
4428:
4426:
4425:
4420:
4415:
4410:
4404:
4402:
4395:
4394:
4392:
4391:
4389:Trapped-ion QC
4386:
4380:
4378:
4372:
4371:
4369:
4368:
4363:
4358:
4353:
4347:
4345:
4343:Quantum optics
4336:
4330:
4329:
4327:
4326:
4321:
4320:
4319:
4312:
4307:
4302:
4297:
4292:
4287:
4282:
4273:
4271:
4263:
4262:
4260:
4259:
4254:
4249:
4248:
4247:
4237:
4236:
4235:
4225:
4224:
4223:
4213:
4208:
4202:
4200:
4192:
4191:
4189:
4188:
4187:
4186:
4182:
4176:
4172:
4161:
4160:
4159:
4149:
4147:Quantum volume
4144:
4138:
4136:
4130:
4129:
4127:
4126:
4121:
4116:
4111:
4106:
4100:
4098:
4090:
4089:
4087:
4086:
4081:
4076:
4071:
4066:
4061:
4056:
4051:
4046:
4041:
4036:
4031:
4026:
4024:Boson sampling
4021:
4016:
4010:
4008:
4002:
4001:
3998:
3997:
3995:
3994:
3989:
3988:
3987:
3982:
3977:
3967:
3962:
3957:
3951:
3949:
3940:
3939:
3934:
3933:
3932:
3922:
3921:
3920:
3910:
3905:
3900:
3895:
3894:
3893:
3888:
3877:
3875:
3869:
3868:
3866:
3865:
3860:
3858:Solovay–Kitaev
3855:
3850:
3845:
3840:
3835:
3830:
3825:
3820:
3815:
3810:
3805:
3800:
3795:
3790:
3784:
3782:
3778:
3777:
3775:
3774:
3773:
3772:
3762:
3761:
3760:
3750:
3745:
3740:
3735:
3734:
3733:
3723:
3718:
3712:
3710:
3706:
3705:
3700:
3698:
3697:
3690:
3683:
3675:
3668:
3667:
3594:
3521:
3467:
3396:
3343:(22): 224109.
3323:
3266:
3249:
3182:
3122:
3043:
3041:
3038:
3035:
3034:
3024:
3023:
3021:
3018:
3017:
3016:
3009:
3006:
2994:
2982:
2979:
2977:
2974:
2973:
2972:
2969:
2966:
2963:
2960:
2950:
2949:
2856:
2854:
2847:
2841:
2838:
2837:
2836:
2822:
2819:
2814:
2803:
2791:
2788:
2785:
2782:
2772:
2761:
2758:
2755:
2752:
2747:
2743:
2722:
2719:
2716:
2713:
2698:
2697:
2685:
2681:
2676:
2673:
2668:
2665:
2662:
2659:
2656:
2653:
2648:
2645:
2640:
2637:
2634:
2631:
2627:
2621:
2618:
2613:
2602:
2601:
2590:
2587:
2583:
2579:
2574:
2571:
2566:
2563:
2560:
2557:
2554:
2551:
2546:
2543:
2538:
2535:
2532:
2527:
2523:
2518:
2514:
2511:
2506:
2503:
2498:
2495:
2492:
2488:
2484:
2479:
2476:
2471:
2468:
2465:
2462:
2459:
2456:
2451:
2448:
2443:
2440:
2437:
2432:
2428:
2423:
2419:
2416:
2411:
2408:
2403:
2392:
2391:
2380:
2377:
2373:
2369:
2365:
2361:
2356:
2353:
2348:
2344:
2340:
2335:
2331:
2326:
2322:
2319:
2316:
2313:
2310:
2306:
2302:
2299:
2294:
2290:
2285:
2281:
2276:
2273:
2267:
2262:
2258:
2255:
2252:
2249:
2246:
2242:
2238:
2235:
2230:
2226:
2221:
2217:
2214:
2208:
2205:
2201:
2196:
2193:
2190:
2187:
2184:
2179:
2175:
2151:
2148:
2144:
2140:
2137:
2132:
2128:
2123:
2119:
2116:
2113:
2110:
2107:
2104:
2101:
2090:
2089:
2078:
2075:
2070:
2067:
2062:
2059:
2056:
2053:
2048:
2045:
2040:
2037:
2032:
2027:
2024:
2019:
2016:
2012:
2008:
2003:
2000:
1995:
1992:
1986:
1983:
1978:
1975:
1969:
1966:
1961:
1957:
1934:
1933:
1922:
1917:
1912:
1909:
1904:
1901:
1897:
1893:
1890:
1887:
1884:
1881:
1866:
1863:
1859:
1858:
1845:
1840:
1836:
1833:
1825:
1821:
1817:
1812:
1807:
1799:
1792:
1789:
1782:
1779:
1776:
1770:
1762:
1758:
1754:
1749:
1744:
1736:
1729:
1726:
1719:
1716:
1713:
1706:
1701:
1698:
1693:
1685:
1678:
1675:
1668:
1665:
1662:
1644:
1643:
1632:
1627:
1619:
1612:
1609:
1602:
1599:
1596:
1593:
1590:
1585:
1577:
1570:
1567:
1560:
1555:
1547:
1540:
1537:
1510:
1507:
1501:
1498:
1492:
1489:
1485:
1481:
1477:
1473:
1467:
1464:
1458:
1455:
1452:
1449:
1446:
1440:
1437:
1431:
1428:
1408:
1402:
1399:
1393:
1390:
1378:instructions.
1356:
1348:
1344:
1340:
1339:
1336:
1333:
1332:
1327:
1323:
1319:
1318:
1313:
1309:
1305:
1304:
1302:
1297:
1291:
1288:
1262:
1259:
1236:
1230:
1227:
1221:
1218:
1203:
1200:
1179:
1176:
1172:
1149:
1145:
1124:
1121:
1117:
1107:, where state
1096:
1093:
1089:
1068:
1065:
1061:
1057:
1053:
1049:
1046:
1043:
1038:
1035:
1031:
1026:
1005:
1002:
998:
994:
990:
986:
983:
978:
975:
971:
966:
945:
942:
938:
917:
914:
910:
896:
895:
802:
800:
793:
787:
784:
771:
768:
764:Clifford gates
735:
732:
729:
726:
723:
711:
710:
699:
696:
691:
687:
683:
680:
677:
672:
668:
664:
661:
657:
651:
644:
641:
633:
629:
624:
620:
616:
613:
610:
605:
601:
597:
594:
591:
586:
582:
576:
572:
568:
565:
559:
556:
550:
547:
544:
539:
535:
531:
528:
525:
520:
516:
512:
509:
484:
479:
476:
473:
469:
463:
459:
455:
435:
432:
427:
423:
419:
416:
413:
408:
404:
400:
397:
393:
380:
377:
342:
339:
321:
317:
305:
304:
290:
283:
280:
271:
267:
261:
257:
253:
247:
244:
218:
215:
212:
209:
206:
203:
200:
174:
167:
164:
137:
134:
96:
95:Pauli encoding
93:
91:
88:
76:Jeremy O'Brien
15:
13:
10:
9:
6:
4:
3:
2:
4578:
4567:
4564:
4563:
4561:
4546:
4538:
4536:
4528:
4527:
4524:
4518:
4515:
4513:
4510:
4508:
4505:
4503:
4500:
4498:
4494:
4491:
4489:
4485:
4481:
4478:
4477:
4475:
4473:
4467:
4457:
4454:
4452:
4449:
4447:
4444:
4442:
4439:
4438:
4436:
4434:
4430:
4424:
4421:
4419:
4416:
4414:
4413:Spin qubit QC
4411:
4409:
4406:
4405:
4403:
4400:
4396:
4390:
4387:
4385:
4382:
4381:
4379:
4377:
4373:
4367:
4364:
4362:
4359:
4357:
4354:
4352:
4349:
4348:
4346:
4344:
4340:
4337:
4331:
4325:
4322:
4318:
4317:
4313:
4311:
4308:
4306:
4303:
4301:
4298:
4296:
4293:
4291:
4288:
4286:
4283:
4281:
4278:
4277:
4275:
4274:
4272:
4270:
4264:
4258:
4255:
4253:
4250:
4246:
4243:
4242:
4241:
4238:
4234:
4231:
4230:
4229:
4226:
4222:
4221:cluster state
4219:
4218:
4217:
4214:
4212:
4209:
4207:
4204:
4203:
4201:
4199:
4193:
4185:
4181:
4177:
4175:
4171:
4167:
4166:
4165:
4162:
4158:
4155:
4154:
4153:
4150:
4148:
4145:
4143:
4140:
4139:
4137:
4131:
4125:
4122:
4120:
4117:
4115:
4112:
4110:
4107:
4105:
4102:
4101:
4099:
4097:
4091:
4085:
4082:
4080:
4077:
4075:
4072:
4070:
4067:
4065:
4062:
4060:
4057:
4055:
4052:
4050:
4047:
4045:
4042:
4040:
4037:
4035:
4032:
4030:
4029:Deutsch–Jozsa
4027:
4025:
4022:
4020:
4017:
4015:
4012:
4011:
4009:
4007:
4003:
3993:
3990:
3986:
3983:
3981:
3978:
3976:
3973:
3972:
3971:
3968:
3966:
3965:Quantum money
3963:
3961:
3958:
3956:
3953:
3952:
3950:
3948:
3944:
3938:
3935:
3931:
3928:
3927:
3926:
3923:
3919:
3916:
3915:
3914:
3911:
3909:
3906:
3904:
3901:
3899:
3896:
3892:
3889:
3887:
3884:
3883:
3882:
3879:
3878:
3876:
3874:communication
3870:
3864:
3861:
3859:
3856:
3854:
3851:
3849:
3846:
3844:
3841:
3839:
3836:
3834:
3831:
3829:
3826:
3824:
3821:
3819:
3816:
3814:
3811:
3809:
3806:
3804:
3801:
3799:
3796:
3794:
3791:
3789:
3786:
3785:
3783:
3779:
3771:
3768:
3767:
3766:
3763:
3759:
3756:
3755:
3754:
3751:
3749:
3746:
3744:
3741:
3739:
3736:
3732:
3729:
3728:
3727:
3724:
3722:
3719:
3717:
3714:
3713:
3711:
3707:
3703:
3696:
3691:
3689:
3684:
3682:
3677:
3676:
3673:
3663:
3659:
3655:
3651:
3647:
3643:
3639:
3635:
3631:
3627:
3622:
3617:
3613:
3609:
3605:
3598:
3595:
3590:
3586:
3582:
3578:
3574:
3570:
3566:
3562:
3558:
3554:
3549:
3544:
3540:
3536:
3532:
3525:
3522:
3517:
3513:
3509:
3505:
3501:
3497:
3492:
3487:
3483:
3479:
3471:
3468:
3463:
3459:
3455:
3451:
3447:
3443:
3439:
3435:
3430:
3425:
3422:(3): 030503.
3421:
3417:
3413:
3405:
3403:
3401:
3397:
3392:
3388:
3384:
3380:
3376:
3372:
3368:
3364:
3360:
3356:
3351:
3346:
3342:
3338:
3334:
3327:
3324:
3319:
3315:
3311:
3307:
3303:
3299:
3294:
3289:
3285:
3281:
3273:
3271:
3267:
3262:
3261:
3253:
3250:
3245:
3241:
3237:
3233:
3228:
3223:
3219:
3215:
3210:
3205:
3202:(2): 023023.
3201:
3197:
3193:
3186:
3183:
3177:
3172:
3168:
3164:
3160:
3156:
3151:
3146:
3143:(1): 015004.
3142:
3138:
3134:
3126:
3123:
3118:
3114:
3109:
3104:
3100:
3096:
3092:
3088:
3084:
3080:
3075:
3070:
3066:
3062:
3058:
3051:
3049:
3045:
3039:
3029:
3026:
3019:
3015:
3012:
3011:
3007:
3005:
3003:
3000:
2992:
2988:
2980:
2975:
2970:
2967:
2964:
2961:
2958:
2954:
2953:
2946:
2943:
2935:
2924:
2921:
2917:
2914:
2910:
2907:
2903:
2900:
2896:
2893: –
2892:
2888:
2887:Find sources:
2881:
2877:
2873:
2867:
2866:
2862:
2857:This section
2855:
2851:
2846:
2845:
2839:
2820:
2817:
2812:
2804:
2786:
2780:
2773:
2756:
2750:
2745:
2717:
2711:
2703:
2702:
2701:
2683:
2674:
2671:
2666:
2663:
2657:
2654:
2646:
2643:
2638:
2635:
2629:
2625:
2619:
2616:
2611:
2604:
2603:
2585:
2572:
2569:
2564:
2561:
2555:
2552:
2544:
2541:
2536:
2533:
2525:
2521:
2512:
2504:
2501:
2496:
2490:
2477:
2474:
2469:
2466:
2460:
2457:
2449:
2446:
2441:
2438:
2430:
2426:
2417:
2409:
2406:
2401:
2394:
2393:
2375:
2367:
2363:
2359:
2354:
2351:
2346:
2342:
2338:
2333:
2329:
2320:
2314:
2308:
2300:
2297:
2292:
2288:
2283:
2279:
2274:
2271:
2265:
2256:
2250:
2244:
2236:
2233:
2228:
2224:
2215:
2206:
2194:
2188:
2182:
2177:
2165:
2164:
2163:
2146:
2138:
2135:
2130:
2126:
2117:
2111:
2105:
2099:
2076:
2073:
2068:
2065:
2060:
2057:
2054:
2051:
2046:
2043:
2038:
2035:
2030:
2025:
2022:
2017:
2014:
2010:
2006:
2001:
1998:
1993:
1990:
1984:
1976:
1967:
1964:
1959:
1947:
1946:
1945:
1943:
1939:
1920:
1915:
1910:
1907:
1902:
1899:
1895:
1891:
1885:
1879:
1872:
1871:
1870:
1864:
1862:
1838:
1834:
1831:
1823:
1819:
1787:
1777:
1768:
1760:
1756:
1724:
1714:
1704:
1699:
1673:
1663:
1653:
1652:
1651:
1649:
1607:
1597:
1591:
1588:
1565:
1558:
1535:
1524:
1523:
1522:
1496:
1487:
1479:
1462:
1453:
1447:
1435:
1426:
1397:
1388:
1379:
1377:
1371:
1354:
1346:
1342:
1334:
1325:
1321:
1311:
1307:
1300:
1295:
1286:
1257:
1225:
1216:
1207:
1201:
1195:
1191:
1174:
1147:
1143:
1119:
1091:
1063:
1055:
1047:
1041:
1033:
1024:
1000:
992:
984:
973:
940:
912:
892:
889:
881:
870:
867:
863:
860:
856:
853:
849:
846:
842:
839: –
838:
834:
833:Find sources:
827:
823:
819:
813:
812:
808:
803:This section
801:
797:
792:
791:
785:
783:
781:
777:
769:
767:
765:
761:
757:
754:-axis of the
753:
749:
733:
730:
727:
724:
721:
689:
685:
681:
678:
675:
670:
666:
659:
649:
639:
622:
618:
614:
611:
608:
603:
599:
592:
584:
580:
574:
570:
566:
554:
545:
537:
533:
529:
526:
523:
518:
514:
507:
500:
499:
498:
482:
477:
474:
471:
461:
457:
425:
421:
417:
414:
411:
406:
402:
395:
378:
376:
372:
370:
361:
357:
354:
352:
348:
340:
338:
335:
319:
315:
288:
278:
269:
265:
259:
255:
251:
242:
232:
231:
230:
229:), such that
216:
213:
210:
207:
204:
201:
198:
191:(for example
190:
172:
162:
132:
120:
118:
114:
110:
106:
101:
94:
89:
87:
85:
81:
77:
73:
68:
66:
62:
58:
54:
50:
46:
42:
38:
34:
30:
26:
22:
4441:Charge qubit
4366:KLM protocol
4315:
4179:
4169:
4083:
3863:Purification
3793:Eastin–Knill
3611:
3607:
3597:
3538:
3534:
3524:
3481:
3477:
3470:
3419:
3415:
3340:
3336:
3326:
3283:
3279:
3259:
3252:
3199:
3195:
3185:
3176:10356/161272
3140:
3136:
3125:
3064:
3060:
3028:
2984:
2981:In chemistry
2938:
2929:
2919:
2912:
2905:
2898:
2886:
2870:Please help
2858:
2699:
2091:
1941:
1937:
1935:
1868:
1860:
1647:
1645:
1380:
1376:quantum gate
1372:
1209:The circuit
1208:
1205:
899:
884:
875:
865:
858:
851:
844:
832:
816:Please help
804:
773:
759:
756:Bloch sphere
751:
747:
712:
382:
373:
366:
355:
344:
336:
306:
121:
102:
98:
69:
49:ground state
28:
24:
18:
4472:programming
4451:Phase qubit
4356:Circuit QED
3828:No-deleting
3770:cloud-based
3067:(1): 4213.
786:Formulation
379:Measurement
105:Hamiltonian
90:Description
4512:libquantum
4446:Flux qubit
4351:Cavity QED
4300:Bacon–Shor
4290:stabilizer
3818:No-cloning
3621:2004.04174
3548:1704.05018
3491:2111.05176
3429:1710.01022
3293:2111.05176
3209:1509.04279
3150:2101.08448
3040:References
2902:newspapers
848:newspapers
57:observable
4418:NV center
3853:Threshold
3833:No-hiding
3798:Gleason's
3662:215548188
3646:0036-8075
3573:1476-4687
3516:243861087
3484:: 1–128.
3454:2058-9565
3375:0021-9606
3350:1208.5986
3318:243861087
3286:: 1–128.
3236:1367-2630
3099:2041-1723
3074:1304.3061
2932:June 2024
2859:does not
2818:π
2813:±
2787:⋅
2757:θ
2746:θ
2742:∇
2718:θ
2672:π
2667:−
2664:θ
2655:−
2644:π
2636:θ
2589:⟩
2586:ϕ
2570:π
2565:−
2562:θ
2542:π
2537:−
2534:θ
2526:†
2513:ϕ
2510:⟨
2497:−
2494:⟩
2491:ϕ
2475:π
2467:θ
2447:π
2439:θ
2431:†
2418:ϕ
2415:⟨
2379:⟩
2376:ϕ
2347:−
2334:†
2321:ϕ
2318:⟨
2312:⟩
2309:ϕ
2293:†
2257:ϕ
2254:⟨
2248:⟩
2245:ϕ
2229:†
2216:ϕ
2213:⟨
2207:θ
2204:∂
2200:∂
2189:θ
2178:θ
2174:∇
2150:⟩
2147:ϕ
2131:†
2118:ϕ
2115:⟨
2106:θ
2061:−
2039:−
2023:θ
2015:−
1994:−
1985:θ
1982:∂
1974:∂
1960:θ
1956:∇
1908:θ
1900:−
1886:θ
1844:⊤
1835:…
1820:θ
1816:∂
1791:→
1788:θ
1775:∂
1757:θ
1753:∂
1728:→
1725:θ
1712:∂
1677:→
1674:θ
1661:Δ
1611:→
1608:θ
1595:∇
1589:−
1569:→
1566:θ
1539:→
1536:θ
1509:⟩
1500:→
1497:θ
1488:ψ
1466:→
1463:θ
1454:ψ
1451:⟨
1439:→
1436:θ
1401:→
1398:θ
1343:θ
1335:⋮
1322:θ
1308:θ
1290:→
1287:θ
1261:→
1258:θ
1229:→
1226:θ
1202:Algorithm
1178:⟩
1175:ψ
1123:⟩
1120:ψ
1095:⟩
1092:ψ
1067:⟩
1064:ψ
1048:ψ
1045:⟨
1042:
1037:⟩
1034:ψ
1004:⟩
1001:ψ
985:ψ
982:⟨
977:⟩
974:ψ
944:⟩
941:ψ
916:⟩
913:ψ
878:June 2024
805:does not
731:⊗
725:⊗
698:⟩
686:θ
679:⋯
667:θ
660:ψ
643:^
619:θ
612:⋯
600:θ
593:ψ
590:⟨
581:α
571:∑
564:⟩
558:^
549:⟨
534:θ
527:⋯
515:θ
458:θ
434:⟩
422:θ
415:⋯
403:θ
396:ψ
316:α
282:^
266:α
256:∑
246:^
214:⊗
208:⊗
202:⊗
166:^
136:^
61:optimizer
4560:Category
4480:OpenQASM
4456:Transmon
4333:Physical
4133:Quantum
4034:Grover's
3808:Holevo's
3781:Theorems
3731:timeline
3721:NISQ era
3654:32855334
3581:28905916
3462:56376912
3391:30699239
3383:23248989
3244:92988541
3117:25055053
3008:See also
2999:Google's
2162:. Thus,
1942:X,Y or Z
714:string
4470:Quantum
4408:Kane QC
4267:Quantum
4195:Quantum
4124:PostBQP
4094:Quantum
4079:Simon's
3872:Quantum
3709:General
3626:Bibcode
3608:Science
3589:4390182
3553:Bibcode
3496:Bibcode
3434:Bibcode
3355:Bibcode
3298:Bibcode
3214:Bibcode
3155:Bibcode
3108:4124861
3079:Bibcode
2989:or the
2916:scholar
2880:removed
2865:sources
1944:, then
1865:Example
862:scholar
826:removed
811:sources
31:) is a
4488:IBM QX
4484:Qiskit
4423:NMR QC
4401:-based
4305:Steane
4276:Codes
4074:Shor's
3980:SARG04
3788:Bell's
3660:
3652:
3644:
3587:
3579:
3571:
3535:Nature
3514:
3460:
3452:
3389:
3381:
3373:
3316:
3242:
3234:
3115:
3105:
3097:
2918:
2911:
2904:
2897:
2889:
1936:where
1646:where
1025:argmin
864:
857:
850:
843:
835:
307:where
53:ansatz
23:, the
4310:Toric
3753:Qubit
3658:S2CID
3616:arXiv
3585:S2CID
3543:arXiv
3512:S2CID
3486:arXiv
3458:S2CID
3424:arXiv
3387:S2CID
3345:arXiv
3314:S2CID
3288:arXiv
3240:S2CID
3204:arXiv
3145:arXiv
3069:arXiv
3020:Notes
2923:JSTOR
2909:books
869:JSTOR
855:books
4502:Cirq
4493:Quil
4399:Spin
4295:Shor
3975:BB84
3908:LOCC
3650:PMID
3642:ISSN
3577:PMID
3569:ISSN
3450:ISSN
3379:PMID
3371:ISSN
3232:ISSN
3113:PMID
3095:ISSN
2895:news
2863:any
2861:cite
2733:and
2092:As,
841:news
809:any
807:cite
74:and
43:and
35:for
4316:gnu
4280:CSS
4157:XEB
4119:QMA
4114:QIP
4109:EQP
4104:BQP
4084:VQE
4039:HHL
3843:PBR
3634:doi
3612:369
3561:doi
3539:549
3504:doi
3482:986
3442:doi
3363:doi
3341:137
3306:doi
3284:986
3222:doi
3171:hdl
3163:doi
3103:PMC
3087:doi
2976:Use
2874:by
1802:old
1739:old
1688:old
1622:old
1580:old
1550:new
1135:is
965:min
820:by
29:VQE
19:In
4562::
4507:Q#
3656:.
3648:.
3640:.
3632:.
3624:.
3610:.
3606:.
3583:.
3575:.
3567:.
3559:.
3551:.
3537:.
3533:.
3510:.
3502:.
3494:.
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