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# Variational quantum algorithms

## Abstract

Applications such as simulating complicated quantum systems or solving large-scale linear algebra problems are very challenging for classical computers, owing to the extremely high computational cost. Quantum computers promise a solution, although fault-tolerant quantum computers will probably not be available in the near future. Current quantum devices have serious constraints, including limited numbers of qubits and noise processes that limit circuit depth. Variational quantum algorithms (VQAs), which use a classical optimizer to train a parameterized quantum circuit, have emerged as a leading strategy to address these constraints. VQAs have now been proposed for essentially all applications that researchers have envisaged for quantum computers, and they appear to be the best hope for obtaining quantum advantage. Nevertheless, challenges remain, including the trainability, accuracy and efficiency of VQAs. Here we overview the field of VQAs, discuss strategies to overcome their challenges and highlight the exciting prospects for using them to obtain quantum advantage.

## Key points

• Variational quantum algorithms (VQAs) are the leading proposal for achieving quantum advantage using near-term quantum computers.

• VQAs have been developed for a wide range of applications, including finding ground states of molecules, simulating dynamics of quantum systems and solving linear systems of equations.

• VQAs share a common structure, where a task is encoded into a parameterized cost function that is evaluated using a quantum computer, and a classical optimizer trains the parameters in the VQA.

• The adaptive nature of VQAs is well suited to handle the constraints of near-term quantum computers.

• Trainability, accuracy and efficiency are three challenges that arise when applying VQAs to large-scale applications, and strategies are currently being developed to address these challenges.

## Relevant articles

• ### Variational quantum eigensolver techniques for simulating carbon monoxide oxidation

Communications Physics Open Access 06 August 2022

• ### Quantum annealing with special drivers for circuit fault diagnostics

Scientific Reports Open Access 08 July 2022

• ### Variational quantum evolution equation solver

Scientific Reports Open Access 25 June 2022

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## References

1. Shor, P. W. Algorithms for quantum computation: discrete logarithms and factoring. In Proc. 35th Annual Symposium on Foundations of Computer Science, 124–134 (IEEE, 1994).

2. Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996).

3. Harrow, A. W., Hassidim, A. & Lloyd, S. Quantum algorithm for linear systems of equations. Phys. Rev. Lett. 103, 150502 (2009).

4. IBM Makes Quantum Computing Available on IBM Cloud to Accelerate Innovation. Press release at https://www-03.ibm.com/press/us/en/pressrelease/49661.wss (2016).

5. Adedoyin, A. et al. Quantum algorithm implementations for beginners. Preprint at https://arxiv.org/abs/1804.03719 (2018).

6. Preskill, J. Quantum computing in the NISQ era and beyond. Quantum 2, 79 (2018).

7. Arute, F. et al. Quantum supremacy using a programmable superconducting processor. Nature 574, 505–510 (2019).

8. Zhong, H.-S. et al. Quantum computational advantage using photons. Science 370, 1460–1463 (2020).

9. Bittel, L. & Kliesch, M. Training variational quantum algorithms is np-hard — even for logarithmically many qubits and free fermionic systems. Preprint at https://arxiv.org/abs/2101.07267 (2021).

10. Kingma, D. P. & Ba, J. Adam: a method for stochastic optimization. In Proc. 3rd International Conference on Learning Representations (ICLR) (ICLR, 2015).

11. Kübler, J. M., Arrasmith, A., Cincio, L. & Coles, P. J. An adaptive optimizer for measurement-frugal variational algorithms. Quantum 4, 263 (2020).

12. Sweke, R. et al. Stochastic gradient descent for hybrid quantum–classical optimization. Quantum 4, 314 (2020).

13. McArdle, S. et al. Variational ansatz-based quantum simulation of imaginary time evolution. NPJ Quantum Inf. 5, 75 (2019).

14. Stokes, J., Izaac, J., Killoran, N. & Carleo, G. Quantum natural gradient. Quantum 4, 269 (2020).

15. Koczor, B. & Benjamin, S. C. Quantum natural gradient generalised to non-unitary circuits. Preprint at https://arxiv.org/abs/1912.08660 (2019).

16. Wilson, M. et al. Optimizing quantum heuristics with meta-learning. Preprint at https://arxiv.org/abs/1908.03185 (2019).

17. Spall, J. C. Multivariate stochastic approximation using a simultaneous perturbation gradient approximation. IEEE Trans. Automat. Control 37, 332–341 (1992).

18. Nakanishi, K. M., Fujii, K. & Todo, S. Sequential minimal optimization for quantum–classical hybrid algorithms. Phys. Rev. Res. 2, 043158 (2020).

19. Parrish, R. M., Iosue, J. T., Ozaeta, A. & McMahon, P. L. A Jacobi diagonalization and Anderson acceleration algorithm for variational quantum algorithm parameter optimization. Preprint at https://arxiv.org/abs/1904.03206 (2019).

20. Huembeli, P. & Dauphin, A. Characterizing the loss landscape of variational quantum circuits. Quantum Sci. Technol. 6, 025011 (2021).

21. Harrow, A. & Napp, J. Low-depth gradient measurements can improve convergence in variational hybrid quantum–classical algorithms. Phys. Rev. Lett. 126, 140502 (2021).

22. Kandala, A. et al. Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. Nature 549, 242–246 (2017).

23. Gard, B. T. et al. Efficient symmetry-preserving state preparation circuits for the variational quantum eigensolver algorithm. NPJ Quantum Inf. 6, 10 (2020).

24. Otten, M., Cortes, C. L. & Gray, S. K. Noise-resilient quantum dynamics using symmetry-preserving ansatzes. Preprint at https://arxiv.org/abs/1910.06284 (2019).

25. Tkachenko, N. V. et al. Correlation-informed permutation of qubits for reducing ansatz depth in VQE. PRX Quantum 2, 020337 (2021).

26. Kokail, C. et al. Self-verifying variational quantum simulation of lattice models. Nature 569, 355–360 (2019).

27. Bravo-Prieto, C., Lumbreras-Zarapico, J., Tagliacozzo, L. & Latorre, J. I. Scaling of variational quantum circuit depth for condensed matter systems. Quantum 4, 272 (2020).

28. Taube, A. G. & Bartlett, R. J. New perspectives on unitary coupled-cluster theory. Int. J. Quantum Chem. 106, 3393–3401 (2006).

29. Peruzzo, A. et al. A variational eigenvalue solver on a photonic quantum processor. Nat. Commun. 5, 4213 (2014).

30. Bravyi, S. B. & Kitaev, A. Y. Fermionic quantum computation. Ann. Phys. 298, 210–226 (2002).

31. Lee, J., Huggins, W. J., Head-Gordon, M. & Whaley, K. B. Generalized unitary coupled cluster wave functions for quantum computation. J. Chem. Theory Comput. 15, 311–324 (2019).

32. Motta, M. et al. Low rank representations for quantum simulation of electronic structure. Preprint at https://arxiv.org/abs/1808.02625 (2018).

33. Matsuzawa, Y. & Kurashige, Y. Jastrow-type decomposition in quantum chemistry for low-depth quantum circuits. J. Chem. Theory Comput. 16, 944–952 (2020).

34. Kivlichan, I. D. et al. Quantum simulation of electronic structure with linear depth and connectivity. Phys. Rev. Lett. 120, 110501 (2018).

35. Setia, K., Bravyi, S., Mezzacapo, A. & Whitfield, J. D. Superfast encodings for fermionic quantum simulation. Phys. Rev. Res. 1, 033033 (2019).

36. Farhi, E., Goldstone, J. & Gutmann, S. A quantum approximate optimization algorithm. Preprint at https://arxiv.org/abs/1411.4028 (2014).

37. Hadfield, S. et al. From the quantum approximate optimization algorithm to a quantum alternating operator ansatz. Algorithms 12, 34 (2019).

38. Lloyd, S. Quantum approximate optimization is computationally universal. Preprint at https://arxiv.org/abs/1812.11075 (2018).

39. Morales, M. E., Biamonte, J. & Zimborás, Z. On the universality of the quantum approximate optimization algorithm. Quantum Inf. Process. 19, 291 (2020).

40. Wang, Z., Rubin, N. C., Dominy, J. M. & Rieffel, E. G. XY mixers: analytical and numerical results for the quantum alternating operator ansatz. Phys. Rev. A 101, 012320 (2020).

41. Wecker, D., Hastings, M. B. & Troyer, M. Progress towards practical quantum variational algorithms. Phys. Rev. A 92, 042303 (2015).

42. Wiersema, R. et al. Exploring entanglement and optimization within the Hamiltonian variational ansatz. Phys. Rev. X Quantum 1, 020319 (2020).

43. Ho, W. W. & Hsieh, T. H. Efficient variational simulation of non-trivial quantum states. SciPost Phys. 6, 029 (2019).

44. Grimsley, H. R., Economou, S. E., Barnes, E. & Mayhall, N. J. An adaptive variational algorithm for exact molecular simulations on a quantum computer. Nat. Commun. 10, 3007 (2019).

45. Tang, H. L. et al. qubit-ADAPT-VQE: an adaptive algorithm for constructing hardware-efficient ansatze on a quantum processor. PRX Quantum 2, 020310 (2021).

46. Yordanov, Y. S., Armaos, V., Barnes, C. H. & Arvidsson-Shukur, D. R. Iterative qubit-excitation based variational quantum eigensolver. Preprint at https://arxiv.org/abs/2011.10540 (2020).

47. Zhu, L. et al. An adaptive quantum approximate optimization algorithm for solving combinatorial problems on a quantum computer. Preprint at https://arxiv.org/abs/2005.10258 (2020).

48. Rattew, A. G., Hu, S., Pistoia, M., Chen, R. & Wood, S. A domain-agnostic, noise-resistant, hardware-efficient evolutionary variational quantum eigensolver. Preprint at https://arxiv.org/abs/1910.09694 (2019).

49. Chivilikhin, D. et al. MoG-VQE: multiobjective genetic variational quantum eigensolver. Preprint at https://arxiv.org/abs/2007.04424 (2020).

50. Cincio, L., Rudinger, K., Sarovar, M. & Coles, P. J. Machine learning of noise-resilient quantum circuits. Phys. Rev. X Quantum 2, 010324 (2021).

51. Cincio, L., Subaşı, Y., Sornborger, A. T. & Coles, P. J. Learning the quantum algorithm for state overlap. New J. Phys. 20, 113022 (2018).

52. Du, Y., Huang, T., You, S., Hsieh, M.-H. & Tao, D. Quantum circuit architecture search: error mitigation and trainability enhancement for variational quantum solvers. Preprint at https://arxiv.org/abs/2010.10217 (2020).

53. Zhang, S.-X., Hsieh, C.-Y., Zhang, S. & Yao, H. Differentiable quantum architecture search. Preprint at https://arxiv.org/abs/2010.08561 (2020).

54. Bilkis, M., Cerezo, M., Verdon, G., Coles, P. J. & Cincio, L. A semi-agnostic ansatz with variable structure for quantum machine learning. Preprint at https://arxiv.org/abs/2103.06712 (2021).

55. Rattew, A. G., Hu, S., Pistoia, M., Chen, R. & Wood, S. A domain-agnostic, noise-resistant, hardware-efficient evolutionary variational quantum eigensolver. Preprint at https://arxiv.org/abs/1910.09694 (2019).

56. Yang, Z.-C., Rahmani, A., Shabani, A., Neven, H. & Chamon, C. Optimizing variational quantum algorithms using Pontryagin’s minimum principle. Phys. Rev. X 7, 021027 (2017).

57. Magann, A. B. et al. From pulses to circuits and back again: a quantum optimal control perspective on variational quantum algorithms. Phys. Rev. X Quantum 2, 010101 (2021).

58. Choquette, A. et al. Quantum-optimal-control-inspired ansatz for variational quantum algorithms. Phys. Rev. Res. 3, 023092 (2021).

59. Li, J., Yang, X., Peng, X. & Sun, C.-P. Hybrid quantum–classical approach to quantum optimal control. Phys. Rev. Lett. 118, 150503 (2017).

60. Lu, D. et al. Enhancing quantum control by bootstrapping a quantum processor of 12 qubits. NPJ Quantum Inf. 3, 45 (2017).

61. O’Malley, P. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016).

62. Takeshita, T. et al. Increasing the representation accuracy of quantum simulations of chemistry without extra quantum resources. Phys. Rev. X 10, 011004 (2020).

63. Valiant, L. G. Quantum circuits that can be simulated classically in polynomial time. SIAM J. Comput. 31, 1229–1254 (2002).

64. Terhal, B. M. & DiVincenzo, D. P. Classical simulation of noninteracting-fermion quantum circuits. Phys. Rev. A 65, 032325 (2002).

65. Jozsa, R. & Miyake, A. Matchgates and classical simulation of quantum circuits. Proc. Math. Phys. Eng. Sci. 464, 3089–3106 (2008).

66. Mizukami, W. et al. Orbital optimized unitary coupled cluster theory for quantum computer. Phys. Rev. Res. 2, 033421 (2020).

67. Sokolov, I. O. et al. Quantum orbital-optimized unitary coupled cluster methods in the strongly correlated regime: can quantum algorithms outperform their classical equivalents? J. Chem. Phys. 152, 124107 (2020).

68. McClean, J. R., Kimchi-Schwartz, M. E., Carter, J. & de Jong, W. A. Hybrid quantum–classical hierarchy for mitigation of decoherence and determination of excited states. Phys. Rev. A 95, 042308 (2017).

69. Parrish, R. M., Hohenstein, E. G., McMahon, P. L. & Martínez, T. J. Quantum computation of electronic transitions using a variational quantum eigensolver. Phys. Rev. Lett. 122, 230401 (2019).

70. Parrish, R. M. & McMahon, P. L. Quantum filter diagonalization: quantum eigendecomposition without full quantum phase estimation. Preprint at https://arxiv.org/abs/1909.08925 (2019).

71. Huggins, W. J., Lee, J., Baek, U., O’Gorman, B. & Whaley, K. B. A non-orthogonal variational quantum eigensolver. New J. Phys. 22, 073009 (2020).

72. Stair, N. H., Huang, R. & Evangelista, F. A. A multireference quantum Krylov algorithm for strongly correlated electrons. J. Chem. Theory Comput. 16, 2236–2245 (2020).

73. Bharti, K. & Haug, T. Iterative quantum assisted eigensolver. Preprint at https://arxiv.org/abs/2010.05638 (2020).

74. Bharti, K. & Haug, T. Quantum assisted simulator. Preprint at https://arxiv.org/abs/2011.06911 (2020).

75. Markov, I. L. & Shi, Y. Simulating quantum computation by contracting tensor networks. SIAM J. Comput. 38, 963–981 (2008).

76. Kim, I. H. & Swingle, B. Robust entanglement renormalization on a noisy quantum computer. Preprint at https://arxiv.org/abs/1711.07500 (2017).

77. Kim, I. H. Holographic quantum simulation. Preprint at https://arxiv.org/abs/1702.02093 (2017).

78. Liu, J.-G., Zhang, Y.-H., Wan, Y. & Wang, L. Variational quantum eigensolver with fewer qubits. Phys. Rev. Res. 1, 023025 (2019).

79. Barratt, F. et al. Parallel quantum simulation of large systems on small quantum computers. npj Quantum Inf. 7, 79 (2021).

80. Yuan, X., Sun, J., Liu, J., Zhao, Q. & Zhou, Y. Quantum simulation with hybrid tensor networks. Preprint at https://arxiv.org/abs/2007.00958 (2020).

81. Fujii, K., Mitarai, K., Mizukami, W. & Nakagawa, Y. O. Deep variational quantum eigensolver: a divide-and-conquer method for solving a larger problem with smaller size quantum computers. Preprint at https://arxiv.org/abs/2007.10917 (2020).

82. Mazzola, G., Ollitrault, P. J., Barkoutsos, P. K. & Tavernelli, I. Nonunitary operations for ground-state calculations in near-term quantum computers. Phys. Rev. Lett. 123, 130501 (2019).

83. Martyn, J. & Swingle, B. Product spectrum ansatz and the simplicity of thermal states. Phys. Rev. A 100, 032107 (2019).

84. Yoshioka, N., Nakagawa, Y. O., Mitarai, K. & Fujii, K. Variational quantum algorithm for nonequilibrium steady states. Phys. Rev. Res. 2, 043289 (2020).

85. Verdon, G., Marks, J., Nanda, S., Leichenauer, S. & Hidary, J. Quantum Hamiltonian-based models and the variational quantum thermalizer algorithm. Preprint at https://arxiv.org/abs/1910.02071 (2019).

86. Liu, J., Mao, L., Zhang, P. & Wang, L. Solving quantum statistical mechanics with variational autoregressive networks and quantum circuits. Mach. Learn. Sci. Technol. 2, 025011 (2021).

87. Sim, S., Johnson, P. D. & Aspuru-Guzik, A. Expressibility and entangling capability of parameterized quantum circuits for hybrid quantum–classical algorithms. Adv. Quantum Technol. 2, 1900070 (2019).

88. Nakaji, K. & Yamamoto, N. Expressibility of the alternating layered ansatz for quantum computation. Quantum5, 434 (2021).

89. Schuld, M., Sweke, R. & Meyer, J. J. The effect of data encoding on the expressive power of variational quantum machine learning models. Phys. Rev. A 103, 032430 (2021).

90. Abbas, A. et al. The power of quantum neural networks. Nat. Comput. Sci. 1, 403–409 (2021).

91. Holmes, Z., Sharma, K., Cerezo, M. & Coles, P. J. Connecting ansatz expressibility to gradient magnitudes and barren plateaus. Preprint at https://arxiv.org/abs/2101.02138 (2021).

92. Guerreschi, G. G. & Smelyanskiy, M. Practical optimization for hybrid quantum–classical algorithms. Preprint at https://arxiv.org/abs/1701.01450 (2017).

93. Mitarai, K., Negoro, M., Kitagawa, M. & Fujii, K. Quantum circuit learning. Phys. Rev. A 98, 032309 (2018).

94. Schuld, M., Bergholm, V., Gogolin, C., Izaac, J. & Killoran, N. Evaluating analytic gradients on quantum hardware. Phys. Rev. A 99, 032331 (2019).

95. Bergholm, V. et al. Pennylane: Automatic differentiation of hybrid quantum–classical computations. Preprint at https://arxiv.org/abs/1811.04968 (2018).

96. Mari, A., Bromley, T. R. & Killoran, N. Estimating the gradient and higher-order derivatives on quantum hardware. Phys. Rev. A 103, 012405 (2021).

97. Cerezo, M. & Coles, P. J. Impact of barren plateaus on the Hessian and higher order derivatives. Quantum Sci. Technol. 6, 035006 (2021).

98. Koczor, B. & Benjamin, S. C. Quantum analytic descent. Preprint at https://arxiv.org/abs/2008.13774 (2020).

99. Li, Y. & Benjamin, S. C. Efficient variational quantum simulator incorporating active error minimization. Phys. Rev. X 7, 021050 (2017).

100. Yuan, X., Endo, S., Zhao, Q., Li, Y. & Benjamin, S. C. Theory of variational quantum simulation. Quantum 3, 191 (2019).

101. Endo, S., Sun, J., Li, Y., Benjamin, S. C. & Yuan, X. Variational quantum simulation of general processes. Phys. Rev. Lett. 125, 010501 (2020).

102. Mitarai, K. & Fujii, K. Methodology for replacing indirect measurements with direct measurements. Phys. Rev. Res. 1, 013006 (2019).

103. Biamonte, J. Universal variational quantum computation. Phys. Rev. A 103, L030401 (2021).

104. Abrams, D. S. & Lloyd, S. Quantum algorithm providing exponential speed increase for finding eigenvalues and eigenvectors. Phys. Rev. Lett. 83, 5162–5165 (1999).

105. Aspuru-Guzik, A., Dutoi, A. D., Love, P. J. & Head-Gordon, M. Simulated quantum computation of molecular energies. Science 309, 1704–1707 (2005).

106. Higgott, O., Wang, D. & Brierley, S. Variational quantum computation of excited states. Quantum 3, 156 (2019).

107. Buhrman, H., Cleve, R., Watrous, J. & De Wolf, R. Quantum fingerprinting. Phys. Rev. Lett. 87, 167902 (2001).

108. Jones, T., Endo, S., McArdle, S., Yuan, X. & Benjamin, S. C. Variational quantum algorithms for discovering Hamiltonian spectra. Phys. Rev. A 99, 062304 (2019).

109. Nakanishi, K. M., Mitarai, K. & Fujii, K. Subspace-search variational quantum eigensolver for excited states. Phys. Rev. Res. 1, 033062 (2019).

110. McClean, J. R. et al. Low depth mechanisms for quantum optimization. Preprint at https://arxiv.org/abs/2008.08615 (2020).

111. Garcia-Saez, A. & Latorre, J. Addressing hard classical problems with adiabatically assisted variational quantum eigensolvers. Preprint at https://arxiv.org/abs/1806.02287 (2018).

112. Cerezo, M., Sharma, K., Arrasmith, A. & Coles, P. J. Variational quantum state eigensolver. Preprint at https://arxiv.org/abs/2004.01372 (2020).

113. Wang, D., Higgott, O. & Brierley, S. Accelerated variational quantum eigensolver. Phys. Rev. Lett. 122, 140504 (2019).

114. Wang, G., Koh, D. E., Johnson, P. D. & Cao, Y. Minimizing estimation runtime on noisy quantum computers. Phys. Rev. X Quantum 2, 010346 (2021).

115. Guoming, W., Koh, D. E., Johnson, P. D. & Cao, Yudong. Bayesian inference with engineered likelihood functions for robust amplitude estimation. PRX Quantum 2, 010346 (2021).

116. Nielsen, M. A. & Chuang, I. L. Quantum Computation and Quantum Information, 10th Anniversary Edition (Cambridge Univ. Press, 2011).

117. McLachlan, A. A variational solution of the time-dependent Schrodinger equation. Mol. Phys. 8, 39–44 (1964).

118. Yao, Y.-X. et al. Adaptive variational quantum dynamics simulations. Preprint at https://arxiv.org/abs/2011.00622 (2020).

119. Zhang, Z.-J., Sun, J., Yuan, X. & Yung, M.-H. Low-depth hamiltonian simulation by adaptive product formula. Preprint at https://arxiv.org/abs/2011.05283 (2020).

120. Heya, K., Nakanishi, K. M., Mitarai, K. & Fujii, K. Subspace variational quantum simulator. Preprint at https://arxiv.org/abs/1904.08566 (2019).

121. Cirstoiu, C. et al. Variational fast forwarding for quantum simulation beyond the coherence time. NPJ Quantum Inf. 6, 82 (2020).

122. Gibbs, J. et al. Long-time simulations with high fidelity on quantum hardware. Preprint at https://arxiv.org/abs/2102.04313 (2021).

123. Khatri, S. et al. Quantum-assisted quantum compiling. Quantum 3, 140 (2019).

124. Commeau, B. et al. Variational Hamiltonian diagonalization for dynamical quantum simulation. Preprint at https://arxiv.org/abs/2009.02559 (2020).

125. Moll, N. et al. Quantum optimization using variational algorithms on near-term quantum devices. Quantum Sci. Technol. 3, 030503 (2018).

126. Lin, C. Y.-Y. & Zhu, Y. Performance of qaoa on typical instances of constraint satisfaction problems with bounded degree. Preprint at https://arxiv.org/abs/1601.01744 (2016).

127. Wang, Z., Hadfield, S., Jiang, Z. & Rieffel, E. G. Quantum approximate optimization algorithm for MaxCut: a fermionic view. Phys. Rev. A 97, 022304 (2018).

128. Shaydulin, R., Safro, I. & Larson, J. Multistart methods for quantum approximate optimization. In 2019 IEEE High Performance Extreme Computing Conference (HPEC) (IEEE, 2019); https://ieeexplore.ieee.org/document/8916288/

129. Romero, J. et al. Strategies for quantum computing molecular energies using the unitary coupled cluster ansatz. Quantum Sci. Technol. 4, 014008 (2018).

130. Crooks, G. E. Performance of the quantum approximate optimization algorithm on the maximum cut problem. Preprint at https://arxiv.org/abs/1811.08419 (2018).

131. Wecker, D., Hastings, M. B. & Troyer, M. Training a quantum optimizer. Phys. Rev. A 94, 022309 (2016).

132. Khairy, S., Shaydulin, R., Cincio, L., Alexeev, Y. & Balaprakash, P. Learning to optimize variational quantum circuits to solve combinatorial problems. Proc. AAAI Conf. Artif. Intell. 34, 2367–2375 (2020).

133. Ambainis, A. Variable time amplitude amplification and a faster quantum algorithm for solving systems of linear equations. In 29th Int. Symp. Theoretical Aspects of Computer Science (STACS 2012), 636–647 (Dagstuhl, 2012).

134. Subaşı, Y., Somma, R. D. & Orsucci, D. Quantum algorithms for systems of linear equations inspired by adiabatic quantum computing. Phys. Rev. Lett. 122, 060504 (2019).

135. Childs, A., Kothari, R. & Somma, R. Quantum algorithm for systems of linear equations with exponentially improved dependence on precision. SIAM J. Comput. 46, 1920–1950 (2017).

136. Chakraborty, S., Gilyén, A. & Jeffery, S. The power of block-encoded matrix powers: improved regression techniques via faster Hamiltonian simulation. In Leibniz International Proceedings in Informatics (LIPIcs) Vol. 132, 33:1–33:14 (Dagstuhl, 2019).

137. Scherer, A. et al. Concrete resource analysis of the quantum linear-system algorithm used to compute the electromagnetic scattering cross section of a 2D target. Quantum Inf. Process. 16, 60 (2017).

138. Bravo-Prieto, C. et al. Variational quantum linear solver: a hybrid algorithm for linear systems. Preprint at https://arxiv.org/abs/1909.05820 (2019).

139. Xu, X. et al. Variational algorithms for linear algebra. Preprint at https://arxiv.org/abs/1911.05759 (2019).

140. Huang, H.-Y., Bharti, K. & Rebentrost, P. Near-term quantum algorithms for linear systems of equations. Preprint at https://arxiv.org/abs/1909.07344 (2019).

141. Lubasch, M., Joo, J., Moinier, P., Kiffner, M. & Jaksch, D. Variational quantum algorithms for nonlinear problems. Phys. Rev. A 101, 010301 (2020).

142. Kyriienko, O., Paine, A. E. & Elfving, V. E. Solving nonlinear differential equations with differentiable quantum circuits. Phys. Rev. A 103, 052416 (2021).

143. Anschuetz, E., Olson, J., Aspuru-Guzik, A. & Cao, Y. Variational quantum factoring. In International Workshop on Quantum Technology and Optimization Problems, 74–85 (Springer, 2019).

144. Lloyd, S., Mohseni, M. & Rebentrost, P. Quantum principal component analysis. Nat. Phys. 10, 631–633 (2014).

145. LaRose, R., Tikku, A., O’Neel-Judy, É., Cincio, L. & Coles, P. J. Variational quantum state diagonalization. NPJ Quantum Inf. 5, 57 (2019).

146. Heya, K., Suzuki, Y., Nakamura, Y. & Fujii, K. Variational quantum gate optimization. Preprint at https://arxiv.org/abs/1810.12745 (2018).

147. Jones, T. & Benjamin, S. C. Quantum compilation and circuit optimisation via energy dissipation. Preprint at https://arxiv.org/abs/1811.03147 (2018).

148. Sharma, K., Khatri, S., Cerezo, M. & Coles, P. J. Noise resilience of variational quantum compiling. New J. Phys. 22, 043006 (2020).

149. Carolan, J. et al. Variational quantum unsampling on a quantum photonic processor. Nat. Phys. 16, 322–327 (2020).

150. Johnson, P. D., Romero, J., Olson, J., Cao, Y. & Aspuru-Guzik, A. Qvector: an algorithm for device-tailored quantum error correction. Preprint at https://arxiv.org/abs/1711.02249 (2017).

151. Xu, X., Benjamin, S. C. & Yuan, X. Variational circuit compiler for quantum error correction. Phys. Rev. Appl. 15, 034068 (2021).

152. Biamonte, J. et al. Quantum machine learning. Nature 549, 195–202 (2017).

153. Farhi, E. & Neven, H. Classification with quantum neural networks on near term processors. Preprint at https://arxiv.org/abs/1802.06002 (2018).

154. Schuld, M., Bocharov, A., Svore, K. M. & Wiebe, N. Circuit-centric quantum classifiers. Phys. Rev. A 101, 032308 (2020).

155. Schuld, M. & Killoran, N. Quantum machine learning in feature Hilbert spaces. Phys. Rev. Lett. 122, 040504 (2019).

156. Havlíček, V. et al. Supervised learning with quantum-enhanced feature spaces. Nature 567, 209–212 (2019).

157. Stoudenmire, E. & Schwab, D. J. Supervised learning with tensor networks. Adv. Neural Inf. Proc. Syst. 29, 4799–4807 (2016).

158. Lloyd, S., Schuld, M., Ijaz, A., Izaac, J. & Killoran, N. Quantum embeddings for machine learning. Preprint at https://arxiv.org/abs/2001.03622 (2020).

159. Pérez-Salinas, A., Cervera-Lierta, A., Gil-Fuster, E. & Latorre, J. I. Data re-uploading for a universal quantum classifier. Quantum 4, 226 (2020).

160. Kusumoto, T., Mitarai, K., Fujii, K., Kitagawa, M. & Negoro, M. Experimental quantum kernel machine learning with nuclear spins in a solid. Physica A 541, 123290 (2020).

161. Romero, J., Olson, J. P. & Aspuru-Guzik, A. Quantum autoencoders for efficient compression of quantum data. Quantum Sci. Technol. 2, 045001 (2017).

162. Wan, K. H., Dahlsten, O., Kristjánsson, H., Gardner, R. & Kim, M. Quantum generalisation of feedforward neural networks. NPJ Quantum Inf. 3, 36 (2017).

163. Verdon, G., Pye, J. & Broughton, M. A universal training algorithm for quantum deep learning. Preprint at https://arxiv.org/abs/1806.09729 (2018).

164. Cerezo, M., Sone, A., Volkoff, T., Cincio, L. & Coles, P. J. Cost function dependent barren plateaus in shallow parameterized quantum circuits. Nat. Commun. 12, 1791 (2021).

165. Cao, C. & Wang, X. Noise-assisted quantum autoencoder. Phys. Rev. Appl. 15, 054012 (2021).

166. Pepper, A., Tischler, N. & Pryde, G. J. Experimental realization of a quantum autoencoder: the compression of qutrits via machine learning. Phys. Rev. Lett. 122, 060501 (2019).

167. Verdon, G., Broughton, M. & Biamonte, J. A quantum algorithm to train neural networks using low-depth circuits. Preprint at https://arxiv.org/abs/1712.05304 (2017).

168. Benedetti, M. et al. A generative modeling approach for benchmarking and training shallow quantum circuits. NPJ Quantum Inf. 5, 45 (2019).

169. Du, Y., Hsieh, M.-H., Liu, T. & Tao, D. Expressive power of parameterized quantum circuits. Phys. Rev. Res. 2, 033125 (2020).

170. Liu, J.-G. & Wang, L. Differentiable learning of quantum circuit Born machines. Phys. Rev. A 98, 062324 (2018).

171. Coyle, B., Mills, D., Danos, V. & Kashefi, E. The Born supremacy: quantum advantage and training of an Ising Born machine. NPJ Quantum Inf. 6, 60 (2020).

172. Romero, J. & Aspuru-Guzik, A. Variational quantum generators: generative adversarial quantum machine learning for continuous distributions. Preprint at https://arxiv.org/abs/1901.00848 (2019).

173. Altaisky, M. Quantum neural network. Preprint at https://arxiv.org/abs/quant-ph/0107012 (2001).

174. Beer, K. et al. Training deep quantum neural networks. Nat. Commun. 11, 808 (2020).

175. Cong, I., Choi, S. & Lukin, M. D. Quantum convolutional neural networks. Nat. Phys. 15, 1273–1278 (2019).

176. Franken, L. & Georgiev, B. Explorations in quantum neural networks with intermediate measurements. In Proc. ESANN (2020); https://www.esann.org/sites/default/files/proceedings/2020/ES2020-197.pdf

177. Pesah, A. et al. Absence of barren plateaus in quantum convolutional neural networks. Preprint at https://arxiv.org/abs/2011.02966 (2020).

178. Zhang, K., Hsieh, M.-H., Liu, L. & Tao, D. Toward trainability of quantum neural networks. Preprint at https://arxiv.org/abs/2011.06258 (2020).

179. Zurek, W. H. Quantum Darwinism. Nat. Phys. 5, 181–188 (2009).

180. Arrasmith, A., Cincio, L., Sornborger, A. T., Zurek, W. H. & Coles, P. J. Variational consistent histories as a hybrid algorithm for quantum foundations. Nat. Commun. 10, 3438 (2019).

181. Griffiths, R. B. Consistent Quantum Theory (Cambridge Univ. Press, 2003).

182. Holmes, Z. et al. Barren plateaus preclude learning scramblers. Phys. Rev. Lett. 126, 190501 (2021).

183. Hayden, P. & Preskill, J. Black holes as mirrors: quantum information in random subsystems. J. High Energy Phys. 2007, 120 (2007).

184. Wilde, M. M. Quantum Information Theory (Cambridge Univ. Press, 2013).

185. Rosgen, B. & Watrous, J. On the hardness of distinguishing mixed-state quantum computations. In 20th Annual IEEE Conference on Computational Complexity (CCC’05) 344–354 (IEEE, 2005).

186. Cerezo, M., Poremba, A., Cincio, L. & Coles, P. J. Variational quantum fidelity estimation. Quantum 4, 248 (2020).

187. Bravo-Prieto, C., García-Martín, D. & Latorre, J. I. Quantum singular value decomposer. Phys. Rev. A 101, 062310 (2020).

188. Koczor, B., Endo, S., Jones, T., Matsuzaki, Y. & Benjamin, S. C. Variational-state quantum metrology. New J. Phys. 22, 083038 (2020).

189. Kaubruegger, R. et al. Variational spin-squeezing algorithms on programmable quantum sensors. Phys. Rev. Lett. 123, 260505 (2019).

190. Ma, Z. et al. Adaptive circuit learning for quantum metrology. Preprint at https://arxiv.org/abs/2010.08702 (2020).

191. Beckey, J. L., Cerezo, M., Sone, A. & Coles, P. J. Variational quantum algorithm for estimating the quantum Fisher information. Preprint at https://arxiv.org/abs/2010.10488 (2020).

192. McClean, J. R., Boixo, S., Smelyanskiy, V. N., Babbush, R. & Neven, H. Barren plateaus in quantum neural network training landscapes. Nat. Commun. 9, 4812 (2018).

193. Arrasmith, A., Cerezo, M., Czarnik, P., Cincio, L. & Coles, P. J. Effect of barren plateaus on gradient-free optimization. Preprint at https://arxiv.org/abs/2011.12245 (2020).

194. Harrow, A. W. & Low, R. A. Random quantum circuits are approximate 2-designs. Commun. Math. Phys. 291, 257–302 (2009).

195. Brandao, F. G., Harrow, A. W. & Horodecki, M. Local random quantum circuits are approximate polynomial-designs. Commun. Math. Phys. 346, 397–434 (2016).

196. Uvarov, A., Biamonte, J. D. & Yudin, D. Variational quantum eigensolver for frustrated quantum systems. Phys. Rev. B 102, 075104 (2020).

197. Uvarov, A. & Biamonte, J. On barren plateaus and cost function locality in variational quantum algorithms. J. Phys. A 54, 245301 (2021).

198. Sharma, K., Cerezo, M., Cincio, L. & Coles, P. J. Trainability of dissipative perceptron-based quantum neural networks. Preprint at https://arxiv.org/abs/2005.12458 (2020).

199. Marrero, C. O., Kieferová, M. & Wiebe, N. Entanglement induced barren plateaus. Preprint at https://arxiv.org/abs/2010.15968 (2020).

200. Wang, S. et al. Noise-induced barren plateaus in variational quantum algorithms. Preprint at https://arxiv.org/abs/2007.14384 (2020).

201. Franca, D. S. & Garcia-Patron, R. Limitations of optimization algorithms on noisy quantum devices. Preprint at https://arxiv.org/abs/2009.05532 (2020).

202. Zhou, L., Wang, S.-T., Choi, S., Pichler, H. & Lukin, M. D. Quantum approximate optimization algorithm: performance, mechanism, and implementation on near-term devices. Phys. Rev. X 10, 021067 (2020).

203. Grant, E., Wossnig, L., Ostaszewski, M. & Benedetti, M. An initialization strategy for addressing barren plateaus in parameterized quantum circuits. Quantum 3, 214 (2019).

204. Volkoff, T. & Coles, P. J. Large gradients via correlation in random parameterized quantum circuits. Quantum Sci. Technol. 6, 025008 (2021).

205. Skolik, A., McClean, J. R., Mohseni, M., van der Smagt, P. & Leib, M. Layerwise learning for quantum neural networks. Preprint at https://arxiv.org/abs/2006.14904 (2020).

206. Campos, E., Nasrallah, A. & Biamonte, J. Abrupt transitions in variational quantum circuit training. Phys. Rev. A 103, 032607 (2021).

207. Verdon, G. et al. Learning to learn with quantum neural networks via classical neural networks. Preprint at https://arxiv.org/abs/1907.05415 (2019).

208. Anand, A., Degroote, M. & Aspuru-Guzik, A. Natural evolutionary strategies for variational quantum computation. Preprint at https://arxiv.org/abs/2012.00101 (2020).

209. Cai, Z. Resource estimation for quantum variational simulations of the hubbard model. Phys. Rev. Appl. 14, 014059 (2020).

210. Cade, C., Mineh, L., Montanaro, A. & Stanisic, S. Strategies for solving the Fermi–Hubbard model on near-term quantum computers. Phys. Rev. B 102, 235122 (2020).

211. Jena, A., Genin, S. & Mosca, M. Pauli partitioning with respect to gate sets. Preprint at https://arxiv.org/abs/1907.07859 (2019).

212. Crawford, O. et al. Efficient quantum measurement of Pauli operators in the presence of finite sampling error. Quantum 5, 385 (2021).

213. Verteletskyi, V., Yen, T.-C. & Izmaylov, A. F. Measurement optimization in the variational quantum eigensolver using a minimum clique cover. J. Chem. Phys. 152, 124114 (2020).

214. Izmaylov, A. F., Yen, T.-C., Lang, R. A. & Verteletskyi, V. Unitary partitioning approach to the measurement problem in the variational quantum eigensolver method. J. Chem. Theory Comput. 16, 190–195 (2019).

215. Zhao, A. et al. Measurement reduction in variational quantum algorithms. Phys. Rev. A 101, 062322 (2020).

216. Yen, T.-C., Verteletskyi, V. & Izmaylov, A. F. Measuring all compatible operators in one series of single-qubit measurements using unitary transformations. J. Chem. Theory Comput. 16, 2400–2409 (2020).

217. Gokhale, P. & Chong, F. T. o(n3) measurement cost for variational quantum eigensolver on molecular Hamiltonians. Preprint at https://arxiv.org/abs/1908.11857 (2019).

218. McClean, J. R., Romero, J., Babbush, R. & Aspuru-Guzik, A. The theory of variational hybrid quantum–classical algorithms. New J. Phys. 18, 023023 (2016).

219. Huggins, W. J. et al. Efficient and noise resilient measurements for quantum chemistry on near-term quantum computers. NPJ Quantum Inf. 7, 1–9 (2021).

220. Rubin, N. C., Babbush, R. & McClean, J. Application of fermionic marginal constraints to hybrid quantum algorithms. New J. Phys. 20, 053020 (2018).

221. Arrasmith, A., Cincio, L., Somma, R. D. & Coles, P. J. Operator sampling for shot-frugal optimization in variational algorithms. Preprint at https://arxiv.org/abs/2004.06252 (2020).

222. van Straaten, B. & Koczor, B. Measurement cost of metric-aware variational quantum algorithms. Preprint at https://arxiv.org/abs/2005.05172 (2020).

223. Huang, H.-Y., Kueng, R. & Preskill, J. Predicting many properties of a quantum system from very few measurements. Nat. Phys. 16, 1050–1057 (2020).

224. Hadfield, C., Bravyi, S., Raymond, R. & Mezzacapo, A. Measurements of quantum Hamiltonians with locally-biased classical shadows. Preprint at https://arxiv.org/abs/2006.15788 (2020).

225. Torlai, G., Mazzola, G., Carleo, G. & Mezzacapo, A. Precise measurement of quantum observables with neural-network estimators. Phys. Rev. Res. 2, 022060 (2020).

226. Endo, S., Cai, Z., Benjamin, S. C. & Yuan, X. Hybrid quantum–classical algorithms and quantum error mitigation. J. Phys. Soc. Japan 90, 032001 (2021).

227. Temme, K., Bravyi, S. & Gambetta, J. M. Error mitigation for short-depth quantum circuits. Phys. Rev. Lett. 119, 180509 (2017).

228. Kandala, A. et al. Error mitigation extends the computational reach of a noisy quantum processor. Nature 567, 491–495 (2019).

229. Endo, S., Benjamin, S. C. & Li, Y. Practical quantum error mitigation for near-future applications. Phys. Rev. X 8, 031027 (2018).

230. Cai, Z. Multi-exponential error extrapolation and combining error mitigation techniques for nisq applications. Preprint at https://arxiv.org/abs/2007.01265 (2020).

231. Otten, M. & Gray, S. K. Recovering noise-free quantum observables. Phys. Rev. A 99, 012338 (2019).

232. Endo, S., Zhao, Q., Li, Y., Benjamin, S. & Yuan, X. Mitigating algorithmic errors in a Hamiltonian simulation. Phys. Rev. A 99, 012334 (2019).

233. Sun, J. et al. Mitigating realistic noise in practical noisy intermediate-scale quantum devices. Phys. Rev. Appl. 15, 034026 (2021).

234. Strikis, A., Qin, D., Chen, Y., Benjamin, S. C. & Li, Y. Learning-based quantum error mitigation. Preprint at https://arxiv.org/abs/2005.07601 (2020).

235. Czarnik, P., Arrasmith, A., Coles, P. J. & Cincio, L. Error mitigation with Clifford quantum-circuit data. Preprint at https://arxiv.org/abs/2005.10189 (2020).

236. Lowe, A. et al. Unified approach to data-driven quantum error mitigation. Preprint at https://arxiv.org/abs/2011.01157 (2020).

237. McArdle, S., Yuan, X. & Benjamin, S. Error-mitigated digital quantum simulation. Phys. Rev. Lett. 122, 180501 (2019).

238. Bonet-Monroig, X., Sagastizabal, R., Singh, M. & O’Brien, T. Low-cost error mitigation by symmetry verification. Phys. Rev. A 98, 062339 (2018).

239. McClean, J. R., Jiang, Z., Rubin, N. C., Babbush, R. & Neven, H. Decoding quantum errors with subspace expansions. Nat. Commun. 11, 636 (2020).

240. Koczor, B. Exponential error suppression for near-term quantum devices. Preprint at https://arxiv.org/abs/2011.05942 (2020).

241. Huggins, W. J. et al. Virtual distillation for quantum error mitigation. Preprint at https://arxiv.org/abs/2011.07064 (2020).

242. Bravyi, S., Sheldon, S., Kandala, A., Mckay, D. C. & Gambetta, J. M. Mitigating measurement errors in multi-qubit experiments. Phys. Rev. A 103, 042605 (2021).

243. Su, D. et al. Error mitigation on a near-term quantum photonic device. Quantum 5, 452 (2021).

244. Gentini, L., Cuccoli, A., Pirandola, S., Verrucchi, P. & Banchi, L. Noise-resilient variational hybrid quantum–classical optimization. Phys. Rev. A 102, 052414 (2020).

245. Fontana, E., Cerezo, M., Arrasmith, A., Rungger, I. & Coles, P. J. Optimizing parameterized quantum circuits via noise-induced breaking of symmetries. Preprint at https://arxiv.org/abs/2011.08763 (2020).

246. Xue, C., Chen, Z.-Y., Wu, Y.-C. & Guo, G.-P. Effects of quantum noise on quantum approximate optimization algorithm. Preprint at https://arxiv.org/abs/1909.02196 (2019).

247. Marshall, J., Wudarski, F., Hadfield, S. & Hogg, T. Characterizing local noise in QAOA circuits. IOP SciNotes 1, 025208 (2020).

248. Kim, I. H. Noise-resilient preparation of quantum many-body ground states. Preprint at https://arxiv.org/abs/1703.00032 (2017).

249. Broughton, M. et al. Tensorflow quantum: a software framework for quantum machine learning. Preprint at https://arxiv.org/abs/2003.02989 (2020).

250. Luo, X.-Z., Liu, J.-G., Zhang, P. & Wang, L. Yao. jl: Extensible, efficient framework for quantum algorithm design. Quantum 4, 341 (2020).

251. Sanders, Y. R. et al. Compilation of fault-tolerant quantum heuristics for combinatorial optimization. Phys. Rev. X Quantum 1, 020312 (2020).

## Acknowledgements

M.C. thanks K. Sharma for discussions. M.C. was initially supported by the Laboratory Directed Research and Development (LDRD) programme of Los Alamos National Laboratory (LANL) under project no. 20180628ECR, and later supported by the Center for Nonlinear Studies at LANL. A.A. was initially supported by the LDRD programme of LANL under project no. 20200056DR, and later supported by the US Department of Energy (DOE), Office of Science, Office of High Energy Physics QuantISED programme under contract nos. DE-AC52-06NA25396 and KA2401032. S.C.B. acknowledges financial support from EPSRC Hub grants under the agreement nos. EP/M013243/1 and EP/T001062/1, and from EU H2020-FETFLAG-03-2018 under the grant agreement no. 820495 (AQTION). S.E. was supported by MEXT Quantum Leap Flagship Program (MEXT QLEAP) grant nos. JPMXS0120319794, JPMXS0118068682 and JST ERATO grant no. JPMJER1601. K.F. was supported by Japan Society for the Promotion of Science (JSPS) KAKENHI grant no. 16H02211, JST ERATO JPMJER1601 and JST CREST JPMJCR1673. K.M. was supported by JST PRESTO grant no. JPMJPR2019 and JSPS KAKENHI grant no. 20K22330. K.M. and K.F. were also supported by MEXT QLEAP grant no. JPMXS0118067394 and JPMXS0120319794. X.Y. acknowledges support from the Simons Foundation. L.C. was initially supported by the LDRD programme of LANL under project no. 20190065DR, and later supported by the US DOE, Office of Science, Office of Advanced Scientific Computing Research under the Quantum Computing Application Teams (QCAT) programme. P.J.C. was initially supported by the LANL ASC Beyond Moore’s Law project, and later supported by the US DOE, Office of Science, Office of Advanced Scientific Computing Research, under the Accelerated Research in Quantum Computing (ARQC) programme. Most recently, M.C., L.C. and P.J.C. were supported by the Quantum Science Center (QSC), a National Quantum Information Science Research Center of the US DOE.

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## Glossary

Circuit depth

An integer number that counts the maximum length in the circuit between the input and the output. This length is usually defined in terms of layers of gates acting in parallel.

Ancilla qubits

Auxiliary qubits used during a quantum computation.

A quantum adiabatic transformation involves changing a system’s Hamiltonian slowly enough that the system remains in the ground state. Trotterization approximates this evolution with a series of discrete steps.

Pauli strings

A Pauli string acting on n qubits is defined as a Hermitian operator from the set {1,X,Y,Z}n, where X,Y,Z are Pauli operators, and 1 is the identity on a single qubit.

Trotter

A mathematical method used to approximate the matrix etH, which describes the evolution for a time t under a Hamiltonian H as $${e}^{tH}\sim {({e}^{\frac{tH}{n}})}^{n}$$. The Trotter error is the error induced from the Trotter method approximation.

Quantum optimal control

Quantum optimal control is a theoretical framework that provides tools for the systematic manipulation of quantum dynamical systems.

Rayleigh–Ritz variational principle

Principle stating that the lowest eigenvalue of a Hermitian operator H is upper-bounded by the minimum expectation value 〈ψHψ〉 found by varying the state $$| \psi \rangle$$.

Method used to estimate the real and complex part of the expectation value of a unitary operator over a quantum state.

Max-Cut

The Max-Cut problem aims to find the maximum cut of a graph, that is, the partitioning of a graph’s vertices that cuts through the most edges.

Schur concavity

A Schur concave function f is a funtion $$f:{{\mathbb{R}}}^{d}\to {\mathbb{R}}$$ that for all x and y in $${{\mathbb{R}}}^{d}$$ is such that if x is majorized by y, then f(x) ≥ f(y).

Unitary 2-design

Ensemble of unitaries, such that sampling over their distribution yields the same properties as sampling random unitaries from the Haar distribution measure up to the first two moments.

Born machines

Generative models that represent the probability distribution of classical dataset as quantum pure states.

Restricted Boltzmann machine

Generative computational model that can learn a probability distribution over its set of inputs.

Quantum Fisher information

The quantum Fisher information quantifies the sensitivity of a quantum state to a parameter or a set of parameters.

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Cerezo, M., Arrasmith, A., Babbush, R. et al. Variational quantum algorithms. Nat Rev Phys 3, 625–644 (2021). https://doi.org/10.1038/s42254-021-00348-9

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