Characterization of variational quantum algorithms using free fermions

Gabriel Matos1, Chris N. Self2,3, Zlatko Papić1, Konstantinos Meichanetzidis4,5, and Henrik Dreyer6

1School of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, United Kingdom
2Quantinuum, Partnership House, Carlisle Place, London, SW1P 1BX, United Kingdom
3Blackett Laboratory, Imperial College London, London SW7 2AZ, United Kingdom
4Quantinuum, 17 Beaumont St., Oxford OX1 2NA, United Kingdom
5Department of Computer Science, University of Oxford, Oxford OX1 3QD, United Kingdom
6Quantinuum, Leopoldstrasse 180, 80804 Munich, Germany

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We study variational quantum algorithms from the perspective of free fermions. By deriving the explicit structure of the associated Lie algebras, we show that the Quantum Approximate Optimization Algorithm (QAOA) on a one-dimensional lattice – with and without decoupled angles – is able to prepare all fermionic Gaussian states respecting the symmetries of the circuit. Leveraging these results, we numerically study the interplay between these symmetries and the locality of the target state, and find that an absence of symmetries makes nonlocal states easier to prepare. An efficient classical simulation of Gaussian states, with system sizes up to $80$ and deep circuits, is employed to study the behavior of the circuit when it is overparameterized. In this regime of optimization, we find that the number of iterations to converge to the solution scales linearly with system size. Moreover, we observe that the number of iterations to converge to the solution decreases exponentially with the depth of the circuit, until it saturates at a depth which is quadratic in system size. Finally, we conclude that the improvement in the optimization can be explained in terms of better local linear approximations provided by the gradients.

Variational quantum algorithms employ a classical computer to optimize a quantum one, combining the power of both. However, our understanding about these algorithms is still incomplete. Efforts to study them run not only into the limitation that quantum computers are difficult to simulate, but also that the associated classical optimization is itself very hard. In our work, we study the Quantum Approximate Optimization Algorithm (QAOA), a special instance of these algorithms. We show that when the problem we are trying to solve using QAOA has a linear geometry, the quantum states that can be prepared are precisely those that correspond to "free fermions". These can be efficiently simulated using only a classical computer, allowing us to circumvent the limitations of current quantum computers and more comprehensively study variational algorithms. Among other things, we found that symmetries, which tend to simplify a problem and are generally regarded as beneficial, can in certain cases overly constrain the optimization and make it more difficult. We were also able to characterize a phenomenon called "overparameterization", where the optimization becomes significantly easier by increasing the number of parameters in the algorithm. In this regime, we found that the number of iterations to find a solution decreases exponentially with the number of parameters. Additionally, this number of iterations goes from being polynomial to linear in the size of the system. Our work fully characterises what states can be prepared by QAOA in a linear geometry and furthers the understanding of the optimization associated with variational algorithms.

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[1] Amira Abbas, David Sutter, Christa Zoufal, Aurelien Lucchi, Alessio Figalli, and Stefan Woerner. The power of quantum neural networks. Nature Computational Science, 1 (6): 403–409, 2021. ISSN 2662-8457. 10.1038/​s43588-021-00084-1. URL https:/​/​​10.1038/​s43588-021-00084-1.

[2] V. Akshay, H. Philathong, E. Campos, D. Rabinovich, I. Zacharov, Xiao-Ming Zhang, and J. Biamonte. On circuit depth scaling for quantum approximate optimization, 2022. URL https:/​/​​10.1103/​PhysRevA.106.042438.

[3] F. Albertini and D. D'Alessandro. Notions of controllability for quantum mechanical systems, 2001. URL https:/​/​​abs/​quant-ph/​0106128.

[4] Andrew Arrasmith, Zoë Holmes, M. Cerezo, and Patrick J. Coles. Equivalence of quantum barren plateaus to cost concentration and narrow gorges, 2021. URL https:/​/​​10.1088/​2058-9565/​ac7d06.

[5] Frank Arute, Kunal Arya, Ryan Babbush, Dave Bacon, Joseph C. Bardin, Rami Barends, Sergio Boixo, Michael Broughton, Bob B. Buckley, David A. Buell, Brian Burkett, Nicholas Bushnell, Yu Chen, Zijun Chen, Benjamin Chiaro, Roberto Collins, William Courtney, Sean Demura, Andrew Dunsworth, Edward Farhi, Austin Fowler, Brooks Foxen, Craig Gidney, Marissa Giustina, Rob Graff, Steve Habegger, Matthew P. Harrigan, Alan Ho, Sabrina Hong, Trent Huang, William J. Huggins, Lev Ioffe, Sergei V. Isakov, Evan Jeffrey, Zhang Jiang, Cody Jones, Dvir Kafri, Kostyantyn Kechedzhi, Julian Kelly, Seon Kim, Paul V. Klimov, Alexander Korotkov, Fedor Kostritsa, David Landhuis, Pavel Laptev, Mike Lindmark, Erik Lucero, Orion Martin, John M. Martinis, Jarrod R. McClean, Matt McEwen, Anthony Megrant, Xiao Mi, Masoud Mohseni, Wojciech Mruczkiewicz, Josh Mutus, Ofer Naaman, Matthew Neeley, Charles Neill, Hartmut Neven, Murphy Yuezhen Niu, Thomas E. O’Brien, Eric Ostby, Andre Petukhov, Harald Putterman, Chris Quintana, Pedram Roushan, Nicholas C. Rubin, Daniel Sank, Kevin J. Satzinger, Vadim Smelyanskiy, Doug Strain, Kevin J. Sung, Marco Szalay, Tyler Y. Takeshita, Amit Vainsencher, Theodore White, Nathan Wiebe, Z. Jamie Yao, Ping Yeh, and Adam Zalcman. Hartree-fock on a superconducting qubit quantum computer. Science, 369 (6507): 1084–1089, 2020. 10.1126/​science.abb9811. URL https:/​/​​doi/​abs/​10.1126/​science.abb9811.

[6] George S. Barron, Bryan T. Gard, Orien J. Altman, Nicholas J. Mayhall, Edwin Barnes, and Sophia E. Economou. Preserving symmetries for variational quantum eigensolvers in the presence of noise. Phys. Rev. Applied, 16: 034003, Sep 2021. 10.1103/​PhysRevApplied.16.034003. URL https:/​/​​doi/​10.1103/​PhysRevApplied.16.034003.

[7] Marcello Benedetti, Erika Lloyd, Stefan Sack, and Mattia Fiorentini. Parameterized quantum circuits as machine learning models. Quantum Science and Technology, 4 (4): 043001, nov 2019. 10.1088/​2058-9565/​ab4eb5. URL https:/​/​​10.1088/​2058-9565/​ab4eb5.

[8] Jacob Biamonte. Universal variational quantum computation. Phys. Rev. A, 103: L030401, Mar 2021. 10.1103/​PhysRevA.103.L030401. URL https:/​/​​doi/​10.1103/​PhysRevA.103.L030401.

[9] Denis Bokhan, Alena S. Mastiukova, Aleksey S. Boev, Dmitrii N. Trubnikov, and Aleksey K. Fedorov. Multiclass classification using quantum convolutional neural networks with hybrid quantum-classical learning, 2022. URL https:/​/​​10.3389/​fphy.2022.1069985.

[10] Sami Boulebnane, Xavier Lucas, Agnes Meyder, Stanislaw Adaszewski, and Ashley Montanaro. Peptide conformational sampling using the quantum approximate optimization algorithm, 2022. URL https:/​/​​abs/​2204.01821.

[11] Carlos Bravo-Prieto, Ryan LaRose, M. Cerezo, Yigit Subasi, Lukasz Cincio, and Patrick J. Coles. Variational quantum linear solver, 2019. URL https:/​/​​abs/​1909.05820.

[12] Chris Cade, Lana Mineh, Ashley Montanaro, and Stasja Stanisic. Strategies for solving the fermi-hubbard model on near-term quantum computers. Phys. Rev. B, 102: 235122, Dec 2020. 10.1103/​PhysRevB.102.235122. URL https:/​/​​doi/​10.1103/​PhysRevB.102.235122.

[13] Pasquale Calabrese and John Cardy. Entanglement entropy and quantum field theory. Journal of Statistical Mechanics: Theory and Experiment, 2004 (06): P06002, jun 2004. 10.1088/​1742-5468/​2004/​06/​p06002. URL https:/​/​​10.1088/​1742-5468/​2004/​06/​p06002.

[14] M. Cerezo, Andrew Arrasmith, Ryan Babbush, Simon C. Benjamin, Suguru Endo, Keisuke Fujii, Jarrod R. McClean, Kosuke Mitarai, Xiao Yuan, Lukasz Cincio, and Patrick J. Coles. Variational quantum algorithms. Nature Reviews Physics, 3 (9): 625–644, 2021a. ISSN 2522-5820. 10.1038/​s42254-021-00348-9. URL https:/​/​​10.1038/​s42254-021-00348-9.

[15] M. Cerezo, Akira Sone, Tyler Volkoff, Lukasz Cincio, and Patrick J. Coles. Cost function dependent barren plateaus in shallow parametrized quantum circuits. Nature Communications, 12 (1): 1791, 2021b. ISSN 2041-1723. 10.1038/​s41467-021-21728-w. URL https:/​/​​10.1038/​s41467-021-21728-w.

[16] Alba Cervera-Lierta. Exact Ising model simulation on a quantum computer. Quantum, 2: 114, December 2018. ISSN 2521-327X. 10.22331/​q-2018-12-21-114. URL https:/​/​​10.22331/​q-2018-12-21-114.

[17] Ranyiliu Chen, Benchi Zhao, and Xin Wang. Variational quantum algorithm for schmidt decomposition, 2021. URL https:/​/​​abs/​2109.10785.

[18] Iris Cong, Soonwon Choi, and Mikhail D. Lukin. Quantum convolutional neural networks. Nature Physics, 15 (12): 1273–1278, 2019. ISSN 1745-2481. 10.1038/​s41567-019-0648-8. URL https:/​/​​10.1038/​s41567-019-0648-8.

[19] D. D'Alessandro. Introduction to Quantum Control and Dynamics. Chapman & Hall/​CRC Applied Mathematics & Nonlinear Science. CRC Press, 2007. ISBN 9781584888833. https:/​/​​10.1201/​9781584888833.

[20] Yannick Deller, Sebastian Schmitt, Maciej Lewenstein, Steve Lenk, Marika Federer, Fred Jendrzejewski, Philipp Hauke, and Valentin Kasper. Quantum approximate optimization algorithm for qudit systems with long-range interactions, 2022. URL https:/​/​​abs/​2204.00340.

[21] Andrew C. Doherty and Stephen D. Bartlett. Identifying phases of quantum many-body systems that are universal for quantum computation. Phys. Rev. Lett., 103: 020506, Jul 2009. 10.1103/​PhysRevLett.103.020506. URL https:/​/​​doi/​10.1103/​PhysRevLett.103.020506.

[22] Henrik Dreyer, Mircea Bejan, and Etienne Granet. Quantum computing critical exponents. Phys. Rev. A, 104: 062614, Dec 2021. 10.1103/​PhysRevA.104.062614. URL https:/​/​​doi/​10.1103/​PhysRevA.104.062614.

[23] Yuxuan Du, Zhuozhuo Tu, Xiao Yuan, and Dacheng Tao. Efficient measure for the expressivity of variational quantum algorithms. Phys. Rev. Lett., 128: 080506, Feb 2022. 10.1103/​PhysRevLett.128.080506. URL https:/​/​​doi/​10.1103/​PhysRevLett.128.080506.

[24] Amit Dutta, Gabriel Aeppli, Bikas K. Chakrabarti, Uma Divakaran, Thomas F. Rosenbaum, and Diptiman Sen. Quantum Phase Transitions in Transverse Field Spin Models: From Statistical Physics to Quantum Information. Cambridge University Press, 2015. 10.1017/​CBO9781107706057.

[25] Daniel J. Egger, Claudio Gambella, Jakub Marecek, Scott McFaddin, Martin Mevissen, Rudy Raymond, Andrea Simonetto, Stefan Woerner, and Elena Yndurain. Quantum computing for finance: State-of-the-art and future prospects. IEEE Transactions on Quantum Engineering, 1: 1–24, 2020. 10.1109/​TQE.2020.3030314.

[26] Kilian Ender, Anette Messinger, Michael Fellner, Clemens Dlaska, and Wolfgang Lechner. Modular parity quantum approximate optimization, 2022. URL https:/​/​​10.1103/​PRXQuantum.3.030304.

[27] Edward Farhi and Hartmut Neven. Classification with quantum neural networks on near term processors, 2018. URL https:/​/​​abs/​1802.06002.

[28] Edward Farhi, Jeffrey Goldstone, and Sam Gutmann. A quantum approximate optimization algorithm, 2014. URL https:/​/​​abs/​1411.4028.

[29] Verena Feulner and Michael J. Hartmann. Variational quantum eigensolver ansatz for the $j_1$-$j_2$-model, 2022. URL https:/​/​​abs/​2205.11198.

[30] Keisuke Fujii, Kaoru Mizuta, Hiroshi Ueda, Kosuke Mitarai, Wataru Mizukami, and Yuya O. Nakagawa. Deep variational quantum eigensolver: a divide-and-conquer method for solving a larger problem with smaller size quantum computers, 2020. URL https:/​/​​abs/​2007.10917.

[31] Lena Funcke, Tobias Hartung, Karl Jansen, Stefan Kühn, and Paolo Stornati. Dimensional Expressivity Analysis of Parametric Quantum Circuits. Quantum, 5: 422, March 2021a. ISSN 2521-327X. 10.22331/​q-2021-03-29-422. URL https:/​/​​10.22331/​q-2021-03-29-422.

[32] Lena Funcke, Tobias Hartung, Karl Jansen, Stefan Kühn, Manuel Schneider, and Paolo Stornati. Dimensional expressivity analysis, best-approximation errors, and automated design of parametric quantum circuits, 2021b. URL https:/​/​​abs/​2111.11489.

[33] Bryan T. Gard, Linghua Zhu, George S. Barron, Nicholas J. Mayhall, Sophia E. Economou, and Edwin Barnes. Efficient symmetry-preserving state preparation circuits for the variational quantum eigensolver algorithm. npj Quantum Information, 6 (1): 10, 2020. ISSN 2056-6387. 10.1038/​s41534-019-0240-1. URL https:/​/​​10.1038/​s41534-019-0240-1.

[34] Harper R. Grimsley, Sophia E. Economou, Edwin Barnes, and Nicholas J. Mayhall. An adaptive variational algorithm for exact molecular simulations on a quantum computer. Nature Communications, 10 (1): 3007, 2019. ISSN 2041-1723. 10.1038/​s41467-019-10988-2. URL https:/​/​​10.1038/​s41467-019-10988-2.

[35] Harper R. Grimsley, George S. Barron, Edwin Barnes, Sophia E. Economou, and Nicholas J. Mayhall. Adapt-vqe is insensitive to rough parameter landscapes and barren plateaus, 2022. URL https:/​/​​10.1038/​s41534-023-00681-0.

[36] B. Hall and B.C. Hall. Lie Groups, Lie Algebras, and Representations: An Elementary Introduction. Graduate Texts in Mathematics. Springer, 2003. ISBN 9780387401225.

[37] Tobias Haug, Kishor Bharti, and M.S. Kim. Capacity and quantum geometry of parametrized quantum circuits. PRX Quantum, 2: 040309, Oct 2021. 10.1103/​PRXQuantum.2.040309. URL https:/​/​​doi/​10.1103/​PRXQuantum.2.040309.

[38] Vojtěch Havlíček, Antonio D. Córcoles, Kristan Temme, Aram W. Harrow, Abhinav Kandala, Jerry M. Chow, and Jay M. Gambetta. Supervised learning with quantum-enhanced feature spaces. Nature, 567 (7747): 209–212, 2019. ISSN 1476-4687. 10.1038/​s41586-019-0980-2. URL https:/​/​​10.1038/​s41586-019-0980-2.

[39] M. Hebenstreit, R. Jozsa, B. Kraus, S. Strelchuk, and M. Yoganathan. All pure fermionic non-gaussian states are magic states for matchgate computations. Phys. Rev. Lett., 123: 080503, Aug 2019. 10.1103/​PhysRevLett.123.080503. URL https:/​/​​doi/​10.1103/​PhysRevLett.123.080503.

[40] Rebekah Herrman, Phillip C. Lotshaw, James Ostrowski, Travis S. Humble, and George Siopsis. Multi-angle quantum approximate optimization algorithm, 2021. URL https:/​/​​abs/​2109.11455.

[41] Wen Wei Ho and Timothy H. Hsieh. Efficient variational simulation of non-trivial quantum states. SciPost Phys., 6: 29, 2019. 10.21468/​SciPostPhys.6.3.029. URL https:/​/​​10.21468/​SciPostPhys.6.3.029.

[42] Wen Wei Ho, Cheryne Jonay, and Timothy H. Hsieh. Ultrafast variational simulation of nontrivial quantum states with long-range interactions. Phys. Rev. A, 99: 052332, May 2019. 10.1103/​PhysRevA.99.052332. URL https:/​/​​doi/​10.1103/​PhysRevA.99.052332.

[43] Zoë Holmes, Kunal Sharma, M. Cerezo, and Patrick J. Coles. Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum, 3: 010313, Jan 2022. 10.1103/​PRXQuantum.3.010313. URL https:/​/​​doi/​10.1103/​PRXQuantum.3.010313.

[44] Patrick Huembeli and Alexandre Dauphin. Characterizing the loss landscape of variational quantum circuits. Quantum Science and Technology, 6 (2): 025011, feb 2021. 10.1088/​2058-9565/​abdbc9. URL https:/​/​​10.1088/​2058-9565/​abdbc9.

[45] Ammar Jahin, Andy C. Y. Li, Thomas Iadecola, Peter P. Orth, Gabriel N. Perdue, Alexandru Macridin, M. Sohaib Alam, and Norm M. Tubman. Fermionic approach to variational quantum simulation of kitaev spin models, 2022. URL https:/​/​​10.1103/​PhysRevA.106.022434.

[46] Manpreet Singh Jattana, Fengping Jin, Hans De Raedt, and Kristel Michielsen. Assessment of the variational quantum eigensolver: application to the heisenberg model, 2022. URL https:/​/​​10.3389/​fphy.2022.907160.

[47] Zhang Jiang, Kevin J. Sung, Kostyantyn Kechedzhi, Vadim N. Smelyanskiy, and Sergio Boixo. Quantum algorithms to simulate many-body physics of correlated fermions. Phys. Rev. Applied, 9: 044036, Apr 2018. 10.1103/​PhysRevApplied.9.044036. URL https:/​/​​doi/​10.1103/​PhysRevApplied.9.044036.

[48] Abhinav Kandala, Antonio Mezzacapo, Kristan Temme, Maika Takita, Markus Brink, Jerry M. Chow, and Jay M. Gambetta. Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. Nature, 549 (7671): 242–246, 2017. ISSN 1476-4687. 10.1038/​nature23879. URL https:/​/​​10.1038/​nature23879.

[49] Bobak Toussi Kiani, Seth Lloyd, and Reevu Maity. Learning unitaries by gradient descent, 2020. URL https:/​/​​abs/​2001.11897.

[50] Joonho Kim, Jaedeok Kim, and Dario Rosa. Universal effectiveness of high-depth circuits in variational eigenproblems. Phys. Rev. Research, 3: 023203, Jun 2021. 10.1103/​PhysRevResearch.3.023203. URL https:/​/​​doi/​10.1103/​PhysRevResearch.3.023203.

[51] Ian D. Kivlichan, Jarrod McClean, Nathan Wiebe, Craig Gidney, Alán Aspuru-Guzik, Garnet Kin-Lic Chan, and Ryan Babbush. Quantum simulation of electronic structure with linear depth and connectivity. Phys. Rev. Lett., 120: 110501, Mar 2018. 10.1103/​PhysRevLett.120.110501. URL https:/​/​​doi/​10.1103/​PhysRevLett.120.110501.

[52] Ankit Kulshrestha and Ilya Safro. Beinit: Avoiding barren plateaus in variational quantum algorithms, 2022. URL https:/​/​​abs/​2204.13751.

[53] Martin Larocca, Piotr Czarnik, Kunal Sharma, Gopikrishnan Muraleedharan, Patrick J. Coles, and M. Cerezo. Diagnosing barren plateaus with tools from quantum optimal control, 2021a. URL https:/​/​​10.22331/​q-2022-09-29-824.

[54] Martin Larocca, Nathan Ju, Diego García-Martín, Patrick J. Coles, and M. Cerezo. Theory of overparametrization in quantum neural networks, 2021b. URL https:/​/​​abs/​2109.11676.

[55] Martin Larocca, Frederic Sauvage, Faris M. Sbahi, Guillaume Verdon, Patrick J. Coles, and M. Cerezo. Group-invariant quantum machine learning, 2022. URL https:/​/​​10.1103/​PRXQuantum.3.030341.

[56] Ze-Tong Li, Fan-Xu Meng, Han Zeng, Zai-Chen Zhang, and Xu-Tao Yu. An efficient gradient sensitive alternate framework for variational quantum eigensolver with variable ansatz, 2022. URL https:/​/​​abs/​2205.03031.

[57] Xia Liu, Geng Liu, Jiaxin Huang, and Xin Wang. Mitigating barren plateaus of variational quantum eigensolvers, 2022. URL https:/​/​​abs/​2205.13539.

[58] Owen Lockwood. An empirical review of optimization techniques for quantum variational circuits, 2022. URL https:/​/​​abs/​2202.01389.

[59] Chufan Lyu, Xusheng Xu, Man-Hong Yung, and Abolfazl Bayat. Symmetry enhanced variational quantum eigensolver, 2022. URL https:/​/​​10.22331/​q-2023-01-19-899.

[60] Carlos Ortiz Marrero, Mária Kieferová, and Nathan Wiebe. Entanglement induced barren plateaus, 2020. URL https:/​/​​abs/​2010.15968.

[61] Gabriel Matos, Sonika Johri, and Zlatko Papić. Quantifying the efficiency of state preparation via quantum variational eigensolvers. PRX Quantum, 2: 010309, Jan 2021. 10.1103/​PRXQuantum.2.010309. URL https:/​/​​doi/​10.1103/​PRXQuantum.2.010309.

[62] Glen Bigan Mbeng, Rosario Fazio, and Giuseppe Santoro. Quantum annealing: a journey through digitalization, control, and hybrid quantum variational schemes, 2019. URL https:/​/​​abs/​1906.08948.

[63] Jarrod R. McClean, Sergio Boixo, Vadim N. Smelyanskiy, Ryan Babbush, and Hartmut Neven. Barren plateaus in quantum neural network training landscapes. Nature Communications, 9 (1): 4812, Nov 2018. ISSN 2041-1723. 10.1038/​s41467-018-07090-4. URL https:/​/​​10.1038/​s41467-018-07090-4.

[64] Antonio Anna Mele, Glen Bigan Mbeng, Giuseppe Ernesto Santoro, Mario Collura, and Pietro Torta. Avoiding barren plateaus via transferability of smooth solutions in hamiltonian variational ansatz, 2022. URL https:/​/​​10.1103/​PhysRevA.106.L060401.

[65] Johannes Jakob Meyer, Marian Mularski, Elies Gil-Fuster, Antonio Anna Mele, Francesco Arzani, Alissa Wilms, and Jens Eisert. Exploiting symmetry in variational quantum machine learning, 2022. URL https:/​/​​10.1103/​PRXQuantum.4.010328.

[66] M. E. S. Morales, J. D. Biamonte, and Z. Zimborás. On the universality of the quantum approximate optimization algorithm. Quantum Information Processing, 19 (9): 291, 2020. ISSN 1573-1332. 10.1007/​s11128-020-02748-9. URL https:/​/​​10.1007/​s11128-020-02748-9.

[67] M. Nakahara. Geometry, Topology, and Physics. Geometry, Topology, and Physics. Institute of Physics Pub., 2017. ISBN 9781138413368.

[68] Kouhei Nakaji and Naoki Yamamoto. Expressibility of the alternating layered ansatz for quantum computation. Quantum, 5: 434, April 2021. ISSN 2521-327X. 10.22331/​q-2021-04-19-434. URL https:/​/​​10.22331/​q-2021-04-19-434.

[69] J. Nocedal and S. Wright. Numerical Optimization. Springer Series in Operations Research and Financial Engineering. Springer New York, 2006. ISBN 9780387227429. https:/​/​​10.1007/​b98874.

[70] Ken N. Okada, Keita Osaki, Kosuke Mitarai, and Keisuke Fujii. Identification of topological phases using classically-optimized variational quantum eigensolver, 2022. URL https:/​/​​abs/​2202.02909.

[71] Román Orús, Samuel Mugel, and Enrique Lizaso. Quantum computing for finance: Overview and prospects. Reviews in Physics, 4: 100028, 2019. ISSN 2405-4283. https:/​/​​10.1016/​j.revip.2019.100028. URL https:/​/​​science/​article/​pii/​S2405428318300571.

[72] Michał Oszmaniec, Ninnat Dangniam, Mauro E.S. Morales, and Zoltán Zimborás. Fermion sampling: A robust quantum computational advantage scheme using fermionic linear optics and magic input states. PRX Quantum, 3: 020328, May 2022. 10.1103/​PRXQuantum.3.020328. URL https:/​/​​doi/​10.1103/​PRXQuantum.3.020328.

[73] Matthew Otten, Cristian L. Cortes, and Stephen K. Gray. Noise-resilient quantum dynamics using symmetry-preserving ansatzes, 2019. URL https:/​/​​abs/​1910.06284.

[74] Carlos Outeiral, Martin Strahm, Jiye Shi, Garrett M. Morris, Simon C. Benjamin, and Charlotte M. Deane. The prospects of quantum computing in computational molecular biology. WIREs Computational Molecular Science, 11 (1): e1481, 2021. https:/​/​​10.1002/​wcms.1481. URL https:/​/​​doi/​abs/​10.1002/​wcms.1481.

[75] Guido Pagano, Aniruddha Bapat, Patrick Becker, Katherine S. Collins, Arinjoy De, Paul W. Hess, Harvey B. Kaplan, Antonis Kyprianidis, Wen Lin Tan, Christopher Baldwin, Lucas T. Brady, Abhinav Deshpande, Fangli Liu, Stephen Jordan, Alexey V. Gorshkov, and Christopher Monroe. Quantum approximate optimization of the long-range ising model with a trapped-ion quantum simulator. Proceedings of the National Academy of Sciences, 117 (41): 25396–25401, 2020. 10.1073/​pnas.2006373117. URL https:/​/​​doi/​abs/​10.1073/​pnas.2006373117.

[76] Alberto Peruzzo, Jarrod McClean, Peter Shadbolt, Man-Hong Yung, Xiao-Qi Zhou, Peter J. Love, Alán Aspuru-Guzik, and Jeremy L. O’Brien. A variational eigenvalue solver on a photonic quantum processor. Nature Communications, 5 (1): 4213, 2014. ISSN 2041-1723. 10.1038/​ncomms5213. URL https:/​/​​10.1038/​ncomms5213.

[77] John Preskill. Quantum Computing in the NISQ era and beyond. Quantum, 2: 79, August 2018. ISSN 2521-327X. 10.22331/​q-2018-08-06-79. URL https:/​/​​10.22331/​q-2018-08-06-79.

[78] Robert Raussendorf, Daniel E. Browne, and Hans J. Briegel. Measurement-based quantum computation on cluster states. Phys. Rev. A, 68: 022312, Aug 2003. 10.1103/​PhysRevA.68.022312. URL https:/​/​​doi/​10.1103/​PhysRevA.68.022312.

[79] Francesco Scala, Stefano Mangini, Chiara Macchiavello, Daniele Bajoni, and Dario Gerace. Quantum variational learning for entanglement witnessing, 2022. URL https:/​/​​10.1109/​IJCNN55064.2022.9892080.

[80] Paul M. Schindler, Tommaso Guaita, Tao Shi, Eugene Demler, and J. Ignacio Cirac. A variational ansatz for the ground state of the quantum sherrington-kirkpatrick model, 2022. URL https:/​/​​10.1103/​PhysRevLett.129.220401.

[81] Maria Schuld and Nathan Killoran. Quantum machine learning in feature hilbert spaces. Phys. Rev. Lett., 122: 040504, Feb 2019. 10.1103/​PhysRevLett.122.040504. URL https:/​/​​doi/​10.1103/​PhysRevLett.122.040504.

[82] Kazuhiro Seki, Tomonori Shirakawa, and Seiji Yunoki. Symmetry-adapted variational quantum eigensolver. Phys. Rev. A, 101: 052340, May 2020. 10.1103/​PhysRevA.101.052340. URL https:/​/​​doi/​10.1103/​PhysRevA.101.052340.

[83] Ruslan Shaydulin and Stefan M. Wild. Exploiting symmetry reduces the cost of training qaoa. IEEE Transactions on Quantum Engineering, 2: 1–9, 2021. 10.1109/​TQE.2021.3066275.

[84] Sukin Sim, Peter D. Johnson, and Alán Aspuru-Guzik. Expressibility and entangling capability of parameterized quantum circuits for hybrid quantum-classical algorithms. Advanced Quantum Technologies, 2 (12): 1900070, 2019. https:/​/​​10.1002/​qute.201900070. URL https:/​/​​doi/​abs/​10.1002/​qute.201900070.

[85] James Stokes, Josh Izaac, Nathan Killoran, and Giuseppe Carleo. Quantum natural gradient. Quantum, 4: 269, may 2020. 10.22331/​q-2020-05-25-269. URL https:/​/​​10.22331.

[86] Zheng-Hang Sun, Yong-Yi Wang, Jian Cui, and Heng Fan. Performance of quantum approximate optimization algorithm for preparing non-trivial quantum states: dependence of translational symmetry and improvement, 2022. URL https:/​/​​10.1088/​1367-2630/​acb22c.

[87] Kevin J. Sung, Marko J. Rančić, Olivia T. Lanes, and Nicholas T. Bronn. Preparing majorana zero modes on a noisy quantum processor, 2022. URL https:/​/​​10.1088/​2058-9565/​acb796.

[88] Jacopo Surace and Luca Tagliacozzo. Fermionic gaussian states: an introduction to numerical approaches, 2021. URL https:/​/​​10.21468/​SciPostPhysLectNotes.54.

[89] Xiaoyu Tang, Chufan Lyu, Junning Li, Xusheng Xu, Man-Hong Yung, and Abolfazl Bayat. Variational quantum simulation of long-range interacting systems, 2022. URL https:/​/​​abs/​2203.14281.

[90] Zeyi Tao, Jindi Wu, Qi Xia, and Qun Li. Laws: Look around and warm-start natural gradient descent for quantum neural networks, 2022. URL https:/​/​​abs/​2205.02666.

[91] Barbara M. Terhal and David P. DiVincenzo. Classical simulation of noninteracting-fermion quantum circuits. Phys. Rev. A, 65: 032325, Mar 2002. 10.1103/​PhysRevA.65.032325. URL https:/​/​​doi/​10.1103/​PhysRevA.65.032325.

[92] Jules Tilly, Hongxiang Chen, Shuxiang Cao, Dario Picozzi, Kanav Setia, Ying Li, Edward Grant, Leonard Wossnig, Ivan Rungger, George H. Booth, and Jonathan Tennyson. The variational quantum eigensolver: a review of methods and best practices, 2021. URL https:/​/​​10.1016/​j.physrep.2022.08.003.

[93] Nikolay V. Tkachenko, James Sud, Yu Zhang, Sergei Tretiak, Petr M. Anisimov, Andrew T. Arrasmith, Patrick J. Coles, Lukasz Cincio, and Pavel A. Dub. Correlation-informed permutation of qubits for reducing ansatz depth in the variational quantum eigensolver. PRX Quantum, 2: 020337, Jun 2021a. 10.1103/​PRXQuantum.2.020337. URL https:/​/​​doi/​10.1103/​PRXQuantum.2.020337.

[94] Nikolay V. Tkachenko, James Sud, Yu Zhang, Sergei Tretiak, Petr M. Anisimov, Andrew T. Arrasmith, Patrick J. Coles, Lukasz Cincio, and Pavel A. Dub. Correlation-informed permutation of qubits for reducing ansatz depth in the variational quantum eigensolver. PRX Quantum, 2: 020337, Jun 2021b. 10.1103/​PRXQuantum.2.020337. URL https:/​/​​doi/​10.1103/​PRXQuantum.2.020337.

[95] Takashi Tsuchimochi, Masaki Taii, Taisei Nishimaki, and Seiichiro L. Ten-no. Adaptive construction of shallower quantum circuits with quantum spin projection for fermionic systems, 2022. URL https:/​/​​abs/​2205.07097.

[96] A V Uvarov and J D Biamonte. On barren plateaus and cost function locality in variational quantum algorithms. Journal of Physics A: Mathematical and Theoretical, 54 (24): 245301, may 2021. 10.1088/​1751-8121/​abfac7. URL https:/​/​​10.1088/​1751-8121/​abfac7.

[97] Leslie G. Valiant. Quantum circuits that can be simulated classically in polynomial time. SIAM Journal on Computing, 31 (4): 1229–1254, 2002. 10.1137/​S0097539700377025. URL https:/​/​​10.1137/​S0097539700377025.

[98] Tyler Volkoff and Patrick J Coles. Large gradients via correlation in random parameterized quantum circuits. Quantum Science and Technology, 6 (2): 025008, jan 2021. 10.1088/​2058-9565/​abd891. URL https:/​/​​10.1088/​2058-9565/​abd891.

[99] Samson Wang, Enrico Fontana, M. Cerezo, Kunal Sharma, Akira Sone, Lukasz Cincio, and Patrick J. Coles. Noise-induced barren plateaus in variational quantum algorithms. Nature Communications, 12 (1): 6961, 2021. ISSN 2041-1723. 10.1038/​s41467-021-27045-6. URL https:/​/​​10.1038/​s41467-021-27045-6.

[100] Zhihui Wang, Stuart Hadfield, Zhang Jiang, and Eleanor G. Rieffel. Quantum approximate optimization algorithm for maxcut: A fermionic view. Phys. Rev. A, 97: 022304, Feb 2018. 10.1103/​PhysRevA.97.022304. URL https:/​/​​doi/​10.1103/​PhysRevA.97.022304.

[101] Ada Warren, Linghua Zhu, Nicholas J. Mayhall, Edwin Barnes, and Sophia E. Economou. Adaptive variational algorithms for quantum gibbs state preparation, 2022. URL https:/​/​​abs/​2203.12757.

[102] Johannes Weidenfeller, Lucia C. Valor, Julien Gacon, Caroline Tornow, Luciano Bello, Stefan Woerner, and Daniel J. Egger. Scaling of the quantum approximate optimization algorithm on superconducting qubit based hardware, 2022. URL https:/​/​​10.22331/​q-2022-12-07-870.

[103] Roeland Wiersema, Cunlu Zhou, Yvette de Sereville, Juan Felipe Carrasquilla, Yong Baek Kim, and Henry Yuen. Exploring entanglement and optimization within the hamiltonian variational ansatz. PRX Quantum, 1: 020319, Dec 2020. 10.1103/​PRXQuantum.1.020319. URL https:/​/​​doi/​10.1103/​PRXQuantum.1.020319.

[104] Bennet Windt, Alexander Jahn, Jens Eisert, and Lucas Hackl. Local optimization on pure Gaussian state manifolds. SciPost Phys., 10: 66, 2021. 10.21468/​SciPostPhys.10.3.066. URL https:/​/​​10.21468/​SciPostPhys.10.3.066.

[105] Xu-Dan Xie, Xingyu Guo, Hongxi Xing, Zheng-Yuan Xue, Dan-Bo Zhang, and Shi-Liang Zhu. Variational thermal quantum simulation of the lattice schwinger model, 2022. URL https:/​/​​10.1103/​PhysRevD.106.054509.

[106] Xiaosi Xu, Jinzhao Sun, Suguru Endo, Ying Li, Simon C. Benjamin, and Xiao Yuan. Variational algorithms for linear algebra. Science Bulletin, 66 (21): 2181–2188, 2021. ISSN 2095-9273. https:/​/​​10.1016/​j.scib.2021.06.023. URL https:/​/​​science/​article/​pii/​S2095927321004631.

[107] Zhi-Cheng Yang, Armin Rahmani, Alireza Shabani, Hartmut Neven, and Claudio Chamon. Optimizing variational quantum algorithms using pontryagin's minimum principle. Phys. Rev. X, 7: 021027, May 2017. 10.1103/​PhysRevX.7.021027. URL https:/​/​​doi/​10.1103/​PhysRevX.7.021027.

[108] Xuchen You, Shouvanik Chakrabarti, and Xiaodi Wu. A convergence theory for over-parameterized variational quantum eigensolvers, 2022. URL https:/​/​​abs/​2205.12481.

[109] Feng Zhang, Niladri Gomes, Noah F. Berthusen, Peter P. Orth, Cai-Zhuang Wang, Kai-Ming Ho, and Yong-Xin Yao. Shallow-circuit variational quantum eigensolver based on symmetry-inspired hilbert space partitioning for quantum chemical calculations. Phys. Rev. Research, 3: 013039, Jan 2021. 10.1103/​PhysRevResearch.3.013039. URL https:/​/​​doi/​10.1103/​PhysRevResearch.3.013039.

[110] Kaining Zhang, Min-Hsiu Hsieh, Liu Liu, and Dacheng Tao. Gaussian initializations help deep variational quantum circuits escape from the barren plateau, 2022. URL https:/​/​​abs/​2203.09376.

[111] Zeqiao Zhou, Yuxuan Du, Xinmei Tian, and Dacheng Tao. Qaoa-in-qaoa: solving large-scale maxcut problems on small quantum machines, 2022. URL https:/​/​​abs/​2205.11762.

Cited by

[1] Xie-Hang Yu, Zongping Gong, and J. Ignacio Cirac, "Free-fermion Page curve: Canonical typicality and dynamical emergence", Physical Review Research 5 1, 013044 (2023).

[2] Diego García-Martín, Martin Larocca, and M. Cerezo, "Effects of noise on the overparametrization of quantum neural networks", arXiv:2302.05059, (2023).

[3] Leela Ganesh Chandra Lakkaraju, Srijon Ghosh, Debasis Sadhukhan, and Aditi SenDe, "Mimicking quantum correlation of a long-range Hamiltonian by finite-range interactions", Physical Review A 106 5, 052425 (2022).

The above citations are from SAO/NASA ADS (last updated successfully 2023-06-09 04:02:33). The list may be incomplete as not all publishers provide suitable and complete citation data.

On Crossref's cited-by service no data on citing works was found (last attempt 2023-06-09 04:02:31).