Scaling of variational quantum circuit depth for condensed matter systems
1Departament de Física Quàntica i Astrofísica and Institut de Ciències del Cosmos (ICCUB), Universitat de Barcelona, Martí i Franquès 1, 08028 Barcelona, Spain.
2Barcelona Supercomputing Center, Barcelona, Spain.
3Center for Quantum Technologies, National University of Singapore, Singapore.
4Technology Innovation Institute, Abu Dhabi, UAE.
| Published: | 2020-05-28, volume 4, page 272 |
| Eprint: | arXiv:2002.06210v3 |
| Doi: | https://doi.org/10.22331/q-2020-05-28-272 |
| Citation: | Quantum 4, 272 (2020). |
Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.
Abstract
We benchmark the accuracy of a variational quantum eigensolver based on a finite-depth quantum circuit encoding ground state of local Hamiltonians. We show that in gapped phases, the accuracy improves exponentially with the depth of the circuit. When trying to encode the ground state of conformally invariant Hamiltonians, we observe two regimes. A $\textit{finite-depth}$ regime, where the accuracy improves slowly with the number of layers, and a $\textit{finite-size}$ regime where it improves again exponentially. The cross-over between the two regimes happens at a critical number of layers whose value increases linearly with the size of the system. We discuss the implication of these observations in the context of comparing different variational ansatz and their effectiveness in describing critical ground states.
► BibTeX data
► References
[1] I. Buluta and F. Nori, Science 326, 108 (2009).
https://doi.org/10.1126/science.1177838
[2] K. L. Brown, W. J. Munro, and V. M. Kendon, Entropy 12, 2268 (2010).
https://doi.org/10.3390/e12112268
[3] I. M. Georgescu, S. Ashhab, and F. Nori, Reviews of Modern Physics 86, 153 (2014).
https://doi.org/10.1103/RevModPhys.86.153
[4] Y. Cao, J. Romero, J. P. Olson, M. Degroote, P. D. Johnson, M. Kieferová, I. D. Kivlichan, T. Menke, B. Peropadre, N. P. D. Sawaya, S. Sim, L. Veis, and A. Aspuru-Guzik, Chemical Reviews 119, 10856 (2019).
https://doi.org/10.1021/acs.chemrev.8b00803
[5] D. S. Abrams and S. Lloyd, Phys. Rev. Lett. 83, 5162 (1999).
https://doi.org/10.1103/PhysRevLett.83.5162
[6] D. W. Berry, M. Kieferová, A. Scherer, Y. R. Sanders, G. H. Low, N. Wiebe, C. Gidney, and R. Babbush, npj Quantum Information 4, 22 (2018).
https://doi.org/10.1038/s41534-018-0071-5
[7] F. Verstraete, J. I. Cirac, and J. I. Latorre, Physical Review A 79, 032316 (2009).
https://doi.org/10.1103/PhysRevA.79.032316
[8] S. P. Jordan, K. S. M. Lee, and J. Preskill, Science 336, 1130 (2012).
https://doi.org/10.1126/science.1217069
[9] K. Temme, T. J. Osborne, K. G. Vollbrecht, D. Poulin, and F. Verstraete, Nature 471, 87 (2011).
https://doi.org/10.1038/nature09770
[10] J. Preskill, Quantum 2, 79 (2018).
https://doi.org/10.22331/q-2018-08-06-79
[11] A. Peruzzo, J. McClean, P. Shadbolt, M.-H. Yung, X.-Q. Zhou, P. J. Love, A. Aspuru-Guzik, and J. L. O'Brien, Nature Communications 5, 4213 (2014).
https://doi.org/10.1038/ncomms5213
[12] C. Kokail, C. Maier, R. van Bijnen, T. Brydges, M. K. Joshi, P. Jurcevic, C. A. Muschik, P. Silvi, R. Blatt, C. F. Roos, and P. Zoller, Nature 569, 355 (2019).
https://doi.org/10.1038/s41586-019-1177-4
[13] O. Higgott, D. Wang, and S. Brierley, Quantum 3, 156 (2019).
https://doi.org/10.22331/q-2019-07-01-156
[14] T. Jones, S. Endo, S. McArdle, X. Yuan, and S. C. Benjamin, Phys. Rev. A 99, 062304 (2019).
https://doi.org/10.1103/PhysRevA.99.062304
[15] Y. Li and S. C. Benjamin, Phys. Rev. X 7, 021050 (2017).
https://doi.org/10.1103/PhysRevX.7.021050
[16] J. Romero, J. P. Olson, and A. Aspuru-Guzik, Quantum Science and Technology 2, 045001 (2017).
https://doi.org/10.1088/2058-9565/aa8072
[17] S. Khatri, R. LaRose, A. Poremba, L. Cincio, A. T. Sornborger, and P. J. Coles, Quantum 3, 140 (2019).
https://doi.org/10.22331/q-2019-05-13-140
[18] A. Arrasmith, L. Cincio, A. T. Sornborger, W. H. Zurek, and P. J. Coles, Nature communications 10, 3438 (2019).
https://doi.org/10.1038/s41467-019-11417-0
[19] R. LaRose, A. Tikku, É. O'Neel-Judy, L. Cincio, and P. J. Coles, npj Quantum Information 5, 1 (2018).
https://doi.org/10.1038/s41534-019-0167-6
[20] C. Bravo-Prieto, D. García-Martín, and J. I. Latorre, (2019a), arXiv:1905.01353 [quant-ph].
arXiv:1905.01353
[21] C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, and P. J. Coles, (2019b), arXiv:1909.05820 [quant-ph].
arXiv:1909.05820
[22] C. Cirstoiu, Z. Holmes, J. Iosue, L. Cincio, P. J. Coles, and A. Sornborger, (2019), arXiv:1910.04292 [quant-ph].
arXiv:1910.04292
[23] K. Sharma, S. Khatri, M. Cerezo, and P. J. Coles, New Journal of Physics (2020).
https://iopscience.iop.org/article/10.1088/1367-2630/ab784c
[24] J. Carolan, M. Mohseni, J. Olson, M. Prabhu, C. Chen, D. Bunandar, Y. Niu, N. Harris, F. Wong, M. Hochberg, S. Lloyd, and D. Englund, Nature Physics 95, 1 (2020).
https://doi.org/10.1038/s41567-019-0747-6
[25] S. McArdle, T. Jones, S. Endo, Y. Li, S. C. Benjamin, and X. Yuan, npj Quantum Information 5, 1 (2019).
https://doi.org/10.1038/s41534-019-0187-2
[26] A. Pérez-Salinas, A. Cervera-Lierta, E. Gil-Fuster, and J. I. Latorre, Quantum 4, 226 (2020).
https://doi.org/10.22331/q-2020-02-06-226
[27] C. M. Dawson and M. A. Nielsen, Quantum Info. Comput. 6, 81 (2006).
http://dl.acm.org/citation.cfm?id=2011679.2011685
[28] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information: 10th Anniversary Edition (Cambridge University Press, 2010).
https://doi.org/10.1017/CBO9780511976667
[29] A. Kitaev, A. Shen, and M. Vyalyi, Classical and Quantum Computation (Am. Math. Soc., Providence, Rhode Island, 2002).
https://doi.org/10.1090/gsm/047
[30] A. W. Harrow, B. Recht, and I. L. Chuang, Journal of Mathematical Physics 43, 4445 (2002).
https://doi.org/10.1063/1.1495899
[31] F. Wegner, Annalen der Physik 506, 77 (1994).
https://doi.org/10.1002/andp.19945060203
[32] S. D. Głazek and K. G. Wilson, Phys. Rev. D 48, 5863 (1993).
https://doi.org/10.1103/PhysRevD.48.5863
[33] S. D. Glazek, Phys. Rev. D 49, 4214 (1994).
https://doi.org/10.1103/PhysRevD.49.4214
[34] S. Dusuel and G. S. Uhrig, J. Phys. A: Math. Gen. 37, 9275 (2004).
https://doi.org/10.1088/0305-4470/37/39/014
[35] M. B. Hastings and X.-G. Wen, Phys. Rev. B 72, 045141 (2005).
https://doi.org/10.1103/PhysRevB.72.045141
[36] Y. Huang and X. Chen, Phys. Rev. B 91, 195143 (2015).
https://doi.org/10.1103/PhysRevB.91.195143
[37] J. I. Cirac, D. Perez-Garcia, N. Schuch, and F. Verstraete, J. Stat. Mech. 2017, 083105 (2017).
https://doi.org/10.1088/1742-5468/aa7e55
[38] P. Kos, M. Ljubotina, and T. Prosen, Phys. Rev. X 8, 021062 (2018).
https://doi.org/10.1103/PhysRevX.8.021062
[39] E. H. Lieb and D. W. Robinson, Comm. Math. Phys. 28, 251 (1972).
http://projecteuclid.org/euclid.cmp/1103858407
[40] M. B. Hastings, Journal of Statistical Mechanics: Theory and Experiment 2007, P08024 (2007).
https://doi.org/10.1088/1742-5468/2007/08/P08024
[41] J. Eisert, M. Cramer, and M. B. Plenio, Reviews of Modern Physics 82, 277 (2010).
https://doi.org/10.1103/RevModPhys.82.277
[42] N. Laflorencie, Physics Reports Quantum entanglement in condensed matter systems, 646, 1 (2016).
https://doi.org/10.1016/j.physrep.2016.06.008
[43] D. Aharonov, D. Gottesman, S. Irani, and J. Kempe, Commun. Math. Phys. 287, 41 (2009).
https://doi.org/10.1007/s00220-008-0710-3
[44] T. J. Osborne, Rep. Prog. Phys. 75, 022001 (2012).
https://doi.org/10.1088/0034-4885/75/2/022001
[45] C. Holzhey, F. Larsen, and F. Wilczek, Nucl. Phys. B 424, 443 (1994).
https://doi.org/10.1016/0550-3213(94)90402-2
[46] C. Callan and F. Wilczek, Physics Letters B 333, 55 (1994).
https://doi.org/10.1016/0370-2693(94)91007-3
[47] J. I. Latorre, E. Rico, and G. Vidal, Quantum Info. Comput. 4, 48 (2004).
http://dl.acm.org/citation.cfm?id=2011572.2011576
[48] P. Calabrese and J. Cardy, J. Stat. Mech. 2004, P06002 (2004).
https://doi.org/10.1088/1742-5468/2004/06/P06002
[49] E. Farhi, J. Goldstone, and S. Gutmann, (2014), arXiv:1411.4028 [quant-ph].
arXiv:1411.4028
[50] G. B. Mbeng, R. Fazio, and G. Santoro, (2019), arXiv:1906.08948 [quant-ph].
arXiv:1906.08948
[51] T. D. Schultz, D. C. Mattis, and E. H. Lieb, Rev. Mod. Phys. 36, 856 (1964).
https://doi.org/10.1103/RevModPhys.36.856
[52] A. A. Belavin, A. M. Polyakov, and A. B. Zamolodchikov, Journal of Statistical Physics 34, 763 (1984).
https://doi.org/10.1007/BF01009438
[53] M. Henkel, Conformal Invariance and Critical Phenomena, Theoretical and Mathematical Physics (Springer-Verlag, Berlin Heidelberg, 1999).
https://doi.org/10.1007/978-3-662-03937-3
[54] F. H. L. Essler, H. Frahm, F. Göhmann, A. Klümper, and V. E. Korepin, The One-Dimensional Hubbard Model (Cambridge University Press, 2005).
https://doi.org/10.1017/CBO9780511534843
[55] A. García-Saez and J. I. Latorre, (2018), arXiv:1806.02287 [quant-ph].
arXiv:1806.02287
[56] I. Affleck, Phys. Rev. Lett. 56, 746 (1986).
https://doi.org/10.1103/PhysRevLett.56.746
[57] J. L. Cardy, Nuclear Physics B 270, 186 (1986).
https://doi.org/10.1016/0550-3213(86)90552-3
[58] L. Tagliacozzo, T. R. de Oliveira, S. Iblisdir, and J. I. Latorre, Phys. Rev. B 78, 024410 (2008).
https://doi.org/10.1103/PhysRevB.78.024410
[59] F. Pollmann, S. Mukerjee, A. M. Turner, and J. E. Moore, Phys. Rev. Lett. 102, 255701 (2009).
https://doi.org/10.1103/PhysRevLett.102.255701
[60] B. Pirvu, G. Vidal, F. Verstraete, and L. Tagliacozzo, Phys. Rev. B 86, 075117 (2012).
https://doi.org/10.1103/PhysRevB.86.075117
[61] V. Stojevic, J. Haegeman, I. P. McCulloch, L. Tagliacozzo, and F. Verstraete, Phys. Rev. B 91, 035120 (2015).
https://doi.org/10.1103/PhysRevB.91.035120
[62] L. Vanderstraeten, M. Mariën, J. Haegeman, N. Schuch, J. Vidal, and F. Verstraete, Phys. Rev. Lett. 119, 070401 (2017).
https://doi.org/10.1103/PhysRevLett.119.070401
[63] S. Bravyi, M. B. Hastings, and F. Verstraete, Phys. Rev. Lett. 97, 050401 (2006).
https://doi.org/10.1103/PhysRevLett.97.050401
[64] T. Nishino, K. Okunishi, and M. Kikuchi, Physics Letters A 213, 69 (1996).
https://doi.org/10.1016/0375-9601(96)00128-4
[65] F. Verstraete and J. I. Cirac, Phys. Rev. B 73, 094423 (2006).
https://doi.org/10.1103/PhysRevB.73.094423
[66] G. Evenbly and G. Vidal, in Strongly correlated systems (Springer, 2013) pp. 99–130.
https://doi.org/10.1007/978-3-642-35106-8_4
[67] M. Collura, L. Dell'Anna, T. Felser, and S. Montangero, (2019), arXiv:1905.11351 [quant-ph].
arXiv:1905.11351
[68] M. A. Nielsen, M. R. Dowling, M. Gu, and A. C. Doherty, Science 311, 1133 (2006).
https://doi.org/10.1126/science.1121541
[69] M. R. Dowling and M. A. Nielsen, Quantum Info. Comput. 8, 861–899 (2008).
https://dl.acm.org/doi/10.5555/2016985.2016986
[70] R. H. Byrd, P. Lu, J. Nocedal, and C. Zhu, SIAM J. Sci. Comput. 16, 1190 (1995).
https://doi.org/10.1137/0916069
[71] P. Virtanen et al., Nature Methods (2020).
https://doi.org/10.1038/s41592-019-0686-2
[72] J. R. Johansson, P. D. Nation, and F. Nori, Computer Physics Communications 184, 1234 (2013).
https://doi.org/10.1016/j.cpc.2012.11.019
[73] J. R. McClean, S. Boixo, V. N. Smelyanskiy, R. Babbush, and H. Neven, Nature Communications 9, 4812 (2018).
https://doi.org/10.1038/s41467-018-07090-4
[74] M. Cerezo, A. Sone, T. Volkoff, L. Cincio, and P. J. Coles, (2020), arXiv:2001.00550 [quant-ph].
arXiv:2001.00550
Cited by
[1] Chufan Lyu, Xiaoyu Tang, Junning Li, Xusheng Xu, Man-Hong Yung, and Abolfazl Bayat, "Variational quantum simulation of long-range interacting systems", New Journal of Physics 25 5, 053022 (2023).
[2] Tomasz Szołdra, Rick Mukherjee, and Peter Schmelcher, "Scalable preparation of matrix product states with sequential and brick wall quantum circuits", Quantum Science and Technology 11 2, 025056 (2026).
[3] Kishor Bharti, Alba Cervera-Lierta, Thi Ha Kyaw, Tobias Haug, Sumner Alperin-Lea, Abhinav Anand, Matthias Degroote, Hermanni Heimonen, Jakob S. Kottmann, Tim Menke, Wai-Keong Mok, Sukin Sim, Leong-Chuan Kwek, and Alán Aspuru-Guzik, "Noisy intermediate-scale quantum algorithms", Reviews of Modern Physics 94 1, 015004 (2022).
[4] Yue Ruan, Zhiqiang Yuan, Xiling Xue, and Zhihao Liu, "Quantum approximate optimization for combinatorial problems with constraints", Information Sciences 619, 98 (2023).
[5] Gabriele Cenedese, Maria Bondani, Alexei Andreanov, Matteo Carrega, Giuliano Benenti, and Dario Rosa, "Shallow quantum circuits are robust hunters for quantum many-body scars", The European Physical Journal Plus 140 6, 517 (2025).
[6] Yunjun Yu, Guoping Hu, Caicheng Liu, Junjie Xiong, and Ziyang Wu, "Prediction of Solar Irradiance One Hour Ahead Based on Quantum Long Short-Term Memory Network", IEEE Transactions on Quantum Engineering 4, 1 (2023).
[7] G. Xu, Y. B. Guo, X. Li, K. Wang, Z. Fan, Z. S. Zhou, H. J. Liao, and T. Xiang, "Concurrent quantum eigensolver for multiple low-energy eigenstates", Physical Review A 107 5, 052423 (2023).
[8] Zhi-Yuan Wei and Daniel Malz, "State preparation with parallel-sequential circuits", Quantum 10, 2079 (2026).
[9] Mirko Consiglio, Wayne J Chetcuti, Carlos Bravo-Prieto, Sergi Ramos-Calderer, Anna Minguzzi, José I Latorre, Luigi Amico, and Tony J G Apollaro, "Variational quantum eigensolver for SU(N) fermions", Journal of Physics A: Mathematical and Theoretical 55 26, 265301 (2022).
[10] Chufan Lyu, Xusheng Xu, Man-Hong Yung, and Abolfazl Bayat, "Symmetry enhanced variational quantum spin eigensolver", Quantum 7, 899 (2023).
[11] Carlos Bravo-Prieto, Julien Baglio, Marco Cè, Anthony Francis, Dorota M. Grabowska, and Stefano Carrazza, "Style-based quantum generative adversarial networks for Monte Carlo events", Quantum 6, 777 (2022).
[12] Luca Tagliacozzo, "Optimal simulation of quantum dynamics", Nature Physics 18 9, 970 (2022).
[13] Zheng-Hang Sun, Yong-Yi Wang, Jian Cui, and Heng Fan, "Improving the performance of quantum approximate optimization for preparing non-trivial quantum states without translational symmetry", New Journal of Physics 25 1, 013015 (2023).
[14] David Amaro, Carlo Modica, Matthias Rosenkranz, Mattia Fiorentini, Marcello Benedetti, and Michael Lubasch, "Filtering variational quantum algorithms for combinatorial optimization", Quantum Science and Technology 7 1, 015021 (2022).
[15] O. V. Borzenkova, G. I. Struchalin, A. S. Kardashin, V. V. Krasnikov, N. N. Skryabin, S. S. Straupe, S. P. Kulik, and J. D. Biamonte, "Variational simulation of Schwinger's Hamiltonian with polarization qubits", Applied Physics Letters 118 14, 144002 (2021).
[16] Thomas Ayral, Pauline Besserve, Denis Lacroix, and Edgar Andres Ruiz Guzman, "Quantum computing with and for many-body physics", The European Physical Journal A 59 10, 227 (2023).
[17] P. Chandarana, N. N. Hegade, K. Paul, F. Albarrán-Arriagada, E. Solano, A. del Campo, and Xi Chen, "Digitized-counterdiabatic quantum approximate optimization algorithm", Physical Review Research 4 1, 013141 (2022).
[18] Nazlı Uğur Köylüoğlu, Swarnadeep Majumder, Mirko Amico, Sarah Mostame, Ewout van den Berg, M. A. Rajabpour, Zlatko Minev, and Khadijeh Najafi, "Measuring central charge on a universal quantum processor", Nature Communications 17 1, 305 (2026).
[19] Pranav Chandarana, Koushik Paul, Mikel Garcia-de-Andoin, Yue Ban, Mikel Sanz, and Xi Chen, "Photonic counterdiabatic quantum optimization algorithm", Communications Physics 7 1, 315 (2024).
[20] Chenfeng Cao, Filippo Maria Gambetta, Ashley Montanaro, and Raul A. Santos, "Unveiling quantum phase transitions from traps in variational quantum algorithms", npj Quantum Information 11 1, 93 (2025).
[21] Ha-Vu Phan, Tan-Hung Pham, Khang B. Tran, Quoc-Thang Phan, Quoc Dung Phan, and Yuan-Kang Wu, 2025 IEEE Industry Applications Society Annual Meeting (IAS) 1 (2025) ISBN:978-1-6654-5776-7.
[22] John S. Van Dyke, Karunya Shirali, George S. Barron, Nicholas J. Mayhall, Edwin Barnes, and Sophia E. Economou, "Scaling adaptive quantum simulation algorithms via operator pool tiling", Physical Review Research 6 1, L012030 (2024).
[23] John P. T. Stenger, C. Stephen Hellberg, and Daniel Gunlycke, "Implementing Jastrow-Gutzwiller operators on a quantum computer using the cascaded variational quantum eigensolver algorithm", Physical Review A 107 6, 062606 (2023).
[24] 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", Physics Reports 986, 1 (2022).
[25] Sebastián Roca-Jerat, Teresa Sancho-Lorente, Juan Román-Roche, and David Zueco, "Circuit Complexity through phase transitions: Consequences in quantum state preparation", SciPost Physics 15 5, 186 (2023).
[26] Yu-Qin Chen, Shi-Xin Zhang, Chang-Yu Hsieh, and Shengyu Zhang, "Non-Hermitian ground-state-searching algorithm enhanced by a variational toolbox", Physical Review A 107 4, 042418 (2023).
[27] Andrew Patterson, Hongxiang Chen, Leonard Wossnig, Simone Severini, Dan Browne, and Ivan Rungger, "Quantum state discrimination using noisy quantum neural networks", Physical Review Research 3 1, 013063 (2021).
[28] Ivan Medina, Alexandre Drinko, Guilherme I. Correr, Pedro C. Azado, and Diogo O. Soares-Pinto, "Variational-quantum-eigensolver–inspired optimization for spin-chain work extraction", Physical Review A 110 1, 012443 (2024).
[29] András Grabarits, Kasturi Ranjan Swain, Mahsa Seyed Heydari, Pranav Chandarana, Fernando J. Gómez-Ruiz, and Adolfo del Campo, "Quantum chaos in random Ising networks", Physical Review Research 7 1, 013146 (2025).
[30] Yihao Liu, Min-Quan He, and Z.D. Wang, "Variational quantum eigensolvers with quantum Gaussian filters for solving ground-state problems in quantum many-body systems", Physics Letters A 555, 130766 (2025).
[31] Elias Zapusek, Kristina Kirova, Walter Hahn, Michael Marthaler, and Florentin Reiter, "Variational quantum thermalizers based on weakly-symmetric nonunitary multi-qubit operations", Quantum Science and Technology 11 2, 025006 (2026).
[32] Mourad Halla, "Quantum natural gradient with geodesic corrections for small shallow quantum circuits", Physica Scripta 100 5, 055121 (2025).
[33] Dingjie Lu, Zhao Wang, Jun Liu, Yangfan Li, Wei-Bin Ewe, and Zhuangjian Liu, "Quantum finite element algorithm for solving Euler–Bernoulli and heat transfer PDEs with Dirichlet, Neumann, and Robin boundary conditions", Quantum Science and Technology 11 1, 015051 (2026).
[34] Bernhard Jobst, Adam Smith, and Frank Pollmann, "Finite-depth scaling of infinite quantum circuits for quantum critical points", Physical Review Research 4 3, 033118 (2022).
[35] Stavros Efthymiou, Sergi Ramos-Calderer, Carlos Bravo-Prieto, Adrián Pérez-Salinas, Diego García-Martín, Artur Garcia-Saez, José Ignacio Latorre, and Stefano Carrazza, " Qibo: a framework for quantum simulation with hardware acceleration", Quantum Science and Technology 7 1, 015018 (2022).
[36] Nhat A. Nghiem and Tzu-Chieh Wei, "Quantum algorithm for solving nonlinear algebraic equations", Physical Review Applied 24 5, 054005 (2025).
[37] Chufan Lyu, Victor Montenegro, and Abolfazl Bayat, "Accelerated variational algorithms for digital quantum simulation of many-body ground states", Quantum 4, 324 (2020).
[38] Daniel Huerga, "Variational Quantum Simulation of Valence-Bond Solids", Quantum 6, 874 (2022).
[39] Benedikt Fauseweh and Jian-Xin Zhu, "Quantum computing Floquet energy spectra", Quantum 7, 1063 (2023).
[40] Bernhard Jobst, Kevin Shen, Carlos A. Riofrío, Elvira Shishenina, and Frank Pollmann, "Efficient MPS representations and quantum circuits from the Fourier modes of classical image data", Quantum 8, 1544 (2024).
[41] Teresa Sancho-Lorente, Juan Román-Roche, and David Zueco, "Quantum kernels to learn the phases of quantum matter", Physical Review A 105 4, 042432 (2022).
[42] 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 (2021).
[43] Andrey Kardashin, Anastasiia Pervishko, Jacob Biamonte, and Dmitry Yudin, "Numerical hardware-efficient variational quantum simulation of a soliton solution", Physical Review A 104 2, L020402 (2021).
[44] Daniel Gunlycke, C. Stephen Hellberg, and John P. T. Stenger, "Cascaded variational quantum eigensolver algorithm", Physical Review Research 6 1, 013238 (2024).
[45] Bingzhi Zhang and Quntao Zhuang, "Energy-dependent barren plateau in bosonic variational quantum circuits", Quantum Science and Technology 10 1, 015009 (2025).
[46] Sun Woo Park, Hyunju Lee, Byung Chun Kim, Youngho Woo, and Kyungtaek Jun, 2021 International Conference on Information and Communication Technology Convergence (ICTC) 1357 (2021) ISBN:978-1-6654-2383-0.
[47] Alexey Uvarov, Jacob D. Biamonte, and Dmitry Yudin, "Variational quantum eigensolver for frustrated quantum systems", Physical Review B 102 7, 075104 (2020).
[48] Eli Chertkov, Justin Bohnet, David Francois, John Gaebler, Dan Gresh, Aaron Hankin, Kenny Lee, David Hayes, Brian Neyenhuis, Russell Stutz, Andrew C. Potter, and Michael Foss-Feig, "Holographic dynamics simulations with a trapped-ion quantum computer", Nature Physics 18 9, 1074 (2022).
[49] Johannes Herrmann, Sergi Masot Llima, Ants Remm, Petr Zapletal, Nathan A. McMahon, Colin Scarato, François Swiadek, Christian Kraglund Andersen, Christoph Hellings, Sebastian Krinner, Nathan Lacroix, Stefania Lazar, Michael Kerschbaum, Dante Colao Zanuz, Graham J. Norris, Michael J. Hartmann, Andreas Wallraff, and Christopher Eichler, "Realizing quantum convolutional neural networks on a superconducting quantum processor to recognize quantum phases", Nature Communications 13 1, 4144 (2022).
[50] Shi-Xin Zhang and Shuai Yin, "Universal imaginary-time critical dynamics on a quantum computer", Physical Review B 109 13, 134309 (2024).
[51] Bingzhi Zhang and Quntao Zhuang, "Fast decay of classification error in variational quantum circuits", Quantum Science and Technology 7 3, 035017 (2022).
[52] Erik J. Gustafson, Juha Tiihonen, Diana Chamaki, Farshud Sorourifar, J. Wayne Mullinax, Andy C. Y. Li, Filip B. Maciejewski, Nicolas P. D. Sawaya, Jaron T. Krogel, David E. Bernal Neira, and Norm M. Tubman, "Surrogate optimization of variational quantum circuits", Proceedings of the National Academy of Sciences 122 36, e2408530122 (2025).
[53] 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 2, 020319 (2020).
[54] 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 (2021).
[55] Borja Requena, Gorka Muñoz-Gil, Maciej Lewenstein, Vedran Dunjko, and Jordi Tura, "Certificates of quantum many-body properties assisted by machine learning", Physical Review Research 5 1, 013097 (2023).
[56] C. Tabares, A. Muñoz de las Heras, L. Tagliacozzo, D. Porras, and A. González-Tudela, "Variational Quantum Simulators Based on Waveguide QED", Physical Review Letters 131 7, 073602 (2023).
[57] Rubén Darío Guerrero, "Bee-yond the plateau: Training QNNs with swarm algorithms", The Journal of Chemical Physics 162 1, 011101 (2025).
[58] Alina Joch, Götz S Uhrig, and Benedikt Fauseweh, "Entanglement-informed construction of variational quantum circuits", Quantum Science and Technology 10 3, 035032 (2025).
[59] Elies Gil-Fuster, Jens Eisert, and Carlos Bravo-Prieto, "Understanding quantum machine learning also requires rethinking generalization", Nature Communications 15 1, 2277 (2024).
[60] Alejandro Sopena, Max Hunter Gordon, Diego García-Martín, Germán Sierra, and Esperanza López, "Algebraic Bethe Circuits", Quantum 6, 796 (2022).
[61] Liyun Su, Man Wu, and Fenglan Li, "A variational quantum circuits architecture with multi-head attention for chaotic time series prediction", Complex & Intelligent Systems 11 8, 347 (2025).
[62] Qingyu Li, Chiranjib Mukhopadhyay, and Abolfazl Bayat, "Fermionic simulators for enhanced scalability of variational quantum simulation", Physical Review Research 5 4, 043175 (2023).
[63] Dheeraj Peddireddy, Utkarsh Priyam, and Vaneet Aggarwal, "Noisy tensor-ring approximation for computing gradients of a variational quantum eigensolver for combinatorial optimization", Physical Review A 108 4, 042429 (2023).
[64] Leo Joon Il Moon, Mandar M Sohoni, Michael A Shimizu, Praveen Viswanathan, Kevin Zhang, Eun-Ah Kim, and Peter L McMahon, "Hamiltonian-reconstruction distance as a success metric for the variational quantum eigensolver", Quantum Science and Technology 11 2, 025035 (2026).
[65] Gabriel Matos, Sonika Johri, and Zlatko Papić, "Quantifying the Efficiency of State Preparation via Quantum Variational Eigensolvers", PRX Quantum 2 1, 010309 (2021).
[66] Mirko Consiglio, "Variational Quantum Algorithms for Many-Body Systems", arXiv:2502.11985, (2025).
[67] David Rogerson and Ananda Roy, "Quantum Circuit Optimization using Differentiable Programming of Tensor Network States", arXiv:2408.12583, (2024).
[68] David Quiroga, Jason Han, and Anastasios Kyrillidis, "Quantum EigenGame for excited state calculation", arXiv:2503.13644, (2025).
[69] Ahmad Sohrabi, "Deep Teleportation: Quantum Simulation of Conscious Report in Attentional Blink", arXiv:2512.18585, (2025).
The above citations are from Crossref's cited-by service (last updated successfully 2026-09-08 12:28:23) and SAO/NASA ADS (last updated successfully 2026-09-08 12:28:24). The list may be incomplete as not all publishers provide suitable and complete citation data.
This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.