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] 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).
[2] 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).
[3] 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).
[4] 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).
[5] Yue Ruan, Zhiqiang Yuan, Xiling Xue, and Zhihao Liu, "Quantum approximate optimization for combinatorial problems with constraints", Information Sciences 619, 98 (2023).
[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] Chufan Lyu, Victor Montenegro, and Abolfazl Bayat, "Accelerated variational algorithms for digital quantum simulation of many-body ground states", Quantum 4, 324 (2020).
[8] Daniel Huerga, "Variational Quantum Simulation of Valence-Bond Solids", Quantum 6, 874 (2022).
[9] Benedikt Fauseweh and Jian-Xin Zhu, "Quantum computing Floquet energy spectra", Quantum 7, 1063 (2023).
[10] 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).
[11] 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).
[12] 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).
[13] Chufan Lyu, Xusheng Xu, Man-Hong Yung, and Abolfazl Bayat, "Symmetry enhanced variational quantum spin eigensolver", Quantum 7, 899 (2023).
[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 (2021).
[15] 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).
[16] Luca Tagliacozzo, "Optimal simulation of quantum dynamics", Nature Physics 18 9, 970 (2022).
[17] 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).
[18] 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).
[19] 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).
[20] 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).
[21] 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).
[22] 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.
[23] Alexey Uvarov, Jacob D. Biamonte, and Dmitry Yudin, "Variational quantum eigensolver for frustrated quantum systems", Physical Review B 102 7, 075104 (2020).
[24] 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).
[25] 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).
[26] 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).
[27] Bingzhi Zhang and Quntao Zhuang, "Fast decay of classification error in variational quantum circuits", Quantum Science and Technology 7 3, 035017 (2022).
[28] 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).
[29] 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).
[30] 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).
[31] 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).
[32] 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).
[33] 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).
[34] 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).
[35] 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).
[36] 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).
[37] Alejandro Sopena, Max Hunter Gordon, Diego García-Martín, Germán Sierra, and Esperanza López, "Algebraic Bethe Circuits", Quantum 6, 796 (2022).
[38] Qingyu Li, Chiranjib Mukhopadhyay, and Abolfazl Bayat, "Fermionic simulators for enhanced scalability of variational quantum simulation", Physical Review Research 5 4, 043175 (2023).
[39] 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).
[40] Gabriel Matos, Sonika Johri, and Zlatko Papić, "Quantifying the Efficiency of State Preparation via Quantum Variational Eigensolvers", PRX Quantum 2 1, 010309 (2021).
The above citations are from Crossref's cited-by service (last updated successfully 2023-11-28 21:12:54) and SAO/NASA ADS (last updated successfully 2023-11-28 21:12:55). 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.