Scaling of variational quantum circuit depth for condensed matter systems

Carlos Bravo-Prieto1,2, Josep Lumbreras-Zarapico1, Luca Tagliacozzo1, and José I. Latorre1,3,4

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.

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Updated version: The authors have uploaded version v4 of this work to the arXiv which may contain updates or corrections not contained in the published version v3. The authors left the following comment on the arXiv:
11 + 4 pages, 5 figures

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.

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► 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

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[8] Daniel Huerga, "Variational Quantum Simulation of Valence-Bond Solids", Quantum 6, 874 (2022).

[9] 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).

[10] 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).

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