Accelerated variational algorithms for digital quantum simulation of many-body ground states

Chufan Lyu, Victor Montenegro, and Abolfazl Bayat

Institute of Fundamental and Frontier Sciences, University of Electronic Science and Technology of China, Chengdu 610051, China

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.


One of the key applications for the emerging quantum simulators is to emulate the ground state of many-body systems, as it is of great interest in various fields from condensed matter physics to material science. Traditionally, in an analog sense, adiabatic evolution has been proposed to slowly evolve a simple Hamiltonian, initialized in its ground state, to the Hamiltonian of interest such that the final state becomes the desired ground state. Recently, variational methods have also been proposed and realized in quantum simulators for emulating the ground state of many-body systems. Here, we first provide a quantitative comparison between the adiabatic and variational methods with respect to required quantum resources on digital quantum simulators, namely the depth of the circuit and the number of two-qubit quantum gates. Our results show that the variational methods are less demanding with respect to these resources. However, they need to be hybridized with a classical optimization which can converge slowly. Therefore, as the second result of the paper, we provide two different approaches for speeding the convergence of the classical optimizer by taking a good initial guess for the parameters of the variational circuit. We show that these approaches are applicable to a wide range of Hamiltonian and provide significant improvement in the optimization procedure.

Many features of complex systems can only be understood as collective behavior of several particles, the so called many-body systems. Low-energy, especially the ground state, properties are of high importance in physics as they are responsible for phenomena such as magnetism, molecular formation and the emergence of new phase of matter.
Simulating quantum behavior at the many-body level is beyond the capability of classical computer due to the requirement of exponentially large resources. The true solution is to exploit another quantum system with better controllability, namely quantum simulator, to emulate the behavior of the complex system of interest. One can exploit adiabatic approach for simulating the ground state of many-body systems on quantum simulators, which are now emerging in various physical setups. However, this demands high quality hardware which cannot be achieved in existing quantum simulators. Therefore, a hybrid variational method, called variational quantum eigensolver (VQE), has already been proposed to simplify the quantum hardware at the price of the addition of a classical optimizer.
In this paper, we first provide a quantitative comparison for the hardware needed in the adiabatic and the VQE approaches. Our results show that the VQE can indeed significantly simplify the quantum circuit. Then we focus on accelerating the convergence of the classical optimizer. We have developed two methods to improve the initial guess for the parameters of the optimizer. This tends to start the optimization procedure closer to the optimal case and thus significantly speedup the convergence. We have shown that our protocol is applicable over a wide range of physical models.

► BibTeX data

► References

[1] F. Arute, K. Arya, R. Babbush, D. Bacon, J. C. Bardin, R. Barends, S. Boixo, M. Broughton, B. B. Buckley, D. A. Buell, B. Burkett, N. Bushnell, and et al. Hartree-fock on a superconducting qubit quantum computer, 2020. URL https:/​/​​abs/​2004.04174.

[2] A. Aspuru-Guzik and P. Walther. Photonic quantum simulators. Nature Physics, 8 (4): 285–291, Apr 2012. 10.1038/​nphys2253.

[3] R. Barends, A. Shabani, L. Lamata, J. Kelly, A. Mezzacapo, U. L. Heras, R. Babbush, A. G. Fowler, B. Campbell, Y. Chen, and et al. Digitized adiabatic quantum computing with a superconducting circuit. Nature, 534 (7606): 222–226, Jun 2016. 10.1038/​nature17658.

[4] P. K. Barkoutsos, J. F. Gonthier, I. Sokolov, N. Moll, G. Salis, A. Fuhrer, M. Ganzhorn, D. J. Egger, M. Troyer, A. Mezzacapo, S. Filipp, and I. Tavernelli. Quantum algorithms for electronic structure calculations: Particle-hole hamiltonian and optimized wave-function expansions. Phys. Rev. A, 98: 022322, Aug 2018. 10.1103/​PhysRevA.98.022322.

[5] A. Bayat and S. Bose. Information-transferring ability of the different phases of a finite xxz spin chain. Phys. Rev. A, 81: 012304, Jan 2010. 10.1103/​PhysRevA.81.012304.

[6] A. Bayat, P. Sodano, and S. Bose. Negativity as the entanglement measure to probe the kondo regime in the spin-chain kondo model. Phys. Rev. B, 81: 064429, Feb 2010. 10.1103/​PhysRevB.81.064429.

[7] H. Bernien, S. Schwartz, A. Keesling, H. Levine, A. Omran, H. Pichler, S. Choi, A. S. Zibrov, M. Endres, M. Greiner, and et al. Probing many-body dynamics on a 51-atom quantum simulator. Nature, 551 (7682): 579–584, Nov 2017. 10.1038/​nature24622.

[8] X. Bonet-Monroig, R. Sagastizabal, M. Singh, and T. E. O'Brien. Low-cost error mitigation by symmetry verification. Phys. Rev. A, 98: 062339, Dec 2018. 10.1103/​PhysRevA.98.062339.

[9] P. Bordia, H. Lüschen, S. Scherg, S. Gopalakrishnan, M. Knap, U. Schneider, and I. Bloch. Probing slow relaxation and many-body localization in two-dimensional quasiperiodic systems. Phys. Rev. X, 7: 041047, Nov 2017. 10.1103/​PhysRevX.7.041047.

[10] M. Born and V. Fock. Beweis des adiabatensatzes. Zeitschrift für Physik, 51 (3): 165–180, Mar 1928. 10.1007/​BF01343193.

[11] S. Bose. Quantum communication through an unmodulated spin chain. Phys. Rev. Lett., 91: 207901, Nov 2003. 10.1103/​PhysRevLett.91.207901.

[12] C. Bravo-Prieto, J. Lumbreras-Zarapico, L. Tagliacozzo, and J. I. Latorre. Scaling of variational quantum circuit depth for condensed matter systems. Quantum, 4: 272, May 2020. 10.22331/​q-2020-05-28-272.

[13] I. Buluta and F. Nori. Quantum simulators. Science, 326 (5949): 108–111, 2009. 10.1126/​science.1177838.

[14] G. Carleo and M. Troyer. Solving the quantum many-body problem with artificial neural networks. Science, 355 (6325): 602–606, Feb 2017. 10.1126/​science.aag2302.

[15] J. Carrasquilla and R. G. Melko. Machine learning phases of matter. Nature Physics, 13 (5): 431–434, Feb 2017. 10.1038/​nphys4035.

[16] G. D. Chiara, S. Montangero, P. Calabrese, and R. Fazio. Entanglement entropy dynamics of heisenberg chains. Journal of Statistical Mechanics: Theory and Experiment, 2006 (03): P03001–P03001, mar 2006. 10.1088/​1742-5468/​2006/​03/​p03001.

[17] K. Ch'ng, J. Carrasquilla, R. G. Melko, and E. Khatami. Machine learning phases of strongly correlated fermions. Phys. Rev. X, 7: 031038, Aug 2017. 10.1103/​PhysRevX.7.031038.

[18] J. Cirac and P. Zoller. Goals and opportunities in quantum simulation. Nature Physics, 8: 264–266, 04 2012. 10.1038/​nphys2275.

[19] J. I. Colless, V. V. Ramasesh, D. Dahlen, M. S. Blok, M. E. Kimchi-Schwartz, J. R. McClean, J. Carter, W. A. de Jong, and I. Siddiqi. Computation of molecular spectra on a quantum processor with an error-resilient algorithm. Phys. Rev. X, 8: 011021, Feb 2018. 10.1103/​PhysRevX.8.011021.

[20] D. J. Craik. Magnetism: principles and applications. Wiley, 2003.

[21] S. Endo, S. C. Benjamin, and Y. Li. Practical quantum error mitigation for near-future applications. Phys. Rev. X, 8: 031027, Jul 2018. 10.1103/​PhysRevX.8.031027.

[22] Y. Endoh, G. Shirane, R. J. Birgeneau, P. M. Richards, and S. L. Holt. Dynamics of an $s=\frac{1}{2}$, one-dimensional heisenberg antiferromagnet. Phys. Rev. Lett., 32: 170–173, Jan 1974. 10.1103/​PhysRevLett.32.170.

[23] E. Farhi, J. Goldstone, and S. Gutmann. A quantum approximate optimization algorithm, 2014. URL https:/​/​​abs/​1411.4028.

[24] U. Farooq, A. Bayat, S. Mancini, and S. Bose. Adiabatic many-body state preparation and information transfer in quantum dot arrays. Phys. Rev. B, 91: 134303, Apr 2015. 10.1103/​PhysRevB.91.134303.

[25] W. M. C. Foulkes, L. Mitas, R. J. Needs, and G. Rajagopal. Quantum monte carlo simulations of solids. Rev. Mod. Phys., 73: 33–83, Jan 2001. 10.1103/​RevModPhys.73.33.

[26] T. Fukuhara, A. Kantian, M. Endres, M. Cheneau, P. Schauß, S. Hild, D. Bellem, U. Schollwöck, T. Giamarchi, C. Gross, and et al. Quantum dynamics of a mobile spin impurity. Nature Physics, 9 (4): 235–241, Feb 2013a. 10.1038/​nphys2561.

[27] T. Fukuhara, P. Schauß, M. Endres, S. Hild, M. Cheneau, I. Bloch, and C. Gross. Microscopic observation of magnon bound states and their dynamics. Nature, 502 (7469): 76–79, Sep 2013b. 10.1038/​nature12541.

[28] I. M. Georgescu, S. Ashhab, and F. Nori. Quantum simulation. Rev. Mod. Phys., 86: 153–185, Mar 2014. 10.1103/​RevModPhys.86.153.

[29] J. Gray, L. Banchi, A. Bayat, and S. Bose. Machine-learning-assisted many-body entanglement measurement. Phys. Rev. Lett., 121: 150503, Oct 2018. 10.1103/​PhysRevLett.121.150503.

[30] C. Gross and I. Bloch. Quantum simulations with ultracold atoms in optical lattices. Science, 357 (6355): 995–1001, 2017. 10.1126/​science.aal3837.

[31] N. Hatano and M. Suzuki. Finding exponential product formulas of higher orders. In Quantum annealing and other optimization methods, pages 37–68. Springer, 2005. 10.1007/​11526216_2.

[32] T. Helgaker, P. Jorgensen, and J. Olsen. Molecular electronic-structure theory. John Wiley & Sons, 2014. 10.1002/​9781119019572.

[33] C. Hempel, C. Maier, J. Romero, J. McClean, T. Monz, H. Shen, P. Jurcevic, B. P. Lanyon, P. Love, R. Babbush, A. Aspuru-Guzik, R. Blatt, and C. F. Roos. Quantum chemistry calculations on a trapped-ion quantum simulator. Phys. Rev. X, 8: 031022, Jul 2018. 10.1103/​PhysRevX.8.031022.

[34] T. Hensgens, T. Fujita, L. Janssen, X. Li, C. J. Van Diepen, C. Reichl, W. Wegscheider, S. Das Sarma, and L. M. K. Vandersypen. Quantum simulation of a fermi–hubbard model using a semiconductor quantum dot array. Nature, 548 (7665): 70–73, Aug 2017. 10.1038/​nature23022.

[35] U. L. Heras, A. Mezzacapo, L. Lamata, S. Filipp, A. Wallraff, and E. Solano. Digital quantum simulation of spin systems in superconducting circuits. Phys. Rev. Lett., 112: 200501, May 2014. 10.1103/​PhysRevLett.112.200501.

[36] Y. Herasymenko and T. E. O'Brien. A diagrammatic approach to variational quantum ansatz construction, 2019. URL https:/​/​​abs/​1907.08157v2.

[37] O. Higgott, D. Wang, and S. Brierley. Variational Quantum Computation of Excited States. Quantum, 3: 156, July 2019. 10.22331/​q-2019-07-01-156.

[38] F. Jensen. Introduction to computational chemistry. John wiley & sons, 2017.

[39] R. O. Jones. Density functional theory: Its origins, rise to prominence, and future. Rev. Mod. Phys., 87: 897–923, Aug 2015. 10.1103/​RevModPhys.87.897.

[40] T. Jones, S. Endo, S. McArdle, X. Yuan, and S. C. Benjamin. Variational quantum algorithms for discovering hamiltonian spectra. Phys. Rev. A, 99: 062304, Jun 2019. 10.1103/​PhysRevA.99.062304.

[41] A. Kandala, A. Mezzacapo, K. Temme, M. Takita, M. Brink, J. M. Chow, and J. M. Gambetta. Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. Nature, 549 (7671): 242–246, Sep 2017. 10.1038/​nature23879.

[42] A. Kandala, K. Temme, A. D. Córcoles, A. Mezzacapo, J. M. Chow, and J. M. Gambetta. Error mitigation extends the computational reach of a noisy quantum processor. Nature, 567 (7749): 491–495, Mar 2019. 10.1038/​s41586-019-1040-7.

[43] D. P. Kingma and J. Ba. Adam: A method for stochastic optimization, 2014. URL https:/​/​​abs/​1412.6980.

[44] 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 et al. Self-verifying variational quantum simulation of lattice models. Nature, 569 (7756): 355–360, May 2019. 10.1038/​s41586-019-1177-4.

[45] B. P. Lanyon, C. Hempel, D. Nigg, M. Muller, R. Gerritsma, F. Zahringer, P. Schindler, J. T. Barreiro, M. Rambach, G. Kirchmair, and et al. Universal digital quantum simulation with trapped ions. Science, 334 (6052): 57–61, Sep 2011. 10.1126/​science.1208001.

[46] Y. Li and S. C. Benjamin. Efficient variational quantum simulator incorporating active error minimization. Phys. Rev. X, 7: 021050, Jun 2017. 10.1103/​PhysRevX.7.021050.

[47] Z. Li, M.-H. Yung, H. Chen, D. Lu, J. D. Whitfield, X. Peng, A. Aspuru-Guzik, and J. Du. Solving quantum ground-state problems with nuclear magnetic resonance. Scientific Reports, 1 (1): 88, Sep 2011. 10.1038/​srep00088.

[48] S. Lloyd. Universal quantum simulators. Science, 273 (5278): 1073–1078, 1996. 10.1126/​science.273.5278.1073.

[49] C. Lyu. Accelerated variational algorithms for digital quantum simulation of the many-body ground states. https:/​/​​cfenglv/​PQIT, 2020.

[50] F. B. Maciejewski, Z. Zimborás, and M. Oszmaniec. Mitigation of readout noise in near-term quantum devices by classical post-processing based on detector tomography. Quantum, 4: 257, Apr. 2020. 10.22331/​q-2020-04-24-257.

[51] S. McArdle, T. Jones, S. Endo, Y. Li, S. C. Benjamin, and X. Yuan. Variational ansatz-based quantum simulation of imaginary time evolution. npj Quantum Information, 5 (1): 75, Sep 2019. 10.1038/​s41534-019-0187-2.

[52] J. R. McClean, J. Romero, R. Babbush, and A. Aspuru-Guzik. The theory of variational hybrid quantum-classical algorithms. New Journal of Physics, 18 (2): 023023, feb 2016. 10.1088/​1367-2630/​18/​2/​023023.

[53] J. R. McClean, M. E. Kimchi-Schwartz, J. Carter, and W. A. de Jong. Hybrid quantum-classical hierarchy for mitigation of decoherence and determination of excited states. Phys. Rev. A, 95: 042308, Apr 2017. 10.1103/​PhysRevA.95.042308.

[54] H.-J. Mikeska and A. K. Kolezhuk. One-dimensional magnetism. In U. Schollwöck, J. Richter, D. J. J. Farnell, and R. F. Bishop, editors, Quantum Magnetism, pages 1–83. Springer Berlin Heidelberg, 2004. 10.1007/​BFb0119591.

[55] N. Moll, P. Barkoutsos, L. S. Bishop, J. M. Chow, A. Cross, D. J. Egger, S. Filipp, A. Fuhrer, J. M. Gambetta, M. Ganzhorn, A. Kandala, A. Mezzacapo, P. Müller, W. Riess, G. Salis, J. Smolin, I. Tavernelli, and K. Temme. Quantum optimization using variational algorithms on near-term quantum devices. Quantum Science and Technology, 3 (3): 030503, jun 2018. 10.1088/​2058-9565/​aab822.

[56] M. Motta, C. Sun, A. T. K. Tan, M. J. O'Rourke, E. Ye, A. J. Minnich, F. G. S. L. Brandão, and G. K.-L. Chan. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics, 16 (2): 205–210, Feb 2020. 10.1038/​s41567-019-0704-4.

[57] G. Nenciu. On the adiabatic theorem of quantum mechanics. Journal of Physics A: Mathematical and General, 13 (2): L15–L18, feb 1980. 10.1088/​0305-4470/​13/​2/​002.

[58] M. A. Nielsen and I. L. Chuang. Quantum Computation and Quantum Information: 10th Anniversary Edition. Cambridge University Press, 2010. 10.1017/​CBO9780511976667.

[59] Y. Nomura, A. S. Darmawan, Y. Yamaji, and M. Imada. Restricted boltzmann machine learning for solving strongly correlated quantum systems. Phys. Rev. B, 96: 205152, Nov 2017. 10.1103/​PhysRevB.96.205152.

[60] P. O'Malley, R. Babbush, I. Kivlichan, J. Romero, J. McClean, R. Barends, J. Kelly, P. Roushan, A. Tranter, N. Ding, and et al. Scalable quantum simulation of molecular energies. Phys. Rev. X, 6 (3), Jul 2016. 10.1103/​physrevx.6.031007.

[61] A. Peruzzo, J. McClean, P. Shadbolt, M.-H. Yung, X.-Q. Zhou, P. J. Love, A. Aspuru-Guzik, and J. L. O'Brien. A variational eigenvalue solver on a photonic quantum processor. Nature Communications, 5 (1): 4213, Jul 2014. 10.1038/​ncomms5213.

[62] D. Petrosyan, G. M. Nikolopoulos, and P. Lambropoulos. State transfer in static and dynamic spin chains with disorder. Phys. Rev. A, 81: 042307, Apr 2010. 10.1103/​PhysRevA.81.042307.

[63] V. N. Premakumar and R. Joynt. Error mitigation in quantum computers subject to spatially correlated noise, 2018. URL https:/​/​​abs/​1812.07076.

[64] S. J. Reddi, S. Kale, and S. Kumar. On the convergence of adam and beyond, 2019. URL https:/​/​​abs/​1904.09237.

[65] J. Romero, R. Babbush, J. R. McClean, C. Hempel, P. J. Love, and A. Aspuru-Guzik. Strategies for quantum computing molecular energies using the unitary coupled cluster ansatz. Quantum Science and Technology, 4 (1): 014008, oct 2018. 10.1088/​2058-9565/​aad3e4.

[66] P. Roushan, C. Neill, J. Tangpanitanon, V. M. Bastidas, A. Megrant, R. Barends, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, and et al. Spectroscopic signatures of localization with interacting photons in superconducting qubits. Science, 358 (6367): 1175–1179, Nov 2017. 10.1126/​science.aao1401.

[67] S. Sachdev. Quantum Phase Transitions. American Cancer Society, 2007. 10.1002/​9780470022184.hmm108.

[68] Y. Salathé, M. Mondal, M. Oppliger, J. Heinsoo, P. Kurpiers, A. Potočnik, A. Mezzacapo, U. Las Heras, L. Lamata, E. Solano, S. Filipp, and A. Wallraff. Digital quantum simulation of spin models with circuit quantum electrodynamics. Phys. Rev. X, 5: 021027, Jun 2015. 10.1103/​PhysRevX.5.021027.

[69] J. Salfi, J. A. Mol, R. Rahman, G. Klimeck, M. Y. Simmons, L. C. L. Hollenberg, and S. Rogge. Quantum simulation of the hubbard model with dopant atoms in silicon. Nature Communications, 7 (1): 11342, Apr 2016a. 10.1038/​ncomms11342.

[70] J. Salfi, M. Tong, S. Rogge, and D. Culcer. Quantum computing with acceptor spins in silicon. Nanotechnology, 27 (24): 244001, May 2016b. 10.1088/​0957-4484/​27/​24/​244001.

[71] U. Schollwöck. The density-matrix renormalization group. Rev. Mod. Phys., 77: 259–315, Apr 2005. 10.1103/​RevModPhys.77.259.

[72] M. Schreiber, S. S. Hodgman, P. Bordia, H. P. Lüschen, M. H. Fischer, R. Vosk, E. Altman, U. Schneider, and I. Bloch. Observation of many-body localization of interacting fermions in a quasirandom optical lattice. Science, 349 (6250): 842–845, 2015. 10.1126/​science.aaa7432.

[73] Y. Shen, X. Zhang, S. Zhang, J.-N. Zhang, M.-H. Yung, and K. Kim. Quantum implementation of the unitary coupled cluster for simulating molecular electronic structure. Phys. Rev. A, 95: 020501, Feb 2017. 10.1103/​PhysRevA.95.020501.

[74] E. S. Sørensen, M.-S. Chang, N. Laflorencie, and I. Affleck. Impurity entanglement entropy and the kondo screening cloud. Journal of Statistical Mechanics: Theory and Experiment, 2007 (01): L01001–L01001, jan 2007a. 10.1088/​1742-5468/​2007/​01/​l01001.

[75] E. S. Sørensen, M.-S. Chang, N. Laflorencie, and I. Affleck. Quantum impurity entanglement. Journal of Statistical Mechanics: Theory and Experiment, 2007 (08): P08003–P08003, aug 2007b. 10.1088/​1742-5468/​2007/​08/​p08003.

[76] D. Suter and G. A. Álvarez. Colloquium: Protecting quantum information against environmental noise. Rev. Mod. Phys., 88: 041001, Oct 2016. 10.1103/​RevModPhys.88.041001.

[77] K. Temme, S. Bravyi, and J. M. Gambetta. Error mitigation for short-depth quantum circuits. Phys. Rev. Lett., 119: 180509, Nov 2017. 10.1103/​PhysRevLett.119.180509.

[78] F. Vatan and C. Williams. Optimal quantum circuits for general two-qubit gates. Phys. Rev. A, 69: 032315, Mar 2004. 10.1103/​PhysRevA.69.032315.

[79] J. Wang, F. Sciarrino, A. Laing, and M. G. Thompson. Integrated photonic quantum technologies. Nature Photonics, 14 (5): 273–284, Oct 2019. 10.1038/​s41566-019-0532-1.

[80] L. Wang. Discovering phase transitions with unsupervised learning. Phys. Rev. B, 94: 195105, Nov 2016. 10.1103/​PhysRevB.94.195105.

[81] Y. Wang, G. Li, and X. Wang. Variational quantum gibbs state preparation with a truncated taylor series, 2020. URL https:/​/​​abs/​2005.08797.

[82] D. Wecker, M. B. Hastings, and M. Troyer. Progress towards practical quantum variational algorithms. Phys. Rev. A, 92: 042303, Oct 2015. 10.1103/​PhysRevA.92.042303.

[83] J. D. Whitfield, J. Biamonte, and A. Aspuru-Guzik. Simulation of electronic structure hamiltonians using quantum computers. Molecular Physics, 109 (5): 735–750, Mar 2011. 10.1080/​00268976.2011.552441.

[84] K. Xu, J.-J. Chen, Y. Zeng, Y.-R. Zhang, C. Song, W. Liu, Q. Guo, P. Zhang, D. Xu, H. Deng, K. Huang, and et al. Emulating many-body localization with a superconducting quantum processor. Phys. Rev. Lett., 120: 050507, Feb 2018. 10.1103/​PhysRevLett.120.050507.

[85] Z. Yan, Y.-R. Zhang, M. Gong, Y. Wu, Y. Zheng, S. Li, C. Wang, F. Liang, J. Lin, Y. Xu, C. Guo, L. Sun, C.-Z. Peng, K. Xia, and et al. Strongly correlated quantum walks with a 12-qubit superconducting processor. Science, 364 (6442): 753–756, 2019. 10.1126/​science.aaw1611.

[86] S. Yang, A. Bayat, and S. Bose. Spin-state transfer in laterally coupled quantum-dot chains with disorders. Phys. Rev. A, 82: 022336, Aug 2010. 10.1103/​PhysRevA.82.022336.

[87] Y. Ye, Z.-Y. Ge, Y. Wu, S. Wang, M. Gong, Y.-R. Zhang, Q. Zhu, R. Yang, S. Li, F. Liang, J. Lin, Y. Xu, C. Guo, L. Sun, and et al. Propagation and localization of collective excitations on a 24-qubit superconducting processor. Phys. Rev. Lett., 123: 050502, Jul 2019. 10.1103/​PhysRevLett.123.050502.

[88] X. Yuan, S. Endo, Q. Zhao, Y. Li, and S. C. Benjamin. Theory of variational quantum simulation. Quantum, 3: 191, oct 2019. 10.22331/​q-2019-10-07-191.

[89] J. Zhang, G. Pagano, P. W. Hess, A. Kyprianidis, P. Becker, H. Kaplan, A. V. Gorshkov, Z.-X. Gong, and C. Monroe. Observation of a many-body dynamical phase transition with a 53-qubit quantum simulator. Nature, 551 (7682): 601–604, Nov 2017. 10.1038/​nature24654.

Cited by

On Crossref's cited-by service no data on citing works was found (last attempt 2020-09-22 16:11:48). On SAO/NASA ADS no data on citing works was found (last attempt 2020-09-22 16:11:49).