Variational quantum solver employing the PDS energy functional

Bo Peng and Karol Kowalski

Physical and Computational Science Division, Pacific Northwest National Laboratory, Richland, Washington 99354, United States of America

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Recently a new class of quantum algorithms that are based on the quantum computation of the connected moment expansion has been reported to find the ground and excited state energies. In particular, the Peeters-Devreese-Soldatov (PDS) formulation is found variational and bearing the potential for further combining with the existing variational quantum infrastructure. Here we find that the PDS formulation can be considered as a new energy functional of which the PDS energy gradient can be employed in a conventional variational quantum solver. In comparison with the usual variational quantum eigensolver (VQE) and the original static PDS approach, this new variational quantum solver offers an effective approach to navigate the dynamics to be free from getting trapped in the local minima that refer to different states, and achieve high accuracy at finding the ground state and its energy through the rotation of the trial wave function of modest quality, thus improves the accuracy and efficiency of the quantum simulation. We demonstrate the performance of the proposed variational quantum solver for toy models, H$_2$ molecule, and strongly correlated planar H$_4$ system in some challenging situations. In all the case studies, the proposed variational quantum approach outperforms the usual VQE and static PDS calculations even at the lowest order. We also discuss the limitations of the proposed approach and its preliminary execution for model Hamiltonian on the NISQ device.

In this paper, we developed a new effective cost function for the variational quantum solver, which in principle can not only help avoid some Barren Plateaus, but also converge to ground state energies outside of the ansatz class, and thus outperforms the conventional one in some challenging cases.

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[1] S. Amari. Natural gradient works efficiently in learning. Neural Computation, 10 (2): 251–276, 1998. https:/​/​​10.1162/​089976698300017746.

[2] 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. https:/​/​​10.1126/​science.abb9811.

[3] Ryan Babbush, Nathan Wiebe, Jarrod McClean, James McClain, Hartmut Neven, and Garnet Kin-Lic Chan. Low-depth quantum simulation of materials. Phys. Rev. X, 8: 011044, 2018. https:/​/​​10.1103/​PhysRevX.8.011044.

[4] Rodney J. Bartlett, Stanisław A. Kucharski, and Jozef Noga. Alternative coupled-cluster ansätze II. the unitary coupled-cluster method. Chem. Phys. Lett., 155 (1): 133–140, 1989. https:/​/​​10.1016/​s0009-2614(89)87372-5.

[5] Dominic W Berry, Graeme Ahokas, Richard Cleve, and Barry C Sanders. Efficient quantum algorithms for simulating sparse hamiltonians. Comm. Math. Phys., 270 (2): 359–371, 2007. https:/​/​​10.1007/​s00220-006-0150-x.

[6] Dominic W. Berry, Andrew M. Childs, Richard Cleve, Robin Kothari, and Rolando D. Somma. Simulating hamiltonian dynamics with a truncated taylor series. Phys. Rev. Lett., 114: 090502, 2015. https:/​/​​10.1103/​PhysRevLett.114.090502.

[7] Tatiana A. Bespalova and Oleksandr Kyriienko. Hamiltonian operator approximation for energy measurement and ground state preparation. preprint, arXiv:2009.03351, 2020. URL https:/​/​​abs/​2009.03351.

[8] N. Bogolubov. On the theory of superfluidity. J. Phys., 11: 23–32, 1947. https:/​/​​10.1142/​9789814612524_0001.

[9] Sergey Bravyi, Jay M. Gambetta, Antonio Mezzacapo, and Kristan Temme. Tapering off qubits to simulate fermionic hamiltonians. preprint, arXiv:1701.08213, 2017. URL https:/​/​​abs/​1701.08213.

[10] M. Cerezo, Akira Sone, Tyler Volkoff, Lukasz Cincio, and Patrick J. Coles. Cost function dependent barren plateaus in shallow parametrized quantum circuits. Nat. Commun., 12 (1), 2021. https:/​/​​10.1038/​s41467-021-21728-w.

[11] Marco Cerezo and Patrick J Coles. Impact of barren plateaus on the hessian and higher order derivatives. preprint, arXiv:2008.07454, 2020. URL https:/​/​​abs/​2008.07454. https:/​/​​10.1088/​2058-9565/​abf51a.

[12] Andrew M Childs. On the relationship between continuous-and discrete-time quantum walk. Comm. Math. Phys., 294 (2): 581–603, 2010. https:/​/​​10.1007/​s00220-009-0930-1.

[13] Andrew M. Childs and Nathan Wiebe. Hamiltonian simulation using linear combinations of unitary operations. Quantum Inf. Comput., 12 (11-12): 0901–0924, 2012. https:/​/​​10.26421/​qic12.11-12.

[14] Daniel Claudino, Bo Peng, Nicholas P. Bauman, Karol Kowalski, and Travis S. Humble. Improving the accuracy and efficiency of quantum connected moments expansions. Quantum Sci. Technol., accepted, 2021. https:/​/​​10.1088/​2058-9565/​ac0292.

[15] Richard Cleve, Artur Ekert, Chiara Macchiavello, and Michele Mosca. On quantum algorithms. Proc. R. Soc. Lond. A, 454 (1969): 339–354, 1998. https:/​/​​10.1002/​(SICI)1099-0526(199809/​10)4:1<33::AID-CPLX10>3.0.CO;2-U.

[16] 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, 2018. https:/​/​​10.1103/​PhysRevX.8.011021.

[17] Francesco A Evangelista, Garnet Kin-Lic Chan, and Gustavo E Scuseria. Exact parameterization of fermionic wave functions via unitary coupled cluster theory. J. Chem. Phys., 151 (24): 244112, 2019. https:/​/​​10.1063/​1.5133059.

[18] Vassilios Fessatidis, Jay D Mancini, Robert Murawski, and Samuel P Bowen. A generalized moments expansion. Phys. Lett. A, 349 (5): 320–323, 2006. https:/​/​​10.1016/​j.physleta.2005.09.039.

[19] Vassilios Fessatidis, Frank A Corvino, Jay D Mancini, Robert K Murawski, and John Mikalopas. Analytic properties of moments matrices. Phys. Lett. A, 374 (28): 2890–2893, 2010. https:/​/​​10.1016/​j.physleta.2010.05.010.

[20] R. P. Feynman. Slow electrons in a polar crystal. Phys. Rev., 97: 660–665, 1955. https:/​/​​10.1103/​PhysRev.97.660.

[21] András Gilyén, Yuan Su, Guang Hao Low, and Nathan Wiebe. Quantum singular value transformation and beyond: Exponential improvements for quantum matrix arithmetics. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, STOC 2019, page 193–204, New York, NY, USA, 2019. Association for Computing Machinery. ISBN 9781450367059. https:/​/​​10.1145/​3313276.3316366.

[22] Pranav Gokhale, Olivia Angiuli, Yongshan Ding, Kaiwen Gui, Teague Tomesh, Martin Suchara, Margaret Martonosi, and Frederic T. Chong. $o(n^3)$ measurement cost for variational quantum eigensolver on molecular hamiltonians. IEEE Trans. Qunatum Eng., 1: 1–24, 2020. https:/​/​​10.1109/​TQE.2020.3035814.

[23] Jérôme F. Gonthier, Maxwell D. Radin, Corneliu Buda, Eric J. Doskocil, Clena M. Abuan, and Jhonathan Romero. Identifying challenges towards practical quantum advantage through resource estimation: the measurement roadblock in the variational quantum eigensolver. preprint, arXiv:2012.04001, 2020. URL https:/​/​​abs/​2012.04001.

[24] Harper R Grimsley, Sophia E Economou, Edwin Barnes, and Nicholas J Mayhall. An adaptive variational algorithm for exact molecular simulations on a quantum computer. Nat. Commun., 10 (1): 1–9, 2019. https:/​/​​10.1038/​s41467-019-10988-2.

[25] Gian Giacomo Guerreschi and Mikhail Smelyanskiy. Practical optimization for hybrid quantum-classical algorithms. preprint, arXiv:1701.01450, 2017. URL https:/​/​​abs/​1701.01450.

[26] T. Häner, D. S. Steiger, M. Smelyanskiy, and M. Troyer. High performance emulation of quantum circuits. In SC '16: Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis, pages 866–874, Nov 2016. https:/​/​​10.1109/​SC.2016.73.

[27] Mark R Hoffmann and Jack Simons. A unitary multiconfigurational coupled-cluster method: Theory and applications. J. Chem. Phys., 88 (2): 993–1002, 1988. https:/​/​​10.1063/​1.454125.

[28] William J. Huggins, Jarrod R. McClean, Nicholas C. Rubin, Zhang Jiang, Nathan Wiebe, K. Birgitta Whaley, and Ryan Babbush. Efficient and noise resilient measurements for quantum chemistry on near-term quantum computers. npj Quantum Inf., 7 (1): 23, 2021. https:/​/​​10.1038/​s41534-020-00341-7.

[29] William James Huggins, Joonho Lee, Unpil Baek, Bryan O'Gorman, and K Birgitta Whaley. A non-orthogonal variational quantum eigensolver. New J. Phys., 22: 073009, 2020. https:/​/​​10.1088/​1367-2630/​ab867b.

[30] Artur F. Izmaylov, Tzu-Ching Yen, Robert A. Lang, and Vladyslav Verteletskyi. Unitary partitioning approach to the measurement problem in the variational quantum eigensolver method. J. Chem. Theory Comput., 16 (1): 190–195, 2020. https:/​/​​10.1021/​acs.jctc.9b00791.

[31] 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: 242–246, 2017. https:/​/​​10.1038/​nature23879.

[32] Abhinav Kandala, Kristan Temme, Antonio D Corcoles, Antonio Mezzacapo, Jerry M Chow, and Jay M Gambetta. Error mitigation extends the computational reach of a noisy quantum processor. Nature, 567: 491–495, 2019. https:/​/​​10.1038/​s41586-019-1040-7.

[33] Peter J Knowles. On the validity and applicability of the connected moments expansion. Chem. Phys. Lett., 134 (6): 512–518, 1987. https:/​/​​10.1016/​0009-2614(87)87184-1.

[34] Karol Kowalski and Bo Peng. Quantum simulations employing connected moments expansions. J. Chem. Phys., 153 (20): 201102, 2020. https:/​/​​10.1063/​5.0030688.

[35] Werner Kutzelnigg. Error analysis and improvements of coupled-cluster theory. Theor. Chim. Acta, 80 (4-5): 349–386, 1991. https:/​/​​10.1007/​BF01117418.

[36] Oleksandr Kyriienko. Quantum inverse iteration algorithm for programmable quantum simulators. npj Quantum Inf., 6 (1): 1–8, 2020. https:/​/​​10.1038/​s41534-019-0239-7.

[37] Joonho Lee, William J. Huggins, Martin Head-Gordon, and K. Birgitta Whaley. Generalized unitary coupled cluster wave functions for quantum computation. J. Chem. Theory Comput., 15 (1): 311–324, 2019. https:/​/​​10.1021/​acs.jctc.8b01004.

[38] A Luis and J Peřina. Optimum phase-shift estimation and the quantum description of the phase difference. Phys. Rev. A, 54 (5): 4564, 1996. https:/​/​​10.1103/​PhysRevA.54.4564.

[39] Jay D Mancini, Yu Zhou, and Peter F Meier. Analytic properties of connected moments expansions. Int. J. Quantum Chem., 50 (2): 101–107, 1994. https:/​/​​10.1002/​qua.560500203.

[40] Jay D Mancini, William J Massano, Janice D Prie, and Yu Zhuo. Avoidance of singularities in moments expansions: a numerical study. Phys. Lett. A, 209 (1-2): 107–112, 1995. https:/​/​​10.1016/​0375-9601(95)00757-2.

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

[42] Sam McArdle, Tyson Jones, Suguru Endo, Ying Li, Simon C Benjamin, and Xiao Yuan. Variational ansatz-based quantum simulation of imaginary time evolution. npj Quantum Inf., 5 (1): 1–6, 2019. https:/​/​​10.1038/​s41534-019-0187-2.

[43] Sam McArdle, Suguru Endo, Alan Aspuru-Guzik, Simon C Benjamin, and Xiao Yuan. Quantum computational chemistry. Rev. Mod. Phys., 92 (1): 015003, 2020. https:/​/​​10.1103/​RevModPhys.92.015003.

[44] Jarrod R McClean, Jonathan Romero, Ryan Babbush, and Alán Aspuru-Guzik. The theory of variational hybrid quantum-classical algorithms. New J. Phys., 18 (2): 023023, 2016. https:/​/​​10.1088/​1367-2630/​18/​2/​023023.

[45] Jarrod R McClean, Sergio Boixo, Vadim N Smelyanskiy, Ryan Babbush, and Hartmut Neven. Barren plateaus in quantum neural network training landscapes. Nat. Commun., 9 (1): 1–6, 2018. https:/​/​​10.1038/​s41467-018-07090-4.

[46] Mario Motta, Chong Sun, Adrian TK Tan, Matthew J O’Rourke, Erika Ye, Austin J Minnich, Fernando GSL Brandão, and Garnet Kin-Lic Chan. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nat. Phys., 16 (2): 205–210, 2020. https:/​/​​10.1038/​s41567-019-0704-4.

[47] Michael A. Nielsen and Isaac L. Chuang. Quantum Computation and Quantum Information: 10th Anniversary Edition. Cambridge University Press, New York, NY, USA, 10th edition, 2011. ISBN 1107002176, 9781107002173. https:/​/​​10.1017/​CBO9780511976667.

[48] Robert M Parrish and Peter L McMahon. Quantum filter diagonalization: Quantum eigendecomposition without full quantum phase estimation. preprint, arXiv:1909.08925, 2019. URL https:/​/​​abs/​1909.08925.

[49] François M Peeters and Jozef T Devreese. Upper bounds for the free energy. a generalisation of the bogolubov inequality and the feynman inequality. J. Phys. A: Math. Gen., 17 (3): 625, 1984. https:/​/​​10.1088/​0305-4470/​17/​3/​024.

[50] 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. Nat. Commun., 5: 4213, 2014. https:/​/​​10.1038/​ncomms5213.

[51] Arthur Pesah, M Cerezo, Samson Wang, Tyler Volkoff, Andrew T Sornborger, and Patrick J Coles. Absence of barren plateaus in quantum convolutional neural networks. preprint, arXiv:2011.02966, 2020. URL https:/​/​​abs/​2011.02966.

[52] David Poulin, Alexei Kitaev, Damian S. Steiger, Matthew B. Hastings, and Matthias Troyer. Quantum algorithm for spectral measurement with a lower gate count. Phys. Rev. Lett., 121: 010501, 2018. https:/​/​​10.1103/​PhysRevLett.121.010501.

[53] John Preskill. Quantum computing in the nisq era and beyond. Quantum, 2: 79, 2018. https:/​/​​10.22331/​q-2018-08-06-79.

[54] Janice D Prie, D Schwall, Jay D Mancini, D Kraus, and William J Massano. On the relation between the connected-moments expansion and the lanczos variational scheme. Nuov. Cim. D, 16 (5): 433–448, 1994. https:/​/​​10.1007/​BF02463732.

[55] Jonathan Romero, Ryan Babbush, Jarrod R McClean, Cornelius Hempel, Peter J Love, and Alán Aspuru-Guzik. Strategies for quantum computing molecular energies using the unitary coupled cluster ansatz. Quantum Sci. Technol., 4 (1): 014008, 2018. https:/​/​​10.1088/​2058-9565/​aad3e4.

[56] Nicholas C Rubin, Ryan Babbush, and McClean Jarrod. Application of fermionic marginal constraints to hybrid quantum algorithms. New J. Phys., 20: 053020, 2018. https:/​/​​10.1088/​1367-2630/​aab919.

[57] Maria Schuld, Ville Bergholm, Christian Gogolin, Josh Izaac, and Nathan Killoran. Evaluating analytic gradients on quantum hardware. Phys. Rev. A, 99: 032331, 2019. https:/​/​​10.1103/​PhysRevA.99.032331.

[58] Jacob T. Seeley, Martin J. Richard, and Peter J. Love. The bravyi-kitaev transformation for quantum computation of electronic structure. J. Chem. Phys., 137 (22): 224109, 2012. https:/​/​​10.1063/​1.4768229.

[59] Kazuhiro Seki and Seiji Yunoki. Quantum power method by a superposition of time-evolved states. PRX Quantum, 2: 010333, 2021. https:/​/​​10.1103/​PRXQuantum.2.010333.

[60] Yangchao Shen, Xiang Zhang, Shuaining Zhang, Jing-Ning Zhang, Man-Hong Yung, and Kihwan Kim. Quantum implementation of the unitary coupled cluster for simulating molecular electronic structure. Phys. Rev. A, 95: 020501, 2017. https:/​/​​10.1103/​PhysRevA.95.020501.

[61] Peter W Shor. Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM Rev., 41 (2): 303–332, 1999. https:/​/​​10.1137/​S0036144598347011.

[62] Andrey V Soldatov. Generalized variational principle in quantum mechanics. Int. J. Mod. Phys. B, 9 (22): 2899–2936, 1995. https:/​/​​10.1142/​S0217979295001087.

[63] James Stokes, Josh Izaac, Nathan Killoran, and Giuseppe Carleo. Quantum Natural Gradient. Quantum, 4: 269, 2020. https:/​/​​10.22331/​q-2020-05-25-269.

[64] Andrew G Taube and Rodney J Bartlett. New perspectives on unitary coupled-cluster theory. Int. J. Quantum Chem., 106 (15): 3393–3401, 2006. https:/​/​​10.1002/​qua.21198.

[65] Nazakat Ullah. Removal of the singularity in the moment-expansion formalism. Phys. Rev. A, 51 (3): 1808, 1995. https:/​/​​10.1103/​PhysRevA.51.1808.

[66] Alexey Uvarov and Jacob Biamonte. On barren plateaus and cost function locality in variational quantum algorithms. J. Phys. A: Math. and Theo., 54: 245301, 2021. https:/​/​​10.1088/​1751-8121/​abfac7.

[67] Harish J. Vallury, Michael A. Jones, Charles D. Hill, and Lloyd C. L. Hollenberg. Quantum computed moments correction to variational estimates. Quantum, 4: 373, 2020. https:/​/​​10.22331/​q-2020-12-15-373.

[68] Samson Wang, Enrico Fontana, Marco Cerezo, Kunal Sharma, Akira Sone, Lukasz Cincio, and Patrick J Coles. Noise-induced barren plateaus in variational quantum algorithms. preprint, arXiv:2007.14384, 2020. URL https:/​/​​abs/​2007.14384.

[69] Dave Wecker, Matthew B. Hastings, and Matthias Troyer. Progress towards practical quantum variational algorithms. Phys. Rev. A, 92: 042303, 2015. https:/​/​​10.1103/​PhysRevA.92.042303.

[70] David Wierichs, Christian Gogolin, and Michael Kastoryano. Avoiding local minima in variational quantum eigensolvers with the natural gradient optimizer. Phys. Rev. Research, 2: 043246, 2020. https:/​/​​10.1103/​PhysRevResearch.2.043246.

[71] Naoki Yamamoto. On the natural gradient for variational quantum eigensolver. preprint, arXiv:1909.05074, 2019. URL https:/​/​​abs/​1909.05074.

[72] Tzu-Ching Yen, Vladyslav Verteletskyi, and Artur F. Izmaylov. Measuring all compatible operators in one series of single-qubit measurements using unitary transformations. J. Chem. Theory Comput., 16 (4): 2400–2409, 2020. https:/​/​​10.1021/​acs.jctc.0c00008.

[73] Xiao Yuan, Suguru Endo, Qi Zhao, Ying Li, and Simon C. Benjamin. Theory of variational quantum simulation. Quantum, 3: 191, 2019. https:/​/​​10.22331/​q-2019-10-07-191.

Cited by

[1] Daniel Claudino, Bo Peng, Nicholas P. Bauman, Karol Kowalski, and Travis S. Humble, "Improving the accuracy and efficiency of quantum connected moments expansions", arXiv:2103.09124.

[2] Daniel Claudino, Alexander J. McCaskey, and Dmitry I. Lyakh, "A backend-agnostic, quantum-classical framework for simulations of chemistry in C++", arXiv:2105.01619.

[3] Dmitry A. Fedorov, Bo Peng, Niranjan Govind, and Yuri Alexeev, "VQE Method: A Short Survey and Recent Developments", arXiv:2103.08505.

[4] Edgar Andres Ruiz Guzman and Denis Lacroix, "Predicting ground state, excited states and long-time evolution of many-body systems from short-time evolution on a quantum computer", arXiv:2104.08181.

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