Fluid fermionic fragments for optimizing quantum measurements of electronic Hamiltonians in the variational quantum eigensolver

Seonghoon Choi, Ignacio Loaiza, and Artur F. Izmaylov

Department of Physical and Environmental Sciences, University of Toronto Scarborough, Toronto, Ontario M1C 1A4, Canada
Chemical Physics Theory Group, Department of Chemistry, University of Toronto, Toronto, Ontario M5S 3H6, Canada

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Measuring the expectation value of the molecular electronic Hamiltonian is one of the challenging parts of the variational quantum eigensolver. A widely used strategy is to express the Hamiltonian as a sum of measurable fragments using fermionic operator algebra. Such fragments have an advantage of conserving molecular symmetries that can be used for error mitigation. The number of measurements required to obtain the Hamiltonian expectation value is proportional to a sum of fragment variances. Here, we introduce a new method for lowering the fragments' variances by exploiting flexibility in the fragments' form. Due to idempotency of the occupation number operators, some parts of two-electron fragments can be turned into one-electron fragments, which then can be partially collected in a purely one-electron fragment. This repartitioning does not affect the expectation value of the Hamiltonian but has non-vanishing contributions to the variance of each fragment. The proposed method finds the optimal repartitioning by employing variances estimated using a classically efficient proxy for the quantum wavefunction. Numerical tests on several molecules show that repartitioning of one-electron terms lowers the number of measurements by more than an order of magnitude.

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[1] 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, 1–7 (2014).

[2] Jarrod R. McClean, Jonathan Romero, Ryan Babbush, and Alán Aspuru-Guzik. ``The theory of variational hybrid quantum-classical algorithms''. New J. Phys. 18, 023023 (2016).

[3] Ilya G. Ryabinkin, Robert A. Lang, Scott N. Genin, and Artur F. Izmaylov. ``Iterative qubit coupled cluster approach with efficient screening of generators''. J. Chem. Theory Comput. 16, 1055–1063 (2020).

[4] Marco 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''. Nat. Rev. Phys. 3, 625–644 (2021).

[5] Abhinav Anand, Philipp Schleich, Sumner Alperin-Lea, Phillip W. K. Jensen, Sukin Sim, Manuel Díaz-Tinoco, Jakob S. Kottmann, Matthias Degroote, Artur F. Izmaylov, and Alán Aspuru-Guzik. ``A quantum computing view on unitary coupled cluster theory''. Chem. Soc. Rev. 51, 1659–1684 (2022).

[6] John Preskill. ``Quantum computing in the NISQ era and beyond''. Quantum 2, 79 (2018).

[7] Dominic W Berry, Craig Gidney, Mario Motta, Jarrod R. McClean, and Ryan Babbush. ``Qubitization of arbitrary basis quantum chemistry leveraging sparsity and low rank factorization''. Quantum 3, 208 (2019).

[8] Mario Motta, Erika Ye, Jarrod R. McClean, Zhendong Li, Austin J. Minnich, Ryan Babbush, and Garnet Kin-Lic Chan. ``Low rank representations for quantum simulation of electronic structure''. npj Quantum Inf. 7, 1–7 (2021).

[9] 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, 1–9 (2021).

[10] Tzu-Ching Yen and Artur F. Izmaylov. ``Cartan subalgebra approach to efficient measurements of quantum observables''. PRX Quantum 2, 040320 (2021).

[11] Jeffrey Cohn, Mario Motta, and Robert M. Parrish. ``Quantum filter diagonalization with compressed double-factorized hamiltonians''. PRX Quantum 2, 040352 (2021).

[12] Sergey B. Bravyi and Alexei Yu. Kitaev. ``Fermionic quantum computation''. Ann. Phys. 298, 210–226 (2002).

[13] Jacob T. Seeley, Martin J. Richard, and Peter J. Love. ``The Bravyi-Kitaev transformation for quantum computation of electronic structure''. J. Chem. Phys. 137, 224109 (2012).

[14] Ian D. Kivlichan, Jarrod McClean, Nathan Wiebe, Craig Gidney, Alán Aspuru-Guzik, Garnet Kin-Lic Chan, and Ryan Babbush. ``Quantum simulation of electronic structure with linear depth and connectivity''. Phys. Rev. Lett. 120, 110501 (2018).

[15] Ophelia Crawford, Barnaby van Straaten, Daochen Wang, Thomas Parks, Earl Campbell, and Stephen Brierley. ``Efficient quantum measurement of Pauli operators in the presence of finite sampling error''. Quantum 5, 385 (2021).

[16] Jérôme F. Gonthier, Maxwell D. Radin, Corneliu Buda, Eric J. Doskocil, Clena M. Abuan, and Jhonathan Romero. ``Measurements as a roadblock to near-term practical quantum advantage in chemistry: Resource analysis''. Phys. Rev. Research 4, 033154 (2022).

[17] Andrew Jena, Scott Genin, and Michele Mosca. ``Pauli partitioning with respect to gate sets'' (2019). arXiv:1907.07859.

[18] Hsin-Yuan Huang, Richard Kueng, and John Preskill. ``Predicting many properties of a quantum system from very few measurements''. Nat. Phys. 16, 1050–1057 (2020).

[19] Charles Hadfield, Sergey Bravyi, Rudy Raymond, and Antonio Mezzacapo. ``Measurements of quantum hamiltonians with locally-biased classical shadows''. Commun. Math. Phys. 391, 951–967 (2022).

[20] Stefan Hillmich, Charles Hadfield, Rudy Raymond, Antonio Mezzacapo, and Robert Wille. ``Decision diagrams for quantum measurements with shallow circuits''. In 2021 IEEE International Conference on Quantum Computing and Engineering (QCE). Pages 24–34. (2021).

[21] Hsin-Yuan Huang, Richard Kueng, and John Preskill. ``Efficient estimation of Pauli observables by derandomization''. Phys. Rev. Lett. 127, 030503 (2021).

[22] Bujiao Wu, Jinzhao Sun, Qi Huang, and Xiao Yuan. ``Overlapped grouping measurement: A unified framework for measuring quantum states'' (2021). arXiv:2105.13091.

[23] Charles Hadfield. ``Adaptive Pauli shadows for energy estimation'' (2021). arXiv:2105.12207.

[24] 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, 2400–2409 (2020).

[25] Tzu-Ching Yen, Aadithya Ganeshram, and Artur F Izmaylov. ``Deterministic improvements of quantum measurements with grouping of compatible operators, non-local transformations, and covariance estimates'' (2022). arXiv:2201.01471.

[26] Seonghoon Choi, Tzu-Ching Yen, and Artur F. Izmaylov. ``Improving quantum measurements by introducing “ghost” pauli products''. J. Chem. Theory Comput. 18, 7394–7402 (2022).

[27] Scott Aaronson and Daniel Gottesman. ``Improved simulation of stabilizer circuits''. Phys. Rev. A 70, 052328 (2004).

[28] Zachary Pierce Bansingh, Tzu-Ching Yen, Peter D. Johnson, and Artur F. Izmaylov. ``Fidelity overhead for nonlocal measurements in variational quantum algorithms''. J. Phys. Chem. A 126, 7007–7012 (2022).

[29] Guillermo García-Pérez, Matteo A.C. Rossi, Boris Sokolov, Francesco Tacchino, Panagiotis Kl. Barkoutsos, Guglielmo Mazzola, Ivano Tavernelli, and Sabrina Maniscalco. ``Learning to measure: Adaptive informationally complete generalized measurements for quantum algorithms''. PRX Quantum 2, 040342 (2021).

[30] Laurin E. Fischer, Daniel Miller, Francesco Tacchino, Panagiotis Kl. Barkoutsos, Daniel J. Egger, and Ivano Tavernelli. ``Ancilla-free implementation of generalized measurements for qubits embedded in a qudit space'' (2022). arXiv:2203.07369.

[31] Adam Glos, Anton Nykänen, Elsi-Mari Borrelli, Sabrina Maniscalco, Matteo A. C. Rossi, Zoltán Zimborás, and Guillermo García-Pérez. ``Adaptive POVM implementations and measurement error mitigation strategies for near-term quantum devices'' (2022). arXiv:2208.07817.

[32] Ilya G. Ryabinkin, Scott N. Genin, and Artur F. Izmaylov. ``Constrained variational quantum eigensolver: Quantum computer search engine in the fock space''. J. Chem. Theory Comput. 15, 249–255 (2019).

[33] 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, 1084–1089 (2020).

[34] Ignacio Loaiza, Alireza Marefat Khah, Nathan Wiebe, and Artur F. Izmaylov. ``Reducing molecular electronic hamiltonian simulation cost for linear combination of unitaries approaches'' (2022). arXiv:2208.08272.

[35] Vera von Burg, Guang Hao Low, Thomas Häner, Damian S. Steiger, Markus Reiher, Martin Roetteler, and Matthias Troyer. ``Quantum computing enhanced computational catalysis''. Phys. Rev. Research 3, 033055 (2021).

[36] Joonho Lee, Dominic W. Berry, Craig Gidney, William J. Huggins, Jarrod R. McClean, Nathan Wiebe, and Ryan Babbush. ``Even more efficient quantum computations of chemistry through tensor hypercontraction''. PRX Quantum 2, 030305 (2021).

[37] Seonghoon Choi, Ignacio Loaiza, and Artur F. Izmaylov. ``Data for: Fluid fermionic fragments for optimizing quantum measurements of electronic Hamiltonians in the variational quantum eigensolver''. url: doi.org/​10.5281/​zenodo.7335451.

[38] X. Bonet-Monroig, R. Sagastizabal, M. Singh, and T. E. O'Brien. ``Low-cost error mitigation by symmetry verification''. Phys. Rev. A 98, 062339 (2018).

[39] Suguru Endo, Zhenyu Cai, Simon C. Benjamin, and Xiao Yuan. ``Hybrid quantum-classical algorithms and quantum error mitigation''. J. Phys. Soc. Japan 90, 032001 (2021).

[40] Zhenyu Cai, Ryan Babbush, Simon C. Benjamin, Suguru Endo, William J. Huggins, Ying Li, Jarrod R. McClean, and Thomas E. O'Brien. ``Quantum error mitigation'' (2022). arXiv:2210.00921.

[41] Jarrod R. McClean, Mollie E. Kimchi-Schwartz, Jonathan Carter, and Wibe A. de Jong. ``Hybrid quantum-classical hierarchy for mitigation of decoherence and determination of excited states''. Phys. Rev. A 95, 042308 (2017).

[42] William J Huggins, Joonho Lee, Unpil Baek, Bryan O’Gorman, and K Birgitta Whaley. ``A non-orthogonal variational quantum eigensolver''. New J. Phys. 22, 073009 (2020).

[43] Nicholas H. Stair, Renke Huang, and Francesco A. Evangelista. ``A multireference quantum krylov algorithm for strongly correlated electrons''. J. Chem. Theory Comput. 16, 2236–2245 (2020).

[44] Xavier Bonet-Monroig, Ryan Babbush, and Thomas E. O'Brien. ``Nearly optimal measurement scheduling for partial tomography of quantum states''. Phys. Rev. X 10, 031064 (2020).

[45] Andrew Zhao, Nicholas C. Rubin, and Akimasa Miyake. ``Fermionic partial tomography via classical shadows''. Phys. Rev. Lett. 127, 110504 (2021).

[46] Alán Aspuru-Guzik, Anthony D. Dutoi, Peter J. Love, and Martin Head-Gordon. ``Simulated quantum computation of molecular energies''. Science 309, 1704–1707 (2005).

[47] Seth Lloyd. ``Universal quantum simulators''. Science 273, 1073–1078 (1996).

[48] Luis A. Martínez-Martínez, Tzu-Ching Yen, and Artur F. Izmaylov. ``Assessment of various hamiltonian partitionings for the electronic structure problem on a quantum computer using the trotter approximation'' (2022). arXiv:2210.10189.

[49] Masuo Suzuki. ``Fractal decomposition of exponential operators with applications to many-body theories and monte carlo simulations''. Phys. Lett. A 146, 319–323 (1990).

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