Efficient quantum measurement of Pauli operators in the presence of finite sampling error

Ophelia Crawford1, Barnaby van Straaten1, Daochen Wang1,2, Thomas Parks1, Earl Campbell1,3, and Stephen Brierley1

1Riverlane, Cambridge, UK
2Joint Center for Quantum Information and Computer Science, University of Maryland, College Park, USA
3Department of Physics and Astronomy, University of Sheffield, Sheffield, UK

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Estimating the expectation value of an operator corresponding to an observable is a fundamental task in quantum computation. It is often impossible to obtain such estimates directly, as the computer is restricted to measuring in a fixed computational basis. One common solution splits the operator into a weighted sum of Pauli operators and measures each separately, at the cost of many measurements. An improved version collects mutually commuting Pauli operators together before measuring all operators within a collection simultaneously. The effectiveness of doing this depends on two factors. Firstly, we must understand the improvement offered by a given arrangement of Paulis in collections. In our work, we propose two natural metrics for quantifying this, operating under the assumption that measurements are distributed optimally among collections so as to minimise the overall finite sampling error. Motivated by the mathematical form of these metrics, we introduce $\large{S}$ORTED $\large{I}$NSERTION, a collecting strategy that exploits the weighting of each Pauli operator in the overall sum. Secondly, to measure all Pauli operators within a collection simultaneously, a circuit is required to rotate them to the computational basis. In our work, we present two efficient circuit constructions that suitably rotate any collection of $\boldsymbol{k}$ independent commuting $\boldsymbol{n}$-qubit Pauli operators using at most $\boldsymbol{kn-k(k+1)/2}$ and $\boldsymbol{O(kn/\log k)}$ two-qubit gates respectively. Our methods are numerically illustrated in the context of the Variational Quantum Eigensolver, where the operators in question are molecular Hamiltonians. As measured by our metrics, $\large{S}$ORTED $\large{I}$NSERTION outperforms four conventional greedy colouring algorithms that seek the minimum number of collections.

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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. Nature Communications, 5 (1): 4213, 2014. 10.1038/​ncomms5213.

[2] John Preskill. Quantum computing in the NISQ era and beyond. Quantum, 2: 79, August 2018. ISSN 2521-327X. 10.22331/​q-2018-08-06-79.

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

[4] P. J. J. O'Malley, R. Babbush, I. D. Kivlichan, J. Romero, J. R. McClean, R. Barends, J. Kelly, P. Roushan, A. Tranter, N. Ding, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, A. G. Fowler, E. Jeffrey, E. Lucero, A. Megrant, J. Y. Mutus, M. Neeley, C. Neill, C. Quintana, D. Sank, A. Vainsencher, J. Wenner, T. C. White, P. V. Coveney, P. J. Love, H. Neven, A. Aspuru-Guzik, and J. M. Martinis. Scalable quantum simulation of molecular energies. Phys. Rev. X, 6: 031007, July 2016. 10.1103/​PhysRevX.6.031007.

[5] 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 (7671): 242–246, 2017. 10.1038/​nature23879.

[6] Sam McArdle, Suguru Endo, Alán Aspuru-Guzik, Simon C. Benjamin, and Xiao Yuan. Quantum computational chemistry. Rev. Mod. Phys., 92: 015003, Mar 2020. 10.1103/​RevModPhys.92.015003.

[7] Ilya G. Ryabinkin, Scott N. Genin, and Artur F. Izmaylov. Constrained variational quantum eigensolver: Quantum computer search engine in the Fock space. Journal of Chemical Theory and Computation, 15 (1): 249–255, 01 2019. 10.1021/​acs.jctc.8b00943.

[8] 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 Science and Technology, 4 (1): 014008, Oct 2018. 10.1088/​2058-9565/​aad3e4.

[9] Daochen Wang, Oscar Higgott, and Stephen Brierley. Accelerated variational quantum eigensolver. Phys. Rev. Lett., 122: 140504, Apr 2019. 10.1103/​PhysRevLett.122.140504.

[10] 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, Apr 2017. 10.1103/​PhysRevA.95.042308.

[11] Raffaele Santagati, Jianwei Wang, Antonio A. Gentile, Stefano Paesani, Nathan Wiebe, Jarrod R. McClean, Sam Morley-Short, Peter J. Shadbolt, Damien Bonneau, Joshua W. Silverstone, David P. Tew, Xiaoqi Zhou, Jeremy L. O’Brien, and Mark G. Thompson. Witnessing eigenstates for quantum simulation of Hamiltonian spectra. Science Advances, 4 (1), 2018. 10.1126/​sciadv.aap9646.

[12] 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.

[13] Kentaro Heya, Ken M Nakanishi, Kosuke Mitarai, and Keisuke Fujii. Subspace variational quantum simulator. arXiv e-prints, Apr 2019. https:/​/​arxiv.org/​abs/​1904.08566.

[14] Tyson Jones, Suguru Endo, Sam McArdle, Xiao Yuan, and Simon C. Benjamin. Variational quantum algorithms for discovering Hamiltonian spectra. Phys. Rev. A, 99: 062304, Jun 2019. 10.1103/​PhysRevA.99.062304.

[15] Oscar Higgott, Daochen Wang, and Stephen Brierley. Variational quantum computation of excited states. Quantum, 3: 156, July 2019. ISSN 2521-327X. 10.22331/​q-2019-07-01-156.

[16] Vladyslav Verteletskyi, Tzu-Ching Yen, and Artur F. Izmaylov. Measurement optimization in the variational quantum eigensolver using a minimum clique cover. The Journal of Chemical Physics, 152 (12): 124114, 2020. 10.1063/​1.5141458.

[17] Andrew Jena, Scott Genin, and Michele Mosca. Pauli partitioning with respect to gate sets. arXiv e-prints, July 2019. https:/​/​arxiv.org/​abs/​1907.07859.

[18] Tzu-Ching Yen, Vladyslav Verteletskyi, and Artur F. Izmaylov. Measuring all compatible operators in one series of single-qubit measurements using unitary transformations. Journal of Chemical Theory and Computation, 16 (4): 2400–2409, 04 2020. 10.1021/​acs.jctc.0c00008.

[19] William J. Huggins, Jarrod McClean, Nicholas Rubin, Zhang Jiang, Nathan Wiebe, K. Birgitta Whaley, and Ryan Babbush. Efficient and noise resilient measurements for quantum chemistry on near-term quantum computers. arXiv e-prints, July 2019. https:/​/​arxiv.org/​abs/​1907.13117.

[20] Pranav Gokhale, Olivia Angiuli, Yongshan Ding, Kaiwen Gui, Teague Tomesh, Martin Suchara, Margaret Martonosi, and Frederic T. Chong. Minimizing state preparations in variational quantum eigensolver by partitioning into commuting families. arXiv e-prints, July 2019. https:/​/​arxiv.org/​abs/​1907.13623.

[21] Andrew Zhao, Andrew Tranter, William M. Kirby, Shu Fay Ung, Akimasa Miyake, and Peter J. Love. Measurement reduction in variational quantum algorithms. Phys. Rev. A, 101: 062322, Jun 2020. 10.1103/​PhysRevA.101.062322.

[22] P. Gokhale, O. Angiuli, Y. Ding, K. Gui, T. Tomesh, M. Suchara, M. Martonosi, and F. T. Chong. $O(N^3)$ measurement cost for variational quantum eigensolver on molecular Hamiltonians. IEEE Transactions on Quantum Engineering, 1: 1–24, 2020. 10.1109/​TQE.2020.3035814.

[23] 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. Journal of Chemical Theory and Computation, 16 (1): 190–195, 01 2020. 10.1021/​acs.jctc.9b00791.

[24] Artur F. Izmaylov, Tzu-Ching Yen, and Ilya G. Ryabinkin. Revising the measurement process in the variational quantum eigensolver: is it possible to reduce the number of separately measured operators? Chem. Sci., 10: 3746–3755, 2019. 10.1039/​C8SC05592K.

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

[26] Nicholas C Rubin, Ryan Babbush, and Jarrod McClean. Application of fermionic marginal constraints to hybrid quantum algorithms. New Journal of Physics, 20 (5): 053020, May 2018. 10.1088/​1367-2630/​aab919.

[27] Alexandre Blais, Jay Gambetta, A. Wallraff, D. I. Schuster, S. M. Girvin, M. H. Devoret, and R. J. Schoelkopf. Quantum-information processing with circuit quantum electrodynamics. Phys. Rev. A, 75: 032329, Mar 2007. 10.1103/​PhysRevA.75.032329.

[28] Anthony Laing, Alberto Peruzzo, Alberto Politi, Maria Rodas Verde, Matthaeus Halder, Timothy C. Ralph, Mark G. Thompson, and Jeremy L. O'Brien. High-fidelity operation of quantum photonic circuits. Applied Physics Letters, 97 (21): 211109, 2010. 10.1063/​1.3497087.

[29] R. Barends, J. Kelly, A. Megrant, A. Veitia, D. Sank, E. Jeffrey, T. C. White, J. Mutus, A. G. Fowler, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, C. Neill, P. O'Malley, P. Roushan, A. Vainsencher, J. Wenner, A. N. Korotkov, A. N. Cleland, and John M. Martinis. Superconducting quantum circuits at the surface code threshold for fault tolerance. Nature, 508 (7497): 500–503, 2014. 10.1038/​nature13171.

[30] C. J. Ballance, T. P. Harty, N. M. Linke, M. A. Sepiol, and D. M. Lucas. High-fidelity quantum logic gates using trapped-ion hyperfine qubits. Phys. Rev. Lett., 117: 060504, Aug 2016. 10.1103/​PhysRevLett.117.060504.

[31] Joseph L. Allen, Robert Kosut, Jaewoo Joo, Peter Leek, and Eran Ginossar. Optimal control of two qubits via a single cavity drive in circuit quantum electrodynamics. Phys. Rev. A, 95: 042325, Apr 2017. 10.1103/​PhysRevA.95.042325.

[32] Norbert M. Linke, Dmitri Maslov, Martin Roetteler, Shantanu Debnath, Caroline Figgatt, Kevin A. Landsman, Kenneth Wright, and Christopher Monroe. Experimental comparison of two quantum computing architectures. Proceedings of the National Academy of Sciences, 114 (13): 3305–3310, 2017. ISSN 0027-8424. 10.1073/​pnas.1618020114.

[33] G Wendin. Quantum information processing with superconducting circuits: a review. Reports on Progress in Physics, 80 (10): 106001, sep 2017. 10.1088/​1361-6633/​aa7e1a.

[34] Matthew Reagor, Christopher B. Osborn, Nikolas Tezak, Alexa Staley, Guenevere Prawiroatmodjo, Michael Scheer, Nasser Alidoust, Eyob A. Sete, Nicolas Didier, Marcus P. da Silva, Ezer Acala, Joel Angeles, Andrew Bestwick, Maxwell Block, Benjamin Bloom, Adam Bradley, Catvu Bui, Shane Caldwell, Lauren Capelluto, Rick Chilcott, Jeff Cordova, Genya Crossman, Michael Curtis, Saniya Deshpande, Tristan El Bouayadi, Daniel Girshovich, Sabrina Hong, Alex Hudson, Peter Karalekas, Kat Kuang, Michael Lenihan, Riccardo Manenti, Thomas Manning, Jayss Marshall, Yuvraj Mohan, William O’Brien, Johannes Otterbach, Alexander Papageorge, Jean-Philip Paquette, Michael Pelstring, Anthony Polloreno, Vijay Rawat, Colm A. Ryan, Russ Renzas, Nick Rubin, Damon Russel, Michael Rust, Diego Scarabelli, Michael Selvanayagam, Rodney Sinclair, Robert Smith, Mark Suska, Ting-Wai To, Mehrnoosh Vahidpour, Nagesh Vodrahalli, Tyler Whyland, Kamal Yadav, William Zeng, and Chad T. Rigetti. Demonstration of universal parametric entangling gates on a multi-qubit lattice. Science Advances, 4 (2), 2018. 10.1126/​sciadv.aao3603.

[35] V. M. Schäfer, C. J. Ballance, K. Thirumalai, L. J. Stephenson, T. G. Ballance, A. M. Steane, and D. M. Lucas. Fast quantum logic gates with trapped-ion qubits. Nature, 555 (7694): 75–78, 2018. 10.1038/​nature25737.

[36] A. E. Webb, S. C. Webster, S. Collingbourne, D. Bretaud, A. M. Lawrence, S. Weidt, F. Mintert, and W. K. Hensinger. Resilient entangling gates for trapped ions. Phys. Rev. Lett., 121: 180501, Nov 2018. 10.1103/​PhysRevLett.121.180501.

[37] Harry Levine, Alexander Keesling, Ahmed Omran, Hannes Bernien, Sylvain Schwartz, Alexander S. Zibrov, Manuel Endres, Markus Greiner, Vladan Vuletić, and Mikhail D. Lukin. High-fidelity control and entanglement of Rydberg-atom qubits. Phys. Rev. Lett., 121: 123603, Sep 2018. 10.1103/​PhysRevLett.121.123603.

[38] Y. He, S. K. Gorman, D. Keith, L. Kranz, J. G. Keizer, and M. Y. Simmons. A two-qubit gate between phosphorus donor electrons in silicon. Nature, 571 (7765): 371–375, 2019a. 10.1038/​s41586-019-1381-2.

[39] W. Huang, C. H. Yang, K. W. Chan, T. Tanttu, B. Hensen, R. C. C. Leon, M. A. Fogarty, J. C. C. Hwang, F. E. Hudson, K. M. Itoh, A. Morello, A. Laucht, and A. S. Dzurak. Fidelity benchmarks for two-qubit gates in silicon. Nature, 569 (7757): 532–536, 2019. 10.1038/​s41586-019-1197-0.

[40] Reinhold Blumel, Nikodem Grzesiak, and Yunseong Nam. Power-optimal, stabilized entangling gate between trapped-ion qubits. arXiv e-prints, May 2019. https:/​/​arxiv.org/​abs/​1905.09292.

[41] Y. He, S. K. Gorman, D. Keith, L. Kranz, J. G. Keizer, and M. Y. Simmons. A two-qubit gate between phosphorus donor electrons in silicon. Nature, 571 (7765): 371–375, 2019b. 10.1038/​s41586-019-1381-2.

[42] Maarten Van den Nest, Jeroen Dehaene, and Bart De Moor. Graphical description of the action of local Clifford transformations on graph states. Phys. Rev. A, 69: 022316, Feb 2004. 10.1103/​PhysRevA.69.022316.

[43] Scott Aaronson and Daniel Gottesman. Improved simulation of stabilizer circuits. Phys. Rev. A, 70: 052328, Nov 2004. 10.1103/​PhysRevA.70.052328.

[44] Ketan N. Patel, Igor L. Markov, and John P. Hayes. Optimal synthesis of linear reversible circuits. Quantum Info. Comput., 8 (3): 282–294, March 2008. ISSN 1533-7146. 10.26421/​QIC8.3-4.

[45] Hsin-Yuan Huang, Richard Kueng, and John Preskill. Predicting many properties of a quantum system from very few measurements. Nature Physics, 16 (10): 1050–1057, Jun 2020. ISSN 1745-2481. 10.1038/​s41567-020-0932-7.

[46] Charles Hadfield, Sergey Bravyi, Rudy Raymond, and Antonio Mezzacapo. Measurements of quantum Hamiltonians with locally-biased classical shadows. arXiv e-prints, June 2020. https:/​/​arxiv.org/​abs/​2006.15788.

[47] Jonas M. Kübler, Andrew Arrasmith, Lukasz Cincio, and Patrick J. Coles. An adaptive optimizer for measurement-frugal variational algorithms. Quantum, 4: 263, May 2020. ISSN 2521-327X. 10.22331/​q-2020-05-11-263.

[48] Andrew Arrasmith, Lukasz Cincio, Rolando D. Somma, and Patrick J. Coles. Operator sampling for shot-frugal optimization in variational algorithms. arXiv e-prints, April 2020. https:/​/​arxiv.org/​abs/​2004.06252.

[49] J.B. Conway. A Course in Functional Analysis. Graduate Texts in Mathematics. Springer New York, 1994. ISBN 9780387972459.

[50] John Watrous. The Theory of Quantum Information. Cambridge University Press, 2018.

[51] Adrian Kosowski and Krzysztof Manuszewski. Classical coloring of graphs. In Marek Kubale, editor, Graph Colorings, chapter 1. American Mathematical Society, 2004. ISBN 978-0-8218-7942-9. 10.1090/​conm/​352.

[52] David Zuckerman. Linear degree extractors and the inapproximability of max clique and chromatic number. In Proceedings of the Thirty-Eighth Annual ACM Symposium on Theory of Computing (STOC), pages 681–690, New York, NY, USA, 2006. Association for Computing Machinery. ISBN 1595931341. 10.1145/​1132516.1132612.

[53] A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. A. Sloane. Quantum error correction and orthogonal geometry. Phys. Rev. Lett., 78: 405–408, Jan 1997. 10.1103/​PhysRevLett.78.405.

[54] Daniel Gottesman. Stabilizer codes and quantum error correction. PhD thesis, California Institute of Technology, Jan 1997.

[55] Michael A. Nielsen and Isaac L. Chuang. Quantum Computation and Quantum Information: 10th Anniversary Edition. Cambridge University Press, New York, NY, USA, 2011. ISBN 978-1107002173.

[56] M. Hein, J. Eisert, and H. J. Briegel. Multiparty entanglement in graph states. Phys. Rev. A, 69: 062311, Jun 2004. 10.1103/​PhysRevA.69.062311.

[57] André Bouchet. Recognizing locally equivalent graphs. Discrete Mathematics, 114 (1): 75 – 86, 1993. ISSN 0012-365X. 10.1016/​0012-365X(93)90357-Y.

[58] Jiaqing Jiang, Xiaoming Sun, Shang-Hua Teng, Bujiao Wu, Kewen Wu, and Jialin Zhang. Optimal space-depth trade-off of CNOT circuits in quantum logic synthesis. In Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 213–229, 2020. 10.1137/​1.9781611975994.13.

[59] Cristopher Moore and Martin Nilsson. Parallel quantum computation and quantum codes. SIAM J. Comput., 31 (3): 799–815, March 2002. ISSN 0097-5397. 10.1137/​S0097539799355053.

[60] Eric Dennis, Alexei Kitaev, Andrew Landahl, and John Preskill. Topological quantum memory. Journal of Mathematical Physics, 43 (9): 4452–4505, 2002. 10.1063/​1.1499754.

[61] Jarrod R McClean, Nicholas C Rubin, Kevin J Sung, Ian D Kivlichan, Xavier Bonet-Monroig, Yudong Cao, Chengyu Dai, E Schuyler Fried, Craig Gidney, Brendan Gimby, Pranav Gokhale, Thomas Häner, Tarini Hardikar, Vojtěch Havlíček, Oscar Higgott, Cupjin Huang, Josh Izaac, Zhang Jiang, Xinle Liu, Sam McArdle, Matthew Neeley, Thomas O'Brien, Bryan O'Gorman, Isil Ozfidan, Maxwell D Radin, Jhonathan Romero, Nicolas P D Sawaya, Bruno Senjean, Kanav Setia, Sukin Sim, Damian S Steiger, Mark Steudtner, Qiming Sun, Wei Sun, Daochen Wang, Fang Zhang, and Ryan Babbush. OpenFermion: the electronic structure package for quantum computers. Quantum Science and Technology, 5 (3): 034014, June 2020. 10.1088/​2058-9565/​ab8ebc.

[62] Sergey B. Bravyi and Alexei Yu. Kitaev. Fermionic quantum computation. Annals of Physics, 298 (1): 210 – 226, 2002. ISSN 0003-4916. 10.1006/​aphy.2002.6254.

[63] Sergey Bravyi, Jay M. Gambetta, Antonio Mezzacapo, and Kristan Temme. Tapering off qubits to simulate fermionic Hamiltonians. arXiv e-prints, Jan 2017. https:/​/​arxiv.org/​abs/​1701.08213.

[64] Aric A. Hagberg, Daniel A. Schult, and Pieter J. Swart. Exploring network structure, dynamics, and function using NetworkX. In Gaël Varoquaux, Travis Vaught, and Jarrod Millman, editors, Proceedings of the 7th Python in Science Conference, pages 11 – 15, 2008.

[65] V.L. Arlazarov, E.A. Dinic, M.A. Kronod, and I.A. Faradez. On economical construction of the transitive closure of an oriented graph. Soviet Mathematics Doklady, pages 1209–10, 1970.

[66] Robert M. Parrish, Lori A. Burns, Daniel G. A. Smith, Andrew C. Simmonett, A. Eugene DePrince, Edward G. Hohenstein, Uğur Bozkaya, Alexander Yu. Sokolov, Roberto Di Remigio, Ryan M. Richard, Jérôme F. Gonthier, Andrew M. James, Harley R. McAlexander, Ashutosh Kumar, Masaaki Saitow, Xiao Wang, Benjamin P. Pritchard, Prakash Verma, Henry F. Schaefer, Konrad Patkowski, Rollin A. King, Edward F. Valeev, Francesco A. Evangelista, Justin M. Turney, T. Daniel Crawford, and C. David Sherrill. Psi4 1.1: An open-source electronic structure program emphasizing automation, advanced libraries, and interoperability. Journal of Chemical Theory and Computation, 13 (7): 3185–3197, 07 2017. 10.1021/​acs.jctc.7b00174.

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[1] Guoming Wang, Dax Enshan Koh, Peter D. Johnson, and Yudong Cao, "Minimizing Estimation Runtime on Noisy Quantum Computers", PRX Quantum 2 1, 010346 (2021).

[2] Jie Liu, Zhenyu Li, and Jinlong Yang, "An efficient adaptive variational quantum solver of the Schrödinger equation based on reduced density matrices", The Journal of Chemical Physics 154 24, 244112 (2021).

[3] Johannes Borregaard, Matthias Christandl, and Daniel Stilck França, "Noise-robust exploration of many-body quantum states on near-term quantum devices", arXiv:1909.04786, npj Quantum Information 7 1, 45 (2021).

[4] Andrew Zhao, Nicholas C. Rubin, and Akimasa Miyake, "Fermionic Partial Tomography via Classical Shadows", Physical Review Letters 127 11, 110504 (2021).

[5] Yong-Xin Yao, Niladri Gomes, Feng Zhang, Cai-Zhuang Wang, Kai-Ming Ho, Thomas Iadecola, and Peter P. Orth, "Adaptive Variational Quantum Dynamics Simulations", PRX Quantum 2 3, 030307 (2021).

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[8] Igor O. Sokolov, Panagiotis Kl. Barkoutsos, Lukas Moeller, Philippe Suchsland, Guglielmo Mazzola, and Ivano Tavernelli, "Microcanonical and finite-temperature ab initio molecular dynamics simulations on quantum computers", Physical Review Research 3 1, 013125 (2021).

[9] Hsin-Yuan Huang, Richard Kueng, and John Preskill, "Efficient Estimation of Pauli Observables by Derandomization", Physical Review Letters 127 3, 030503 (2021).

[10] Bálint Koczor, "Exponential Error Suppression for Near-Term Quantum Devices", Physical Review X 11 3, 031057 (2021).

[11] Tatiana A. Bespalova and Oleksandr Kyriienko, "Hamiltonian Operator Approximation for Energy Measurement and Ground-State Preparation", PRX Quantum 2 3, 030318 (2021).

[12] Sam McArdle, Suguru Endo, Alán Aspuru-Guzik, Simon C. Benjamin, and Xiao Yuan, "Quantum computational chemistry", arXiv:1808.10402, Reviews of Modern Physics 92 1, 015003 (2020).

[13] 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 (NISQ) algorithms", arXiv:2101.08448.

[14] Cristina Cîrstoiu, Zoë Holmes, Joseph Iosue, Lukasz Cincio, Patrick J. Coles, and Andrew Sornborger, "Variational fast forwarding for quantum simulation beyond the coherence time", npj Quantum Information 6, 82 (2020).

[15] Andrew Zhao, Andrew Tranter, William M. Kirby, Shu Fay Ung, Akimasa Miyake, and Peter J. Love, "Measurement reduction in variational quantum algorithms", Physical Review A 101 6, 062322 (2020).

[16] Xavier Bonet-Monroig, Ryan Babbush, and Thomas E. O'Brien, "Nearly Optimal Measurement Scheduling for Partial Tomography of Quantum States", Physical Review X 10 3, 031064 (2020).

[17] Chris Cade, Lana Mineh, Ashley Montanaro, and Stasja Stanisic, "Strategies for solving the Fermi-Hubbard model on near-term quantum computers", Physical Review B 102 23, 235122 (2020).

[18] Barnaby van Straaten and Bálint Koczor, "Measurement Cost of Metric-Aware Variational Quantum Algorithms", PRX Quantum 2 3, 030324 (2021).

[19] Andrew Arrasmith, Lukasz Cincio, Rolando D. Somma, and Patrick J. Coles, "Operator Sampling for Shot-frugal Optimization in Variational Algorithms", arXiv:2004.06252.

[20] Charles Hadfield, Sergey Bravyi, Rudy Raymond, and Antonio Mezzacapo, "Measurements of Quantum Hamiltonians with Locally-Biased Classical Shadows", arXiv:2006.15788.

[21] Zhenyu Cai, "Resource Estimation for Quantum Variational Simulations of the Hubbard Model", Physical Review Applied 14 1, 014059 (2020).

[22] Nobuyuki Yoshioka, Yuya O. Nakagawa, Kosuke Mitarai, and Keisuke Fujii, "Variational quantum algorithm for nonequilibrium steady states", Physical Review Research 2 4, 043289 (2020).

[23] Giacomo Torlai, Guglielmo Mazzola, Giuseppe Carleo, and Antonio Mezzacapo, "Precise measurement of quantum observables with neural-network estimators", Physical Review Research 2 2, 022060 (2020).

[24] Alexander Cowtan, Will Simmons, and Ross Duncan, "A Generic Compilation Strategy for the Unitary Coupled Cluster Ansatz", arXiv:2007.10515.

[25] Pranav Gokhale and Frederic T. Chong, "$O(N^3)$ Measurement Cost for Variational Quantum Eigensolver on Molecular Hamiltonians", arXiv:1908.11857.

[26] Jonas M. Kübler, Andrew Arrasmith, Lukasz Cincio, and Patrick J. Coles, "An Adaptive Optimizer for Measurement-Frugal Variational Algorithms", arXiv:1909.09083.

[27] Bálint Koczor and Simon C. Benjamin, "Quantum Analytic Descent", arXiv:2008.13774.

[28] James Stokes, Josh Izaac, Nathan Killoran, and Giuseppe Carleo, "Quantum Natural Gradient", arXiv:1909.02108.

[29] Lena Funcke, Tobias Hartung, Karl Jansen, Stefan Kühn, Paolo Stornati, and Xiaoyang Wang, "Measurement Error Mitigation in Quantum Computers Through Classical Bit-Flip Correction", arXiv:2007.03663.

[30] Andrew Tranter, Peter J. Love, Florian Mintert, Nathan Wiebe, and Peter V. Coveney, "Ordering of Trotterization: Impact on Errors in Quantum Simulation of Electronic Structure", Entropy 21 12, 1218 (2019).

[31] Tzu-Ching Yen and Artur F. Izmaylov, "Cartan sub-algebra approach to efficient measurements of quantum observables", arXiv:2007.01234.

[32] Ikko Hamamura and Takashi Imamichi, "Efficient evaluation of quantum observables using entangled measurements", npj Quantum Information 6, 56 (2020).

[33] Ilya G. Ryabinkin, Robert A. Lang, Scott N. Genin, and Artur F. Izmaylov, "Iterative Qubit Coupled Cluster approach with efficient screening of generators", arXiv:1906.11192.

[34] Robert A. Lang, Ilya G. Ryabinkin, and Artur F. Izmaylov, "Unitary transformation of the electronic Hamiltonian with an exact quadratic truncation of the Baker-Campbell-Hausdorff expansion", arXiv:2002.05701.

[35] Hsin-Yuan Huang and Richard Kueng, "Predicting Features of Quantum Systems from Very Few Measurements", arXiv:1908.08909.

[36] Shi-Ning Sun, Mario Motta, Ruslan N. Tazhigulov, Adrian T. K. Tan, Garnet Kin-Lic Chan, and Austin J. Minnich, "Quantum Computation of Finite-Temperature Static and Dynamical Properties of Spin Systems Using Quantum Imaginary Time Evolution", arXiv:2009.03542.

[37] 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 POVMs for near-term quantum algorithms", arXiv:2104.00569.

[38] Samuel J. Elman, Adrian Chapman, and Steven T. Flammia, "Free fermions behind the disguise", arXiv:2012.07857.

[39] Chang-yu Hsieh, Qiming Sun, Shengyu Zhang, and Chee Kong Lee, "Unitary-Coupled Restricted Boltzmann Machine Ansatz for Quantum Simulations", arXiv:1912.02988, npj Quantum Information 7, 19 (2019).

[40] Siyuan Niu and Aida Todri-Sanial, "Enabling multi-programming mechanism for quantum computing in the NISQ era", arXiv:2102.05321.

[41] Ewout van den Berg and Kristan Temme, "Circuit optimization of Hamiltonian simulation by simultaneous diagonalization of Pauli clusters", arXiv:2003.13599.

[42] Cambyse Rouzé and Daniel Stilck França, "Learning quantum many-body systems from a few copies", arXiv:2107.03333.

[43] Adrian Chapman and Steven T. Flammia, "Characterization of solvable spin models via graph invariants", arXiv:2003.05465.

[44] Mohsen Heidari and Wojciech Szpankowski, "Quantum State Classification via Quantum Fourier", arXiv:2102.05209.

[45] Chee-Kong Lee, Chang-Yu Hsieh, Shengyu Zhang, and Liang Shi, "Variational Quantum Simulation of Chemical Dynamics with Quantum Computers", arXiv:2110.06143.

[46] Chris Cade and P. Marcos Crichigno, "Complexity of Supersymmetric Systems and the Cohomology Problem", arXiv:2107.00011.

The above citations are from Crossref's cited-by service (last updated successfully 2021-10-22 17:13:43) and SAO/NASA ADS (last updated successfully 2021-10-22 17:13:44). The list may be incomplete as not all publishers provide suitable and complete citation data.

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