Overlapped grouping measurement: A unified framework for measuring quantum states

Bujiao Wu1,2, Jinzhao Sun3,1, Qi Huang4,1, and Xiao Yuan1,2

1Center on Frontiers of Computing Studies, Peking University, Beijing 100871, China
2School of Computer Science, Peking University, Beijing 100871, China
3Clarendon Laboratory, University of Oxford, Parks Road, Oxford OX1 3PU, United Kingdom
4School of Physics, Peking University, Beijing 100871, China

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Abstract

Quantum algorithms designed for realistic quantum many-body systems, such as chemistry and materials, usually require a large number of measurements of the Hamiltonian. Exploiting different ideas, such as importance sampling, observable compatibility, or classical shadows of quantum states, different advanced measurement schemes have been proposed to greatly reduce the large measurement cost. Yet, the underline cost reduction mechanisms seem distinct from each other, and how to systematically find the optimal scheme remains a critical challenge. Here, we address this challenge by proposing a unified framework of quantum measurements, incorporating advanced measurement methods as special cases. Our framework allows us to introduce a general scheme – overlapped grouping measurement, which simultaneously exploits the advantages of most existing methods. An intuitive understanding of the scheme is to partition the measurements into overlapped groups with each one consisting of compatible measurements. We provide explicit grouping strategies and numerically verify its performance for different molecular Hamiltonians with up to 16 qubits. Our numerical result shows significant improvements over existing schemes. Our work paves the way for efficient quantum measurement and fast quantum processing with current and near-term quantum devices.

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[1] Kaifeng Bu, Dax Enshan Koh, Roy J. Garcia, and Arthur Jaffe, "Classical shadows with Pauli-invariant unitary ensembles", npj Quantum Information 10 1, 6 (2024).

[2] Benoît Vermersch, Aniket Rath, Bharathan Sundar, Cyril Branciard, John Preskill, and Andreas Elben, "Enhanced Estimation of Quantum Properties with Common Randomized Measurements", PRX Quantum 5 1, 010352 (2024).

[3] Linqing Peng, Xing Zhang, and Garnet Kin-Lic Chan, "Fermionic Reduced Density Low-Rank Matrix Completion, Noise Filtering, and Measurement Reduction in Quantum Simulations", Journal of Chemical Theory and Computation 19 24, 9151 (2023).

[4] Yanqi Song, Yusen Wu, Shengyao Wu, Dandan Li, Qiaoyan Wen, Sujuan Qin, and Fei Gao, "A quantum federated learning framework for classical clients", Science China Physics, Mechanics & Astronomy 67 5, 250311 (2024).

[5] Kouhei Nakaji, Suguru Endo, Yuichiro Matsuzaki, and Hideaki Hakoshima, "Measurement optimization of variational quantum simulation by classical shadow and derandomization", Quantum 7, 995 (2023).

[6] Matteo Ippoliti, "Classical shadows based on locally-entangled measurements", Quantum 8, 1293 (2024).

[7] Linghua Zhu, Senwei Liang, Chao Yang, and Xiaosong Li, "Optimizing Shot Assignment in Variational Quantum Eigensolver Measurement", Journal of Chemical Theory and Computation 20 6, 2390 (2024).

[8] Daniel McNulty, Filip B. Maciejewski, and Michał Oszmaniec, "Estimating Quantum Hamiltonians via Joint Measurements of Noisy Noncommuting Observables", Physical Review Letters 130 10, 100801 (2023).

[9] Yuma Nakamura, Yoshichika Yano, and Nobuyuki Yoshioka, "Adaptive measurement strategy for quantum subspace methods", New Journal of Physics 26 3, 033028 (2024).

[10] 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", npj Quantum Information 9 1, 14 (2023).

[11] Bujiao Wu and Dax Enshan Koh, "Error-mitigated fermionic classical shadows on noisy quantum devices", npj Quantum Information 10 1, 39 (2024).

[12] William Kirby, Mario Motta, and Antonio Mezzacapo, "Exact and efficient Lanczos method on a quantum computer", Quantum 7, 1018 (2023).

[13] Jiaqi Miao, Chang-Yu Hsieh, and Shi-Xin Zhang, "Neural-network-encoded variational quantum algorithms", Physical Review Applied 21 1, 014053 (2024).

[14] You Zhou and Qing Liu, "Performance analysis of multi-shot shadow estimation", Quantum 7, 1044 (2023).

[15] Weitang Li, Yufei Ge, Shi-Xin Zhang, Yu-Qin Chen, and Shengyu Zhang, "Efficient and Robust Parameter Optimization of the Unitary Coupled-Cluster Ansatz", Journal of Chemical Theory and Computation 20 9, 3683 (2024).

[16] Antoine Michel, Sebastian Grijalva, Loïc Henriet, Christophe Domain, and Antoine Browaeys, "Blueprint for a digital-analog variational quantum eigensolver using Rydberg atom arrays", Physical Review A 107 4, 042602 (2023).

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

[18] Zi-Jian Zhang, Jinzhao Sun, Xiao Yuan, and Man-Hong Yung, "Low-Depth Hamiltonian Simulation by an Adaptive Product Formula", Physical Review Letters 130 4, 040601 (2023).

[19] Dax Enshan Koh and Sabee Grewal, "Classical Shadows With Noise", arXiv:2011.11580, (2020).

[20] Junyu Liu, Zimu Li, Han Zheng, Xiao Yuan, and Jinzhao Sun, "Towards a variational Jordan-Lee-Preskill quantum algorithm", Machine Learning: Science and Technology 3 4, 045030 (2022).

[21] Ting Zhang, Jinzhao Sun, Xiao-Xu Fang, Xiao-Ming Zhang, Xiao Yuan, and He Lu, "Experimental Quantum State Measurement with Classical Shadows", Physical Review Letters 127 20, 200501 (2021).

[22] Dax Enshan Koh and Sabee Grewal, "Classical Shadows With Noise", Quantum 6, 776 (2022).

[23] Bryce Fuller, Charles Hadfield, Jennifer R. Glick, Takashi Imamichi, Toshinari Itoko, Richard J. Thompson, Yang Jiao, Marna M. Kagele, Adriana W. Blom-Schieber, Rudy Raymond, and Antonio Mezzacapo, "Approximate Solutions of Combinatorial Problems via Quantum Relaxations", arXiv:2111.03167, (2021).

[24] Daniel Miller, Laurin E. Fischer, Igor O. Sokolov, Panagiotis Kl. Barkoutsos, and Ivano Tavernelli, "Hardware-Tailored Diagonalization Circuits", arXiv:2203.03646, (2022).

[25] Ariel Shlosberg, Andrew J. Jena, Priyanka Mukhopadhyay, Jan F. Haase, Felix Leditzky, and Luca Dellantonio, "Adaptive estimation of quantum observables", Quantum 7, 906 (2023).

[26] Masaya Kohda, Ryosuke Imai, Keita Kanno, Kosuke Mitarai, Wataru Mizukami, and Yuya O. Nakagawa, "Quantum expectation-value estimation by computational basis sampling", Physical Review Research 4 3, 033173 (2022).

[27] Alexander Gresch and Martin Kliesch, "Guaranteed efficient energy estimation of quantum many-body Hamiltonians using ShadowGrouping", arXiv:2301.03385, (2023).

[28] Zhenhuan Liu, Pei Zeng, You Zhou, and Mile Gu, "Characterizing correlation within multipartite quantum systems via local randomized measurements", Physical Review A 105 2, 022407 (2022).

[29] Weitang Li, Zigeng Huang, Changsu Cao, Yifei Huang, Zhigang Shuai, Xiaoming Sun, Jinzhao Sun, Xiao Yuan, and Dingshun Lv, "Toward Practical Quantum Embedding Simulation of Realistic Chemical Systems on Near-term Quantum Computers", arXiv:2109.08062, (2021).

[30] Seonghoon Choi, Ignacio Loaiza, and Artur F. Izmaylov, "Fluid fermionic fragments for optimizing quantum measurements of electronic Hamiltonians in the variational quantum eigensolver", Quantum 7, 889 (2023).

[31] Andrew Jena, Scott N. Genin, and Michele Mosca, "Optimization of variational-quantum-eigensolver measurement by partitioning Pauli operators using multiqubit Clifford gates on noisy intermediate-scale quantum hardware", Physical Review A 106 4, 042443 (2022).

[32] Tianren Gu, Xiao Yuan, and Bujiao Wu, "Efficient measurement schemes for bosonic systems", Quantum Science and Technology 8 4, 045008 (2023).

[33] Xiao-Ming Zhang, Zixuan Huo, Kecheng Liu, Ying Li, and Xiao Yuan, "Unbiased random circuit compiler for time-dependent Hamiltonian simulation", arXiv:2212.09445, (2022).

[34] Marco Majland, Rasmus Berg Jensen, Mads Greisen Højlund, Nikolaj Thomas Zinner, and Ove Christiansen, "Runtime optimization for vibrational structure on quantum computers: coordinates and measurement schemes", arXiv:2211.11615, (2022).

[35] Yifei Chen, Zhan Yu, Chenghong Zhu, and Xin Wang, "Efficient information recovery from Pauli noise via classical shadow", arXiv:2305.04148, (2023).

[36] Seonghoon Choi and Artur F. Izmaylov, "Measurement optimization techniques for excited electronic states in near-term quantum computing algorithms", arXiv:2302.11421, (2023).

The above citations are from Crossref's cited-by service (last updated successfully 2024-05-26 15:37:23) and SAO/NASA ADS (last updated successfully 2024-05-26 15:37:24). The list may be incomplete as not all publishers provide suitable and complete citation data.