Measurement optimization of variational quantum simulation by classical shadow and derandomization

Kouhei Nakaji1,4, Suguru Endo2, Yuichiro Matsuzaki1, and Hideaki Hakoshima3

1Device Technology Research Institute, National Institute of Advanced Industrial Science and Technology (AIST),1-1-1 Umezono, Tsukuba, Ibaraki 305-8568, Japan.
2NTT Computer and Data Science laboratories, NTT corporation, Musashino, Tokyo 180-8585, Japan
3Center for Quantum Information and Quantum Biology, Osaka University, 1-2 Machikaneyama, Toyonaka, Osaka 560-0043, Japan.
4Current address: Department of Computer Science, University of Toronto, Toronto, Ontario, Canada

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Abstract

Simulating large quantum systems is the ultimate goal of quantum computing. Variational quantum simulation (VQS) gives us a tool to achieve the goal in near-term devices by distributing the computation load to both classical and quantum computers. However, as the size of the quantum system becomes large, the execution of VQS becomes more and more challenging. One of the most severe challenges is the drastic increase in the number of measurements; for example, the number of measurements tends to increase by the fourth power of the number of qubits in a quantum simulation with a chemical Hamiltonian. This work aims to dramatically decrease the number of measurements in VQS by recently proposed shadow-based strategies such as classical shadow and derandomization. Even though previous literature shows that shadow-based strategies successfully optimize measurements in the variational quantum optimization (VQO), how to apply them to VQS was unclear due to the gap between VQO and VQS in measuring observables. In this paper, we bridge the gap by changing the way of measuring observables in VQS and propose an algorithm to optimize measurements in VQS by shadow-based strategies. Our theoretical analysis not only reveals the advantage of using our algorithm in VQS but theoretically supports using shadow-based strategies in VQO, whose advantage has only been given numerically. Additionally, our numerical experiment shows the validity of using our algorithm with a quantum chemical system.

Simulating large quantum systems is the ultimate goal of quantum computing. Variational Quantum Simulation (VQS) is a promising quantum algorithm to realize quantum simulation in the near-term quantum computer. However, executing VQS becomes increasingly challenging as the quantum system grows in size, with one of the most severe challenges being the significant increase in the number of required measurements. To address this issue, we proposed an algorithm to optimize measurements in VQS using measurement optimization techniques like classical shadow and derandomization by changing the way of measuring observables in VQS. We demonstrated the validity of the algorithm using numerical experiments with quantum chemical systems. Additionally, we theoretically revealed the advantage of using shadow-based strategies, such as classical shadow and derandomization, not only in VQS but also in Variational Quantum Optimization (VQO). This study has significant implications for measurement optimization in general variational quantum algorithms.

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