Measurement optimization of variational quantum simu-lation by classical shadow and derandomization

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 dis-tributing 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 mea-surements; 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 optimize measurements in VQS by recently proposed measurement optimization techniques such as classical shadow and derandomization. Even though previous literature shows that the measurement optimization techniques successfully reduce 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. Our numerical experiment shows the validity of using our algorithm with quantum chemical systems. Importantly, we theoretically reveal the advantage of using shadow-based strategies, e.g., classical shadow and derandomization, not only in VQS but also in VQO by calculating the variance of an observable. To the best of our knowledge, the advantage of using shadow -based strategies in VQO was given only numerically in previous research; therefore, this paper also gives significant implications for measurement optimization in general variational quantum algorithms.