Pauli String Partitioning Algorithm with the Ising Model for Simultaneous Measurement

Abstract: We propose an efficient algorithm for partitioning Pauli strings into subgroups, which can be simultaneously measured in a single quantum circuit. Our partitioning algorithm drastically reduces the total number of measurements in a variational quantum eigensolver for a quantum chemistry, one of the most promising applications of quantum computing. The algorithm is based on the Ising model optimization problem, which can be quickly solved using an Ising machine. We develop an algorithm that is applicable to problems with sizes larger than the maximum number of variables that an Ising machine can handle ($n_\text{bit}$) through its iterative use. The algorithm has much better time complexity and solution optimality than other algorithms such as Boppana--Halld\'orsson algorithm and Bron--Kerbosch algorithm, making it useful for the quick and effective reduction of the number of quantum circuits required for measuring the expectation values of multiple Pauli strings. We investigate the performance of the algorithm using the second-generation Digital Annealer, a high-performance Ising hardware, for up to $65,535$ Pauli strings using Hamiltonians of molecules and the full tomography of quantum states. We demonstrate that partitioning problems for quantum chemical calculations can be solved with a time complexity of $O(N)$ for $N\leq n_\text{bit}$ and $O(N^2)$ for $N>n_\text{bit}$ for the worst case, where $N$ denotes the number of candidate Pauli strings and $n_\text{bit}=8,192$ for the second-generation Digital Annealer used in this study. The reduction factor, which is the number of Pauli strings divided by the number of obtained partitions, can be $200$ at maximum.