Sampling-based Quantum Optimization Algorithm with Quantum Relaxation

Abstract: Variational Quantum Algorithm (VQA) is a hybrid algorithm for noisy quantum devices. However, statistical fluctuations and physical noise degrade the solution quality, so it is difficult to maintain applicability for large-scale problems. In contrast, Sampling-based Quantum Algorithms have recently been successfully applied to large-scale quantum chemistry problems. The quantum device is used only for sampling, and the ground state and its energy are estimated on the classical device. In this study, we propose the Sampling-based Quantum Optimization Algorithm (SQOA). Two challenges exist in constructing a Sampling-based Quantum Algorithm for combinatorial optimization. The first challenge is that we need to encode the optimization problem in a non-diagonal Hamiltonian, even though many VQAs encode it into the Ising Hamiltonian, which is diagonal. The second challenge is that we need a method to prepare the input state to be sampled efficiently. We employ the Quantum Relaxation (QR) method for the first challenge, which encodes multiple classical variables in one qubit. It reduces required qubits compared to the Ising Hamiltonian approach. Moreover, we investigate the parameter transferability in the Quantum Alternating Operator Ansatz for QR Hamiltonians for the second challenge. We show that restricting parameters to a linear form exhibits moderate transferability for 3-regular MaxCut problems, similar to transferability observed in the Quantum Approximate Optimization Algorithm. This property allows us to efficiently prepare the input state for a large instance using the parameters from a small instance. We leveraged transferability to create input states and applied SQOA with QR to the MaxCut instances. Transferring parameters from a 20-node problem demonstrates that SQOA with QR provides high-quality solutions for 40-node problems without variational parameter optimization.