Abstract

A promising approach to the practical application of the Quantum Approximate Optimization Algorithm (QAOA) is finding QAOA parameters classically in simulation and sampling the solutions from QAOA with optimized parameters on a quantum computer. Doing so requires repeated evaluations of QAOA energy in simulation. We propose a novel approach for accelerating the evaluation of QAOA energy by leveraging the symmetry of the problem. We show a connection between classical symmetries of the objective function and the symmetries of the terms of the cost Hamiltonian with respect to the QAOA energy. We show how by considering only the terms that are not connected by symmetry, we can significantly reduce the cost of evaluating the QAOA energy. Our approach is general and applies to any known subgroup of symmetries and is not limited to graph problems. Our results are directly applicable to nonlocal QAOA generalization RQAOA. We outline how available fast graph automorphism solvers can be leveraged for computing the symmetries of the problem in practice. We implement the proposed approach on the MaxCut problem using a state-of-the-art tensor network simulator and a graph automorphism solver on a benchmark of 48 graphs with up to 10,000 nodes. Our approach provides an improvement for $p=1$ on $71.7\%$ of the graphs considered, with a median speedup of $4.06$, on a benchmark where $62.5\%$ of the graphs are known to be hard for automorphism solvers.

Highlights

  • Recent advances in quantum hardware [1], [2] pave the way for demonstrating useful quantum advantage in the coming years

  • USING SYMMETRY TO REDUCE THE NUMBER OF OBSERVABLES TO EVALUATE we show how classical symmetries of the objective function reduce the cost of evaluating quantum approximate optimization algorithm (QAOA) energy

  • In this article we presented an approach for accelerating the computation of the QAOA energy by leveraging fast graph symmetry solvers

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Summary

INTRODUCTION

Recent advances in quantum hardware [1], [2] pave the way for demonstrating useful quantum advantage in the coming years. We show that these symmetries introduce classes of equivalence on the terms of the cost Hamiltonian with regard to QAOA energy, making it sufficient to evaluate only the energy of one term from each equivalence class For some problems, such symmetries of the objective function may be known a priori or can be computed by using state-of-the-art graph automorphism solvers, which we demonstrate to be fast compared with the cost of evaluating the energy. Our results are applicable to QAOA and to its nonlocal generalization recursive QAOA (RQAOA) [15]–[17] because of the need to optimize parameters and evaluate the energy of each term to compute correlations We implement this approach numerically for the MaxCut problem using the state-of-the-art tensor network simulator QTensor [11], [18] and using nauty [19] for computing the automorphism group of the graph.

BACKGROUND
OBJECTIVE FUNCTION SYMMETRIES AND QAOA
FAST SOLVERS FOR GRAPH SYMMETRIES
ACCELERATING QAOA TRAINING BY USING FAST GRAPH AUTOMORPHISM SOLVERS
NUMERICAL EXPERIMENTS
Findings
CONCLUSION
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