Symmetries and Dimension Reduction in Quantum Approximate Optimization Algorithm

TL;DR

This paper explores symmetry-based dimension reduction in QAOA, enabling more efficient quantum optimization by focusing on smaller subspaces that retain essential problem features and solutions.

Contribution

It introduces a novel dimension reduction technique for QAOA that preserves the problem Hamiltonian but alters the mixer and initial state, leading to polynomial-sized subspaces.

Findings

Reduced QAOA spaces have polynomial dimensions

Each subspace contains unique classical solutions

Lower bounds on the number of solutions are established

Abstract

In this paper, the Quantum Approximate Optimization Algorithm (QAOA) is analyzed by leveraging symmetries inherent in problem Hamiltonians. We focus on the generalized formulation of optimization problems defined on the sets of $n$-element $d$-ary strings. Our main contribution encompasses dimension reductions for the originally proposed QAOA. These reductions retain the same problem Hamiltonian as the original QAOA but differ in terms of their mixer Hamiltonian, and initial state. The vast QAOA space has a daunting dimension of exponential scaling in $n$, where certain reduced QAOA spaces exhibit dimensions governed by polynomial functions. This phenomenon is illustrated in this paper, by providing partitions corresponding to polynomial dimensions of the corresponding subspaces. As a result, each reduced QAOA partition encapsulates unique classical solutions absent in others, allowing us to establish a lower bound on the number of solutions to the initial optimization problem. Our novel approach opens promising practical advantages in accelerating the algorithm. Restricting the algorithm to Hilbert spaces of smaller dimension may lead to significant acceleration of both quantum and classical simulation of circuits and serve as a tool to cope with barren plateaus problem.