Reducing QAOA Circuit Depth by Factoring out Semi-Symmetries

TL;DR

This paper introduces semi-symmetries in QUBO matrices and an algorithm to factor them out, significantly reducing circuit depth and couplings in QAOA for various combinatorial problems, thus improving quantum algorithm efficiency.

Contribution

The paper presents a novel concept of semi-symmetries and an algorithm to identify and factor them out, reducing QAOA circuit complexity while preserving the problem's energy spectrum.

Findings

Reduced couplings by up to 49%

Reduced circuit depth by up to 41%

Applicable to multiple well-known optimization problems

Abstract

QAOA is a quantum algorithm for solving combinatorial optimization problems. It is capable of searching for the minimizing solution vector $x$ of a QUBO problem $x^TQx$. The number of two-qubit CNOT gates in the QAOA circuit scales linearly in the number of non-zero couplings of $Q$ and the depth of the circuit scales accordingly. Since CNOT operations have high error rates it is crucial to develop algorithms for reducing their number. We, therefore, present the concept of \textit{semi-symmetries} in QUBO matrices and an algorithm for identifying and factoring them out into ancilla qubits. \textit{Semi-symmetries} are prevalent in QUBO matrices of many well-known optimization problems like \textit{Maximum Clique}, \textit{Hamilton Cycles}, \textit{Graph Coloring}, \textit{Vertex Cover} and \textit{Graph Isomorphism}, among others. We theoretically show that our modified QUBO matrix $Q_{mod}$ describes the same energy spectrum as the original $Q$. Experiments conducted on the five optimization problems mentioned above demonstrate that our algorithm achieved reductions in the number of couplings by up to $49\%$ and in circuit depth by up to $41\%$.