A Recursive Lower Bound on the Energy Improvement of the Quantum Approximate Optimization Algorithm

TL;DR

This paper develops an analytical recursive method to estimate the lower bounds on energy improvements in deep QAOA circuits, revealing exponential decay of gains with increasing layers.

Contribution

It introduces a recursive analytical approach to bound the energy improvement in deep QAOA, extending understanding beyond small-depth guarantees.

Findings

The lower bound on QAOA energy gain decreases exponentially with the number of layers.

The bound decreases more rapidly than the actual energy gain, indicating diminishing returns at high depth.

Numerical results confirm the accuracy of the recursive bounds and their exponential decay.

Abstract

The Quantum Approximate Optimization Algorithm (QAOA) uses a quantum computer to implement a variational method with $2p$ layers of alternating unitary operators, optimized by a classical computer to minimize a cost function. While rigorous performance guarantees exist for the QAOA at small depths $p$, the behavior at large depths remains less clear, though simulations suggest exponentially fast convergence for certain problems. In this work, we gain insights into the deep QAOA using an analytic expansion of the cost function around transition states. Transition states are constructed recursively: from a local minima of the QAOA with $p$ layers we obtain transition states of the QAOA with $p+1$ layers, which are stationary points characterized by a unique direction of negative curvature. We construct an analytic estimate of the negative curvature and the corresponding direction in parameter space at each transition state. Expansion of the QAOA cost function along the negative direction to the quartic order gives a lower bound of the QAOA cost function improvement. We provide physical intuition behind the analytic expressions for the local curvature and quartic expansion coefficient. Our numerical study confirms the accuracy of our approximations, and reveals that the obtained bound and the true value of the QAOA cost function gain have a characteristic exponential decrease with the number of layers $p$, with the bound decreasing more rapidly. Our study establishes an analytical method for recursively studying the QAOA applicable in the regime of high circuit depth.