Spin-Boson Mapping of the Quantum Approximate Optimization Algorithm

TL;DR

This paper maps high-depth QAOA states for the SK model to a spin-boson system, enabling efficient simulation and optimization of QAOA performance beyond previous computational limits.

Contribution

It introduces a spin-boson mapping for QAOA, allowing scalable simulation and analysis of high-depth regimes in the infinite-size limit.

Findings

QAOA converges to a spin-boson state in the infinite-size limit.

Numerical evidence shows QAOA achieves near-optimal energy with depth O(n/ε^{1.13}).

QAOA attains approximately 2.2% error at depth 160, surpassing prior methods.

Abstract

The Quantum Approximate Optimization Algorithm (QAOA) achieves monotonically improving performance with circuit depth $p$, yet the study of the high-depth regime has been obstructed by the exponential in $p$ cost of existing exact evaluation techniques. In this Letter, we prove that, in the infinite-size limit, the depth-$p$ QAOA state for the Sherrington-Kirkpatrick (SK) model converges to the state of a spin coupled to $p$ bosonic modes. We simulate the spin-boson system using matrix product states and provide numerical evidence that QAOA obtains a $(1-\epsilon)$ approximation to the optimal energy of the SK model with circuit depth $O(n/\epsilon^{1.13})$ in the average case. The modest computational cost of our approach allows us to optimize QAOA parameters and observe that QAOA achieves $\varepsilon\lesssim 2.2\%$ at $p=160$ in the infinite-size limit, extending far beyond $p\leq 20$ accessible to prior exact methods. Our mapping provides a many-body route to study and optimize high-depth QAOA in regimes previously inaccessible to exact evaluation.