Generalized Probabilistic Approximate Optimization Algorithm

TL;DR

This paper introduces PAOA, a flexible variational Monte Carlo algorithm for Ising models, demonstrating its advantages over QAOA and simulated annealing in solving complex spin-glass problems.

Contribution

The paper formalizes PAOA, a new variational Monte Carlo framework that generalizes prior methods, enabling fast sampling on modern probabilistic hardware with a principled variational foundation.

Findings

PAOA outperforms QAOA on 26-spin Sherrington-Kirkpatrick model.

PAOA extends simulated annealing by optimizing multiple temperature profiles.

Implementation on FPGA-based hardware demonstrates practical efficiency.

Abstract

We introduce a generalized \textit{Probabilistic Approximate Optimization Algorithm (PAOA)}, a classical variational Monte Carlo framework that extends and formalizes prior work by Weitz \textit{et al.}~\cite{Combes_2023}, enabling parameterized and fast sampling on present-day Ising machines and probabilistic computers. PAOA operates by iteratively modifying the couplings of a network of binary stochastic units, guided by cost evaluations from independent samples. We establish a direct correspondence between derivative-free updates and the gradient of the full Markov flow over the exponentially large state space, showing that PAOA admits a principled variational formulation. Simulated annealing emerges as a limiting case under constrained parameterizations, and we implement this regime on an FPGA-based probabilistic computer with on-chip annealing to solve large 3D spin-glass problems. Benchmarking PAOA against QAOA on the canonical 26-spin Sherrington-Kirkpatrick model with matched parameters reveals superior performance for PAOA. We show that PAOA naturally extends simulated annealing by optimizing multiple temperature profiles, leading to improved performance over SA on heavy-tailed problems such as SK-L\'evy.