# Generalized Probabilistic Approximate Optimization Algorithm

## TL;DR

The paper introduces a new optimization algorithm that improves on simulated annealing and runs on probabilistic computers, showing better performance on complex problems.

## Contribution

The generalized PAOA extends simulated annealing by optimizing multiple temperature profiles and enables fast sampling on probabilistic computers.

## Key findings

- PAOA outperforms QAOA on the 26-spin Sherrington–Kirkpatrick model with matched parameters.
- Simulated annealing is a limiting case of PAOA under constrained parameterizations.
- PAOA improves performance over SA on heavy-tailed problems like SK–Lévy.

## Abstract

We introduce the generalized Probabilistic Approximate Optimization Algorithm (PAOA), a classical variational Monte Carlo framework that extends and formalizes the recently introduced PAOA, 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–Lévy.

Finding solutions in rugged energy landscapes is hard. Here, authors introduce a generalized Probabilistic Approximate Optimization Algorithm, a classical variational Monte Carlo method that reshapes the landscape and runs on probabilistic computers, recovers simulated annealing, and learns multi-temperature schedules.