Approximate Boltzmann distributions in quantum approximate optimization
Phillip C. Lotshaw, George Siopsis, James Ostrowski, Rebekah Herrman, Rizwanul Alam, Sarah Powers, Travis S. Humble
Abstract
Approaches to compute or estimate the output probability distributions from the quantum approximate optimization algorithm (QAOA) are needed to assess the likelihood it will obtain a quantum computational advantage. We analyze output from QAOA circuits solving 7200 random MaxCut instances, with $n=14--23$ qubits and depth parameter $p\ensuremath{\le}12$, and find that the average basis state probabilities follow approximate Boltzmann distributions: The average probabilities scale exponentially with their energy (cut value), with a peak at the optimal solution. We describe the rate of exponential scaling or effective temperature in terms of a series with a leading-order term $T\ensuremath{\sim}{C}_{\mathrm{min}}/n\sqrt{p}$, with ${C}_{\mathrm{min}}$ the optimal solution energy. Using this scaling, we generate approximate output distributions with up to 38 qubits and find these give accurate accounts of important performance metrics in cases we can simulate exactly.