Low-depth Clifford circuits approximately solve MaxCut

Journal Article · Thu Jun 20 00:00:00 EDT 2024 · Physical Review Research

DOI:https://doi.org/10.1103/PhysRevResearch.6.023294· OSTI ID:2375894

Muñoz-Arias, Manuel H.; Kourtis, Stefanos; Blais, Alexandre

We introduce a quantum-inspired approximation algorithm for MaxCut based on low-depth Clifford circuits. We start by showing that the solution unitaries found by the adaptive quantum approximation optimization algorithm (ADAPT-QAOA) for the MaxCut problem on weighted fully connected graphs are (almost) Clifford circuits. Motivated by this observation, we devise an approximation algorithm for MaxCut, ADAPT-Clifford, that searches through the Clifford manifold by combining a minimal set of generating elements of the Clifford group. Our algorithm finds an approximate solution of MaxCut on an $N$ -vertex graph by building a depth $O\left(N\right)$ Clifford circuit. The algorithm has runtime complexity $O\left(N^2\right)$ and $O\left(N^3\right)$ for sparse and dense graphs, respectively, and space complexity $O\left(N^2\right)$ , with improved solution quality achieved at the expense of more demanding runtimes. We implement ADAPT-Clifford and characterize its performance on graphs with positive and signed weights. The case of signed weights is illustrated with the paradigmatic Sherrington-Kirkpatrick model, for which our algorithm finds solutions with ground-state mean energy density corresponding to $\sim 94\%$ of the Parisi value in the thermodynamic limit. The case of positive weights is investigated by comparing the cut found by ADAPT-Clifford with the cut found with the Goemans-Williamson (GW) algorithm. For both sparse and dense instances we provide copious evidence that, up to hundreds of nodes, ADAPT-Clifford finds cuts of lower energy than GW.

Published by the American Physical Society 2024

• Sponsoring Organization: USDOE
• OSTI ID: 2375894
• Journal Information: Physical Review Research, Journal Name: Physical Review Research Journal Issue: 2 Vol. 6; ISSN 2643-1564; ISSN PPRHAI
• Publisher: American Physical Society
• Country of Publication: United States
• Language: English