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, \emph{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(N) Clifford circuit, with worst-case runtime and space
complexities O(N6) and O(N2), respectively. 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 ∼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. Since good approximate solutions to MaxCut can be
efficiently found within the Clifford manifold, we hope our results will
motivate to rethink the approach so far used to search for quantum speedup in
combinatorial optimization problems.Comment: 15 pages, 9 figures, 5 pages appendi