Computational Phase Transitions in Binary Compressed Sensing: Quantum Annealing Inside the Relaxation Gap

We map the computational phase transition boundary in binary compressed sensing and identify a regime where D-Wave's quantum annealer recovers signals in a region where all tested classical methods fail, including Approximate Message Passing (AMP), which achieves the Bayes-optimal recovery threshold asymptotically for Gaussian matrices. In 19,775 experiments (n in {32, 64}, nine classical solvers, two D-Wave modes), we find that quantum annealing recovers sparse binary signals in the relaxation gap - the regime below the Donoho-Tanner l1 phase transition where the l0 solution exists but convex relaxations fail. At n=32, k=5, m/n=0.19, D-Wave achieves 7% exact recovery while AMP and eight other solvers score 0% across 250 combined trials Fisher exact p=0.018. At n=64, embedding overhead limits the QPU, but D-Wave's hybrid solver remains competitive with AMP. Energy landscape analysis reveals that the QUBO ground state contains the true signal, but incorrect solutions occupy shallower local basins that trap classical search - a structure consistent with quantum tunneling dynamics. To our knowledge, this constitutes preliminary finite-size evidence that quantum annealing succeeds in a narrow regime where all tested classical methods, including the Bayes-optimal AMP, fail within a well-characterized combinatorial inference problem. Confirmation at larger n, higher trial counts, and with stronger classical controls remains an open problem.