Do Quantum AIs Dream in Paths? Path-Integral Slow Thinking through Grover Interference

Summary

The paper proposes 'quantum slow thinking': a trainable quantum reasoning circuit prepares a coherent superposition of all K^M = 3^8 action sequences on a 2x3 sliding-puzzle configuration graph (120 nodes, horizon M=8), an exact verifier partitions the ensemble into accepted and rejected sectors, and Grover amplitude amplification recombines them before measurement. Because the amplified success probability G_q(p) = sin^2(q arcsin sqrt(p)) is non-monotonic, with first maximum at the interior value p*(q) = sin^2(pi/2q), training toward this interior target avoids the monotonic pressure toward unit success that drives policy collapse in classical RLVR. Two models are trained with exact classical gradients: table-based QuCoNet (720 parameters) and a 29k-parameter prefix-conditioned transformer generating the coins. At q=9 circuit applications (four Grover rounds, B=32), the models reach held-out accuracies of 0.407 and 0.496 versus 0.126 for the strongest of 27 classical controls (3.2x and 3.9x), and fixed-size-policy capacity scales as q^1.74 versus q^0.62 under classical repetition. The authors candidly report that an untrained uniform policy read through the same amplification remains the strongest held-out reference on this solution-dense benchmark, and that a two-sided classical loss importing the target preserves at least as much path diversity — diversity stems from the interior target, while verified gains require coherent amplification. Limitations: all results are exact classical statevector simulations (no hardware), coherent-verifier costs are excluded from matched budgets (sensitivity analysis provided), and the benchmark is small, Markovian, and exactly verifiable.