Lattice-quantile estimation of π and convex-region integrals from coined two-dimensional quantum walks

Abstract

Monte Carlo integration is fundamentally limited by the M^(-1/2) rate that the Cramer-Rao bound imposes on any sample-mean estimator of an expectation value, regardless of how the samples are drawn. Coined discrete-time quantum walks (DTQWs) are known to spread ballistically - their position variance scales as T^2 against the diffusive T of classical random walks - yet this faster spreading has not been exploited for numerical integration. We show that coupling the ballistic scaling of a 2D DTQW to the Hardy-Huxley asymptotic for Gauss circle lattice counts produces estimators whose dominant error is a deterministic number-theoretic residual controlled by walk depth T, not a statistical fluctuation controlled by sample count M. The construction replaces the empirical mean of a sample-mean estimator with the ratio N(R-hat)/R-hat^2 of a lattice count to the square of a radial position quantile, a structural change that sidesteps the Cramer-Rao barrier. A single batch of measurements then propagates through classically precomputed multipliers to cover an entire family of integrals simultaneously. We develop the framework for convex smooth domains via Kraetzel's lattice asymptotic and for smooth integrals with convex or annular super-level sets via Cavalieri's principle, and provide a parameter-free identity for the bias floor (validated to within 1.5x across all tested depths). Every experiment is benchmarked against the classical random walk with the identical estimator to isolate the quantum contribution; the framework is oracle-free in the QAE sense (no controlled unitary encoding the integrand is required) and structurally distinct from quantum amplitude estimation and Szegedy-walk approaches. These ratios compare measurement counts at fixed precision and do not include quantum circuit execution cost.