2026/04/22 by Chusei Kiumi
#quant-ph
The Berry phase encodes the geometry of a closed Hamiltonian path, complementing the dynamical phase determined by the accumulated energy. We uncover an adiabatic error-cancellation principle arising from this geometric character, distinguishing Berry-phase estimation from energy estimation. To extract the Berry phase while canceling the dynamical phase, we combine finite-runtime evolutions generated by ± H along the loop. This construction also exactly cancels the leading O(T-1) and all higher odd-order nonoscillatory phase errors. Richardson extrapolation further reduces the residual error to an oscillatory contribution of order O(‖ H(0)‖2Δ(0)-4T-2), whose amplitude is controlled by endpoint data. Beyond this deterministic cancellation, we show that runtime randomization suppresses the remaining oscillatory contribution, reducing the bias after r levels of Richardson extrapolation to O(T-2(r+1)) for any fixed r. By combining these error-cancellation principles, we obtain a randomized Hadamard-test algorithm for Berry phase estimation over the full range [0,2π). The resulting improvement in runtime scaling turns the geometric origin of the cancellation into an algorithmic advantage, reducing the required coherent evolution time without increasing the asymptotic sample complexity. These features make Berry phase estimation a promising candidate for practical quantum computation in the early fault-tolerant regime.