Heisenberg-limited Bayesian phase estimation with low-depth digital quantum circuits

Optimal phase estimation protocols require complex state preparation and readout schemes, generally unavailable or unscalable in many quantum platforms. We develop and analyze a scheme that achieves near-optimal precision up to a constant overhead for Bayesian phase estimation, using simple digital quantum circuits with depths scaling logarithmically with the number of qubits. We find that for Gaussian prior phase distributions with arbitrary widths, the optimal initial state can be approximated with products of Greenberger-Horne-Zeilinger states with varying number of qubits. Using local, adaptive measurements optimized for the prior distribution and the initial state, we show that Heisenberg scaling is achievable and that the proposed scheme outperforms known schemes in the literature that utilize a similar set of initial states. For an example prior width, we present a detailed comparison and find that is also possible to achieve Heisenberg scaling with a scheme that employs non-adaptive measurements, with the right allocation of copies per GHZ state and single-qubit rotations. We also propose an efficient phase unwinding protocol to extend the dynamic range of the proposed scheme, and show that it outperforms existing protocols by achieving an enhanced precision with a smaller number of additional atoms. Lastly, we discuss the impact of noise and imperfect gates.