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.

💡 Research Summary

The paper addresses the long‑standing challenge of achieving Heisenberg‑limited precision in Bayesian phase estimation using only low‑depth digital quantum circuits, a requirement for many near‑term quantum platforms such as tweezer‑based optical clocks, superconducting qubits, and trapped‑ion processors. The authors develop a protocol that works for arbitrary Gaussian prior widths and scales logarithmically with the total number of qubits N, thereby offering a practical route to near‑optimal metrology without the need for deep, highly entangling operations.

The central construct is a “GHZ‑block” state: the N‑qubit register is partitioned into several copies (m_i) of GHZ states of size 2^{k_i}. The overall state is |ψ⟩ = ∏_i (|0⟩^{⊗2^{k_i}} + |1⟩^{⊗2^{k_i}})^{⊗m_i}/√2, with the constraint Σ_i m_i 2^{k_i}=N. By numerically scanning all possible partitions for a given N and prior width δϕ, the authors identify the configuration that minimizes the Bayesian mean‑squared error (BMSE). This optimization is performed using the “optimal quantum interferometer” (OQI) algorithm introduced in earlier work, which simultaneously yields the optimal initial state, measurement POVM, and estimator.

Two measurement strategies are compared. The first, taken from earlier literature, is a non‑adaptive “bit‑by‑bit” scheme that measures each GHZ block in σ_x and σ_y bases and reconstructs the binary expansion of the phase. The second, novel to this work, is a local adaptive measurement: after each block’s parity measurement, the posterior distribution is updated and the measurement basis for the next block is chosen to maximize information gain. Simulations show that the adaptive scheme reaches Heisenberg scaling (Δϕ ∝ 1/N) up to a constant overhead for any prior width, whereas the non‑adaptive scheme either falls short of the standard quantum limit or requires a very specific set of qubit numbers (e.g., N = 2, 9, 26, 63, …).

A further contribution is a “phase‑unwinding” protocol that extends the dynamic range beyond the intrinsic 2π/N periodicity of GHZ states. By allocating a fraction of atoms as “slow atoms” that acquire fractional phases ϕ/2, ϕ/4, …, the authors effectively create a multi‑scale phase representation. The slow‑atom ensemble is combined with the GHZ‑block register using an optimized allocation rule, dramatically reducing the number of additional atoms needed to achieve a given sensitivity compared with existing multi‑scale schemes.

Noise robustness is examined by modeling decoherence (rate γ) and gate infidelity (error ε). The results indicate that partitions dominated by small GHZ blocks (2–4 qubits) are far more tolerant to both types of error than those relying on large GHZ states, because error propagation scales with block size. Consequently, the protocol remains advantageous even when realistic gate fidelities (≈99.5 %) and dephasing times are taken into account.

In summary, the paper delivers four key advances: (1) a near‑optimal Bayesian estimator that works for any Gaussian prior width; (2) a construction of the optimal initial state using only logarithmic‑depth circuits; (3) an adaptive measurement framework that attains Heisenberg scaling with a modest constant overhead; and (4) an efficient phase‑unwinding technique that expands the usable phase interval while requiring fewer ancillary atoms. These results bridge the gap between theoretical metrological limits and the capabilities of current quantum hardware, paving the way for Heisenberg‑limited atomic clocks and other precision sensors built on low‑depth digital quantum platforms.