Achieving the Heisenberg limit using fault-tolerant quantum error correction

TL;DR

This work investigates achieving Heisenberg-limit precision in quantum metrology under fully noisy, fault-prone operations. It introduces a fault-tolerant protocol that encodes probes with a repetition code, uses repeated syndrome measurements, and employs a fault-tolerant logical parity readout to suppress both signal-accumulation errors and QEC-related SPAM errors. A nonzero error threshold is established below which HL scaling is preserved, with only logarithmic overhead in circuit depth due to $O(\log n)$ repetition rounds. The findings demonstrate that HL-precision sensing can be robust to realistic, noisy QEC operations in Pauli-Z signal estimation under bit-flip noise, marking a bridge between theoretical optimality and practical metrology experiments.

Abstract

Quantum effect enables enhanced estimation precision in metrology, with the Heisenberg limit (HL) representing the ultimate limit allowed by quantum mechanics. Although the HL is generally unattainable in the presence of noise, quantum error correction (QEC) can recover the HL in various scenarios. A notable example is estimating a Pauli-$Z$ signal under bit-flip noise using the repetition code, which is both optimal for metrology and robust against noise. However, previous protocols often assume noise affects only the signal accumulation step, while the QEC operations -- including state preparation and measurement -- are noiseless. To overcome this limitation, we study fault-tolerant quantum metrology where all qubit operations are subject to noise. We focus on estimating a Pauli-$Z$ signal under bit-flip noise, together with state preparation and measurement errors in all QEC operations. We propose a fault-tolerant metrological protocol where a repetition code is prepared via repeated syndrome measurements, followed by a fault-tolerant logical measurement. We demonstrate the existence of an error threshold, below which errors are effectively suppressed and the HL is attained.

Figures (14)

• Figure 1: The general framework of quantum metrology. The probe, prepared in known initial state $\varrho$, interacts with the signal via a unitary $\mathcal{U}_\theta$. The final state $\varrho_\theta$ is then measured, from which the parameter $\theta$ is estimated. In our setting, both the state preparation and the measurement process are subject to noise.
• Figure 2: (a) Illustration of the distance-5 repetition code. Data qubits are represented by open circles. (b) The quantum circuit for measuring $Z$ syndrome.
• Figure 3: The MWPM decoding problem for a distance-5 repetition code (with measurement errors). The $X$-error matching graph is shown with $5$ rounds of syndrome measurement. Each detector represents the difference syndrome i.e., the difference (modulo 2) between the syndrome measurement in time step $t$ and $(t-1)$, and ensures that any single measurement error results in two syndrome defects. We assume all syndrome outcomes can be flipped with probability $\mathsf{q}$, even in the last round of the syndrome measurement.
• Figure 4: (a) The top panel shows the observation of two consecutive defects. These defects may be caused by a single-qubit error (middle panel), which occurs with probability $\mathsf{p}$, or by two consecutive measurement errors (bottom panel), which occurs with probability $\mathsf{p}^2$. (b) All possible error mechanisms that occur with probability $\mathsf{p}^2$---each leading to a failure of the matching algorithm.
• Figure 5: Illustration showing local chain that has end points in the same time-like boundary, while non-local chain that has end points in the opposite time-like boundary. The yellow lines show the error associated with the respective chain (shown in orange).

Theorems & Definitions (5)

• Definition 1: Circuit-level bit-flip noise model
• Theorem 1: Error-threshold for Heisenberg-limited sensing
• Lemma 2: Independence of data and syndrome errors
• proof
• Definition 2