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arXiv:2608.00171

Fault-Tolerant Heisenberg-Limited Quantum Sensing

Lorcan O. Conlon, Yu-Xin Wang, Erfan Abbasgholinejad, Victor V. Albert, Michael J. Gullans, Alexey V. Gorshkov

quant-phSubmitted Jul 31, 2026• Announced Aug 4, 2026• 0 citations
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Abstract

Quantum sensors hold great promise for achieving better sensitivity in the measurement of physical quantities compared to their classical counterparts. However, the conditions under which quantum advantage in sensing can be achieved are rather restrictive, and most quantum enhancements in sensing are lost in the presence of noise, errors, or a poorly calibrated system. To overcome these limitations, we are motivated to import ideas from fault-tolerant quantum computing to quantum sensing. Specifically, we consider a qubit noise model where the probability of phase-flip errors is exponentially smaller (in qubit number) compared to the probability of bit-flip errors that occur with probability $p$. For this noise structure, we demonstrate that, given a total sensing time $T$, Heisenberg scaling can be attained for times up to $T\propto 1/p^{(N+1)/2}$ for a $N$-qubit repetition code, in contrast with $T\propto 1/p$ without using a fault-tolerant sensing protocol.

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Quantum sensors hold great promise for achieving better sensitivity in the measurement of physical quantities compared to their classical counterparts. However, the conditions under which quantum advantage in sensing can be achieved are rather restrictive, and most quantum enhancements in sensing are lost in the presence of noise, errors, or a poorly calibrated system. To overcome these limitations, we are motivated to import ideas from fault-tolerant quantum computing to quantum sensing. Specifically, we consider a qubit noise model where the probability of phase-flip errors is exponentially smaller (in qubit number) compared to the probability of bit-flip errors that occur with probability ppp. For this noise structure, we demonstrate that, given a total sensing time TTT, Heisenberg scaling can be attained for times up to T∝1/p(N+1)/2T\propto 1/p^{(N+1)/2}T∝1/p(N+1)/2 for a NNN-qubit repetition code, in contrast with T∝1/pT\propto 1/pT∝1/p without using a fault-tolerant sensing protocol.
Quantum sensors have been shown to enable more precise measurements than classical sensors. By using entanglement and coherent evolution, the estimation error can be made to scale better in interrogation time, photon count, and other resources, achieving Heisenberg-limited scaling. However, environmental noise and hardware errors typically destroy the coherence of the sensitive quantum states. This motivates the use of fault-tolerant techniques to create more robust quantum sensing protocols. Such fault-tolerant quantum sensing approaches have been shown to enhance sensitivity for qubit systems, but many practical sensors require bosonic probes. In this paper, we propose a fault-tolerant protocol using bosons to estimate displacements in the presence of dephasing noise. We construct an optimal error-correcting code under a mean photon-number constraint. Controlled displacements couple the probe to a repetition-encoded qubit register, while repeated boson-parity measurements and resets remove probe noise without erasing the stored logical phase. Under circuit-level bit-flip noise, a distance-ddd code suppresses logical failures to order p(d+1)/2p^{(d+1)/2}p(d+1)/2. We show that, for a decay rate γAγ_AγA​, a total sensing time TTT and a mean photon number budget NNN, and for sufficiently small residual signals, the Fisher information scales as NT2NT^2NT2 over interrogation times of order γA−(d+1)/2γ_A^{-(d+1)/2}γA−(d+1)/2​. This is a significant improvement over the non-fault-tolerant approach that achieves Heisenberg scaling only for a total time O(γA−1)\mathcal O(γ_A^{-1})O(γA−1​).
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-ZZZ 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-ZZZ 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.
Qubits with strongly biased noise, in which phase-flip errors are orders of magnitude more frequent than bit-flips, arise both naturally, as in electron and nuclear spins, and by engineering, as in stabilized cat qubits. This noise structure holds the promise of reducing the daunting hardware overhead of fault-tolerant quantum computing, but exploiting it requires physical operations that do not convert frequent phase-flips into rare bit-flips. In this review, we analyze the most prominent fault-tolerant protocols for biased-noise qubits, organized according to the available set of such bias-preserving operations. When this set is restricted to the CZ gate together with preparation and measurement in the X basis, we show that the complexity of the required syndrome extraction gadgets essentially cancels the benefit of the noise bias: at experimentally relevant error rates, one may as well ignore the bias and rely on standard error correction designed for depolarizing noise. The situation changes drastically when a bias-preserving CX gate is available: the hierarchy of errors can then be reflected in the structure of the code, with frequent phase-flips corrected by a dedicated high-threshold code and rare bit-flips by concatenation with a high-rate code. The same hierarchy also enables hardware-efficient preparation of magic states. Finally, as a bias-preserving CX is forbidden in naturally biased platforms and challenging in engineered ones, we present a measurement-based architecture in which a high-fidelity quantum non-demolition readout of multi-qubit Pauli Z operators takes its place, extending these overhead reductions to a much broader range of physical platforms.
Entangled quantum probes can achieve Heisenberg-limited measurement precision, but this advantage is typically destroyed by noise. We address this issue by introducing a framework that we call encoded quantum signal processing, which unifies quantum error detection and quantum signal processing into an effective single-qubit framework, and provides a paradigm for constructing logical sensors that are robust to noise while remaining sensitive to the signal of interest. We show that encoding sensor qubits into a repetition code and using syndrome measurements as a signal-processing primitive restores Heisenberg scaling under realistic noise, without applying recovery operations. We prove that product-state sensing with syndrome post-processing is fundamentally limited to standard quantum limit (SQL) scaling, and develop four protocols that overcome this barrier through entanglement or sequential signal amplification, achieving Heisenberg-limited precision with exponential error suppression in code distance. For spatially inhomogeneous fields, Bayesian marginalization preserves Heisenberg scaling provided noise decreases sufficiently with system size. The underlying mechanism, which we formalize as encoded quantum signal processing, reduces multi-qubit metrology to an effective single-qubit problem where syndrome measurement implements nonlinear signal transformations. Numerical simulations validate the theoretical predictions: syndrome-based inference achieves near-Heisenberg scaling at noise levels where bare probes approach the SQL, and a concatenated protocol maintains this scaling under joint transverse noise and longitudinal inhomogeneities.
We explore the feasibility of fault-tolerant quantum computation using the bit-flip repetition code in a biased noise channel where only the bit-flip error can occur. While several logic gates can potentially produce phase-flip errors even in such a channel, we propose bias-preserving implementation of S , H , controlled- Z (CZ), and R z gates. We demonstrate that our scheme improves the computational precision in several tasks such as the time evolution of quantum systems and variational quantum eigensolver. Published by the American Physical Society 2024
Extensive quantum error correction is necessary in order to perform a useful computation on a noisy quantum computer. Moreover, quantum error correction must be implemented based on imperfect parity check measurements that may return incorrect outcomes or inject additional faults into the qubits. To achieve fault-tolerant error correction, Shor proposed to repeat the sequence of parity check measurements until the same outcome is observed sufficiently many times. Then, one can use this information to perform error correction. A basic implementation of this fault tolerance strategy requiresΩ(rd2)\Omega (r d^{2})Ω(rd2)parity check measurements for a distance-dddcode defined byrrrparity checks. For some specific highly structured quantum codes, Bombin has shown that single-shot fault-tolerant quantum error correction is possible using onlyrrrmeasurements. In this work, we consider a phenomenological noise model for parity check measurements assuming that each bit of a codeword and the measurement outcome suffer from independent bit flips with some error rateppp. For this model, we demonstrate that fault-tolerant quantum error correction can be achieved usingO(dlog⁡(d))O(d \log (d))O(dlog(d))measurements for any code with distanced≥Ω(nα)d \geq \Omega (n^\alpha)d≥Ω(nα)for some constantα>0\alpha > 0α>0. Moreover, we prove the existence of a sub-single-shot fault-tolerant quantum error correction scheme using fewer thanrrrmeasurements. In some cases, the number of parity check measurements required for fault-tolerant quantum error correction is exponentially smaller than the number of parity checks defining the code. The short measurement sequences constructed generally have high weight and our phenomenological noise model is not realistic in this regime. Our error correction strategy could find applications to small codes and LDPC codes if one can manage to keep the weight of the measured parity checks low.