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arXiv:2503.15743 · IEEE Transactions on Quantum Engineering

Explaining Robust Quantum Metrology by Counting Codewords

Oskar Novak, Narayanan Rengaswamy

Published Jan 1, 2026• 0 citations
arXivHTMLPDF

Abstract

Quantum sensing holds great promise for high-precision magnetic field measurements. However, its performance is significantly limited by noise. The investigation of active quantum error correction to address this noise led to the Hamiltonian-Not-in-Lindblad-Span (HNLS) condition. This states that Heisenberg scaling is achievable if and only if the signal Hamiltonian is orthogonal to the span of the Lindblad operators describing the noise. In this work, we consider a robust quantum metrology setting where the probe state is inspired from CSS codes for noise resilience but there is no active error correction performed. After the state picks up the signal, we measure the code's$\hat{X}$stabilizers to infer the magnetic field parameter,$\theta$. Given$N$copies of the probe state, we derive the probability that all stabilizer measurements return$+1$, which depends on$\theta$. The uncertainty in$\theta$(estimated from these measurements) is bounded by a new quantity, theRobustness Bound, which ties the Quantum Fisher Information of the measurement to the number of weight-2 codewords of the dual code. Through this novel lens of coding theory, we show that for nontrivial CSS code states the HNLS condition still governs the Heisenberg scaling in our robust metrology setting. Our finding suggests fundamental limitations in the use of linear quantum codes for dephased magnetic field sensing applications both in the near-term robust sensing regime and in the long-term fault tolerant era. We also extend our results to general scenarios beyond dephased magnetic field sensing.

DOI
10.1109/tqe.2026.3664680

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The goal of quantum metrology is to estimate an unknown parameter with better-than-classical precision and, ideally, to attain Heisenberg scaling. In realistic settings, however, noise can substantially reduce this advantage and, in many cases, restore standard-quantum-limit-like performance. Preserving the quantum enhancement therefore requires noise-robust strategies, which can be implemented using quantum error correction. In this work, we develop and analyze a quantum-metrology protocol based on Calderbank-Shor-Steane (CSS) codes. The main result is that, for noise perpendicular to the sensing Hamiltonian, discrete-time error correction preserves Heisenberg-like temporal scaling over a finite interrogation-time window whose duration depends on the correction frequency. It is further shown that recovery applied only after the sensing evolution provides no metrological advantage over the corresponding uncorrected noisy protocol in the setup considered here. An analytical framework is developed for arbitrary CSS codes and applied to the Steane and Shor codes, for which the discrete-time corrected dynamics, the Heisenberg-like-to-SQL-like crossover, and the optimal interrogation time are determined and numerically analysed.
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.
Quantum metrology is a promising practical use case for quantum technologies, where physical quantities can be measured with unprecedented precision. In lieu of quantum error correction procedures, near term quantum devices are expected to be noisy, and we have to make do with noisy probe states. We prove that, for a set of carefully chosen symmetric probe states that lie within certain quantum error correction codes, quantum metrology exhibits an advantage over classical metrology even after the probe states are corrupted by a constant number of erasure and dephasing errors. These probe states prove useful for robust metrology not only in the NISQ regime, but also in the asymptotic setting where they achieve Heisenberg scaling. This brings us closer towards making robust quantum metrology a technological reality.
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.
Quantum metrology surpasses the classical precision limit by encoding signals in a probe state such that signal contributions are indistinguishable and their phases accumulate coherently as a collective response. In realistic settings, scalable quantum metrology requires quantum error correction to protect the probe against noise. However, quantum error correction relies on syndrome information that distinguishes errors for identification and correction. Because signals and errors act on the same physical degrees of freedom, there is a structural incompatibility between signal-sensitivity and noise-protection in quantum-error-corrected metrology. We quantify this incompatibility by establishing trade-offs between code distance and the quantum Fisher information of code states for non-degenerate codes, quantum low-density parity-check codes, and generalized Shor codes. We bypass this limitation with asymmetric quantum error correction, in which protection is relaxed along the signal direction while being maintained in complementary directions. We construct such codes for local sensing Hamiltonians, restoring Heisenberg-limited precision while retaining a growing distance and hence protection against local perturbations in complementary directions. Strongly asymmetric quantum low-density parity-check and concatenated asymmetric constructions make the framework sparse, scalable, and continuously tunable. The associated probe states can be prepared by constant-depth adaptive circuits with optimal resource scaling.