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.
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.
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 error correction (QEC) codes have emerged as a powerful tool to protect quantum-enhanced metrology against noise. However, the ability to correct errors alone does not guarantee high metrological sensitivity, as the encoded states may become insensitive to the parameter of interest. Here we show that this limitation can be overcome by exploiting an intrinsic freedom of QEC codes: for a fixed set of correctable errors, the Knill-Laflamme conditions admit an equivalence class of encodings. When the correctable noise possesses unitary symmetries, these symmetries generate continuous transformations within this class, allowing systematic optimization of the encoding to increase the quantum Fisher information while preserving the correctable set of noise. Based on this observation, we develop a symmetry-based optimization approach and derive criteria identifying when such optimization can enhance metrological sensitivity. In particular, for stabilizer-sum Hamiltonians, symmetry optimization can convert a code with vanishing QFI into one achieving the standard quantum limit in general or even Heisenberg scaling in specific cases, illustrating the power of symmetry optimization for QEC-assisted quantum metrology.
We introduce a quantum metrology scheme in which partial quantum error correction suppresses local noise while preserving sensitivity beyond the standard quantum limit. The probes are superpositions of energetically distinct stabilizer sectors, whose sensing coherence is protected by measuring only a subset of local checks. We propose an adaptive weight strategy to balance two competing effects. Increasing the stabilizer-generator weight suppresses parallel noise to O(pδ), with δ increasing with the weight, but also increases susceptibility to perpendicular noise. Our main theorem establishes a sufficient noise condition under which adapting this weight to the system size preserves superlinear quantum Fisher information under simultaneous independent X and Z noise. We illustrate this strategy using repetition, Bacon-Shor, and toric codes, and analyze noise both before and after phase imprinting. In all examples, the error-correction checks and phase-imprinting operators are local and require no nonlocal connectivity.
Quantum error correction has become an indispensable tool for restoring Heisenberg-limited precision in noisy quantum metrology. Existing protocols, however, almost exclusively focus on correcting noise during the signal-encoding stage and implicitly assume that the probe is measured immediately after sensing. In many quantum information processing tasks, the encoded probe must instead be stored before subsequent quantum operations, during which environmental noise can significantly degrade the accumulated metrological information. Here, we propose a correlated-noise correction (CNC) protocol for protecting quantum probes during the storage stage. By correlating probe errors with auxiliary qubits through fixed two-body entangling gates, memory errors are converted into measurable syndromes that are extracted only once after storage. We show that the protocol naturally extends from single-qubit to multi-qubit probes and protects the stored quantum Fisher information against dephasing, bit-flip, and amplitude-damping noise. Furthermore, we demonstrate that preserving the quantum Fisher information does not necessarily require restoring the entire quantum state when the probe is measured immediately after storage, whereas full state recovery becomes essential for subsequent rounds of quantum signal processing. Our results establish correlated-noise correction as a practical framework for protecting metrological information during quantum memory and provide a useful building block for sensing-enabled quantum information processing.
Quantum error correction has been successfully applied to enhance the precision of parameter estimation in the presence of noise. Nonetheless, existing methods require a number of noiseless, controllable ancillae and lack efficient encoding and decoding procedures. In this Letter, we demonstrate that subsystem error correction provides a new direction that can substantially simplify the metrological protocol. We derive general conditions under which subsystem stabilizer codes achieve the Heisenberg limit and show that, for broad classes of noise, this can be realized by syndrome-free protocols using at most a single ancilla qubit. Furthermore, we extend this framework to dynamical error correction and show that Floquet codes can protect time-dependent metrological signals in reaching the Heisenberg limit.
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-d code suppresses logical failures to order p(d+1)/2. We show that, for a decay rate γA, a total sensing time T and a mean photon number budget N, and for sufficiently small residual signals, the Fisher information scales as NT2 over interrogation times of order 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).