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 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.
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.
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).
Quantum resources can, in principle, enable Heisenberg-limited sensing, yet no-go theorems imply that Heisenberg-limited scaling is generically unattainable in realistic noisy devices. While quantum error correction (QEC) can suppress noise, its use in quantum sensing is constrained by stringent requirements, including prior noise characterization, restrictive signal-noise compatibility conditions, and measurement-based syndrome extraction with global control. Here we introduce a QEC protocol based on indefinite causal order (ICO), providing the first application of ICO to QEC. By coherently placing auxiliary controls and noisy evolution in an indefinite causal order, the resulting noncommutative interference enables an auxiliary system to herald and correct errors in real time, avoiding the entanglement encoding and entanglement readout required by traditional QEC. Furthermore, within the time-reversal regime of the Hamiltonian, our protocol extends correctability to parallel noises where traditional protocols may fail. We rigorously establish the protocol for single-noise and multinoise scenarios and demonstrate its performance in single-qubit, many-body, and continuous-variable platforms. We further identify regimes in which error correction can be implemented entirely by unitary control, without measurements. Our results reveal ICO as a powerful resource for metrological QEC and provide a broadly applicable framework for noise-resilient quantum information processing.
Quantum resources can, in principle, enable Heisenberg-limited (HL) sensing, yet no-go theorems imply that HL scaling is generically unattainable in realistic noisy devices. While quantum error correction (QEC) can suppress noise, its use in quantum sensing is constrained by stringent requirements, including prior noise characterization, restrictive signal-noise compatibility conditions, and measurement-based syndrome extraction with global control. Here we introduce an ICO-based QEC protocol, providing the first application of indefinite causal order (ICO) to QEC. By coherently placing auxiliary controls and noisy evolution in an indefinite causal order, the resulting noncommutative interference enables an auxiliary system to herald and correct errors in real time, thereby circumventing the limitations of conventional QEC and restoring HL scaling. We rigorously establish the protocol for single- and multi-noise scenarios and demonstrate its performance in single-qubit, many-body, and continuous-variable platforms. We further identify regimes in which error correction can be implemented entirely by unitary control, without measurements. Our results reveal ICO as a powerful resource for metrological QEC and provide a broadly applicable framework for noise-resilient 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.