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arXiv:2212.06285

Finite-round quantum error correction on symmetric quantum sensors

Yingkai Ouyang, Gavin K. Brennen

Published Dec 12, 2022• 4 citations
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Abstract

In quantum sensing using $N$ entangled probes, the variance of the estimated signal strength $\hatθ$ scales like $ Θ(N^{-2})$ at the Heisenberg limit, which is a quadratic improvement over the standard quantum limit, and is the maximum quantum advantage over classical methods. This limit remains elusive, however, because of the inevitable presence of noise decohering quantum sensors. Here, we introduce a quantum sensing protocol \texttt{ECSense} based on permutation-invariant quantum error correction (QEC) codes that support tunable code parameters to suit the physical noise model. We show that when the signal duration is much shorter than decoherence times, such that errors only accumulate during the $N$ qubit probe state preparation and idle stage, then the estimate's variance of $Θ( N^{-3/2})$ is achievable in the presence of $Θ(\sqrt{N})$ errors while the Heisenberg limit is achieved when the number of errors is a constant. In the more challenging setting where errors also occur during signal accumulation, we prove using a non-Markovian QEC strategy, that even for a linear number of deletion errors, a variance approaching the Heisenberg limit is still achievable. We illustrate a concrete way to implement our protocol on near-term quantum hardware using cavity-assisted geometric phase gates.

DOI
10.48550/arxiv.2212.06285

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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 investigate the precision limits and optimal protocols for sensing single qubit signals in the presence of erasure noise. We study a hierarchy of precision limits achievable with metrological strategies of differing complexity, and identify the optimal protocol for each. The detectability of erasure noise is shown to lead to enhanced precision limits and simplified sensing protocols. For energy gap estimation, we demonstrate that a simple product-state strategy with continuous erasure detection yields significant improvements, outperforming optimal passive entangled protocols over a broad range of finite system sizes. We show that for other single-qubit signals, quantum error correction provides a substantial advantage by correcting the dominant erasure processes, and can restore Heisenberg-limited precision in certain erasure configurations. As a byproduct of our analysis, we find erasure-conversion schemes for qubits subject to thermal noise that attain the corresponding ultimate precision limits.
Quantum error correction (QEC) is theoretically capable of achieving the ultimate estimation limits in noisy quantum metrology. However, existing quantum error-correcting codes designed for noisy quantum metrology generally exploit entanglement between one probe and one noiseless ancilla of the same dimension, and the requirement of noiseless ancillas is one of the major obstacles to implementing the QEC metrological protocol in practice. Here we successfully lift this requirement by explicitly constructing two types of multiprobe quantum error-correcting codes, where the first one utilizes a negligible amount of ancillas and the second one is ancilla free. Specifically, we consider Hamiltonian estimation under Markovian noise and show that (i) when the Heisenberg limit (HL) is achievable our codes can achieve the HL and its optimal asymptotic coefficient and (ii) when only the standard quantum limit (SQL) is achievable (even with arbitrary adaptive quantum strategies) the optimal asymptotic coefficient of the SQL is also achievable by our codes under slight modifications.