Explaining Robust Quantum Metrology by Counting Codewords

TL;DR

Addressing noisy quantum metrology, the paper shows that robust metrology using CSS codeword superpositions without active error correction yields a robustness bound tied to the dual code weight distribution. By leveraging MacWilliams identities, it demonstrates that HL scaling is possible only when the signal Hamiltonian is Not in the Lindblad Span, with dephasing channels for nontrivial linear codes typically forbidding HL. HL can be recovered in noise channels orthogonal to the Hamiltonian (X-channel) or in pure bit-flip scenarios, while general mixed channels maintain HL impossibility unless contrived conditions hold. This work forges a principled link between coding theory and quantum metrology, placing fundamental limits on linear stabilizer codes for robust sensing and guiding protocol design through stabilizer syndrome measurements.

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, $θ$. Given $N$ copies of the probe state, we derive the probability that all stabilizer measurements return $+1$, which depends on $θ$. The uncertainty in $θ$ (estimated from these measurements) is bounded by a new quantity, the \textit{Robustness 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.

Figures (3)

• Figure 1: The circuit describing our robust sensing protocol. First, we prepare $N$ copies of the encoded $N$-qubit probe state $\rho$. Then, each copy evolves due to the sensing Hamiltonian for time $dt$. Finally, all the $\hat{X}$ stabilizers are measured. This is repeated for several increasing time steps.
• Figure 2: Cramer-Rao bound as a function of time for the $N=7$ GHZ state. Here, we see that only in the X-Channel case does the GHZ achieve the HL \ref{['bpcfi']}, with $W_{C^{\perp},2}=\binom{N}{2}$ in \ref{['QFIpurs']}. In the Dephasing and Hadamard Channel cases, it is not even reaching the SQL \ref{['bpcfi']}, with $W_{C^{\perp},2}=0$ in \ref{['QFIpurs']}.
• Figure 3: Cramer-Rao bound as a function of time for the $\ket{0}_{L}$ state of the Steane code. The Steane code, due to its low $\mathcal{Q}_{\rm pure}$ given by \ref{['QFIpurs']}, can only achieve the SQL for the X-Channel Case, as predicted by Theorem 2.

Theorems & Definitions (7)

• Remark 1
• Theorem 1
• Theorem 2
• Theorem 3: Impossibility of Heisenberg-Scaling for Noisy Linear Probes
• Remark 2
• Lemma 4
• Lemma 5