Optimal quantum metrology protocols with erasure qubits

Abstract

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 continuous erasure detection strategy yields significant improvements, outperforming optimal entangled protocols even for large numbers of qubits. 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.

Figures (13)

• Figure 1: (a) Schematic of the erasure qubit. Each level of the qubit probe $\ket{i}$ ($i=1,2$) can decay to an erasure state $\ket{e_j}$ with rate $\gamma_{i,j}$. (b) Product protocols (class (i)): initializing and measuring each qubit individually. (c) Entangled protocols (class (ii)): includes also ancilla-free entangled states and measurements. (d) Sequential protocols (class (iii)): the most general entangled and adaptive strategies, in which we allow any quantum fast control and arbitrary entanglement of the probes with ancillas. (e) Continuous erasure detection protocols (class (iv)): the qubits are initialized into a product state and subjected to restricted controls, allowing for erasure detection and a fast reset. The achievable precision in classes (i)–(iv) is bounded by $\mathfrak{I}_N^{(\text{prod})}$, $\mathfrak{I}_N^{(\text{ent})}$, $\mathfrak{I}^{(\text{ult})}$ (asymptotically in large $N T$), and $\mathfrak{I}_N^{(\text{cd})}$, respectively.
• Figure 2: Hierarchy of precision limits for $G=\sigma_{z}.$ (a) Main plot: optimal QFI rate with different strategies as a function of $\Gamma_{\min}/\Gamma_{\max}$. This figure presents the gap between the ultimate limit (orange line), optimal product strategies (blue line) and optimal entangled strategies for different $N$ (red solid and dotted lines correspond to exact and approximate values respectively supp). Continuous erasure detection (green line) outperforms entangled strategies up to a certain $N$, where this crossover depends on the decay rates. Inset: optimal QFI with the different strategies as a function of $N$ for the symmetric case $\Gamma_{\max}=\Gamma_{\min}$. Continuous erasure detection outperforms entangled strategies up to $N \sim 120$. (b) Summary of the precision limits achieved with the different metrological strategies for $G =\sigma_z.$
• Figure 3: Hierarchy of precision limits for $G=\sigma_{x}$ under single erasure noise, as sketched in (a). (b) Optimal QFI rate as a function of $N$. The optimal product strategies for $\omega=0.1 \gamma$ and $0.01\gamma$ (bright and dark blue lines, respectively) approach $16/\gamma$ in the limit $\omega\rightarrow 0.$ In contrast, the optimal continuous detection strategy (dashed green line) attains this value for any $\omega$. While these protocols are limited to SQL scaling, the optimal entangled strategies ($\omega=0.1 \gamma,\: 0.01 \gamma$, given by the pink, red lines respectively) achieve a Heisenberg scaling in $N$, but not in $T$. The ultimate bound (orange line) is the noiseless HL. (c) Summary of the precision limits.
• Figure S1: (a) Sketch of the erasure configuration: $G=\sigma_{x}$ with two erasure states such that $\gamma_{\min}:=\gamma_{1,1}=\gamma_{2,2}<\gamma_{\max}:=\gamma_{1,2}=\gamma_{2,1}.$ (b) $\mathfrak{I}^{(\text{ult})}$ given this erasure configuration, it can be seen that the precision limits lie between the lower and upper bounds of Eq. \ref{['eq:sx_general_ecqfi_bounds']} depending on $\gamma_{\max}/\gamma_{\min}$.
• Figure S2: $\mathfrak{I}^{(\text{ult})}$ for $G=\cos{\theta}\sigma_z+\sin{\theta}\sigma_x$ as a function of $\gamma_{\min}/\gamma_{\max}$. (a) Exact bounds for several values of $\theta$. (b) For $\theta=\pi/8$ (blue) and $3\pi/8$ (red), comparison of the exact bound (solid) with the convex-sum lower bound of Eq. \ref{['eq:convex_sum']} (dashed), and with the analytic upper bound of Eq. \ref{['eq:zero_off_diag']} (dotted).