Exponential advantage in quantum sensing of correlated parameters

Quantum 10, 1963 (2026) · DOI: 10.22331/q-2026-01-14-1963 · arXiv: 2504.21745

License: CC BY 4.0.

Abstract

Conventionally in quantum sensing, the goal is to estimate one or more unknown parameters that are assumed to be deterministic—that is, they do not change between shots of the quantum-sensing protocol. We instead consider the setting where the parameters are stochastic: each shot of the quantum-sensing protocol senses parameter values that come from independent random draws. In this work, we explore three examples where the stochastic parameters are correlated and show how using entanglement provides a benefit in classification or estimation tasks: (1) a two-parameter classification task, for which an entangled sensor can achieve a large improvement in classification accuracy over an unentangled sensor given a low shot budget (e.g., 97% versus 80% given a budget of 50 shots); (2) an N-parameter estimation task and a classification variant of it, for which an entangled sensor requires just a constant number (independent of N) shots to achieve the same error as an unentangled sensor using exponentially many (∼2^N) shots—with numerical experiments showing that an entangled sensor needs several orders of magnitude fewer shots than an unentangled sensor even in cases when N is small; (3) classifying the magnetization of a spin chain in thermal equilibrium, where the individual spins fluctuate but the total spin in one direction is conserved—this gives a practical setting in which stochastic parameters are correlated in a way that an entangled sensor can be designed to exploit, and numerical experiments showing that even with small N, an entangled sensor can achieve the same accuracy as an unentangled sensor but use ∼100× fewer shots. We also present a theoretical framework for assessing, for a given choice of entangled sensing protocol and distributions to discriminate between, how much advantage the entangled sensor would have over an unentangled sensor. Our work motivates the further study of sensing correlated stochastic parameters using entangled quantum sensors—and since classical sensors by definition cannot be entangled, our work shows the possibility for entangled quantum sensors to achieve an exponential advantage in sample complexity over classical sensors, in contrast to the typical quadratic advantage.

Figures

23 panels with data across 5 figures. Each panel page shows the plot, its columns and its files; each data.csv begins with a header naming the paper, the panel, the source, the license and the provenance route.

Fig. 1

Illustrative figure, no extractable data. Shown in the paper PDF.

Fig. 2

• panel (e): Classification accuracy (%) against the number of samples $S$ of the quantum sensor, for the entangled sensor (black) and the unentangled sensor (green), each point joined to the next by a line. data.csv

Fig. 3

• panel (d): Mean squared error of the estimate of the sum of phases against the number of samples, for the unentangled sensor with $N = 2$ to 8 qubits (coloured curves) and the entangled sensor, the same for any $N$ (black). The dotted line marks an MSE of $10^{-4}$. data.csv
• panel (e): Number of samples required to achieve an MSE of $10^{-4}$ against the number of qubits, for the unentangled sensor (coloured points on a grey dashed line) and the entangled sensor (black points, black dashed line). data.csv
• panel (h): Classification accuracy (%) against the number of samples, for the unentangled sensor with $N = 2$ to 8 qubits (coloured curves) and the entangled sensor, the same for any $N$ (black). The dotted line marks 95%. data.csv
• panel (i): Number of samples required to achieve 95% classification accuracy against the number of qubits, for the unentangled sensor (coloured points on a grey dashed line) and the entangled sensor (black points, black dashed line). data.csv

Fig. 4

• panel (c): Number of samples required to reach 95% classification accuracy against the number of qubits, for the unentangled sensor at $\hbar J\beta = 0.1$, 1.0, 2.0 and 5.0 (coloured dashed lines) and the entangled sensor at any temperature (black). data.csv

Fig. 5

• panel (a-1): Left map of panel (a): absolute value of the characteristic function $\chi$ for the example of the sum of stochastic phases with 2 qubits, over $k_1$ and $k_2$ from $-1.2$ to $1.2$. data.csv
• panel (a-2): Right map of panel (a): absolute value of the feature matrix $F$ of the same 2-qubit example, indexed by the bit-string states $|00\rangle$ to $|11\rangle$. data.csv
• panel (b): Separation value $\Delta_{\mathrm{A,B}}$ against the number of qubits for the sum-of-phases task, for the unentangled sensor (coloured points on a grey dashed line) and the entangled sensor (black points, black dashed line). data.csv
• panel (c-1): Feature matrix map of panel (c) for $\hbar J\beta = 0.1$, $N = 2$: absolute value of the feature matrix $F$ over the bit-string states $|0\dots0\rangle$ to $|1\dots1\rangle$ (rows $|a\rangle$ top to bottom, columns $|b\rangle$ left to right), in reds from 0 to the map's maximum. data.csv
• panel (c-2): Feature matrix map of panel (c) for $\hbar J\beta = 0.1$, $N = 3$: absolute value of the feature matrix $F$ over the bit-string states $|0\dots0\rangle$ to $|1\dots1\rangle$ (rows $|a\rangle$ top to bottom, columns $|b\rangle$ left to right), in reds from 0 to the map's maximum. data.csv
• panel (c-3): Feature matrix map of panel (c) for $\hbar J\beta = 0.1$, $N = 4$: absolute value of the feature matrix $F$ over the bit-string states $|0\dots0\rangle$ to $|1\dots1\rangle$ (rows $|a\rangle$ top to bottom, columns $|b\rangle$ left to right), in reds from 0 to the map's maximum. data.csv
• panel (c-4): Feature matrix map of panel (c) for $\hbar J\beta = 0.1$, $N = 5$: absolute value of the feature matrix $F$ over the bit-string states $|0\dots0\rangle$ to $|1\dots1\rangle$ (rows $|a\rangle$ top to bottom, columns $|b\rangle$ left to right), in reds from 0 to the map's maximum. data.csv
• panel (c-5): Feature matrix map of panel (c) for $\hbar J\beta = 0.1$, $N = 6$: absolute value of the feature matrix $F$ over the bit-string states $|0\dots0\rangle$ to $|1\dots1\rangle$ (rows $|a\rangle$ top to bottom, columns $|b\rangle$ left to right), in reds from 0 to the map's maximum. data.csv
• panel (c-6): Feature matrix map of panel (c) for $\hbar J\beta = 0.1$, $N = 7$: absolute value of the feature matrix $F$ over the bit-string states $|0\dots0\rangle$ to $|1\dots1\rangle$ (rows $|a\rangle$ top to bottom, columns $|b\rangle$ left to right), in reds from 0 to the map's maximum. data.csv
• panel (c-7): Feature matrix map of panel (c) for $\hbar J\beta = 5.0$, $N = 2$: absolute value of the feature matrix $F$ over the bit-string states $|0\dots0\rangle$ to $|1\dots1\rangle$ (rows $|a\rangle$ top to bottom, columns $|b\rangle$ left to right), in blues from 0 to the map's maximum. data.csv
• panel (c-8): Feature matrix map of panel (c) for $\hbar J\beta = 5.0$, $N = 3$: absolute value of the feature matrix $F$ over the bit-string states $|0\dots0\rangle$ to $|1\dots1\rangle$ (rows $|a\rangle$ top to bottom, columns $|b\rangle$ left to right), in blues from 0 to the map's maximum. data.csv
• panel (c-9): Feature matrix map of panel (c) for $\hbar J\beta = 5.0$, $N = 4$: absolute value of the feature matrix $F$ over the bit-string states $|0\dots0\rangle$ to $|1\dots1\rangle$ (rows $|a\rangle$ top to bottom, columns $|b\rangle$ left to right), in blues from 0 to the map's maximum. data.csv
• panel (c-10): Feature matrix map of panel (c) for $\hbar J\beta = 5.0$, $N = 5$: absolute value of the feature matrix $F$ over the bit-string states $|0\dots0\rangle$ to $|1\dots1\rangle$ (rows $|a\rangle$ top to bottom, columns $|b\rangle$ left to right), in blues from 0 to the map's maximum. data.csv
• panel (c-11): Feature matrix map of panel (c) for $\hbar J\beta = 5.0$, $N = 6$: absolute value of the feature matrix $F$ over the bit-string states $|0\dots0\rangle$ to $|1\dots1\rangle$ (rows $|a\rangle$ top to bottom, columns $|b\rangle$ left to right), in blues from 0 to the map's maximum. data.csv
• panel (c-12): Feature matrix map of panel (c) for $\hbar J\beta = 5.0$, $N = 7$: absolute value of the feature matrix $F$ over the bit-string states $|0\dots0\rangle$ to $|1\dots1\rangle$ (rows $|a\rangle$ top to bottom, columns $|b\rangle$ left to right), in blues from 0 to the map's maximum. data.csv
• panel (c-13): Right plot of panel (c): separation value $\Delta_{\mathrm{A,B}}$ against the number of qubits for the XXZ magnetization task, for the entangled sensor at any temperature (black) and the unentangled sensor at $\hbar J\beta = 0.1$ (red) and 5.0 (blue). data.csv

Fig. 6

Illustrative figure, no extractable data. Shown in the paper PDF.

Fig. 7

• panel (1): Number of samples required to reach 95% classification accuracy against $|C|$, for the unentangled sensor (green) and the entangled sensor (black); the grey dashed line is the entangled curve multiplied by 4. The printed $\times 4$ arrow is not drawn. data.csv

Fig. 8

Illustrative figure, no extractable data. Shown in the paper PDF.

Fig. 9

Illustrative figure, no extractable data. Shown in the paper PDF.

Fig. 10

Illustrative figure, no extractable data. Shown in the paper PDF.

Fig. 11

Illustrative figure, no extractable data. Shown in the paper PDF.

Cite

Sridhar Prabhu, Vladimir Kremenetski, Saeed A. Khan, Ryotatsu Yanagimoto, Peter L. McMahon. Exponential advantage in quantum sensing of correlated parameters. Quantum 10, 1963 (2026). https://doi.org/10.22331/q-2026-01-14-1963

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