Exponential advantage in quantum sensing of correlated parameters

TL;DR

The paper tackles quantum sensing when the parameters are stochastic and correlated, showing that entanglement can yield an exponential reduction in the number of samples needed for classification and estimation tasks. It develops a general framework based on characteristic functions and a feature matrix to determine when an entangled sensor outperforms any product-state sensor, with rigorous propositions and theorems outlining hard/easy regimes. The authors demonstrate concrete instances, including a two-qubit Gaussian-distribution task and an exponential-sample-savings scenario for summing correlated phases with a GHZ sensor, as well as a conserved-quantity sensing example in an XXZ spin-chain. They further discuss a theoretical path to extrapolate: under suitable correlations and encoding strategies, entangled sensing can surpass classical limits by providing exponential sample-efficiency, with implications for biological sensing, communications, and many-body physics. The work thus motivates developing entangled quantum sensors for correlated stochastic parameters and offers a framework to identify and design protocols that maximize sample efficiency.

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 there is an advantage in the low-shot regime; (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 accuracy as an unentangled sensor using exponentially many (${\sim}2^N$) shots; (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. 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 (11)

• Figure 1: Quantum sensing of deterministic versus stochastic parametersa) Conventional quantum sensing tasks estimate or classify an (most generally) $N$-dimensional parameter vector $\bm{\theta}$ using samples obtained from a quantum sensor. For each sample, the same deterministic$\bm{\theta}$ is received by the quantum sensor. b) In quantum sensing of stochastic parameters, which we consider in this work, the $N$-dimensional parameter vector $\bm{\theta}$ received by the quantum sensor is sampled from a probability distribution $\mathcal{P}_{\Phi}(\bm{\theta})$. Each sample obtained from the quantum sensor experiences a different, stochastic$\bm{\theta}$. Estimates made using quantum sensor measurement outcomes in this setting consequently depend on the properties of the underlying distribution $\mathcal{P}_{\Phi}(\bm{\theta})$. In this work, we show the advantages of entanglement in the quantum sensor in this latter setting.
• Figure 2: Example of two-qubit stochastic sensing task, discriminating two Gaussian distributions of correlated parameters $\bm{\theta}=(\theta_1,\theta_2)$. a) Conventional sensing for binary distribution discrimination. A fixed sample $\bm{\theta}_{L}$ sampled from one of two Gaussian distributions $L=\rm A, \rm B$ is received by the quantum sensor $S$ times. The sensing interaction described by the unitary operator $U_{\rm sense}(\bm{\theta}) = e^{-\frac{i}{2}(\theta_1 \hat{\sigma}^z_1 + \theta_2 \hat{\sigma}^z_2)}$. Using measurement outcomes averaged over $S$ samples, a maximum likelihood estimator (MLE) is used to determine which class the samples $\bm{\theta}_{L}$ originated from. b) Same task as a), but where the parameters are stochastic. Now each new shot of the quantum-sensing protocol receives a new stochastic vector $\bm{\theta}_L$. c) Circuit diagram for the unentangled sensor, which leads to a qubit distribution that is spread out over the Bloch sphere due to the stochasticity of the parameters. d) Circuit diagram for the entangled sensor (see Appendix \ref{['s: corrgauss']} for more details). By preparing an appropriate entangled state (here the Bell state), the entangled sensor is able to measure directly along the low-noise axis $\theta_1-\theta_2$. The decoding unitary disentangles the two qubits and maps the information of the class to the state of the first qubit, which is measured. The second qubit is always in the ground state $\ket{0}$, and hence does not need to be measured. The delocalization of the first qubit state is then suppressed by the ability of the entangled state to avoid noise along the large noise axis $\theta_1+\theta_2$. e) Classification accuracy performance for the schemes in c) and d). The entangled sensor outperforms the unentangled sensor, achieving a higher classification accuracy for a given number of shots. Insets show the histogram of the MLE prediction, for 50 samples of the quantum sensor. Values less than $0.5$ are predicted to be in Class A. Values greater than $0.5$ are predicted to be in Class B.
• Figure 3: Exponential advantage for estimation and classification tasks involving sensing $N$ correlated stochastic parameters with $N$ qubits.a) Random phases are sampled an $N$-parameter probability distribution whose marginal distributions are uniform over a full period $\mathcal{U}(0,2\pi)$, but which has a fixed value of $\sum_i\theta_i$. We consider an estimation task where the goal is to estimate this sum (which is assumed to be close to $0$) and a binary classification task where the two classes to be distinguished have different values $\pm C$ of this constraint, where $C=0.3$ for the simulation results. b) Optimal sensing protocol using an unentangled $N$-qubit quantum sensor for the estimation task. Measurement outcomes $\{x_k\}$ computed using $S$ shots estimate bit-string probabilities of the qubits. The estimate of $C$ is obtained by applying a linear layer on these results $\{x_k\}$. c) Optimal sensing protocol using an entangled $N$-qubit GHZ-state. The measurement outcomes is the estimate of qubit excitation probability of a single qubit; which is then used to obtain the estimate of the sum of phases (see Appendix \ref{['s: exponential advantage']}). d) Mean squared error (MSE) of the estimate to the true value as a function of the number of samples of the quantum-sensing protocol. The performance of the entangled-sensing protocol is independent of $N$. e) Samples required to achieve an MSE of $10^{-4}$, for the entangled and unentangled quantum-sensing protocols. The sample requirement scales exponentially in $N$ for the unentangled quantum-sensing protocol. f) Optimal sensing protocol using an unentangled $N$-qubit quantum sensor for the binary classification task. The qubit states are spread out entirely over the Bloch spheres prior to measurement. g) Optimal sensing protocol using an entangled $N$-qubit GHZ-state. Like before, all the information is encoded in a single qubit. Qubit states before measurement now show no spreading over the Bloch sphere for different random samples. h) Classification accuracy as a function of the number of samples of the quantum-sensing protocol for different number of qubits. The performance of the entangled-sensing protocol is independent of $N$. i) Samples required to achieve a classification accuracy of $95\%$, for the entangled and unentangled quantum-sensing protocols. The sample requirement scales exponentially in $N$ for the unentangled quantum-sensing protocol.
• Figure 4: Discriminating two distributions with different total magnetization of classically interacting spinsa) An $N$-qubit quantum sensor senses the magnetic field of a system of $N$ classical spins at temperature $T$, interacting via the XXZ Hamiltonian. Qubit $i$ of the quantum sensor is only sensitive to the local Z component of the magnetic field, which in turn is proportional to the $S_i^z$ of the $i^{\rm th}$ spin. The XXZ interaction conserves the total spin component in the Z direction. b) The classification task is then to distinguish between two values of the conserved quantity $\sum_i S_i^z = \pm M$. Classical spin system configurations follow the Boltzmann distribution at finite $T$, and are sampled using a Metropolis Monte Carlo algorithm. c) Number of samples required by the unentangled and entangled protocols to reach 95% classification accuracy. Fluctuations in $S_i^z$ values among samples results in the unentangled sensor requiring a number of samples dependent on $N$ and the $T$. Increasing $T$ increases the amount of local fluctuations, making the task harder for the unentangled sensor. In contrast, the entangled sensor is insensitive to local fluctuations and requires a constant number of samples independent of $N$ and $T$. In these simulations, we set the value of anisotropy to be $\Delta = 0.75$. The value of $\gamma \eta \tau$ sets the scaling between the $S^z_i$ values the phase $\theta_i$ experienced by the qubit. We set this value to be $\pi$, such that the total range of phase that can be experienced is between $-\pi$ and $\pi$ corresponding to the $S^z_i$ values ranging between $-1$ and $1$.
• Figure 5: Framework for sensing advantage of stochastic parametersa) The feature matrix is constructed based on the difference of the characteristic functions and the choice of $U_{\rm sense}$. In particular, $U_{\rm sense}$ determines at what points we take the difference of the two characteristic functions. For our local choice of $U_{\rm sense}$ with $\hat{G}_j = 1/2 (\hat{\sigma}_j^z + \mathbb{\hat{I}})$ in Eq. \ref{['eq:usense general']}, the point we evaluate at is simply given by the difference of the vector of bits from the pair of states $\ket{i},\ket{j}$ in the eigenbasis of $U_{\rm sense}$. Notably, many different pairs of bit-strings can produce the same $\bm{k}$ (for example, $2^N$ pairs of identical states produce an all-zero vector for $\bm{k}$), so $U_{\rm sense}$ can evaluate the same point in the characteristic functions many times. b) We can define the separation value as the difference in the expected value of a projector $\hat{O}$ between the two distributions of the parameters $\bm{\theta}$. For a given choice of $\hat{O}$ and $\rho_{\text{probe}}$, the number of shots required to distinguish the two distributions scales at least as $\sim 1/\Delta_{\rm A,\rm B}$. This separation value $\Delta_{\rm A,\rm B}$ is equivalent to the sum over the elements of the Schur product between the matrices of $\rho_{\text{probe}},F,\hat{O}$ written in the eigenbasis of $U_{\rm sense}$. Due to the limited structure of product state probes and unentangled measurement bases, sufficiently concentrated feature matrices cannot be efficiently distinguished by product states, as they can only weigh most entries exponentially little, and so produce an exponentially small separation value. Conversely, an entangled probe state and measurement basis can pick out specific entries to weigh substantially more than others, and this produces a non-vanishing separation value. For our example in the plot, we present the separation value vs. number of qubits and feature matrices for an entangled sensor (GHZ-state probe and measurement basis) and an unentangled sensor (Hadamard probe state - the optimal unentangled probe - and product basis). The task is the same as in Sec \ref{['sec:exp']}. c) Plotting the feature matrices for XXZ tasks at different temperatures. The concentration of features occurs more strongly and quickly for the higher temperature case, though both eventually plateau due a bounded amount of noise possible with a fixed temperature. On the right, we display the best possible separation value achievable using entangled and unentangled probe states for different temperatures.