Quantum hypothesis testing via robust quantum control

TL;DR

This work develops and compares optimal and robust quantum control strategies for quantum hypothesis testing in open systems, focusing on distinguishing magnetic-field-induced dynamics under environmental noise. By employing GRAPE and its robust variant SAGRAPE, the authors optimize piecewise-constant control pulses to maximize state distinguishability and minimize the Helstrom-bound error $P_e^H$, analyzing both parallel and transverse dephasing and spontaneous emission. The study demonstrates that, while nominal optimal control often improves performance, a robust control design—optimized over a signal-noise window—yields superior robustness to imperfect signals and frequently the lowest average error $⟨P_e^H⟩$ across parameter regimes. These results advance robust quantum hypothesis testing with practical implications for quantum sensing and communication, and point to future work on broader noise sources, multi-objective optimization, and extension to multi-qubit systems with potential gains from quantum Chernoff-type bounds.

Abstract

Quantum hypothesis testing plays a pivotal role in quantum technologies, making decisions or drawing conclusions about quantum systems based on observed data. Recently, quantum control techniques have been successfully applied to quantum hypothesis testing, enabling the reduction of error probabilities in the task of distinguishing magnetic fields in presence of environmental noise. In real-world physical systems, such control is prone to various channels of inaccuracies. Therefore improving the robustness of quantum control in the context of quantum hypothesis testing is crucial. In this work, we utilize optimal control methods to compare scenarios with and without accounting for the effects of signal frequency inaccuracies. For parallel dephasing and spontaneous emission, the optimal control inherently demonstrates a certain level of robustness, while in the case of transverse dephasing with an imperfect signal, it may result in a higher error probability compared to the uncontrolled scheme. To overcome these limitations, we introduce a robust control approach optimized for a range of signal noise, demonstrating superior robustness beyond the predefined tolerance window. On average, both the optimal control and robust control show improvements over the uncontrolled schemes for various dephasing or decay rates, with the robust control yielding the lowest error probability.

Figures (6)

• Figure 1: Schematics of the binary hypothesis testing with quantum control for the detection of an unknown magnetic field in a qubit system. The null hypothesis ($H_0$) represents the background, while the alternative hypothesis ($H_1$) signifies the presence of the magnetic field. A control ($C$) is applied to enhance the distinguishability of states $\rho_0$ and $\rho_1$, allowing them to evolve under the respective influences of $H_0$ and $H_1$.
• Figure 2: The error probability under dephasing dynamics and spontaneous emission. Panels (a) and (d) correspond to parallel dephasing, panels (b) and (e) represent transverse dephasing, while panels (c) and (f) illustrate spontaneous emission. In panels (a)-(c), the error probability is shown as a function of the target time $T$ at a dephasing rate of $\gamma=0.1$. In panels (d)-(f), the error probability is depicted as a function of the dephasing rate $\gamma$ at a fixed target time of $T=10$. The solid lines and blue circles represent the error probability $P_e^H$ under controls found using the GRAPE and SAGRAPE methods, respectively, when a Helstrom measurement is employed. The green dash-dot lines indicate $P_e^H$ under free evolution ("no control"). The black dashed lines and blue triangles depict the error probability $P_e$ obtained when a fixed local measurement is employed (remarked in the legend as "FL").
• Figure 3: The trajectories of quantum states on the Bloch sphere are shown for the null and alternative hypotheses with optimal controls. The Helstrom measurement is used in (a)-(c), while the fixed local measurement is used in (d)-(f). The cases of parallel dephasing (Panels (a,d)), transverse dephasing (Panels (b,e)), and spontaneous emission (Panels (c,f)) are shown in different columns. The dephasing rate is set to $\gamma=0.1$ and the target time is $T=19$. Data points with darker colors correspond to the quantum states at later time $t$. The blue and red arrows denote the final states $\rho_0(T)$ and $\rho_1(T)$, respectively.
• Figure 4: The pulse profiles optimized with or without the signal noise using the GRAPE method. In (a)-(c), the pulse profiles are for the perfect signal, while in (d)-(f), the pulse profiles are designed to have robustness against the signal noise $d\omega\in[-0.1,0.1]$. The cases of parallel dephasing (Panels (a,d)), transverse dephasing (Panels (b,e)), and spontaneous emission (Panels (c,f)) are shown in different columns. The target time for the optimization is set to $T=10$.
• Figure 5: Robustness of the optimal controls in the presence of signal noise. In (a)-(c), the error probability, $P_e^H$, is shown as a function of the signal noise $d\omega$, which is evenly sampled in the range $[-0.1,0.1]$. In (d)-(f), the average error probability, $\langle{P_e^H}\rangle$, over the interval $[-\pi/2T,\pi/2T]$ is plotted as a function of the dephasing rate $\gamma$. The target time for the optimization is set to $T=10$.