Reinforcement Learning with Neural Networks for Quantum Multiple Hypothesis Testing

Sarah Brandsen1, Kevin D. Stubbs2, and Henry D. Pfister2,3

1Department of Physics, Duke University, Durham, North Carolina 27708, USA.
2Department of Mathematics, Duke University, Durham, North Carolina 27708, USA
3Department of Electrical and Computer Engineering, Duke University, Durham, North Carolina 27708, USA

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Abstract

Reinforcement learning with neural networks (RLNN) has recently demonstrated great promise for many problems, including some problems in quantum information theory. In this work, we apply RLNN to quantum hypothesis testing and determine the optimal measurement strategy for distinguishing between multiple quantum states $\{ \rho_{j} \}$ while minimizing the error probability. In the case where the candidate states correspond to a quantum system with many qubit subsystems, implementing the optimal measurement on the entire system is experimentally infeasible.

We use RLNN to find locally-adaptive measurement strategies that are experimentally feasible, where only one quantum subsystem is measured in each round. We provide numerical results which demonstrate that RLNN successfully finds the optimal local approach, even for candidate states up to 20 subsystems. We additionally demonstrate that the RLNN strategy meets or exceeds the success probability for a modified locally greedy approach in each random trial.

While the use of RLNN is highly successful for designing adaptive local measurement strategies, in general a significant gap can exist between the success probability of the optimal locally-adaptive measurement strategy and the optimal collective measurement. We build on previous work to provide a set of necessary and sufficient conditions for collective protocols to strictly outperform locally adaptive protocols. We also provide a new example which, to our knowledge, is the simplest known state set exhibiting a significant gap between local and collective protocols. This result raises interesting new questions about the gap between theoretically optimal measurement strategies and practically implementable measurement strategies.

Reinforcement learning with neural networks (RLNN) has recently demonstrated great promise for many problems, including some problems in quantum information theory. In this work, we apply RLNN to quantum hypothesis testing, where one is given a set of multiple quantum states $\{\rho_{j} \}$ and needs to maximize the probability of guessing the correct state by finding the optimal quantum measurement.

In general, the quantum states may correspond to a large quantum system composed of multiple smaller subsystems and the optimal measurement may require simultaneously measuring all of the quantum subsystems. However, simultaneous measurements on a large number of quantum systems are typically not experimentally feasible to implement. The main result of this work is using RLNN to develop experimentally practical, locally-adaptive methods for quantum hypothesis testing where only one quantum subsystem is measured in each round. We provide numerical results which demonstrate that RLNN successfully finds the optimal local approach, even for candidate states up to 20 subsystems. Furthermore, we demonstrate that these optimal locally adaptive strategies are robust under noise.

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