Projected Dynamic Programming for Sequential Quantum State Discrimination

Sequential Quantum State Discrimination (SQSD) can be naturally framed as a sequential decision-making problem: at each time step, an agent must decide whether to perform an additional measurement to gather more information or to conclude with an optimal decision based on the current belief. In this paper, we formally cast SQSD into a static-hidden-state Partially Observable Markov Decision Process (POMDP) framework. We demonstrate that this formulation precisely subsumes the conventional minimum-error discrimination (MED) scheme as a special one-step case. Furthermore, we apply a regular grid-based discretization to the continuous belief simplex and approximate the possibly continuous measurement space using a finite library. Then we provide rigorous mathematical bounds on the resulting errors and analyze the computational complexity for both offline planning and online execution. Our analysis confirms that the inherent trade-off between accuracy and complexity, as well as the curse of dimensionality regarding the number of hypotheses, are also prominently observed in the quantum regime. Finally, we provide a working example of binary state discrimination to derive explicit forms of various functions and present numerical simulations for trine state discrimination to visualize the sequential structure of our POMDP-based SQSD.