# PSPACE ⊆ BQP

The complexity class PSPACE includes all computational problems that can be solved by a classical computer with polynomial memory. All PSPACE problems are known to be solvable by a quantum computer too with polynomial memory and are, therefore, known to be in BQPSPACE. Here, we present a polynomial time quantum algorithm for a PSPACE-complete problem, implying that PSPACE is a subset of the class BQP of all problems solvable by a quantum computer in polynomial time. In particular, we outline a BQP algorithm for the PSPACE-complete problem of evaluating a full binary NAND tree. An existing best of quadratic speedup is achieved using quantum walks for this problem, which is still exponential in the problem size. By contrast, we achieve an exponential speedup for the problem, allowing for solving it in polynomial time. There are many real-world applications of our result, such as strategy games like chess or Go. As an example, in quantum sensing, the problem of quantum illumination, that is treated as that of channel discrimination, is PSPACE-complete. Our work implies that quantum channel discrimination, and therefore, quantum illumination, can be performed by a quantum computer in polynomial time.

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