Multi-layer quantum search and inclusion of NP into BQP
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Abstract
In this work, we present a multi-layer quantum search method that generates an exponential speedup of the standard Grover's algorithm. As direct applications, any NP problems can be solved efficiently on a quantum circuit with only polynomial gate complexity. In particular, such multi-layer search can solve the factoring problem with an exponential speedup, providing an alternative to Shor's algorithm. Our results show that the exponential speedup of quantum circuits is ubiquitous, and Grover's search is much more powerful than that has been demonstrated. With no contradiction to the quadratic optimality of single-layer query complexity, the great potential of Grover's search is fully released by such multi-layer search design.