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Nested quantum search and NP-complete problems

N. J. Cerf·L. K. Grover·C. P. Williams
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Abstract

A quantum algorithm is known that solves an unstructured search problem in a number of iterations of order $\sqrt{d}$, where $d$ is the dimension of the search space, whereas any classical algorithm necessarily scales as $O(d)$. It is shown here that an improved quantum search algorithm can be devised that exploits the structure of a tree search problem by nesting this standard search algorithm. The number of iterations required to find the solution of an average instance of a constraint satisfaction problem scales as $\sqrt{d^α}$, with a constant $α<1$ depending on the nesting depth and the problem considered. When applying a single nesting level to a problem with constraints of size 2 such as the graph coloring problem, this constant $α$ is estimated to be around 0.62 for average instances of maximum difficulty. This corresponds to a square-root speedup over a classical nested search algorithm, of which our presented algorithm is the quantum counterpart.

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N. J. Cerf, L. K. Grover, C. P. Williams. 1998-06-23. Nested quantum search and NP-complete problems. https://doi.org/10.1103/physreva.61.032303

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