Quantum variants of lattice sieve algorithms are often used to assess the security of lattice based cryptographic constructions. In this work we provide a heuristic, non-asymptotic, analysis of the cost of several algorithms for near neighbour search on high dimensional spheres. These algorithms are used in lattice sieves. We design quantum circuits for near neighbour algorithms and provide software that numerically optimises algorithm parameters according to various cost metrics. Using this software we estimate the cost of classical and quantum near neighbour search on spheres. We find that quantum search may provide a small speedup in dimensions of cryptanalytic interest, but only under exceedingly optimistic physical and algorithmic assumptions.

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