  By applying Grover's quantum search algorithm to the lattice algorithms of
Micciancio and Voulgaris, Nguyen and Vidick, Wang et al., and Pujol and
Stehl\'{e}, we obtain improved asymptotic quantum results for solving the
shortest vector problem. With quantum computers we can provably find a shortest
vector in time $2^{1.799n + o(n)}$, improving upon the classical time
complexity of $2^{2.465n + o(n)}$ of Pujol and Stehl\'{e} and the $2^{2n +
o(n)}$ of Micciancio and Voulgaris, while heuristically we expect to find a
shortest vector in time $2^{0.312n + o(n)}$, improving upon the classical time
complexity of $2^{0.384n + o(n)}$ of Wang et al. These quantum complexities
will be an important guide for the selection of parameters for post-quantum
cryptosystems based on the hardness of the shortest vector problem.
