Accelerating Quantum Algorithms with Precomputation

Real-world applications of computing can be extremely time-sensitive. It would be valuable if we could accelerate such tasks by performing some of the work ahead of time. Motivated by this, we propose a cost model for quantum algorithms that allows quantum precomputation; i.e., for a polynomial amount of "free" computation before the input to an algorithm is fully specified, and methods for taking advantage of it. We analyze two families of unitaries that are asymptotically more efficient to implement in this cost model than in the standard one. The first example of quantum precomputation, based on density matrix exponentiation, could offer an exponential advantage under certain conditions. The second example uses a variant of gate teleportation to achieve a quadratic advantage when compared with implementing the unitaries directly. These examples hint that quantum precomputation may offer a new arena in which to seek quantum advantage.