Efficient quantum circuits for arbitrary sparse unitaries

TL;DR: A simple unified proof that the quantum Turing machine model, the quantum circuit model, anyonic models, permutational quantum computation, and discrete time quantum walks as special cases are all contained in BQP is obtained.

Abstract: Arbitrary exponentially large unitaries cannot be implemented efficiently by quantum circuits. However, we show that quantum circuits can efficiently implement any unitary provided it has at most polynomially many nonzero entries in any row or column, and these entries are efficiently computable. One can formulate a model of computation based on the composition of sparse unitaries which includes the quantum Turing machine model, the quantum circuit model, anyonic models, permutational quantum computation, and discrete time quantum walks as special cases. Thus, we obtain a simple unified proof that these models are all contained in BQP. Furthermore, our general method for implementing sparse unitaries simplifies several existing quantum algorithms.