Efficient quantum circuits for arbitrary sparse unitaries

TL;DR

This paper presents a method to efficiently implement sparse unitaries in quantum circuits, unifying various quantum computation models and simplifying existing algorithms.

Contribution

It introduces a general approach for implementing polynomially sparse unitaries, encompassing multiple quantum models and providing a unified proof of their inclusion in BQP.

Findings

Efficient implementation of sparse unitaries with polynomially many nonzero entries.

Unified framework covering various quantum computation models.

Simplification of several existing quantum algorithms.

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.