## A quantum computer amenable sparse matrix equation solver

##### Abstract

Quantum computation offers a promising alternative to classical computing methods in many
areas of numerical science, with algorithms that make use of the unique way in which quantum
computers store and manipulate data often achieving dramatic improvements in performance over
their classical counterparts. The potential efficiency of quantum computers is particularly
important for numerical simulations, where the capabilities of classical computing systems are
often insufficient for the analysis of real-world problems. In this work, we study problems
involving the solution of matrix equations, for which there currently exists no efficient, general
quantum procedure. We develop a generalization of the Harrow/Hassidim/Lloyd algorithm by
providing an alternative unitary for eigenphase estimation. This unitary, which we have adopted
from research in the area of quantum walks, has the advantage of being well defined for any
arbitrary matrix equation, thereby allowing the solution procedure to be directly implemented on
quantum hardware for any well-conditioned system. The procedure is most useful for sparse
matrix equations, as it allows for the inverse of a matrix to be applied with O(N_nz log (N))
complexity, where N is the number of unknowns, and N_nz is the total number of nonzero elements
in the system matrix. This efficiency is independent of the matrix structure, and hence the
quantum procedure can outperform classical methods for many common system types. We show
this using the example of sparse approximate inverse (SPAI) preconditioning, which involves the
application of matrix inverses for matrices with N_nz = O(N). While these matrices are indeed
sparse, it is often found that their inverses are quite dense, and classical methods can require as
much as O(N^3) time to apply an inverse preconditioner.