Efficient Variational Quantum Linear Solver for Structured Sparse Matrices

We develop a novel approach for efficiently applying the variational quantum linear solver (VQLS) in the context of structured sparse matri-ces. Such matrices frequently arise during numerical solution of partial differential equations which are ubiquitous in science and engineering. Conventionally, Pauli basis is used for the linear combination of uni-taries (LCU) decomposition of the underlying matrix to facilitate evaluation of the global/local VQLS cost functions. However, the number of LCU terms can scale quadratically with respect to the matrix size <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$N$</tex> in the worst case. Using the heat equation as an example, we show that by using an alternate basis one can better exploit the sparsity and underlying structure of matrix leading to number of tensor product terms which scale only logarithmically with respect to the matrix size. Given this new basis is comprised of non-unitary operators, we employ the concept of unitary completion to design efficient quantum circuits for computing the global/local VQLS cost functions. With one additional <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$n$</tex>- Toffoli gate (where, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$n= \log N$</tex>) and one ancilla qubit, our method could provide an exponential time advantage over conventionally used Pauli basis for some applications. We perform a rigorous analysis of our approach and compare with other related concepts in literature including unitary dilation and Bell basis measurement, and discuss its pros/cons.