Matrix encoding method in variational algorithm of calculating eigenvalues and generalized eigenvalues

TL;DR

This paper introduces a quantum variational algorithm that encodes matrix elements into quantum states to compute eigenvalues and generalized eigenvalues efficiently.

Contribution

It presents a novel matrix encoding method within a variational quantum algorithm for eigenvalue problems, with specific circuit complexity analysis.

Findings

The algorithm encodes matrix elements into pure quantum states.

It uses ancilla measurement to construct the loss function probabilistically.

The circuit depth and size are optimized to O(N^2 log N) and O(log N).

Abstract

We propose a variational method for constructing the eigenvalues and generalized eigenvalues for an arbitrary $N\times N$ complex matrix. The quantum part of our algorithm is based on encoding the matrix elements into the pure state of a quantum system and expressing the loss function with optimization parameters in terms of certain probability amplitudes in the superposition state. The principal step of this algorithm is the measurement of the ancilla state that removes all extra terms from the above superposition and allows to probabilistically construct the required loss function along with its derivatives with respect to the optimization parameters. These output data are used to find the new values of optimization parameters for the next iteration of the loss function in the gradient optimization method. The depth and size of the circuit for this algorithm are, respectively, $O(N^2 \log N)$ and $O(\log N)$.