Efficient algorithm for generating Pauli coordinates for an arbitrary linear operator

TL;DR

This paper introduces an efficient algorithm for transforming linear operators into Pauli coordinate basis, significantly reducing computational complexity from O(N^4) to O(N^2 log N), with applications in quantum computing.

Contribution

The paper presents a novel algorithm that reduces basis transformation complexity for linear operators in quantum computing from O(N^4) to O(N^2 log N).

Findings

Algorithm reduces transformation complexity to O(N^2 log N).

Demonstrated application to Hamiltonian for relativistic bosons.

Enabled efficient quantum eigensolver computations.

Abstract

Several linear algebra routines for quantum computing use a basis of tensor products of identity and Pauli operators to describe linear operators, and obtaining the coordinates for any given linear operator from its matrix representation requires a basis transformation, which for an $\mathrm N\times\mathrm N$ matrix generally involves $\mathcal O(\mathrm N^4)$ arithmetic operations. Herein, we present an efficient algorithm that for our particular basis transformation only involves $\mathcal O(\mathrm N^2\log_2\mathrm N)$ operations. Because this algorithm requires fewer than $\mathcal O(\mathrm N^3)$ operations, for large $\mathrm N$, it could be used as a preprocessing step for quantum computing algorithms for certain applications. As a demonstration, we apply our algorithm to a Hamiltonian describing a system of relativistic interacting spin-zero bosons and calculate the ground-state energy using the variational quantum eigensolver algorithm on a quantum computer.