Tridiagonal matrix decomposition for Hamiltonian simulation on a quantum computer

TL;DR

This paper introduces an efficient method for representing tridiagonal matrices in the Pauli basis, enabling Hamiltonian simulation circuits on quantum computers without relying on oracles, thus reducing complexity.

Contribution

The work presents a novel procedure for decomposing tridiagonal matrices into Pauli strings with a scalable approach that avoids oracle use in quantum Hamiltonian simulation.

Findings

Gate complexity is lower than oracle-based methods for fewer than 15 qubits.

Requires half the number of qubits compared to traditional approaches.

Applicable to various Hamiltonians derived from tridiagonal matrices.

Abstract

The construction of quantum circuits to simulate Hamiltonian evolution is central to many quantum algorithms. State-of-the-art circuits are based on oracles whose implementation is often omitted, and the complexity of the algorithm is estimated by counting oracle queries. However, in practical applications, an oracle implementation contributes a large constant factor to the overall complexity of the algorithm. The key finding of this work is the efficient procedure for representation of a tridiagonal matrix in the Pauli basis, which allows one to construct a Hamiltonian evolution circuit without the use of oracles. The procedure represents a general tridiagonal matrix $2^n \times 2^n$ by systematically determining all Pauli strings present in the decomposition, dividing them into commuting subsets. The efficiency is in the number of commuting subsets $O(n)$. The method is demonstrated using the one-dimensional wave equation, verifying numerically that the gate complexity as function of the number of qubits is lower than the oracle based approach for $n < 15$ and requires half the number of qubits. This method is applicable to other Hamiltonians based on the tridiagonal matrices.