Reducing molecular electronic Hamiltonian simulation cost for Linear Combination of Unitaries approaches

TL;DR

This paper explores methods to optimize the Linear Combination of Unitaries (LCU) approach for simulating molecular electronic Hamiltonians on quantum computers, focusing on minimizing the 1-norm of coefficients to reduce circuit complexity.

Contribution

It introduces new techniques for generating LCU decompositions, including symmetry-based reduction and extension to the interaction picture, to lower the 1-norm and improve simulation efficiency.

Findings

Lowest possible LCU 1-norm is half of the spectral range.

Practical techniques can generate near-optimal LCU decompositions.

Interaction picture extension significantly reduces the 1-norm.

Abstract

We consider different Linear Combination of Unitaries (LCU) decompositions for molecular electronic structure Hamiltonians. Using these LCU decompositions for Hamiltonian simulation on a quantum computer, the main figure of merit is the 1-norm of their coefficients, which is associated with the quantum circuit complexity. It is derived that the lowest possible LCU 1-norm for a given Hamiltonian is half of its spectral range. This lowest norm decomposition is practically unattainable for general Hamiltonians; therefore, multiple practical techniques to generate LCU decompositions are proposed and assessed. A technique using symmetries to reduce the 1-norm further is also introduced. In addition to considering LCU in the Schr\"odinger picture, we extend it to the interaction picture, which substantially further reduces the 1-norm.