Compressing Many-Body Fermion Operators Under Unitary Constraints

TL;DR

This paper presents a numerical algorithm for efficiently factorizing many-body fermion operators into sums of squared one-body operators, reducing circuit complexity for quantum simulations.

Contribution

The introduced numerical method improves upon analytical decompositions, often significantly reducing the number of terms needed for fermionic operator factorization.

Findings

The algorithm has complexity comparable to basis transformations.

It often yields fewer squared one-body operators than analytical methods.

Demonstrated application in approximating unitary coupled cluster operators.

Abstract

The most efficient known quantum circuits for preparing unitary coupled cluster states and applying Trotter steps of the arbitrary basis electronic structure Hamiltonian involve interleaved sequences of fermionic Gaussian circuits and Ising interaction type circuits. These circuits arise from factorizing the two-body operators generating those unitaries as a sum of squared one-body operators that are simulated using product formulas. We introduce a numerical algorithm for performing this factorization that has an iteration complexity no worse than single particle basis transformations of the two-body operators and often results in many times fewer squared one-body operators in the sum of squares compared to the analytical decompositions. As an application of this numerical procedure, we demonstrate that our protocol can be used to approximate generic unitary coupled cluster operators and prepare the necessary high-quality initial states for techniques (like ADAPT-VQE) that iteratively construct approximations to the ground state.