Algebraic Compression of Quantum Circuits for Hamiltonian Evolution

TL;DR

This paper introduces an algebraic method to compress quantum circuits for Hamiltonian evolution, reducing circuit depth for specific models, which enhances efficiency in quantum simulations.

Contribution

The paper presents a novel algebraic compression algorithm that simplifies Trotterized quantum circuits for certain Hamiltonians, achieving fixed-depth evolution.

Findings

Successfully compresses Trotter steps into a single quantum gate block

Demonstrates fixed-depth evolution for specific spin models

Applies to adiabatic state preparation of the transverse field Ising model

Abstract

Unitary evolution under a time dependent Hamiltonian is a key component of simulation on quantum hardware. Synthesizing the corresponding quantum circuit is typically done by breaking the evolution into small time steps, also known as Trotterization, which leads to circuits whose depth scales with the number of steps. When the circuit elements are limited to a subset of SU(4) -- or equivalently, when the Hamiltonian may be mapped onto free fermionic models -- several identities exist that combine and simplify the circuit. Based on this, we present an algorithm that compresses the Trotter steps into a single block of quantum gates. This results in a fixed depth time evolution for certain classes of Hamiltonians. We explicitly show how this algorithm works for several spin models, and demonstrate its use for adiabatic state preparation of the transverse field Ising model.