Quantum circuit for exponentiation of Hamiltonians: an algorithmic description based on tensor products

TL;DR

This paper presents an efficient quantum algorithm for exponentiating Hamiltonians expressed as tensor products of Pauli operators, enabling simpler quantum circuit implementations for simulating quantum system dynamics.

Contribution

It introduces a novel, straightforward method to construct compact quantum circuits for Hamiltonian exponentiation based on tensor products of Pauli operators.

Findings

Developed an algorithmic approach for Hamiltonian exponentiation

Constructed compact quantum circuits for tensor product Hamiltonians

Enhanced efficiency and simplicity of quantum simulation circuits

Abstract

Exponentiation of Hamiltonians refers to a mathematical operation to a Hamiltonian operator, typically in the form e^(-i.t.H), where H is the Hamiltonian and t is a time parameter. This operation is fundamental in quantum mechanics, particularly to evolve quantum systems over time according to the Schrodinger equation. In quantum algorithms, such as Adiabatic methods and QAOA, exponentiation enables efficient simulation of a system dynamics. It involves constructing quantum circuits that approximate this exponential operation. When H=\sum_(p=1)^n H_p , each H_p is defined using the Pauli operator basis, which includes the well-known X, Y, Z and Id gates, i.e., H_p=U_1\otimes U_2\otimes \otimes U_n and U_k\in{Id,X,Y,Z}. In this article, we explore the exponentiation of H_p, specifically e^(-i.t.U_1 \otimes U_2\otimes \otimes U_n ), by introducing an algorithmic approach. We demonstrate a straightforward and efficient method to construct compact circuits that are easy to implement.