Exploiting anticommutation in Hamiltonian simulation

TL;DR

This paper reveals that anti-commutative relations in Hamiltonians can actually simplify quantum simulations, contrary to traditional beliefs, and introduces methods to leverage this for improved accuracy and efficiency.

Contribution

It demonstrates that mutually anti-commutative Hamiltonians are easier to simulate and develops modified linear combination of unitaries methods exploiting anti-commutation.

Findings

Anti-commutative relations can reduce simulation hardness.

Proposed methods improve simulation accuracy for electronic Hamiltonians.

Anti-commutation properties can be exploited to lower algorithmic error.

Abstract

Quantum computing can efficiently simulate Hamiltonian dynamics of many-body quantum physics, a task that is generally intractable with classical computers. The hardness lies at the ubiquitous anti-commutative relations of quantum operators, in corresponding with the notorious negative sign problem in classical simulation. Intuitively, Hamiltonians with more commutative terms are also easier to simulate on a quantum computer, and anti-commutative relations generally cause more errors, such as in the product formula method. Here, we theoretically explore the role of anti-commutative relation in Hamiltonian simulation. We find that, contrary to our intuition, anti-commutative relations could also reduce the hardness of Hamiltonian simulation. Specifically, Hamiltonians with mutually anti-commutative terms are easy to simulate, as what happens with ones consisting of mutually commutative terms. Such a property is further utilized to reduce the algorithmic error or the gate complexity in the truncated Taylor series quantum algorithm for general problems. Moreover, we propose two modified linear combinations of unitaries methods tailored for Hamiltonians with different degrees of anti-commutation. We numerically verify that the proposed methods exploiting anti-commutative relations could significantly improve the simulation accuracy of electronic Hamiltonians. Our work sheds light on the roles of commutative and anti-commutative relations in simulating quantum systems.