Towards a NISQ Algorithm to Simulate Hermitian Matrix Exponentiation

TL;DR

This paper proposes a NISQ-compatible heuristic quantum algorithm for simulating Hermitian matrix exponentiation using low-depth parametrized circuits, enabling practical applications on current quantum hardware.

Contribution

It introduces two strategies for generating low-depth PQCs for Hermitian exponentiation and an ancilla-assisted circuit for process characterization, suitable for NISQ devices.

Findings

Numerical simulations demonstrate feasibility on Bell, GHZ, and Crotonic acid states.

The proposed methods require only accessible gates and low circuit depth.

Results outperform traditional methods like product formula and density matrix exponentiation.

Abstract

A practical fault-tolerant quantum computer is worth looking forward to as it provides applications that outperform their known classical counterparts. However, millions of interacting qubits with stringent criteria are required, which is intractable with current quantum technologies. As it would take decades to make it happen, exploiting the power of noisy intermediate-scale quantum(NISQ) devices, which already exist, is becoming one of current goals. Matrix exponentiation, especially hermitian matrix exponentiation, is an essential element for quantum information processing. In this article, a heuristic method is reported as simulating a hermitian matrix exponentiation using parametrized quantum circuit(PQC). To generate PQCs for simulations, two strategies, each with its own advantages, are proposed, and can be deployed on near future quantum devices. Compared with the method such as product formula and density matrix exponentiation, the PQCs provided in our method require only low depth circuit and easily accessible gates, which benefit experimental realizations. Furthermore, in this paper, an ancilla-assisted parameterized quantum circuit is proposed to characterize and compress a unitary process, which is likely to be applicable to realizing applications on NISQ hardwares, such as phase estimation, principal component analyses, and matrix inversion. To support the feasibility of our method, numerical experiments were investigated via simulating evolutions by Bell state, GHZ state and Hamiltonian of Crotonic acid, which show an experimental friendly result when compared with their conventional methods. As pursuing a fault-tolerant quantum computer is still challenging and takes decades, our work, which gives a NISQ device friendly way, contributes to the field of NISQ algorithms and provides a possibility, exploiting the power with current quantum technology.