Efficient algorithm for fidelity estimation of two quantum states

TL;DR

The paper tackles fidelity estimation between density operators, with a focus on commuting mixed states, where $F(\rho_1,\rho_2)=\mathrm{Tr}\sqrt{\rho_1\rho_2}$. It introduces a resource-efficient quantum protocol that combines Density Matrix Exponentiation (LMR) and Improved Quantum Phase Estimation with a Mach-Zehnder interferometer to estimate this fidelity without full quantum state tomography. The main contribution is a concrete algorithm with overall time complexity $O\left(\frac{N^2}{\epsilon^7}\right)$, along with a discussion showing potential efficiency gains when $\epsilon$ is only polynomially small. This approach offers a practical path for benchmarking quantum devices in the NISQ era by enabling efficient fidelity estimation for commuting states.

Abstract

The fidelity estimation between two quantum states is crucial for quantum computation and information science. However, an efficacious method for this, especially for mixed states and higher-dimensional density matrices, remains elusive. While there are many existing algorithms on computing the fidelity between two pure states, there is not much work on how to obtain the fidelity between two mixed states. Here, we propose an efficient quantum algorithm for the fidelity estimation, based primarily on the density matrix exponentiation and interferometeric scheme for mixed states, with a time complexity of $O(N^2/ε^7)$, where $N$ is the system size and $ε$ is a precision error. Our algorithm may serve as a resource-efficient technique to deduce fidelity of any two (pure or mixed) unknown or known quantum states, when the density matrices of the quantum states commute with each other.

Figures (1)

• Figure 1: Circuit diagram of the Mach-Zehnder interferometer.