Variational Quantum Algorithms for Trace Distance and Fidelity Estimation

TL;DR

This paper presents novel hybrid quantum-classical algorithms, VTDE and VFE, for estimating trace distance and fidelity between quantum states on near-term devices, overcoming traditional exponential complexity barriers.

Contribution

The work introduces two new variational algorithms, VTDE and VFE, that efficiently estimate quantum state distances without assumptions on input states, suitable for near-term quantum hardware.

Findings

High accuracy in numerical simulations

Successful experimental implementations

Avoidance of barren plateau issues

Abstract

Estimating the difference between quantum data is crucial in quantum computing. However, as typical characterizations of quantum data similarity, the trace distance and quantum fidelity are believed to be exponentially-hard to evaluate in general. In this work, we introduce hybrid quantum-classical algorithms for these two distance measures on near-term quantum devices where no assumption of input state is required. First, we introduce the Variational Trace Distance Estimation (VTDE) algorithm. We in particular provide the technique to extract the desired spectrum information of any Hermitian matrix by local measurement. A novel variational algorithm for trace distance estimation is then derived from this technique, with the assistance of a single ancillary qubit. Notably, VTDE could avoid the barren plateau issue with logarithmic depth circuits due to a local cost function. Second, we introduce the Variational Fidelity Estimation (VFE) algorithm. We combine Uhlmann's theorem and the freedom in purification to translate the estimation task into an optimization problem over a unitary on an ancillary system with fixed purified inputs. We then provide a purification subroutine to complete the translation. Both algorithms are verified by numerical simulations and experimental implementations, exhibiting high accuracy for randomly generated mixed states.