# Variational Quantum Algorithms for Trace Distance and Fidelity Estimation

arXiv: https://arxiv.org/abs/2012.05768 · HTML: https://mlsanity.com/paper/2012.05768/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Dec 2020.

## Built on by (most cited first, 10 of 10)

- [QuEst: Graph Transformer for Quantum Circuit Reliability Estimation](https://mlsanity.com/paper/2210.16724/) (2022)
- [Quantum Pufferfish Privacy: A Flexible Privacy Framework for Quantum Systems](https://mlsanity.com/paper/2306.13054/) (2023)
- [On exploring the potential of quantum auto-encoder for learning quantum systems](https://mlsanity.com/paper/2106.15432/) (2021)
- [Optimal quantum dataset for learning a unitary transformation](https://mlsanity.com/paper/2203.00546/) (2022)
- [Quantum Neural Estimation of Entropies](https://mlsanity.com/paper/2307.01171/) (2023)
- [Practical distributed quantum information processing with LOCCNet](https://mlsanity.com/paper/2101.12190/) (2021)
- [Estimating quantum relative entropies on quantum computers](https://mlsanity.com/paper/2501.07292/) (2025)
- [Learning Encodings by Maximizing State Distinguishability: Variational Quantum Error Correction](https://mlsanity.com/paper/2506.11552/) (2025)
- [Adaptive Fidelity Estimation for Quantum Programs with Graph-Guided Noise Awareness](https://mlsanity.com/paper/2601.14713/) (2026)
- [Entropy Estimation in Multi-Qutrit Systems via Variational and Classical Neural Networks](https://mlsanity.com/paper/2606.20504/) (2026)

## Abstract (arXiv)

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.