# Beyond the swap test: optimal estimation of quantum state overlap

## TL;DR

This paper develops optimal collective measurement strategies for estimating the overlap between two unknown quantum states, outperforming traditional swap tests, especially for small overlaps and in noisy conditions.

## Contribution

It introduces the optimal measurement framework for quantum state overlap estimation, surpassing swap tests in accuracy and robustness, with detailed analysis of performance and noise resilience.

## Key findings

- Optimal measurements reduce mean square error compared to swap tests.
- Performance advantage increases with system dimension and for small overlaps.
- Optimal measurement is less invasive and more noise-robust than swap tests.

## Abstract

We study the estimation of the overlap between two unknown pure quantum states of a finite dimensional system, given $M$ and $N$ copies of each type. This is a fundamental primitive in quantum information processing that is commonly accomplished from the outcomes of $N$ swap-tests, a joint measurement on one copy of each type whose outcome probability is a linear function of the squared overlap. We show that a more precise estimate can be obtained by allowing for general collective measurements on all copies. We derive the statistics of the optimal measurement and compute the optimal mean square error in the asymptotic pointwise and finite Bayesian estimation settings. Besides, we consider two strategies relying on the estimation of one or both the states, and show that, although they are suboptimal, they outperform the swap test. In particular, the swap test is extremely inefficient for small values of the overlap, which become exponentially more likely as the dimension increases. Finally, we show that the optimal measurement is less invasive than the swap test and study the robustness to depolarizing noise for qubit states.