Evaluating Pauli errors on cluster states by weighted distances

TL;DR

This paper investigates how weighted distance measures can effectively quantify the impact of single-qubit Pauli errors on cluster states, revealing differences undetectable by standard metrics and emphasizing their importance in many-body quantum systems.

Contribution

It introduces the use of weighted Bures length and weighted Hilbert-Schmidt distance to better assess error effects on cluster states, highlighting their advantages over standard distances.

Findings

Weighted distances reveal differences in error impact not seen with standard metrics.

The effect of errors depends on the specific Pauli rotation and qubit position.

Weighted distances are useful for monitoring properties of many-body quantum systems.

Abstract

We address the problem of evaluating the difference between quantum states before and after being affected by errors encoded in unitary transformations. Standard distance functions, e.g., the Bures length, are not fully adequate for such a task. Weighted distances are instead appropriate information measures to quantify distinguishability of multipartite states. Here, we employ the previously introduced weighted Bures length and the newly defined weighted Hilbert-Schmidt distance to quantify how much single-qubit Pauli errors alter cluster states. We find that different errors of the same dimension change cluster states in a different way, i.e., their detectability is in general different. Indeed, they transform an ideal cluster state into a state whose weighted distance from the input depends on the specific chosen Pauli rotation, as well as the position of the affected qubit in the graph related to the state. As these features are undetected by using standard distances, the study proves the usefulness of weighted distances to monitor key but elusive properties of many-body quantum systems.