Qudit Shadow Estimation Based on the Clifford Group and the Power of a Single Magic Gate

TL;DR

This paper analyzes the sample complexity of qudit shadow estimation using the Clifford group, revealing that the overhead compared to qubits scales linearly with the local dimension, and introduces a magic gate to enhance efficiency.

Contribution

It clarifies the sample complexity for qudit shadow estimation and proposes a simple circuit with a magic gate to significantly improve efficiency.

Findings

Overhead of qudit shadow estimation is proportional to the local dimension d.

Adding a single magic gate can eliminate the d overhead, matching qubit performance.

The stabilizer states deviate exponentially from a 3-design with respect to the third moment operator.

Abstract

Shadow estimation is a sample-efficient protocol for learning the properties of a quantum system using randomized measurements, but the current understanding of qudit shadow estimation is quite limited compared with the qubit setting. Here we clarify the sample complexity of qudit shadow estimation based on the Clifford group, where the local dimension $d$ is an odd prime. Notably, we show that the overhead of qudit shadow estimation over the qubit counterpart is only $\mathcal{O}(d)$, independent of the qudit number $n$, although the set of stabilizer states may deviate exponentially from a 3-design with respect to the third moment operator. Furthermore, by adding one layer of magic gates, we propose a simple circuit that can significantly boost the efficiency. Actually, a single magic gate can already eliminate the $\mathcal{O}(d)$ overhead in qudit shadow estimation and bridge the gap from the qubit setting.