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

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.

arXiv comment9+20 pages, 6+6 figures, and 2 tables; title updated, published in PRL. See also the companion paper "Third moments of qudit Clifford orbits and 3-designs based on magic orbits (arXiv:2410.13575)"

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Cited by in corpus (4)

• Stabilizer Rényi Entropy and Conformal Field Theory
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• Classical Shadows with Improved Median-of-Means Estimation
• Quantum Advantage via Efficient Post-processing on Qudit Classical Shadow tomography