Clifford entropy

TL;DR

The paper introduces Clifford entropy as a measure of how close a unitary is to a Clifford unitary, explores its properties, bounds, and implications for quantum circuit depth, with both theoretical and numerical insights.

Contribution

It defines and analyzes Clifford entropy, establishing its properties, bounds, and relevance to quantum circuit complexity, extending stabilizer entropy concepts.

Findings

Clifford entropy vanishes for Clifford unitaries.

Upper bounds on Clifford entropy are derived and numerically estimated.

Clifford entropy relates to circuit depth via concentration of measure in high dimensions.

Abstract

We introduce the Clifford entropy, a measure of how close an arbitrary unitary is to a Clifford unitary, which generalizes the stabilizer entropy for states. We show that this quantity vanishes if and only if a unitary is Clifford, is invariant under composition with Clifford unitaries, and is subadditive under tensor products. Rewriting the Clifford entropy in terms of the stabilizer entropy of the corresponding Choi state allows us to derive an upper bound: that this bound is not tight follows from considering the properties of symmetric informationally complete sets. Nevertheless we are able to numerically estimate the maximum in low dimensions, comparing it to the average over all unitaries, which we derive analytically. Finally, harnessing a concentration of measure result, we show that as the dimension grows large, with probability approaching unity, the ratio between the Clifford entropy of a Haar random unitary and that of a fixed magic gate gives a lower bound on the depth of a doped Clifford circuit which realizes the former in terms of the latter. In fact, numerical evidence suggests that this result holds reliably even in low dimensions. We conclude with several directions for future research.