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Clifford entropy

Gianluca Cuffaro, Matthew B. Weiss

TL;DR

The paper introduces the $\alpha$-Clifford entropy $H_\alpha(U)$ as a quantitative measure of how close a unitary $U$ is to being Clifford, mirroring the stabilizer entropy for states. It proves key resource-theoretic properties—faithfulness, Clifford invariance, and subadditivity—and relates $H_\alpha(U)$ to the stabilizer entropy $\mathcal{M}_\alpha(\rho_\Phi)$ of the Choi state $\rho_\Phi$ of the unitary channel, yielding a non-tight upper bound. It further analyzes the Haar-average behavior, showing $\mathbb{E}_U[H_2(U)]=1- O(d^{-2})$, and employs concentration of measure to bound the $T$-count of a $T$-doped Clifford circuit realizing a Haar random unitary via $t(U) \ge \frac{H_2(U)}{H_2(T)}$ with high probability as dimension grows. Numerical results in low dimensions support the theory and suggest the bound’s practical reliability beyond asymptotics. The work opens directions to connect Clifford entropy with other magic measures, extend to channels, and explore SIC-structure implications for unitary dynamics and scrambling.

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.

Clifford entropy

TL;DR

The paper introduces the -Clifford entropy as a quantitative measure of how close a unitary is to being Clifford, mirroring the stabilizer entropy for states. It proves key resource-theoretic properties—faithfulness, Clifford invariance, and subadditivity—and relates to the stabilizer entropy of the Choi state of the unitary channel, yielding a non-tight upper bound. It further analyzes the Haar-average behavior, showing , and employs concentration of measure to bound the -count of a -doped Clifford circuit realizing a Haar random unitary via with high probability as dimension grows. Numerical results in low dimensions support the theory and suggest the bound’s practical reliability beyond asymptotics. The work opens directions to connect Clifford entropy with other magic measures, extend to channels, and explore SIC-structure implications for unitary dynamics and scrambling.

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.
Paper Structure (10 sections, 20 theorems, 77 equations, 2 figures)

This paper contains 10 sections, 20 theorems, 77 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Weyl-Heisenberg group
  4. The stabilizer entropy
  5. A measure of Cliffordness
  6. Operational interpretation
  7. Conclusion
  8. Average purity of SIC subsystems
  9. Haar average
  10. Bounding subadditivity violation

Key Result

Lemma 2

The matrix $\mathfrak{D}_{\textbf{ab}}(U)= |\mathfrak{C}_\textbf{ab}(U)|^2$ is bistochastic iff $U$ is unitary.

Figures (2)

  • Figure 1: The average of $H_2$ estimated by sampling $25,000$ unitaries is plotted against the analytical value according to Theorem \ref{['H2_haar']}, with perfect agreement. Above appears an estimate of the largest attainable value of $H_2$ obtained by numerically maximizing $H_2$ over the unitary group using the BFGS optimizer. The optimizer was run $100$ times in each dimension.
  • Figure 2: The frequency of subadditivity violation as a function of dimension estimated by sampling $25,000$ pairs of unitaries $U,V$ and calculating the proportion of time that $H_2(UV) \ge H_2(U) + H_2(V)$. This estimate was calculated separately $25$ times and then averaged. After $d=3$, the value is zero up to machine precision.

Theorems & Definitions (37)

  • Definition 1: $\alpha$-Clifford entropy
  • Lemma 2
  • proof
  • Theorem 3: Faithfulness
  • proof
  • Theorem 4: Clifford invariance
  • proof
  • Theorem 5: Subadditivity under tensor product
  • proof
  • Theorem 6
  • ...and 27 more