Stabilizer Entropy of Subspaces

TL;DR

This work investigates how embedding a quantum system into a larger subspace affects nonstabilizerness, quantified by stabilizer entropy (SE), via the average magic (ASE) gap between intrinsic and extrinsic SE. It provides exact and computable formulas for the intrinsic and extrinsic SE averages, derives a Weingarten-calculus framework to analyze subspace embeddings, and demonstrates, through extensive numerics, that zero and even negative ASE gaps are realizable for particular subspaces and dimension combinations. The results reveal that judicious choices of embedding subspaces can reduce resource costs in simulations and potentially enable more efficient quantum or classical simulations, albeit with caveats about encoding/decoding overhead and entanglement. The work also furnishes a practical, open-source toolkit for computing ASE gaps and extremizing SE over subspaces, enabling broader exploration of embedding costs across quantum architectures and symmetry-constrained systems.

Abstract

We consider the costs and benefits of embedding the states of one quantum system within those of another. Such embeddings are ubiquitous, e.g., in error correcting codes and in symmetry-constrained systems. In particular we investigate the impact of embeddings in terms of the resource theory of nonstabilizerness (also known as magic) quantified via the stabilizer entropy (SE). We analytically and numerically study the stabilizer entropy gap or magic gap: the average gap between the SE of a quantum state realized within a subspace of a larger system and the SE of the quantum state considered on its own. We find that while the stabilizer entropy gap is typically positive, requiring the injection of magic, both zero and negative magic gaps are achievable. This suggests that certain choices of embedding subspace provide strong resource advantages over others. We provide formulas for the average nonstabilizerness of a subspace given its corresponding projector and sufficient conditions for realizing zero or negative gaps: in particular, certain classes of stabilizer codes provide paradigmatic examples of the latter. Through numerical optimization, we find subspaces which achieve both minimal and maximal average SE for a variety of dimensions, and compute the magic gap for specific error-correcting codes and symmetry-induced subspaces. Our results suggest that a judicious choice of embedding can lead to greater efficiency in both classical and quantum simulations.

Figures (8)

• Figure 1: On the left: the ASE gap, averaged over subspaces, plotted for $d_B=32$ and for $d_S$ from $2$ to $d_B-1$. In blue, we treat the $\mathcal{H}_{d_B}$ as a single qudit, and in orange we treat it as $n=5$ qubits. On the right: for different qudit dimensions $d_B$, the difference between the standard deviation $\sigma_{d_B}$ of the ASE of $\mathcal{H}_{d_B}$ and the standard deviation $\sigma_{d_S}$ of the ASE averaged over $d_S$-dimensional subspaces. The former was calculated by averaging over 750 random states in $\mathcal{H}_{d_B}$, the latter by averaging over 750 random subspaces.
• Figure 2: For a variety of big Hilbert space dimensions $d_B$, we plot for each small dimension $d_S$ from $2$ to $d_B-1$ the minimal average SE achievable over all choices of subspace as well as the maximum. The dashed horizontal line depicts the average SE of the big space, while the dotted line depicts the average SE of the small space. When the dotted line intersects the red, zero magic gap is achieved.
• Figure 3: The same in several multiqudit cases where the larger Hilbert space is $\mathcal{H}_d^{\otimes n}$. The orange X's depict the ASE of the small space considered as $n$-qubits when the subspace dimension $d_S$ is a power of 2.
• Figure 4: For different values of $n$ and $d$, we plot in blue the minimal achievable ASE of a subspace of dimension $d_S$. In red, we plot the ASE obtained by adding in the optimal complement (in even superposition) for each of 750 random states. In green, we plot the ASE obtained by adding in the optimal fixed complement (in even superposition), again for each of 750 random states. In the latter two cases, the standard deviations are plotted as error bars.
• Figure 5: For different values of $n$ and $d$, we plot the average relative change in ASE over 100 random choices of subspaces with dimension $d_S$, where the ASE is calculated by adding in the optimal fixed complement (in even superposition, for 750 random states).

Theorems & Definitions (5)

• Theorem 1: ASE gap for random subspaces
• proof
• Theorem 2: ASE gap for stabilizer codespaces
• Theorem 3
• proof : \ref{['zerogap']}