Stabilizer Rényi Entropy in Quantum Circuits

• Stabilizer Rényi Entropy is a measure of nonstabilizerness (magic) in quantum many-body states, derived from squared Pauli expectation values.
• It employs ZX-calculus and transfer-matrix techniques to obtain exact solutions in dual-unitary XXZ models, ensuring both theoretical and experimental tractability.
• Long-range SRE isolates nonlocal magic that resists removal by shallow Clifford circuits, serving as a precise diagnostic of global quantum resource content.

Stabilizer Rényi entropy (SRE) is a central invariant quantifying nonstabilizerness (“magic”) in quantum many-body states. As a resource monotone under Clifford operations, SRE characterizes the degree to which a quantum state fails to be simulable within the stabilizer formalism, thereby probing the computational universality and complexity of quantum systems. SRE generalizes the standard Rényi entropy to the probability distributions obtained from the squared Pauli expectation values, and admits both theoretical tractability and direct experimental estimation. Its nonlocal, long-range variants further isolate irreducible magic inaccessible to short-depth stabilizer circuits, providing a sharp diagnostic of global non-Clifford resource content. The following sections systematically expound the definition, computational techniques, exact solutions, and physical implications of SRE, with a focus on the dual-unitary XXZ model and long-range SRE as introduced in recent literature (López et al., 2024).

1. Formal Definition and Long-Range SRE

For an $N$-qubit state $|\psi\rangle$ on $\left(\mathbb{C}^2\right)^{\otimes N}$, the $n$-th order stabilizer purity is

$$\zeta_n(|\psi\rangle) = \frac{1}{2^N} \sum_{P \in \mathcal{P}_N} \langle \psi|P|\psi\rangle^{2n}$$

where $\mathcal{P}_N$ is the full Pauli group. The (order-$n$) SRE is

$$M_n(|\psi\rangle) = \frac{1}{1-n} \ln\left[ \zeta_n(|\psi\rangle) \right]$$

yielding a density $m_n(|\psi\rangle) = \lim_{N\to\infty} M_n/N$ in the thermodynamic limit.

For a mixed state $\rho$, especially for $|\psi\rangle$0, one defines

$|\psi\rangle$1

and a "purity-corrected" SRE

$|\psi\rangle$2

Defining the long-range SRE for a bipartite state $|\psi\rangle$3: $|\psi\rangle$4 yields a measure of nonlocal magic, i.e., the part of SRE that cannot be reduced to local stabilizer operations and which cannot be removed by shallow Clifford circuits (López et al., 2024).

2. Exact Solution in the Dual-Unitary XXZ Model

SRE is calculated exactly in the dual-unitary XXZ chain, employing a graphical calculus (ZX-calculus) to represent quantum circuits and evaluate the moments analytically. The circuit under study is a brick-wall Floquet XXZ sequence with layers composed of odd-bond gates

$|\psi\rangle$5

Transfer-matrix methods show the dominant eigenvalue of the $|\psi\rangle$6-th moment tensor is

$|\psi\rangle$7

which yields for large $|\psi\rangle$8

$|\psi\rangle$9

For a specific bipartition, the long-range SRE (at time $\left(\mathbb{C}^2\right)^{\otimes N}$0) between blocks $\left(\mathbb{C}^2\right)^{\otimes N}$1 and $\left(\mathbb{C}^2\right)^{\otimes N}$2 of size $\left(\mathbb{C}^2\right)^{\otimes N}$3, separated by $\left(\mathbb{C}^2\right)^{\otimes N}$4 sites on a ring of $\left(\mathbb{C}^2\right)^{\otimes N}$5 qubits, is given in closed form: $\left(\mathbb{C}^2\right)^{\otimes N}$6 where $\left(\mathbb{C}^2\right)^{\otimes N}$7, and $\left(\mathbb{C}^2\right)^{\otimes N}$8 is a combinatorially defined quantity arising from the fusion of ZX spiders (López et al., 2024). The long-range magic is then

$\left(\mathbb{C}^2\right)^{\otimes N}$9

This formula is exactly computable for all times and Rényi orders $n$0 using one-dimensional tensor network contraction.

3. ZX-Calculus Representation and Evaluation Rules

ZX-calculus recasts the evaluation of SRE in terms of tensor network diagrams called "spiders." Each $n$1- or $n$2-spider encodes a family of tensors with fusion, identity removal, Hadamard conjugation, and bialgebra/Hopf rewrite rules. Importantly, the $n$3-replica operator

$n$4

generates the summation over Pauli expectation values for SRE computation. Exploiting the spider graphical rules and projector properties of $n$5 enables an exact collapse of network contractions to tractable transfer matrix calculations, eliminating the exponential complexity in $n$6 for periodically structured circuits (López et al., 2024).

4. Finite-Size Scaling and Numerical Verification

The exact formulas for SRE and long-range SRE in the dual-unitary XXZ model were corroborated numerically using Monte Carlo sampling (up to $n$7 samples) for low Rényi indices ($n$8) and moderate system sizes ($n$9), always matching to $$\zeta_n(|\psi\rangle) = \frac{1}{2^N} \sum_{P \in \mathcal{P}_N} \langle \psi|P|\psi\rangle^{2n}$$0 relative error. At deep circuit depths ($$\zeta_n(|\psi\rangle) = \frac{1}{2^N} \sum_{P \in \mathcal{P}_N} \langle \psi|P|\psi\rangle^{2n}$$1), long-range SRE saturates to $$\zeta_n(|\psi\rangle) = \frac{1}{2^N} \sum_{P \in \mathcal{P}_N} \langle \psi|P|\psi\rangle^{2n}$$2 except at Clifford points ($$\zeta_n(|\psi\rangle) = \frac{1}{2^N} \sum_{P \in \mathcal{P}_N} \langle \psi|P|\psi\rangle^{2n}$$3), where it vanishes. Weak coupling ($$\zeta_n(|\psi\rangle) = \frac{1}{2^N} \sum_{P \in \mathcal{P}_N} \langle \psi|P|\psi\rangle^{2n}$$4) slows saturation to this plateau, with the timescale governed by $$\zeta_n(|\psi\rangle) = \frac{1}{2^N} \sum_{P \in \mathcal{P}_N} \langle \psi|P|\psi\rangle^{2n}$$5. No anomalous finite-size corrections were observed, confirming that the thermodynamic results are accurate for moderate system sizes ($$\zeta_n(|\psi\rangle) = \frac{1}{2^N} \sum_{P \in \mathcal{P}_N} \langle \psi|P|\psi\rangle^{2n}$$6) (López et al., 2024).

5. Physical Interpretation and Generation of Magic

Short-depth (one-layer) gates in such circuits immediately generate local magic density $$\zeta_n(|\psi\rangle) = \frac{1}{2^N} \sum_{P \in \mathcal{P}_N} \langle \psi|P|\psi\rangle^{2n}$$7 of order unity. However, only when the light-cones of spatially separated regions $$\zeta_n(|\psi\rangle) = \frac{1}{2^N} \sum_{P \in \mathcal{P}_N} \langle \psi|P|\psi\rangle^{2n}$$8 and $$\zeta_n(|\psi\rangle) = \frac{1}{2^N} \sum_{P \in \mathcal{P}_N} \langle \psi|P|\psi\rangle^{2n}$$9 overlap does long-range SRE $\mathcal{P}_N$0 become nonzero, indicating the emergence of irreducible magic shared "nonlocally." After this overlap, $\mathcal{P}_N$1 grows to a finite value representing two maximally entangled pairs (EPR pairs) of nonlocal magic that cannot be eliminated by any local (shallow) Clifford circuit. This distinction demonstrates that while local SRE ($\mathcal{P}_N$2) is sensitive to short-range circuit effects, long-range SRE isolates the genuinely global quantum resource in non-Clifford quantum circuits (López et al., 2024).

6. Analytical and Conceptual Significance

Dual-unitarity and integrability are central to the analytic tractability of SRE in this model. The available closed-form results are enabled by the combination of ZX-calculus representation, replica operator projectors, and integrable circuit structure. While similar qualitative plateaux in long-range SRE are expected in generic chaotic dual-unitary models, an exact calculation outside the dual-unitary class remains a significant open challenge. The ZX-calculus formalism for evaluating $\mathcal{P}_N$3 promises a systematic route to evaluating magic entropy in a broader class of tensor network and circuit models, potentially unifying the understanding of nonstabilizerness diagnostics across many-body systems (López et al., 2024).

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References:

• "Exact solution of long-range stabilizer Rényi entropy in the dual-unitary XXZ model" (López et al., 2024)