Stabilizer Rényi Entropy Dynamics

• The topic defines stabilizer Rényi entropy as a measure of nonstabilizerness (quantum magic) and its deviation from the stabilizer regime, crucial for fault-tolerant quantum computing.
• It examines the dynamics of magic spreading through unitary and monitored evolutions, highlighting ballistic, diffusive, and superdiffusive transport in many-body systems.
• Applications include providing T-count lower bounds and optimizing quantum circuits by informing resource allocation and magic state distillation strategies.

Stabilizer Rényi entropy dynamics concerns the temporal behavior and spatial propagation of nonstabilizerness (quantum "magic") in many-body quantum systems as characterized by stabilizer Rényi entropy (SRE) and its variants. SRE is a resource-theoretic monotone for magic, quantifying deviations from the stabilizer (Clifford) regime essential for fault-tolerant quantum computation and for understanding the classical simulability barrier. Recent developments have established SRE as a rigorous diagnostic for magic generation, spreading, equilibration, and persistence under both unitary and monitored (open-system) dynamics.

1. Foundational Definitions

The stabilizer Rényi entropy of order $\alpha$ for a quantum state $\rho$ is defined in terms of its Pauli expectation values. For a pure $L$-qubit state $|\psi\rangle$, the $\alpha$-order SRE is

$$M_\alpha(|\psi\rangle) = \frac{1}{1-\alpha}\log\left[ \frac{1}{2^{L}} \sum_{P\in\mathcal{P}_L} |\langle\psi|P|\psi\rangle|^{2\alpha} \right]\,,$$

with $\mathcal{P}_L$ the Pauli group on $L$ qubits. For mixed states, SRE generalizes via convex roofs or directly on reduced density matrices. The particularly important case $\alpha=2$ (second-order) appears in all recent studies and coincides with

$$M_2(|\psi\rangle) = -\log_2\left[ \frac{1}{4^L} \sum_{P\in\mathcal{P}_L} |\langle\psi|P|\psi\rangle|^{4} \right]\,.$$

For subsystems and mixed states $\rho_L$, SRE can be written in terms of stabilizer purity $W(\rho_L)$ and the 2-Rényi entanglement entropy $S_2(\rho_L)$ (Rattacaso et al., 2023, Russomanno et al., 3 Feb 2025): $M_2(\rho_L) = -\log_2 W(\rho_L) - S_2(\rho_L)\,,$ where $$W(\rho_L) = 2^{-L}\sum_{P\in\mathcal{P}_L} [\mathrm{Tr}(P\rho_L)]^4$$.

SRE is faithful ($M_\alpha=0$ iff stabilizer), additive, Clifford-invariant, and monotonic under stabilizer operations for $\alpha\geq2$ (Zhu et al., 10 Sep 2024).

2. Dynamics: Unitary and Monitored Evolution

The time evolution of SRE probes how nonstabilizerness emerges and spreads under Hamiltonian and circuit dynamics. In integrable quantum quenches, such as the transverse-field Ising chain, $M_2[\rho_L(t)]$ rises from zero (stabilizer initial state), ballistically spreads, and equilibrates to its time-averaged value on a scale $\tau(L)\sim L / v_{\mathrm{LR}}$, set by the Lieb–Robinson velocity (Rattacaso et al., 2023). The spatial extent of magic, or "stabilizer-entropy length," increases linearly with time as $L_{\mathrm{SE}}(t) \simeq v_{\mathrm{SE}} t$, and can outpace entanglement fronts: $$\tau(L) \simeq L / v\, ,\quad L_{\mathrm{SE}}(t) \lesssim L_0 + v_s t\, ,$$ with $v_{\mathrm{SE}}$ potentially exceeding the entanglement velocity.

Monitored (open) evolution, exemplified by Lindbladian or quantum-state-diffusion protocols, induces decay of SRE, but numerical studies show a residual "volume-law" SRE persists at finite measurement rate $\gamma$, fitted by a generalized Lorentzian in $\gamma$: $$\overline{\mathcal{M}_2}(\gamma) = \frac{A_L}{1 + (\gamma/\gamma_{0,L})^{b_L}},\quad A_L\propto L\,,$$ signaling that even in strong measurement regimes, nonzero magic can survive in the thermodynamic limit (Russomanno et al., 3 Feb 2025).

3. Spreading and Transport: Ballistic, Diffusive, and Superdiffusive Profiles

SRE dynamics display rich spatial structures, especially in Clifford circuits and models with engineered randomness. In brickwork random Clifford circuits with localized initial magic, the local single-qubit SRE $M_i(t)$ propagates within a well-defined ballistic light cone ($|i-m| \leq t$), but its normalized distribution spreads diffusively inside this cone (Maity et al., 11 Nov 2025). The normalized profile $a_i(t) = M_i(t) / \mathcal{M}(t)$, to leading order, obeys the discrete diffusion equation

$$a_i(t) \approx \frac{1}{2} [ a_{i-1}(t-1) + a_{i+1}(t-1) ],$$

yielding $\text{width}~\sigma(t) \sim \sqrt{t}$. With restricted Clifford gate sets, spreading can become superdiffusive, with $\sigma(t) \sim t^{\beta}$, $\beta>0.5$.

Despite conservation of global SRE under Clifford evolution, the local SRE decays (exponentially with rate $\Gamma\approx0.44$ for random circuits), as nonstabilizerness is encoded in multipartite entanglement inaccessible to local reductions.

4. Exact Solutions and Computational Strategies

Closed-form and exact results for SRE dynamics have been obtained for dual-unitary circuits, notably the XXZ brickwork models (López et al., 7 May 2024). Here, both short-time SRE density and long-range SRE for special bipartitions can be computed using transfer-matrix and ZX-calculus techniques, with scaling

$$m_\alpha = \frac{1}{2(1 - \alpha)} \ln \left[\frac{1 + \cos^{2\alpha}(2J) + \sin^{2\alpha}(2J)}{2} \right],$$

holding in the thermodynamic limit after one layer. For certain partitions and times, SRE saturates to universal values independent of initial state and details, e.g., $L_{B_0} \to 2\ln 2$ for large separation or depth unless the evolution is strictly Clifford.

Numerically, efficient schemes based on symmetries and conjugation properties of Pauli strings reduce SRE evaluation to polynomial complexity in sector size (e.g., total $S^z$), but exponential in system size otherwise (Russomanno et al., 3 Feb 2025, López et al., 7 May 2024).

5. Amortized SRE and Resource Theory of Magic

The amortized $\alpha$-stabilizer Rényi entropy $M^\mathcal{A}_\alpha(U)$ quantifies the maximal increase in SRE that a unitary $U$ can achieve, optimizing over arbitrary (possibly magical) input states and ancillary systems (Zhu et al., 10 Sep 2024): $$M^\mathcal{A}_\alpha(U) = \sup_{m\ge0} \max_{|\phi\rangle \in \mathcal{H}_{n+m}} [ M_\alpha( (U \otimes I_R)|\phi\rangle ) - M_\alpha(|\phi\rangle) ].$$ This amortization reveals the critical phenomenon that prior input magic can enhance a unitary's magic-generating power, a property not shared by other magic monotones (e.g., robustness of magic, stabilizer extent). Formally, for nonstabilizer input $|\phi\rangle$, $$M_\alpha^\mathcal{A}(U) > M_\alpha^{\mathrm{strict}}(U)$$ if there exists $|\phi\rangle$ with $M_\alpha( |\phi\rangle ) > 0$ achieving a greater gain than any stabilizer input.

Amortized SRE is subadditive for composition and tensor products, making it a bona fide monotone at the level of dynamical maps.

6. Concrete Applications: Fault Tolerance and Quantum Advantage

Amortized SRE provides rigorous lower bounds on $T$-count—minimum non-Clifford resource gates—for quantum circuits and Hamiltonian evolutions: $$t(U) \geq \frac{M_2^\mathcal{A}(U)}{ M_2^\mathcal{A}(T) }, \quad M_2^\mathcal{A}(T)=2-\log_2 3\,.$$ For the quantum Fourier transform (QFT) and Heisenberg chain dynamics, these bounds strictly improve upon stabilizer-nullity results, confirming that SRE-based diagnostics are sharper for nontrivial circuits (Zhu et al., 10 Sep 2024). For example, $t(QFT_3)\geq6$, $t(QFT_4)\geq8$.

In circuit optimization and magic state distillation protocols, the enhancement of magic generation through prior nonstabilizerness motivates hybrid approaches, wherein "seed" magic is injected early and subsequently amplified.

For variational or Trotterized simulation algorithms, knowledge of instantaneous SRE growth rates informs spacetime resource allocation, guiding step sizes and diagnosing proximity to the classically tractable stabilizer subspace.

7. Outlook, Open Problems, and Experimental Prospects

Open questions include the existence and nature of measurement-induced magic transitions, critical scaling forms for SRE in monitored systems, and the microscopic basis for observed functional forms (e.g., generalized Lorentzian fits for monitored SRE decay) (Russomanno et al., 3 Feb 2025).

Experimental access to local SRE profiles is feasible with current platforms via local Pauli tomography, enabling empirical tests of theoretical predictions for ballistic, diffusive, and superdiffusive magic spreading (Maity et al., 11 Nov 2025). SRE measures serve both as resource quantifiers and diagnostic tools in noisy or open-system regimes.

Stabilizer Rényi entropy dynamics thus constitute a unifying framework for quantitative analysis of quantum magic across models, architectures, and operating regimes, tightly linked to theoretical understandings of quantum advantage and resource costs for fault-tolerant quantum information processing.

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Key references:

• "Amortized Stabilizer Rényi Entropy of Quantum Dynamics" (Zhu et al., 10 Sep 2024)
• "Local spreading of stabilizer Rényi entropy in a brickwork random Clifford circuit" (Maity et al., 11 Nov 2025)
• "Stabilizer entropy dynamics after a quantum quench" (Rattacaso et al., 2023)
• "Efficient evaluation of the nonstabilizerness in unitary and monitored quantum many-body systems" (Russomanno et al., 3 Feb 2025)
• "Exact solution of long-range stabilizer Rényi entropy in the dual-unitary XXZ model" (López et al., 7 May 2024)