Diffusive Dynamics of Nonstabilizerness

Abstract: Symmetries shape the quantum-information dynamics of many-body systems, but their effect on nonstabilizerness, the resource complementary to entanglement, is less understood. We compute the stabilizer Rényi entropy, a measure of nonstabilizerness, in $\mathrm{U}(1)$-symmetric one-dimensional random circuits. The disorder-averaged dynamics is captured by a four-replica tensor network, which we evaluate by $S_4$-adapted infinite time-evolving block decimation (iTEBD) directly in the thermodynamic limit. Together with a hydrodynamic argument, our results identify a diffusive universality class for the late-time approach of nonstabilizerness to its random-state value, with the stabilizer Rényi entropy gap closing as $1/t$. The same scaling is verified in an energy-conserving nonintegrable Ising chain. More broadly, our framework provides a hydrodynamic perspective on nonstabilizerness generation and offers insight into the design of approximate Haar-random states in Hamiltonian dynamics.

• The paper presents a rigorous framework showing that magic, measured by stabilizer Rényi entropy, diffuses sub-ballistically under U(1) constraints.
• It employs Haar-averaged superoperators and symmetry-adapted tensor network simulations to reduce computational complexity while preserving charge conservation.
• Results reveal that selection rules limit operator dynamics in charge-conserving circuits, offering a scalable diagnostic for quantum resource dynamics.

Diffusive Dynamics of Nonstabilizerness: Summary and Technical Analysis

Context and Motivation

The paper "Diffusive Dynamics of Nonstabilizerness" (2606.13606) addresses the spatiotemporal evolution of nonstabilizerness ("magic") in quantum many-body systems subjected to random circuit dynamics, with a focus on the regime governed by conserved $\mathrm{U}(1)$ charge. Nonstabilizerness quantifies the departure of a quantum state from the stabilizer manifold, and thus signals its capacity to manifest quantum computational advantage. The manuscript rigorously pursues the dynamical fate of magic under locality and symmetry constraints, unifying resource-theoretic and hydrodynamic perspectives.

Formalism and Analytical Approaches

Haar-Averaged Evolution and Operator Replicas

The central object constructed is the four-replica Haar-averaged superoperator for two-qubit gates in charge-conserving ensembles. The paper develops an explicit tensor structure via representation-theoretic decomposition (Weingarten calculus, $S_4$ symmetric sectors), extracting operational selection rules for the propagation of magic. Charge-conservation reduces the effective local Hilbert space, and replica permutation symmetry enables block-diagonalization for numerical simulation.

Symmetry-Adapted Tensor Network Simulation

Leveraging the $S_4$ symmetry, the authors implement a symmetry-preserving infinite time-evolving block decimation (iTEBD) algorithm for matrix product operator (MPO) dynamics. All truncations are performed sector-wise, retaining exact symmetry and reducing computational complexity. A fast Walsh–Hadamard transform (FWHT) protocol is used to efficiently compute stabilizer Rényi entropy (SRE) in large-scale finite systems.

Main Results

Hydrodynamic Magic Diffusion

The principal numerical and analytical result is the demonstration that, in $\mathrm{U}(1)$-symmetric random circuits, the growth and spatial spreading of magic are governed by a diffusive hydrodynamic regime. SRE, as a measure of nonstabilizerness, exhibits sub-ballistic ($\sim\sqrt{t}$) spreading, in contrast to the ballistic propagation of entanglement entropy in non-symmetric settings [Rakovszky et al. 2019]. The late-time relaxation of SRE follows universal diffusion profiles characterized by system parameters and initial charge configurations.

Selection Rules and Operator Dynamics

Explicit selection rules derived from the Haar-averaged superoperator constrain the set of operator strings capable of supporting magic growth in charge-conserving circuits. The analysis reveals that only Pauli operators matching charge-multiplicity conditions contribute to SRE, establishing a dynamical blockade against certain nonstabilizer resources. The canonical decomposition in the neutral basis elucidates the minimal set of degrees of freedom relevant for the diffusive transport.

Numerical Benchmarks and Algorithmic Advances

The MPO simulations reproduce diffusive scaling across system sizes, initial states, and gate types, confirming theoretical predictions. The FWHT-based Pauli sampling algorithm is demonstrated to yield sub-exponential sampling complexity in large systems, offering a scalable alternative for the evaluation of magic observables in practical contexts.

Comparison with Unconstrained and MBL Regimes

The diffusive dynamics found here qualitatively differ from ballistic entanglement growth and magic spreading in unconstrained random circuits [Turkeshi et al. 2025], and from sub-linear, power-law growth in many-body localized (MBL) systems, where SRE exhibits a slow relaxation governed by $\ell$-bit phenomenology [Falcão et al. 2025b]. The results contribute to a more nuanced resource-theoretic taxonomy, elucidating how conservation laws suppress, enhance, or redirect quantum resource dynamics.

Theoretical Implications and Contradictory Claims

The paper asserts that, in charge-conserving systems, magic does not spread ballistically even if the underlying entanglement entropy does, contradicting previously held assumptions in the absence of symmetry constraints. It provides strong evidence that diffusion, not ballisticity, controls nonstabilizerness in such regimes. The symmetry analysis and charge-selection rules formulated are universally valid in random circuit models with local conservation laws, extending beyond the specifics of $\mathrm{U}(1)$.

Practical Implications and Future Directions

The precise characterization and efficient computation of diffusive magic dynamics have significant implications for noisy quantum simulators, monitored circuits, and hybrid quantum information processing schemes. The algorithmic developments—sector-wise truncation and efficient SRE computation—enable scalable diagnostics of quantum complexity for both theoretical and experimental platforms.

In the broader context of quantum resource theories, the results suggest that conservation laws, by constraining operator evolution, fundamentally impact the accessibility and distribution of computational resources in large-scale quantum systems. Techniques developed here are extendable to other conserved quantities and to hybrid monitored circuits, laying groundwork for the systematic classification of quantum computational resources under dynamical constraints.

Conclusion

"Diffusive Dynamics of Nonstabilizerness" (2606.13606) rigorously demonstrates that magic resources, measured by stabilizer Rényi entropy, exhibit diffusive spatial-spreading and temporal relaxation under $\mathrm{U}(1)$-symmetric random circuit dynamics. The explicit symmetry analysis, tensor network algorithms, and efficient sampling protocols together provide a comprehensive framework for probing and quantifying quantum complexity in many-body dynamics. The findings redefine expectations on resource propagation in systems with conservation laws and offer scalable approaches to investigate quantum resource dynamics in experimentally relevant settings.