A central question in quantum many-body physics asks how computational complexity — specifically the nonstabilizerness, or "magic," resource that distinguishes genuinely quantum computation from Clifford-circuit simulation — emerges, propagates, and saturates in physical systems governed by conservation laws. This paper synthesizes five recent arXiv preprints spanning quant-ph and cond-mat.stat-mech to argue for a candidate structural pattern: **conservation laws impose hydrodynamic memory on magic-resource dynamics, and this memory manifests as diffusive relaxation signatures that are measurable, resource-measure-dependent, and sensitive to encoding geometry**. The argument is a heuristic reading, not a derivation; the sources share formal structures (replica tensor networks, hydrodynamic arguments, stabilizer Rényi entropy) but are not unified by a single theorem. Concretely: (1) nonstabilizerness in U(1)-symmetric random circuits relaxes diffusively as
1/t [corpus:arxiv:2606.13606v1]; (2) participation entropy — a complementary spread measure — exhibits the same
t−1/2 hydrodynamic tail precisely because U(1) conservation law couples it to diffusing density correlations [corpus:arxiv:2606.11561v1]; (3) in monitored circuits, the steady-state magic depends on resource-measure geometry near the stabilizer layer, with mana scaling linearly and stabilizer Rényi entropy quadratically in the injection parameter [corpus:arxiv:2606.13470v1]; (4) non-isometric encodings — unavoidable in finite-energy bosonic codes and in holographic models — impose fundamental accuracy ceilings on logical operations that are quantitatively controlled by the approximate quantum error correction framework [corpus:arxiv:2606.13559v1]; and (5) optimal classical shadow estimation of unitary channels achieves Heisenberg scaling
O(dε−1) using only parallel queries, establishing the query complexity of extracting magic-relevant channel information [corpus:arxiv:2606.13638v1]. The falsification path common to these claims is concrete: if U(1)-symmetric Floquet circuits with tunable symmetry-breaking perturbations show that the
1/t nonstabilizerness gap and the
t−1/2 participation entropy tail decouple as the symmetry is explicitly broken, the proposed hydrodynamic-memory mechanism is refuted. Sources are drawn from approximately 80 recent preprints in quant-ph, math-ph, and cond-mat.stat-mech. --- Authorship: Saluca Agentic AI Research Team (Saluca LLC). AI-drafted from arXiv preprint corpus on the date in the filename. Cited arXiv preprints: 2606.11561v1, 2606.11885v1, 2606.12269v1, 2606.13470v1, 2606.13525v1, 2606.13559v1, 2606.13606v1, 2606.13638v1
We study how conservation laws shape the spreading of quantum coherence in many-body dynamics. Focusing on
U(1)-symmetric random circuits, charge-and-dipole conserving circuits, as well as ergodic Hamiltonian dynamics, we probe coherences both globally, via the participation entropy, and locally, via the relative entropy of coherence. Combining exact vector evolution, matrix product state simulations, and replica tensor networks methods, we find that conservation laws replace the logarithmic saturation of unconstrained circuits with slow hydrodynamic relaxation of the global coherence measures. Locally, symmetry-constrained circuits show a clean rise-peak-fall structure whose peak time grows algebraically with subsystem size. In contrast, ergodic Hamiltonians broaden the peak into an extended plateau at larger subsystems, highlighting a qualitatively distinct mechanism. Coherence thus emerges as a sensitive probe of symmetry-constrained thermalization, linking quantum resource dynamics to many-body transport.
We investigate how transport properties of
U(1)-conserving dynamics impact the growth of quantum resources characterizing the complexity of many-body states. We quantify wave-function delocalization using participation entropy (PE), a measure rooted in the coherence theory of pure states, and assess nonstabilizerness through stabilizer Rényi entropy (SRE). Focusing on the XXZ spin chain initialized in domain-wall state, we demonstrate universal power-law growth of both PE and SRE, with scaling exponents explicitly reflecting the underlying transport regimes, ballistic, diffusive, or KPZ-type superdiffusive. Our results establish a solid connection between quantum resources and transport, providing insights into the dynamics of complexity within symmetry-constrained quantum systems.
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
U(1)-symmetric one-dimensional random circuits. The disorder-averaged dynamics is captured by a four-replica tensor network, which we evaluate by
S4-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.
We study the scrambling of local quantum information in chaotic many-body systems in the presence of a locally conserved quantity like charge or energy that moves diffusively.The interplay between conservation laws and scrambling sheds light on the mechanism by which unitary quantum dynamics, which is reversible, gives rise to diffusive hydrodynamics, which is a slow dissipative process.We obtain our results in a random quantum circuit model that is constrained to have a conservation law.We find that a generic spreading operator consists of two parts: (i) a conserved part which comprises the weight of the spreading operator on the local conserved densities, whose dynamics is described by diffusive charge spreading; this conserved part also acts as a source that steadily emits a flux of (ii) nonconserved operators.This emission leads to dissipation in the operator hydrodynamics, with the dissipative process being the slow conversion of operator weight from local conserved operators to nonconserved, at a rate set by the local diffusion current.The emitted nonconserved parts then spread ballistically at a butterfly speed, thus becoming highly nonlocal and, hence, essentially nonobservable, thereby acting as the "reservoir" that facilitates the dissipation.In addition, we find that the nonconserved component develops a power-law tail behind its leading ballistic front due to the slow dynamics of the conserved components.This implies that the out-of-time-order commutator between two initially separated operators grows sharply upon the arrival of the ballistic front, but, in contrast to systems with no conservation laws, it develops a diffusive tail and approaches its asymptotic late-time value only as a power of time instead of exponentially.We also derive these results within an effective hydrodynamic description which contains multiple coupled diffusion equations.
The scrambling of quantum information in closed many-body systems, as measured by out-of-timeordered correlation functions (OTOCs), has received considerable attention lately.Recently, a hydrodynamical description of OTOCs has emerged from considering random local circuits.Numerical work suggests that aspects of this description are universal to ergodic many-body systems, even without randomness; a conjectured explanation for this is that while the random circuits have noise built into them, deterministic quantum systems, much like classically chaotic ones, "generate their own noise" and look effectively random on sufficient length scales and timescales.In this paper, we extend this approach to systems with locally conserved quantities (e.g., energy).We do this by considering local random unitary circuits with a conserved U(1) charge and argue, with numerical and analytical evidence, that the presence of a conservation law slows relaxation in both time-ordered and out-of-time-ordered correlation functions; both can have a diffusively relaxing component or "hydrodynamic tail" at late times.We verify the presence of such tails also in a deterministic, periodically driven system.We show that for OTOCs, the combination of diffusive and ballistic components leads to a wave front with a specific asymmetric shape, decaying as a power law behind the front.These results also explain existing numerical investigations in non-noisy ergodic systems with energy conservation.Moreover, we consider OTOCs in Gibbs states, parametrized by a chemical potential μ, and apply perturbative arguments to show that for μ ≫ 1 the ballistic front of information spreading can only develop at times exponentially large in μ-with the information traveling diffusively at earlier times.We also develop a new formalism for describing OTOCs and operator spreading, which allows us to interpret the saturation of OTOCs as a form of thermalization on the Hilbert space of operators.