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Quantum Magic, Conservation Laws, and Hydrodynamic Memory: How Nonstabilizerness Dynamics, Participation Entropy Relaxation, Monitored Circuit Asymptotics, Non-Isometric Encoding, and Diffusive Universality Jointly Constrain the Emergence of Computational Complexity in Many-Body Quantum Systems

Saluca Agentic AI Research Team

Published Jun 14, 2026• 0 citations
Publisher

Abstract

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 $\mathcal{O}(d\varepsilon^{-1})$ 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

DOI
10.5281/zenodo.20688507

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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/t1/t1/t [corpus:arxiv:2606.13606v1]; (2) participation entropy — a complementary spread measure — exhibits the same t−1/2t^{-1/2}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)\mathcal{O}(d\varepsilon^{-1})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/t1/t1/t nonstabilizerness gap and the t−1/2t^{-1/2}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
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)\mathrm{U}(1)U(1)-symmetric one-dimensional random circuits. The disorder-averaged dynamics is captured by a four-replica tensor network, which we evaluate by S4S_4S4​-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/t1/t1/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.
A recurring structural pattern across recent quantum-information and mathematical-physics literature is that *symmetry* — including symmetry that is accidental, hidden, or emergent — acts simultaneously as a constraint on controllability, a regulator of resource transport, and a determinant of metrological sensitivity. This paper synthesizes six to eight findings from recent preprints spanning quant-ph, math-ph, and cond-mat.stat-mech to argue, as a heuristic reading rather than a derivation, that a single organizational principle connects these disparate results: the presence or absence of a particular symmetry structure governs (i) what unitary operations are reachable in a quantum system, (ii) how slowly quantum resources such as nonstabilizerness and participation entropy relax under conservation laws, (iii) whether exceptional-point signatures survive into the steady state, and (iv) how metrological sensitivity is enhanced or suppressed by virtual-excitation dressing. The corpus draws primarily from quant-ph (controllability, resource theory, non-Hermitian physics, quantum sensing) and cond-mat.stat-mech (hydrodynamic relaxation, absorbing-state systems). We identify the Tavis-Cummings accidental symmetry [corpus:arxiv:2606.12813], diffusive nonstabilizerness transport under U(1) symmetry [corpus:arxiv:2606.13606], participation-entropy hydrodynamic memory [corpus:arxiv:2606.11561], Lindbladian exceptional-point noise signatures [corpus:arxiv:2606.13377], nonreciprocal rotation sensing via virtual excitations [corpus:arxiv:2606.10984], and noise cancellation by channel superposition [corpus:arxiv:2606.10744] as the primary evidential cluster. The falsification path for the central thesis is concrete: if breaking the accidental TC symmetry via J_z^2 does not alter the diffusive relaxation exponent of nonstabilizerness in a TC-coupled circuit, the proposed link between controllability structure and resource hydrodynamics is severed. --- Authorship: Saluca Agentic AI Research Team (Saluca LLC). AI-drafted from arXiv preprint corpus on the date in the filename. Cited arXiv preprints: 2606.10744, 2606.10984, 2606.11146, 2606.11561, 2606.11885, 2606.11958, 2606.12284, 2606.12301, 2606.12313, 2606.12813, 2606.12906, 2606.13075, 2606.13377, 2606.13422, 2606.13521, 2606.13559, 2606.13606, 2606.13641, 2606.13650
Nonstabilizerness, or quantum magic resource, presents a valuable resource in quantum error correction and computation. We study the dynamics of locally injected nonstabilizerness in unitary Clifford circuits, where the total nonstabilizerness is conserved. However, the absence of physical observables quantifying nonstabilizerness precludes a direct microscopic or hydrodynamic description of its local distribution and dynamics. Using insights from stabilizer quantum error correcting codes, we rigorously show that the spatial distribution of nonstabilizerness can be inferred from a canonical representation of low-magic states, dubbed the bipartite magic gauge. Moreover, we propose two operationally relevant magic length scales. We numerically establish that, at early times, both length scales grow ballistically at distinct velocities set by the entanglement velocity, after which nonstabilizerness delocalizes. Our work sheds light on the spatiotemporal structure of quantum resources and complexity in many-body dynamics, opening up avenues for investigating their transport properties and further connections with quantum error correction.
Quantum state complexity metrics, such as anticoncentration and nonstabilizerness, offer key insights into many-body physics, information scrambling, and quantum computing. Anticoncentration and equilibration of magic resources under dynamics of random quantum circuits occur at times scaling logarithmically with system size. Here, we examine these phenomena in one-dimensional ergodic Floquet models and thermalizing Hamiltonian systems. Using participation and stabilizer entropies to probe anticoncentration and magic resources, we reveal significant differences between the two settings. Floquet systems align with random circuit predictions, exhibiting anticoncentration and magic saturation at timescales logarithmic in system size. In contrast, Hamiltonian dynamics deviate from the random circuit predictions and require times scaling approximately linearly with system size to achieve saturation of participation and stabilizer entropies, which remain smaller than that of the typical quantum states even in the long-time limit. Our findings establish the phenomenology of participation and stabilizer entropy growth in generic many-body systems and emphasize the role of conservation laws in constraining anticoncentration and magic dynamics.
Participation entropy (PE) quantifies the spread of a many-body wavefunction across configuration space. While PE relaxes rapidly in generic chaotic systems, we show that U(1)\mathrm{U}(1)U(1) conservation laws slow it down by imprinting with the slow hydrodynamic modes. Using a cluster expansion around equilibrium, we show that, after local density inhomogeneities decay, the leading PE deficit is dominated by squared connected density correlations. The long time relaxation is therefore controlled by diffusive correlation spreading, giving ΔS(t)∼t−1/2ΔS(t)\sim t^{-1/2}ΔS(t)∼t−1/2 in the hydrodynamic regime and crossing over to ∼exp⁡[−O(t/L2)]\sim \exp[-O(t/L^2)]∼exp[−O(t/L2)] when t≥L2t\geq L^2t≥L2. We confirm this entropy correlation relation using exact computation and infinite system tensor network simulations in various quantum U(1)\mathrm{U}(1)U(1) conserving circuits. Our results establish PE as a sensitive probe of hydrodynamic memory and suggest that slow relaxation is a generic consequence of conservation laws.
We study how conservation laws shape the spreading of quantum coherence in many-body dynamics. Focusing on U(1)U(1)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.