Operator hydrodynamics, OTOCs, and entanglement growth in systems without conservation laws

I) Random circuit

• Random-circuit operator spreading follows a biased-diffusion hydrodynamic picture, with vB below the light-cone velocity and diffusive front broadening.These results yield exact formulas for OTOCs and entanglement growth.Source: §I · pp. 1–2
• The diffusive operator front produces an entanglement velocity vE smaller than the butterfly velocity vB.The paper identifies this inequality as an exact result for the random circuit.Source: §I · p. 2

II) Ergodic Floquet system

• Numerics for ergodic Floquet circuits find similarly diffusive wavefront broadening, motivating a tentative conjecture for generic ergodic one-dimensional spin chains.The conjecture includes corresponding consequences for OTOC and entanglement dynamics.Source: §I · p. 2

III) Clifford circuits

• Clifford circuits lack diffusively broadening fronts and have trivial OTOCs, yet can exhibit linear entanglement growth and infinite-temperature thermalization of local observables.Thus ballistic entanglement growth and thermalization alone do not determine scrambling behavior.Source: §I · p. 2

Scope and motivation

• The study connects operator spreading, OTOCs, entanglement entropy, and late-time local-correlation behavior in systems without local conserved quantities.Hydrodynamics is therefore proposed as a potentially universal feature of one-dimensional ergodic local-unitary systems with bounded local Hilbert spaces.Source: §I · p. 2

Operator basis

• Operators are expanded in a complete orthonormal basis of q2L generalized Pauli strings on the one-dimensional chain.The single-site operators generalize Pauli matrices, with the usual case recovered at q = 2.Source: §II · p. 2

Coarse-grained description

• The full operator-spread coefficients encode time evolution, but coarse-grained endpoint weights suffice for quantities such as entanglement entropies and OTO commutators.The paper focuses especially on total weight carried by strings with a specified right endpoint.Source: §II · p. 2

Operator density

• The right-endpoint density is conserved and is therefore interpreted as an operator density for the time-evolved Pauli string.Its evolution is independent of the initial operator’s internal structure, aside from endpoint-dependent initial conditions.Source: §II · p. 2

Circuit geometry

• The circuits use alternating layers of two-site gates, giving a light-cone velocity vLC = 1 because an operator can spread by at most one site per timestep.The same local-circuit geometry underlies the random and Floquet models.Source: §II · p. 2, Figure 1 · p. 3

Analytical program

• For the random circuit, the averaged endpoint density obeys a biased diffusion equation, which is then used to derive OTOC and entanglement dynamics.The comparison with Floquet circuits tests whether the resulting wavefront behavior extends beyond random gates.Source: §II · p. 2

Relation to prior work

• The paper’s regular circuit geometry produces results distinct from KPZ-like entanglement fluctuations reported for circuits with random gate locations.This comparison suggests that KPZ-like behavior depends on randomness in circuit geometry, not only in individual gates.Source: §III · p. 3

Model definition

• Each circuit layer applies two-site gates on alternating bonds, with every gate independently sampled from the Haar distribution over q2 × q2 unitaries.The regular brickwork geometry differs from circuits with randomly chosen gate locations.Source: §III · p. 3

Goals and scope

• The primary goal is to quantify operator spreading and relate it to entanglement growth in the thermodynamic-limit random circuit.The study also examines late-time local correlations, which approach infinite-temperature values for every circuit realization.Source: §III · p. 3

Analytical and numerical approach

• Averaged operator density performs a biased random walk independent of the operator’s internal structure, enabling exact formulas for OTOCs and entanglement growth.Numerical calculations separately quantify fluctuations among random circuit realizations.Source: §III · p. 3

A. Random walk dynamics of operator density

• For non-trivial µ, ν(τ) vanishes for any τ ≥1, while the average modulus squared, |cµν(τ)|2, can be non-zero; the average operator density is then defined.Its equation of motion does not depend explicitly on µ, which enters only as an initial condition on the operator density.Source: A. Random walk dynamics of operator… · p. 3
• The averaged right-endpoint density obeys a local transition rule in which endpoint weight shifts between neighboring sites under each Haar-random two-site gate.The transition probabilities follow from counting nontrivial two-site operators and equal transition probabilities under Haar averaging.Source: A. Random walk dynamics of operator… · p. 4
• Right endpoints perform a biased random walk with right, left, and stationary probabilities p2, (1−p)2, and the remainder, independently of internal operator structure.Thus all initial Pauli strings with the same right endpoint share the same averaged density profile up to translation.Source: A. Random walk dynamics of operator… · p. 4
• The front moves with vB = (q2−1)/(q2+1) below vLC = 1, while its variance grows diffusively with diffusion constant D = (1−vB^2)/4.As q approaches infinity, vB approaches the light-cone velocity and the front becomes infinitely sharp.Source: A. Random walk dynamics of operator… · p. 4
• Left endpoints propagate with velocity −vB and the same diffusion rate, creating two oppositely moving operator fronts.Most operator weight remains near these fronts, with endpoint uncertainty growing as a diffusive width.Source: A. Random walk dynamics of operator… · p. 4, Figure 2 · p. 4
• The continuum scaling profile of the propagating front is Gaussian, while integrated endpoint weights provide the quantities used to analyze OTOCs.The integrated weights are related to the OTO commutator through the operator weight lying to one side of a site.Source: A. Random walk dynamics of operator… · p. 5

• The OTOC is related to operator weights, with corrections exponentially small in time, and changes appreciably when the operator front reaches s = v_Bt.The averaged OTOC can be expressed using the total weight and density of operator strings around site s.Source: B. Behavior of out-of-time-order… · p. 6
• Outside the light cone the OTOC is exactly zero, while after light-cone entry it begins growing before the main operator front arrives.In the early-growth regime, the OTOC is approximated from the operator density near s.Source: B. Behavior of out-of-time-order… · pp. 5–6
• The early exponential increase has a separation-dependent exponent and limited validity, so it does not establish a fixed Lyapunov regime.The OTOC remains exponentially small in s during this regime, and the exponential behavior narrows for large q.Source: B. Behavior of out-of-time-order… · pp. 5–6
• Near the butterfly front, the OTOC rises from exponentially small in s to an O(1) value across a front of width proportional to √τ.This intermediate regime requires combining the operator-density and OTOC expressions.Source: B. Behavior of out-of-time-order… · p. 6
• After the front passes, the OTOC relaxes exponentially toward its equilibrium value 1, with a late-time exponent independent of s.The figure caption identifies this decay and distinguishes it from the earlier light-cone and front-arrival regimes.Source: B. Behavior of out-of-time-order… · p. 6, Figure 3 · p. 6

• The numerical method represents the evolved operator as a matrix product operator, enabling efficient circuit evolution and extraction of OTOCs and regional operator weights.Two circuit layers are implemented as one TEBD step, reaching bond dimension χ = 20000.Source: C. Fluctuations from circuit to circuit · p. 7
• Circuit-to-circuit fluctuations are largest at the traveling wavefront, become smaller behind it, and reveal a well-defined front in individual circuits.The comparison uses 100 random circuit realizations and evaluates both the OTOC and operator weight.Source: C. Fluctuations from circuit to circuit · p. 7
• The maximum standard deviation of the operator weight decreases approximately as τ^-β with 0.4 < β < 0.5.The decay is measured by maximizing the standard deviation over all spatial positions.Source: C. Fluctuations from circuit to circuit · p. 7

• The second Rényi entropy initially grows linearly and saturates at the maximal value L_A log q at long times.The linear-growth regime applies for τ ≪ L_A.Source: D. Relationship to entanglement… · p. 8
• The entanglement velocity satisfies v_E < v_B for finite q because the operator front broadens diffusively in time.A sharp or time-independent-width front would instead give v_E = v_B.Source: D. Relationship to entanglement… · p. 8
• For large q, the entanglement velocity approaches 1 logarithmically slowly as v_E ∼ 1 − log(2)/log(q).The broader scaling is v_LC − v_E ∝ 1/log q.Source: D. Relationship to entanglement… · pp. 7–8
• Diffusive broadening softens the entropy’s saturation compared with a simple operator-spreading model, while the intermediate saturation regime shrinks as q increases.The identity contribution becomes important when other operators have essentially left the subsystem.Source: D. Relationship to entanglement… · p. 8, Figure 5 · p. 8
• The exact entropy formula captures both typical and average second Rényi entropies, with numerical results showing no significant difference between them.Matrix product state calculations over 100 circuit realizations show extremely good agreement with the exact formula.Source: Figure 5 · p. 8, D. Relationship to entanglement… · p. 8

IV. COMPARISON WITH THE KICKED ISING MODEL

• The kicked Ising model retains a diffusively broadening operator front despite its deterministic, periodic gates and absence of energy conservation.This supports a tentative extension of the diffusive-front picture to generic ergodic one-dimensional spin chains.Source: §IV · pp. 8–9
• The butterfly velocity depends strongly on transverse coupling g and can be much smaller than the light-cone velocity vLC.At g ≈ 0.9, the butterfly velocity approaches vLC, while g = 0 leaves an initially localized Z0 operator unchanged.Source: §IV · p. 9
• Reducing vB below its maximum increases the front width at fixed time, consistent with a biased random walk in which stronger rightward hopping raises vB and suppresses diffusion.The figure varies g from 0.2 through 0.9 while fixing h = 0.809 and T = 1.6.Source: §IV · p. 9, Figure 6 · p. 9
• The wavefront width grows as σ(t) ∝ t^α with 0.5 ≲ α ≲ 0.6, indicating roughly diffusive spreading over time.The width is extracted from the standard deviation of ρR(s) = R(s) − R(s − 1).Source: §IV · p. 9, Figure 7 · p. 10
• Numerics confirm vE < vB for multiple transverse-field couplings g, as expected from diffusive front broadening.The model is an ergodic kicked Ising Floquet circuit when g and h are sufficiently large.Source: §IV · pp. 9–10

V. FRACTAL CLIFFORD CIRCUITS

• Under Clifford evolution, each Pauli string maps to a single Pauli string through a constrained linear transformation rather than a superposition of Pauli strings.The transformation preserves Pauli commutation relations and supports the classification into fractal, glider, and periodic circuits.Source: §V · p. 10
• Clifford OTOCs show persistent late-time oscillations instead of settling to the decay expected in generic thermalizing systems.For Pauli-string probes, the result is state-independent and determined by the commuting phase.Source: §V · p. 10
• In the periodic Clifford circuit, most operators move at 1 site per period, producing near-linear entanglement growth with a long-time rate largely independent of the initial product state.Operators that fail to grow rightward by time t are exponentially rare, with their number scaling roughly as 4^-t.Source: §V · pp. 10–11, Figure 8 · p. 11
• Clifford circuits can exhibit linear entanglement growth and thermalization of local observables, yet lack the spectral and OTOC behavior expected of generic ergodic systems.Their spectrum lacks expected level repulsion and instead has exact recurrences on timescales linear in system size.Source: §V · pp. 10–11
• Linear entanglement growth measures operator extent across many Pauli strings, whereas OTOCs additionally detect whether one operator becomes a complicated superposition.Thus entanglement and OTOCs distinguish different aspects of quantum-information spreading.Source: §V · p. 11

VI. CONCLUSIONS

• Operators in random circuits form a front moving at vB, distinct from the light-cone velocity, while the front broadens diffusively and produces vE < vB.The same hydrodynamic description gives exact OTOC and typical entanglement-growth formulas.Source: §VI · p. 11
• A kicked Ising Floquet chain numerically exhibits a diffusively broadening operator front, motivating a tentative conjecture for generic ergodic one-dimensional systems.Source: §VI · p. 11
• Clifford circuits lack diffusively broadening operator fronts and have pathological OTOCs, despite allowing linear entanglement growth and local-observable thermalization.Thus linear entanglement growth and thermalization do not reliably predict OTOC scrambling.Source: §VI · p. 11

Appendix A: Validity of Eq. (14)

• The Appendix A asymptotics require t →∞ with |x − vBt|/t fixed and nonzero, and become less reliable near the front.The near-front expansion is more useful there.Source: Appendix A · p. 12

Appendix B: Application to OTOC

• In the OTOC application, endpoint contributions dominate while the remaining terms are exponentially suppressed in time under the stated scaling conditions.Source: Appendix B · p. 13

Appendix C: Exact operator spread coefficients

• For one-site Pauli operators, exact spread coefficients reduce to counting single-domain configurations inside the light cone, equivalently a two-particle random walk of domain endpoints.Source: Appendix C · p. 13, Figure 9 · p. 13

1. Useful limits

• For fixed operator endpoints, the large-depth spread weight decays exponentially quickly.Source: §1 · p. 14

Appendix D: Long time correlations

• Nontrivial averaged one- and two-point functions vanish at unequal times, independently of the initial state and the second operator’s label.Source: Appendix D · p. 14
• The variance of these correlations decays exponentially in time for random initial product states.Source: Appendix D · p. 14
• For random product states, the argument is less rigorous and uses operator weight concentrated near ±vBt together with exponentially small expectations of extensive Pauli strings.Source: Appendix D · p. 15

1. Infinite temperature results

• At large t with z/t → 0, the infinite-temperature variance is approximated by a single-site Haar-averaged weight and decays exponentially.Source: §1 · p. 15

Appendix E: Haar Identities

• Haar moments for random unitary matrices provide the identities needed to derive averaged square operator-spread coefficients.Source: Appendix E · p. 15

Appendix F: Recurrence times in Translation invariant Clifford quantum cellular automata (CQCA)

• The associated centered matrix has period c + 2, with the period equal to 2 or 3 according to the trace parameter.Source: Appendix F · p. 15
• For system sizes L = 2^n, translation-invariant centered CQCAs have recurrence times linear in system size, at most trec = 12L.Source: Appendix F · p. 15
• For L = 2^n, the recurrence bound implies U^12L is proportional to the identity.Source: Appendix F · p. 16
• Translation-invariant CQCAs have exponentially large average level degeneracy because their Hilbert spaces grow exponentially while the number of eigenvalues is bounded linearly in system size.Source: Appendix F · p. 16