Operator Spreading in Quantum Systems

Updated 3 June 2026

Operator spreading is the expansion of an initially local quantum operator under time evolution, quantifying information scrambling and chaos.
It is characterized by tools like out-of-time-ordered correlators and operator size distributions, linking microscopic dynamics to macroscopic hydrodynamics.
Researchers use frameworks such as random circuits, hydrodynamic models, and open system dynamics to study the interplay between unitary evolution and decoherence.

Operator spreading is a central concept in the study of quantum information dynamics, quantum chaos, and thermalization in many-body quantum systems. It refers to the growth in spatial support (“size”) of an initially local operator under Heisenberg evolution, and provides a rigorous framework for understanding information scrambling, emergent hydrodynamics, and the structure of quantum noise and decoherence. Operator spreading is deeply connected to key observables such as out-of-time-ordered correlators (OTOCs), operator size distributions, and information-theoretic capacities, and can be quantitatively analyzed using a variety of techniques in both closed and open quantum systems.

1. Formal Definitions and Operator Size

In the Heisenberg picture, a local operator $O$ evolves as $O(t) = e^{iHt} O e^{-iHt}$ , where $H$ is the system Hamiltonian. Initially localized, $O$ typically spreads in operator space, acquiring nonzero support over increasingly nonlocal basis elements — for spin systems, sums of Pauli strings with growing domain size. This process is formalized by:

Expansion in operator basis: $O(t) = \sum_\alpha c_{\alpha}(t) W_{\alpha}$ , where $\{W_\alpha\}$ are multi-site Pauli strings or their generalizations.
Operator size: The “size” $|W_\alpha|$ of a basis element is defined as the number of sites on which it acts nontrivially.
Size distribution: $P(s, t) = \sum_{|W_\alpha| = s} |c_\alpha(t)|^2$ .
Mean and variance: $\langle s(t) \rangle = \sum_s s\,P(s, t)$ and $\mathrm{Var}[s(t)] = \sum_s s^2 P(s, t) - \langle s(t) \rangle^2$ .

Physically, operator spreading quantifies how an initially local perturbation becomes delocalized due to quantum interactions, encoding the microscopic structure of information flow and chaos (Schuster et al., 2021).

2. Signatures, Diagnostics, and Hydrodynamics

Out-of-time-ordered correlators (OTOCs): The squared commutator

$O(t) = e^{iHt} O e^{-iHt}$ 0

measures how non-commutativity spreads, signaling information scrambling and operator growth (Nahum et al., 2017 Geller et al., 2021). For short-range Hamiltonians, OTOCs exhibit a ballistic “light cone” characterized by a butterfly velocity $O(t) = e^{iHt} O e^{-iHt}$ 1, with the front broadening diffusively ( $O(t) = e^{iHt} O e^{-iHt}$ 2 in 1D chaotic or generic integrable chains) (Moudgalya et al., 2018 Lopez-Piqueres et al., 2021 Gopalakrishnan et al., 2018 Khemani et al., 2017).

System	Front Velocity ( $O(t) = e^{iHt} O e^{-iHt}$ 3)	Broadening	Reference
Chaotic spin chains	$O(t) = e^{iHt} O e^{-iHt}$ 4	$O(t) = e^{iHt} O e^{-iHt}$ 5	(Nahum et al., 2017 Lopez-Piqueres et al., 2021)
Integrable models	$O(t) = e^{iHt} O e^{-iHt}$ 6	$O(t) = e^{iHt} O e^{-iHt}$ 7 (diffusive); anomalous scaling $O(t) = e^{iHt} O e^{-iHt}$ 8	(Gopalakrishnan et al., 2018 Lopez-Piqueres et al., 2021)
Free fermions	$O(t) = e^{iHt} O e^{-iHt}$ 9 (maximal)	$H$ 0	(Dias et al., 2021)
QHCG	$H$ 1	Front freezes	(Medenjak, 2022)

In higher dimensions, the operator front becomes a stochastic surface, with broadening governed by Kardar-Parisi-Zhang (KPZ) scaling exponents: $H$ 2 in $H$ 3D, $H$ 4 in $H$ 5D (Nahum et al., 2017).

Hydrodynamic Descriptions: Operator right-weight (distribution of the rightmost nontrivial Pauli in the expansion) and generalized hydrodynamics allow mapping operator spreading to biased diffusion equations and, in integrable models, to ballistic quasiparticle propagation with density-dependent diffusion (McCulloch et al., 2021 Gopalakrishnan et al., 2018).

Operator spreading with conservation laws: Unitary dynamics with locally conserved charges yields two-stage hydrodynamics: a diffusive conserved component and a ballistic non-conserved sector, with operator weight conversion set by local diffusion current (Khemani et al., 2017).

3. Random Circuits and Stochastic Fronts

Random unitary circuits provide an analytically tractable framework for the study of operator spreading and scrambling. In 1D Haar-random circuits, operator spreading is modeled as a biased diffusion of string endpoints: $H$ 6 such that OTOCs are expressed as sums or integrals over the resulting densities. The butterfly velocity $H$ 7 and front broadening $H$ 8 depend on the local dimension $H$ 9 (Nahum et al., 2017 Tan et al., 7 Jan 2025). For general unitary-invariant gate ensembles, the drift-diffusion description persists, with universal features governed by low moments of the gate distribution, a finite “binary-ization” time for Pauli-string weights, and a domain-wall width for convergence to random-matrix-like statistics (Tan et al., 7 Jan 2025).

For long-range interactions decaying as $O$ 0, the butterfly light cone scales as $O$ 1 (ballistic) for $O$ 2, stretched-exponentially for $O$ 3, and algebraically for $O$ 4, as captured by a noisy long-range FKPP equation dual to discrete population dynamics (Zhou et al., 9 May 2025).

4. Experimental and Operational Perspectives

Quantum simulation and tomography: Experiments with trapped ions, Rydberg atom arrays, and superconducting circuits have probed operator spreading and OTOCs directly. Floquet engineering enables controlled measurements of light-cone-like propagation and tunable scrambling (Zhao et al., 2021 Geller et al., 2021).

Weak-measurement quantum tomography protocols reconstruct the unknown quantum state by monitoring a Heisenberg-evolved observable under many-body dynamics. The rate of fidelity growth, Fisher information, Shannon entropy, and rank of the measurement record are directly linked to operator spreading; they scale rapidly for strongly chaotic systems and serve as robust chaos indicators, outperforming Krylov complexity which may lack monotonicity or correlational significance in integrable or finite-size systems (Sahu et al., 2023 Sahu, 2024).

Information-theoretic equivalence: Operator spreading is quantitatively connected to the classical information propagation capacity (Holevo bound) of a quantum channel. The distinguishability between channel outputs is upper- and lower-bounded by commutator norms or trace distances, establishing a tight equivalence between operator growth and information transmission, independent of the system size (Shang et al., 12 May 2025).

5. Open Quantum Systems and Dissipation

In open systems, operator spreading interacts with decoherence, local dissipation, and measurement. The Lindblad master equation governs Heisenberg operator evolution, with the Liouvillian spectrum setting late-time decay. Crucially, bulk dissipation accelerates the decay of autocorrelations: as the operator spreads, more local dissipators act on its support, leading to an initial acceleration regime, a plateau where decay rate is proportional to system size, and eventual crossover to asymptotic decay set by the Liouvillian gap (Shirai et al., 2023 Schuster et al., 2022). The universal structure of decay holds for both Ising and Bose-Hubbard chains and is dominated at intermediate times by the ballistic operator spreading under unitary dynamics (Shirai et al., 2023).

6. Integrability, Quantum Maps, and Special Cases

Integrable systems display ballistic operator fronts, but the mechanism differs from chaotic systems:

In interacting integrable models, fronts are carried by the fastest quasiparticles and broaden diffusively due to equilibrium density fluctuations; the diffusive correction can be computed from generalized hydrodynamics (Gopalakrishnan et al., 2018 Lopez-Piqueres et al., 2021).
Free-fermion circuits or quantum hardcore gases (QHCG) exhibit diffusive or even frozen operator fronts; QHCGs, for certain initial conditions, demonstrate a front that eventually ceases to broaden, in contrast to the persistent diffusive front in chaotic or generic integrable systems (Medenjak, 2022 Dias et al., 2021).
Quantum chaotic maps, such as the perturbed Arnold cat map, exhibit initially classical (Koopman) spreading until the Ehrenfest time, followed by quantum scrambling that saturates at the operator space dimension, diagnosable by the Shannon entropy of operator expansion coefficients and spectral form factor dynamics (Moudgalya et al., 2018).

7. Distinction from Recoverability and Metrological Response

The arrival of operator support as diagnosed by OTOCs or right-weight does not guarantee that quantum information, such as parameter sensitivity, remains locally accessible. An operational hierarchy involving quantum Fisher information (QFI) reveals that while OTOC establishes a strict light cone for support arrival, local recoverability (e.g., single-site or block QFI) may lag behind, especially when gauge or $O$ 5 symmetries are broken, as parameter sensitivity disperses into nonlocal correlations. This separation is quantified by an ordering $O$ 6, indicating a gap between operator support and metrological accessibility (Płodzień et al., 4 May 2026).