Spread Complexity in Quantum Systems

• Spread Complexity is a quantum-state complexity measure defined as the first moment of the Krylov-space probability distribution, capturing the average depth reached during time evolution.
• It is computed using the Lanczos algorithm to construct an optimized Krylov basis from an initial state and Hamiltonian, offering clear insights into dynamical evolution.
• Across diverse systems—from chaotic and integrable models to quench dynamics and phase transitions—spread complexity unveils distinct growth profiles and connections to spectral statistics.

Searching arXiv for papers on spread complexity to ground the article in the current literature. arXiv search query: "all:spread complexity" Spread complexity is a quantum-state complexity measure defined by how far a time-evolved state spreads in an ordered orthonormal basis, with the canonical choice given by the Krylov basis generated from the Hamiltonian and the initial state. In its standard state-based form, it is the first moment of the probability distribution over Krylov-space position, $\mathcal C(t)=\sum_n n\,|\phi_n(t)|^2$, and is often described as Krylov complexity for states. It is distinct from Nielsen circuit complexity, because it is not defined through a geodesic cost on a gate manifold, and it is also distinct from inverse-participation-ratio diagnostics, because it measures a weighted positional moment in Krylov space rather than inverse support. At the same time, it is closely connected to operator Krylov complexity, spectral statistics, state-manifold geometry, and, in several settings, holographic or semiclassical observables (Gautam et al., 2023, Huh et al., 2023).

1. Formal definition and Krylov-space construction

The standard construction begins with an initial state $\ket{\psi_0}$ evolving under a Hamiltonian $H$. For a generic orthonormal basis $\{\ket{B_n}\}$, one defines the basis-dependent spread cost

$$\mathcal C_B(t)=\sum_n n\,|\langle \psi(t)|B_n\rangle|^2.$$

A central structural result is that this cost is minimized by the Krylov basis generated from $(H,\ket{\psi_0})$. The Krylov basis $\{\ket{K_n}\}$ is constructed recursively by the Lanczos algorithm, with $\ket{K_0}=\ket{\psi_0}$ and

$$\ket{K_{n+1}}=\frac{1}{b_{n+1}}\Big[(H-a_n)\ket{K_n}-b_n\ket{K_{n-1}}\Big],$$

where $a_n=\langle K_n|H|K_n\rangle$ and $\ket{\psi_0}$0 enforce normalization. Expanding the time-evolved state as

$\ket{\psi_0}$1

one obtains a discrete Schrödinger equation on the Krylov chain,

$\ket{\psi_0}$2

The corresponding spread complexity is then

$\ket{\psi_0}$3

Physically, $\ket{\psi_0}$4 is the average depth reached by the state in the basis naturally generated by its own dynamics (Gautam et al., 2023).

This formalism makes clear why spread complexity is often called the state analogue of operator Krylov complexity. The role played by operator growth in Liouville space is replaced by state spreading in Hilbert space. The same literature also emphasizes that this quantity is not Nielsen circuit complexity: there is no geodesic optimization over gate paths, and the complexity operator is instead the Krylov-position operator $\ket{\psi_0}$5. For thermofield-double states and other specially structured initial states, the same first-moment definition remains operative, and the Krylov amplitudes $\ket{\psi_0}$6 or $\ket{\psi_0}$7 carry the full dynamical content of the construction (Huh et al., 2023).

2. Quench dynamics and universal growth laws

A major line of work studies spread complexity after a sudden quench. If the initial state is the ground state of $\ket{\psi_0}$8 and the post-quench evolution is governed by $\ket{\psi_0}$9, then the local density of states (LDOS) and the survival probability $H$0 control the early Lanczos data. The universal result is that for $H$1, where $H$2 is the LDOS variance, the survival probability has the expansion $H$3, and the spread complexity always begins quadratically, $H$4. Beyond this perturbative window, the behavior separates according to the decay law of the survival probability: Gaussian LDOS and Gaussian survival preserve quadratic growth, whereas a Breit–Wigner LDOS and exponential survival decay generate approximately linear growth because the relevant Lanczos coefficients become approximately linear in $H$5. In full random matrices from the GOE and in chaotic disordered spin-$H$6 chains, the resulting pattern is an initial quadratic onset, an early linear ramp, and late-time saturation; the full random-matrix case additionally displays a pronounced intermediate-time peak tied to the correlation hole in the survival probability, while the realistic spin chain shows only a weak version of that feature (Gautam et al., 2023).

The quenched Lipkin–Meshkov–Glick model supplies a complementary critical example. In the thermodynamic-limit bosonic description, noncritical quenches produce oscillatory spread complexity with complete revivals, and both the amplitude and the period increase as the final Hamiltonian approaches the equilibrium critical point. The effective number of Krylov basis states contributing above a fixed cutoff, $H$7, grows strongly near criticality and does so differently in the broken and symmetric phases, which allows the two phases to be distinguished dynamically. For a critical quench, the oscillatory regime is replaced by

$H$8

while the associated spread entropy grows logarithmically at late times, $H$9 (Afrasiar et al., 2022).

A more general early-time derivation follows directly from the Krylov-space Schrödinger equation. For $\{\ket{B_n}\}$0, one has $\{\ket{B_n}\}$1, $\{\ket{B_n}\}$2, and $\{\ket{B_n}\}$3, so

$\{\ket{B_n}\}$4

The same analysis is compatible with the “Ehrenfest theorem of complexity,” which expresses $\{\ket{B_n}\}$5 in terms of Lanczos data and Krylov amplitudes. In that sense, the universal quadratic onset is not restricted to quenches in disordered many-body systems, but is a generic short-time property of the state-based Krylov construction itself (Huh et al., 2023).

3. Chaos, integrability, and the limits of diagnosis

Spread complexity has often been used as a probe of chaoticity, but the literature also makes clear that its diagnostic scope is limited. In integrable systems with saddle-dominated scrambling, such as the Lipkin–Meshkov–Glick model and the inverted harmonic oscillator with a stabilizing quartic term, the spread complexity of thermofield-double states exhibits a ramp-peak-slope-plateau pattern that closely resembles behavior previously conjectured for chaotic systems. The early-time regime is still quadratic, and the paper also shows that $\{\ket{B_n}\}$6 reproduces the spectral form factor in these examples. The main conclusion is explicit: spread complexity is a useful probe of state spreading and scrambling-like dynamics, but the mere appearance of a peak or of a ramp-peak-slope-plateau profile is not sufficient to diagnose genuine quantum chaos without additional input such as level statistics or other spectral information (Huh et al., 2023).

In periodically driven spin chains, the same issue appears in Floquet language. There the natural basis is constructed by Arnoldi iteration rather than Hermitian Lanczos tridiagonalization, because the central dynamical object is a unitary Floquet operator. In kicked Ising chains with nonintegrable deformations, chaotic dynamics is characterized by suppressed fluctuations of the Arnoldi coefficients, stronger delocalization in Krylov space, and a larger late-time saturation value of spread complexity. Regular or near-integrable dynamics instead shows larger coefficient fluctuations, stronger quasiperiodic structure, and lower saturation. The paper also identifies a Floquet-specific effect: increasing the drive frequency suppresses the saturation value and makes the Arnoldi data more Lanczos-like, while a special dual-unitary point realizes the maximal linear growth slope $\{\ket{B_n}\}$7 before recurrence (Nizami et al., 2024).

A more minimal spectral-statistical perspective is provided by the study of generic two-level subsystems. There the exact spread complexity is simply

$\{\ket{B_n}\}$8

with $\{\ket{B_n}\}$9 the relevant level spacing, and averaging over level-spacing distributions produces the familiar slope-dip-ramp-plateau structure. The same work shows that higher-order level spacings generate additional iterative peaks. Crucially, this structure is not confined to Wigner–Dyson statistics: suitably chosen higher-order spacings in integrable Poisson spectra can produce qualitatively similar spread-complexity profiles. This establishes, in a particularly transparent setting, that spread complexity is strongly sensitive to spectral statistics but not a universal binary marker of chaos (Astaneh et al., 19 Apr 2025).

The diagnostic role of long-time-averaged spread complexity also extends to semiclassical dynamical transitions. In two-mode Bose–Einstein condensates, the long-time average $$\mathcal C_B(t)=\sum_n n\,|\langle \psi(t)|B_n\rangle|^2.$$0 sharply distinguishes self-trapping from Josephson oscillation as the control parameter crosses an initial-state-dependent critical value, and in a triple-well bosonic model the same quantity is larger in classically chaotic regions than in regular ones. This suggests that spread complexity can function as a coarse measure of dynamically accessible phase-space volume, but again the interpretation is model dependent (Zhou et al., 2024).

4. Phase transitions, localization, and non-unitary extensions

Beyond chaos diagnostics, spread complexity has been used to characterize equilibrium, dynamical, and topological phase structure. In the Kitaev chain, it distinguishes the trivial and topological phases across the transition at $$\mathcal C_B(t)=\sum_n n\,|\langle \psi(t)|B_n\rangle|^2.$$1. Depending on the chosen reference state, the signal appears either as a $$\mathcal C_B(t)=\sum_n n\,|\langle \psi(t)|B_n\rangle|^2.$$2-independent plateau throughout the topological regime or as singular behavior in derivatives of the complexity at the critical point. The analysis is mode resolved through the $$\mathcal C_B(t)=\sum_n n\,|\langle \psi(t)|B_n\rangle|^2.$$3 structure of each $$\mathcal C_B(t)=\sum_n n\,|\langle \psi(t)|B_n\rangle|^2.$$4 sector, and one of the paper’s central conclusions is that spread complexity is sensitive to the topological transition although the detailed signature is circuit dependent (Caputa et al., 2022).

In non-Hermitian many-body localization problems, the notion of spread complexity branches into several inequivalent constructions. One route is the singular-value spread complexity, which replaces the non-Hermitian Hamiltonian by a fictitious Hermitian “singular Hamiltonian” and then uses the usual Lanczos procedure. In the disordered hard-core-boson models studied there, a pronounced pre-saturation peak in the singular-value spread complexity diagnoses the ergodic or chaotic regime, while the peak is suppressed in the MBL regime. A second route uses actual non-Hermitian evolution and Arnoldi-generated Krylov bases. For a TFD-like initial state, the saturation value of the spread complexity is always below $$\mathcal C_B(t)=\sum_n n\,|\langle \psi(t)|B_n\rangle|^2.$$5 when complex eigenvalues are present and approaches $$\mathcal C_B(t)=\sum_n n\,|\langle \psi(t)|B_n\rangle|^2.$$6 only when the spectrum becomes entirely real, so the TFD construction detects the real-complex transition. A third route uses a charge-density-wave initial state and probes localization more directly, with qualitative differences between time-reversal-symmetric and non-time-reversal-symmetric models (Ganguli, 2024).

Measurement-induced non-unitary dynamics requires a further modification. In a tight-binding chain subject to repeated measurements, the effective generator is non-Hermitian, so the relevant Krylov construction uses bi-Lanczos or, for complex-symmetric Hamiltonians, a specialized complex-symmetric Lanczos algorithm. In this setting the normalized spread complexity shows a distinctive growth-decay-saturation profile rather than monotone growth to saturation, and the onset time for complexity growth increases as the interval between measurements decreases. In the limit of vanishing interval, that onset time asymptotes to infinity, providing a complexity-based signature of the quantum Zeno effect (Bhattacharya et al., 2023).

A related but distinct long-time localization perspective appears in the analysis of a disordered interacting spin chain across the ergodic–MBL transition. There the infinite-time Krylov spread complexity $$\mathcal C_B(t)=\sum_n n\,|\langle \psi(t)|B_n\rangle|^2.$$7 scales linearly with the Hilbert-space dimension $$\mathcal C_B(t)=\sum_n n\,|\langle \psi(t)|B_n\rangle|^2.$$8 in the ergodic phase, $$\mathcal C_B(t)=\sum_n n\,|\langle \psi(t)|B_n\rangle|^2.$$9, but only as $(H,\ket{\psi_0})$0 with $(H,\ket{\psi_0})$1 in the MBL phase, so the long-time state occupies only a vanishing fraction of the Krylov chain. The infinite-time Krylov profile is approximately flat in the ergodic phase and stretched exponential in the MBL phase, and a large-deviation analysis shows that MBL complexity is dominated by a vanishing fraction of eigenstates with anomalously large complexity (Pain et al., 26 Mar 2026).

5. Geometric, holographic, and computational reformulations

A distinct line of work interprets spread complexity geometrically. For coherent-state manifolds associated with low-rank algebras such as $(H,\ket{\psi_0})$2, $(H,\ket{\psi_0})$3, and Heisenberg–Weyl, the projective state manifold carries a Kähler structure with Fubini–Study metric

$(H,\ket{\psi_0})$4

Within this setting, the spread complexity in the Krylov basis can be written as a scalar expectation value of symmetry generators, and the same scalar is identified with the dilaton of a matter-free Jackiw–Teitelboim gravity model built on the same state manifold. The resulting picture separates two notions of complexity geometrically: Nielsen complexity is encoded by geodesic distance in the Fubini–Study metric, whereas spread complexity appears as a scalar field on that metric background (Chattopadhyay et al., 2023).

The holographic reinterpretation becomes more explicit for locally excited states in holographic $(H,\ket{\psi_0})$5 CFTs. For states of the form

$(H,\ket{\psi_0})$6

the return amplitude is fixed by conformal symmetry, the Lanczos coefficients take an $(H,\ket{\psi_0})$7 form, and the spread complexity can be computed exactly. In the vacuum-on-line, thermal, and finite-size cases, the time derivative of the spread complexity satisfies

$(H,\ket{\psi_0})$8

where $(H,\ket{\psi_0})$9 is the canonical momentum conjugate to the proper radial distance coordinate in the dual $\{\ket{K_n}\}$0 geometry. The choice of proper radial distance is essential: the relation fails in a generic radial coordinate and becomes exact only in the proper-distance variable (Caputa et al., 2024).

Spread complexity has also been recast as a limiting case of a quantum circuit complexity. In that construction, the basic costly operation is a discrete time-evolution gate $\{\ket{K_n}\}$1, while superposition and beam-splitting operations are taken to be cost free. The minimal synthesis cost of a target state is then the expectation value of an operator $\{\ket{K_n}\}$2, where $\{\ket{K_n}\}$3 is an orthogonal basis generated from powers of $\{\ket{K_n}\}$4. In the limit $\{\ket{K_n}\}$5, $\{\ket{K_n}\}$6, so the circuit complexity converges to the usual spread complexity. One stated advantage is computational: the discrete formulation can be built from return amplitudes at finite time steps and may remain useful when the conventional Lanczos or moment expansion becomes inaccessible (Beetar et al., 8 Jun 2025).

For time-dependent harmonic oscillators, spread complexity acquires an unusually direct physical meaning. In the $\{\ket{K_n}\}$7 Krylov basis of the squeezed evolved vacuum, one finds the exact identity

$\{\ket{K_n}\}$8

so spread complexity is directly determined by phase-space fluctuations. In the same model it is also related to circuit complexity by

$\{\ket{K_n}\}$9

and both quantities are fixed by the mean particle number and its rate of change. This suggests a physical interpretation of spread complexity in terms of excitation content and squeezing, rather than only abstract delocalization in Krylov space (Chowdhury, 3 Nov 2025).

6. Terminology, applications, and interpretive boundaries

Although the standard state-based definition of spread complexity is the first moment of a probability distribution in Krylov space, the term is not completely uniform across the literature. In work on operator growth in open quantum systems, the basic object is instead the Shannon entropy of the operator-population distribution in a chosen operator basis,

$\ket{K_0}=\ket{\psi_0}$0

and the “spread complexity” is defined as $\ket{K_0}=\ket{\psi_0}$1. In that operator-space setting the Krylov basis again minimizes the spread quantity, but the measure is entropic rather than a first moment. This usage is therefore related but not identical to the state-based convention that dominates the present topic (Carolan et al., 2024).

The same conceptual flexibility has enabled more speculative applications. In the study of string scattering amplitudes, a “fictional Hamiltonian” is constructed by taking the extrema of the logarithmic derivative of a scattering amplitude as an effective spectrum, building a TFD state from those extrema, and then computing a Krylov spread complexity from the resulting partition function. In that framework, chaotic highly excited string-state scattering into two or three tachyons produces a pre-saturation peak in the complexity, whereas non-chaotic examples such as leading-Regge-trajectory or Veneziano amplitudes yield oscillatory behavior without such a peak. The construction is explicitly phenomenological rather than a derivation from a microscopic field-theory Hamiltonian (Bhattacharya et al., 2024).

A different application appears in high-energy astrophysical neutrinos. There the standard three-flavor oscillation Hamiltonian defines a Krylov basis for flavor evolution, and the resulting spread complexity leads to a new complexity-based flavor-ratio proposal. The paper argues that such ratios could favor an initial source composition $\ket{K_0}=\ket{\psi_0}$2 over the conventional $\ket{K_0}=\ket{\psi_0}$3 benchmark once IceCube achieves its projected flavor sensitivity, and that the complexity-based observables show a slight but nonzero sensitivity to neutrino mass ordering that standard flavor ratios lose after phase averaging (Dixit et al., 2024).

Taken together, these developments indicate that spread complexity is best understood as a basis-optimized measure of dynamical spreading whose most standard realization is the first moment of the Krylov-space distribution of a time-evolved state. It has proved useful in quenches, Floquet systems, localization problems, non-Hermitian dynamics, topological transitions, coherent-state geometry, holography, and several more exploratory domains. At the same time, the current literature repeatedly shows that its interpretation is contextual: ramp-peak-slope-plateau profiles are not uniquely chaotic, phase-sensitive signatures can depend on the reference state, and even the precise functional called “spread complexity” may differ between state-based and operator-space formulations. This suggests that the quantity is most informative when read together with the spectral, dynamical, or geometric structure that generates its Krylov basis.