Quantum Spread Complexity

• Quantum Spread Complexity is a basis-independent measure that quantifies how a pure quantum state spreads under time evolution using the Krylov basis.
• The Lanczos algorithm is employed to tridiagonalize the Hamiltonian, revealing universal regimes like quadratic growth, linear ramp, peak, and plateau in complexity.
• It serves as a diagnostic tool for quantum chaos, phase transitions, and experimental observables, linking state evolution to work statistics in many-body systems.

Quantum spread complexity is a quantitative measure designed to capture the evolving “cost” or “spread” of a quantum pure state as it evolves under a given Hamiltonian, with the measure minimized by a particular choice of basis: the Krylov basis. This construction leverages the Lanczos algorithm to systematically build an orthonormal sequence from repeated action of the Hamiltonian on an initial state, yielding both an unambiguous, physically meaningful definition of quantum state complexity and potent diagnostics for chaos, phase structure, and information dynamics in quantum many-body systems.

1. Basis-Independent Definition and Krylov Construction

A central goal of quantum state complexity is to provide a basis-independent quantitative measure of how complex, or “spread out,” a quantum state becomes under time evolution. The paradigm introduced by “Quantum chaos and the complexity of spread of states” (Balasubramanian et al., 2022) establishes this as a minimization problem:

• For a given initial state $|\Psi_0\rangle$ and a time-evolved state $|\Psi(t)\rangle = e^{-iHt}|\Psi_0\rangle$, expand $|\Psi(t)\rangle$ in a general orthonormal basis $\{|B_n\rangle\}$,
• Define a cost function for the “spread” as

$$\mathcal{C}(t; \mathcal{B}) = \sum_n n\,|\langle B_n|\Psi(t)\rangle|^2,$$

and minimize this over all bases containing $|\Psi_0\rangle$ as $|B_0\rangle$.

The authors demonstrate that this minimization is uniquely achieved by the Krylov basis, constructed by repeated applications of $H$ onto $|\Psi_0\rangle$ followed by orthogonalization. The result is

$$\mathcal{C}(t) = \sum_n n\,p_n(t),\quad p_n(t) = |\psi_n(t)|^2,$$

where $\psi_n(t)$ are the coefficients in the expansion $|\Psi(t)\rangle = \sum_n \psi_n(t) |K_n\rangle$. This eliminates the basis-arbitrariness of previous definitions and anchors the complexity measure directly to the system’s intrinsic quantum dynamics.

2. Lanczos Algorithm and Tridiagonalization

The Krylov basis construction is operationalized through the Lanczos algorithm, which recasts the Hamiltonian in tridiagonal (or “Hessenberg”) form within the Krylov subspace:

$$H|K_n\rangle = a_n |K_n\rangle + b_{n+1}|K_{n+1}\rangle + b_n |K_{n-1}\rangle,$$

with recursion coefficients $a_n, b_n$ encoding the essential features of the system’s dynamics and spectral structure.

The time evolution in this basis reduces to a tight-binding chain model for the coefficients:

$$i \frac{\partial}{\partial t} \psi_n(t) = a_n \psi_n(t) + b_{n+1} \psi_{n+1}(t) + b_n \psi_{n-1}(t).$$

The roles of $a_n$ and $b_n$ are model-dependent:

• In Lie-algebraic systems (e.g., $SL(2,\mathbb{R})$, $SU(2)$, Heisenberg–Weyl), analytic forms for $a_n$, $b_n$ are often accessible, with $b_n$ frequently growing as $\sqrt{n}$ or linearly with $n$.
• Asymptotic relations, such as $a_n \approx 2b_n +$ const., can diagnose “free-particle-like” complexity growth and other structural regimes in the Krylov chain.

3. Universal Dynamical Regimes and Spectral Signatures

The detailed temporal structure of spread complexity in chaotic systems exhibits features that are tightly correlated with universal random-matrix statistics and corresponding spectral diagnostics:

• Early-time universal growth: For short times, $C(t)$ grows quadratically as $C(t) \approx \sigma_E^2 t^2$, where $\sigma_E^2$ is the energy variance of the initial state.
• Linear “ramp”: After the initial quadratic regime, complexity advances linearly for a parametrically wide window, reflecting the spectral rigidity and correlated “ramp” in the spectral form factor. For thermofield double (TFD) states in large chaotic systems, the ramp persists for exponentially long times in the system’s entropy.
• “Peak” and “slope”: Complexity overshoots, reaching a peak exceeding its long-time value, then decays (“slope”) towards its saturated value.
• Plateau: $C(t)$ stabilizes, forming a plateau controlled by late-time universal spectral correlations.

This four-regime “ramp–peak–slope–plateau” structure is a direct quantum analog of the random matrix spectral form factor’s “slope-dip-ramp-plateau” structure and is driven by universal features such as spectral rigidity and level repulsion.

Regime	Complexity Behavior	Spectral Feature
Early-time	$C(t) \sim t^2$	Gaussian survival amplitude decay
Ramp	$C(t) \sim t$	Linear spectral form factor
Peak	$C(t)$ maximal	Maximal delocalization
Slope/Plateau	$C(t)$ decay/saturates	Spectral rigidity/plateau

4. Analytic and Model-Specific Applications

The framework is applied to a broad range of analytically solvable and numerically tractable models:

• Harmonic and inverted oscillators: The spread complexity for stable oscillators is periodic, while for inverted oscillators (corresponding to “unstable saddles”), the complexity exhibits exponential growth.
• Group-manifold models: For particles on $SL(2,\mathbb{R})$, $SU(2)$, and Heisenberg–Weyl, explicit expressions for Lanczos coefficients lead to closed-form complexity evolution.
• Chaotic many-body models: For the SYK model and random matrix ensembles (GUE, GOE, GSE), the sequence of regimes in the spread complexity is realized, with peak heights and plateau values distinguishing random matrix universality classes.
• Schwarzian theory: In the semiclassical limit, the complexity is governed by the analytically continued partition function and exhibits persistent quadratic growth as $C(t)\sim \sigma_E^2 t^2$ in the free-particle limit.

In all cases, the (Krylov) spread complexity provides both an analytic handle and a numerically robust diagnostic for integrable vs. chaotic dynamics.

5. Thermodynamic and Experimental Perspectives

Spread complexity connects directly to experimentally accessible thermodynamic quantities. In quantum quench protocols, the relevant Lanczos coefficients can be mapped to the cumulants of the work distribution:

$$a_0 = \langle W\rangle, \quad b_1^2 = \langle W^2 \rangle - \langle W \rangle^2,$$

thereby relating early-time complexity growth to the variance in energy change under a quench. This correspondence between complexity growth and work statistics enables experimental probes in cold atom setups, NMR, and trapped ion systems, where measurement of survival probabilities and work distributions is feasible.

6. Relation to Operator Growth and Quantum Chaos

Spread complexity is intimately tied to operator growth and chaos diagnostics:

• Krylov complexity: The average position in Krylov space is a state analog of operator size; the evolution of $C(t)$ encapsulates the spreading of quantum information.
• Distinguishing chaos: Features such as pronounced ramp–peak–slope–plateau sequences and parameter-dependent peak properties sharply distinguish chaotic (Wigner-Dyson) models from integrable ones.
• Limitations: The presence of similar complexity growth features in saddle-dominated, yet integrable, models indicates that while spread complexity is sensitive to chaotic features, it is not in itself a unique identifier for quantum chaos—additional physical input or operator-dependent measures may be necessary.

7. Topological Phases and Phase Transitions

The spread complexity framework enables the identification of topological phase transitions and critical points in quantum many-body systems:

• Topological phases: In models such as the Su–Schrieffer–Heeger (SSH) and Kitaev chains, spread complexity becomes constant (“plateau”) in the topological phase and varies smoothly in the trivial phase, capturing topological protection.
• Critical points: Transitions are marked by nonanalytic behavior (e.g., divergence of complexity derivatives) at critical parameter values, providing a sharp diagnostic of phase boundaries.

Dynamical protocols (e.g., quenches from trivial to topological regimes) manifest distinct temporal signatures in spread complexity, reinforcing its effectiveness as a phase diagnostic.

---

In sum, quantum spread complexity provides a rigorous, physically transparent, and basis-independent measure of quantum state complexity, minimizing ambiguity via the Krylov basis/Lanczos construction. Its evolution encodes both universal and model-specific microphysics across the spectrum from integrability to chaos, links to thermodynamic fluctuations and experimental observables, and serves as an incisive probe of information spreading, thermalization, and phase structure in complex quantum systems (Balasubramanian et al., 2022, Caputa et al., 2022, Caputa et al., 2022).