Quantum Dynamical Krylov Complexity

• The paper introduces a parameter-free Lanczos-based framework that quantifies quantum state expansion in a dynamically adapted Krylov basis.
• It demonstrates that Krylov complexity diagnoses operator growth, phase transitions, and PT-symmetry breaking in both Hermitian and non-Hermitian systems.
• Analytic and numerical results link the emergence of exponential complexity growth to the imaginary part of system frequencies and the onset of quantum chaos.

Quantum dynamical Krylov complexity quantifies the growth and spreading of a state or operator under non-equilibrium quantum dynamics via a canonical, parameter-free construction rooted in the Lanczos (tridiagonalization) algorithm. It provides a minimal and basis-optimal measure of the expansion of a quantum state (or operator) in a dynamically adapted orthonormal Krylov basis. This measure has become central to diagnosing operator growth, information scrambling, integrability-to-chaos transitions, localization phenomena, and is intimately connected to emergent algebraic and geometric structures in both closed and open quantum systems.

1. Definition, Construction, and Mathematical Framework

The dynamical Krylov complexity is constructed as follows. Given a reference quantum state or operator $|\phi_0\rangle$ (for states) or $O_0$ (for operators), one defines the Krylov subspace as

$$\mathcal{K}_n = \mathrm{span}\{|\phi_0\rangle, H|\phi_0\rangle, H^2|\phi_0\rangle, \dots, H^{n-1}|\phi_0\rangle\}$$

for time-independent Hamiltonians $H$ (Rabinovici et al., 8 Jul 2025, Beetar et al., 2023).

The Lanczos algorithm (or the Arnoldi process for non-Hermitian or driven systems) recursively generates an orthonormal Krylov basis $\{|K_n\rangle\}$: $$(H-a_n)|K_n\rangle = b_{n+1}|K_{n+1}\rangle + b_n |K_{n-1}\rangle, \qquad b_{-1}=0, \ |K_{-1}\rangle=0$$ where $a_n = \langle K_n| H | K_n \rangle$, $$b_{n+1} = \| (H-a_n)|K_n\rangle - b_n |K_{n-1}\rangle \|$$ (Beetar et al., 2023, Rabinovici et al., 8 Jul 2025).

The time-evolved state is expanded as

$$|\psi(t)\rangle = e^{-i H t} |\phi_0\rangle = \sum_{n=0}^\infty \phi_n(t) |K_n\rangle$$

and the dynamical Krylov complexity (spread complexity) is defined as the mean chain position: $K(t) = \sum_{n=0}^{\infty} n |\phi_n(t)|^2$ which equals the expectation value of the “position operator” $N = \sum_n n |K_n\rangle \langle K_n|$ (Beetar et al., 2023, Rabinovici et al., 8 Jul 2025).

For operator dynamics, the same construction applies to the Liouvillian superoperator $\mathcal{L}(\cdot) = [H, \cdot]$, with an operator Hilbert space inner product, typically $(A|B) = \mathrm{Tr}[A^\dagger B] / \mathcal{D}$ for dimension $\mathcal{D}$ (Rabinovici et al., 8 Jul 2025, Rabinovici et al., 2022).

The amplitudes obey a discrete Schrödinger-like equation: $$i\dot\phi_n(t) = a_n \phi_n(t) + b_{n+1}\phi_{n+1}(t) + b_n \phi_{n-1}(t)$$ with $\phi_{-1}(t) \equiv 0$, $\phi_0(0)=1$, $\phi_{n>0}(0)=0$ (Beetar et al., 2023).

2. Algebraic Structure, Exact Results, and Coherent State Realization

In systems with underlying Lie algebra symmetry, particularly $su(2)$, $su(1,1)$, or more general non-compact/non-Hermitian symmetries, the Hamiltonian can often be rewritten entirely within a finite-dimensional algebra spanned by generators like

$K_0, \ K_{\pm}$

The case of two coupled oscillators with balanced gain and loss, as in the PT-symmetric Bateman model, provides a canonical illustration: the system’s dynamics is captured in terms of $su(1,1)$ generators, and all recursion relations for the Krylov amplitudes and coefficients become fully tractable (Beetar et al., 2023).

For highest-weight reference states, analytic expressions for return amplitudes and Krylov amplitudes can be constructed using coherent states and Gauss decomposition formulas. In closed form,

$$R(t) = e^{i(2c_0+c h)t} \left( \cosh\omega t + i\frac{c}{\omega}\sinh\omega t \right)^{-h}$$

with the corresponding Lanczos coefficients $b_n^2 = k_2 n^2 + k_1 n$, $a_n = c_0 + c_1 n$, and explicit expressions for all intermediate quantities (Beetar et al., 2023, Chowdhury et al., 2024).

This analytic framework links the abstract Krylov dynamics to concrete operator growth and spectral structure, showing that the tridiagonal structure enforces an emergent integrable “tight-binding” chain in Krylov space (Rabinovici et al., 8 Jul 2025, Patramanis et al., 4 Feb 2026).

3. Dynamical Phases, Operator Growth, and PT-Symmetry Breaking

Krylov complexity provides a sharp dynamical probe of phase structure in non-Hermitian systems, especially those admitting PT symmetry. In the PT-symmetric regime (here, the “Rabi phase”), the Hamiltonian’s spectrum is real, corresponding to perfect cancellation of gain and loss. The resulting Krylov complexity is strictly bounded and periodic: $$K(t) \sim \sin^2 (\Omega t), \qquad \text{bounded, no exponential growth}$$ Upon entering the PT-broken regime, either through weak or ultra-strong coupling, the spectrum acquires complex eigenvalues, resulting in exponential complexity growth: $$K(t) \sim \sinh^2 (\Delta t) \sim \frac{1}{2} e^{2\Delta t}, \quad \Delta = |\mathrm{Im}\,\omega|$$ At the phase transition boundary, $K(t)$ exhibits a critical crossover from bounded to unbounded growth, exactly diagnosing the PT-symmetry breaking transition via the behavior of the associated frequencies in the $su(1,1)$ algebraic form (Beetar et al., 2023).

4. Asymptotics, Growth Rates, and Transition Signatures

The analytic form of $K(t)$ allows extraction of late-time growth rates. In phases with real frequency parameter $\omega>0$,

$$K(t) \xrightarrow{t \to \infty} \frac{h(\omega^2+c^2)}{4\omega^2} e^{2\omega t}$$

For imaginary $\omega = i\Delta$,

$$K(t) \sim h(\Delta^2-c^2) \sinh^2(\Delta t) \sim \frac{1}{2} h(\Delta^2-c^2) e^{2\Delta t}$$

This formalism shows the direct link between the nature of the energy spectrum (real vs. complex) and complexity growth, quantitatively identifying the exponential growth rate with the imaginary part of the frequency (Beetar et al., 2023).

Numerical calculations verify these asymptotics and provide detailed phase diagrams, showing strict periodic oscillations in the Rabi region and exponential “runaway” in both weak and ultra-strong PT-broken regimes.

5. Diagnostic Power and Relation to Other Complexity Notions

Krylov complexity not only tracks operator growth but also sharply distinguishes phase transitions and dynamical regimes inaccessible to standard observables. For example, in PT-symmetric systems, the complexity crossover exactly matches the onset of PT-symmetry breaking, with the bounded-to-exponential growth transition serving as a dynamical order parameter for the phase transition (Beetar et al., 2023).

More broadly, Krylov complexity serves as a “minimal spreading” diagnostic, mathematically shown to be optimal among all basis-independent spread complexities up to finite times, via universality theorems (Rabinovici et al., 8 Jul 2025). In many-body and open quantum systems, Krylov complexity reveals both integrability-to-chaos transitions and effects of dissipation, localization, and decoherence (Rabinovici et al., 2022, Rabinovici et al., 8 Jul 2025).

In systems with an $su(1,1)$ algebraic structure, the growth rate of Krylov complexity directly encodes the emergent Lyapunov exponent of the dynamics, and for non-Hermitian PT-symmetric Hamiltonians, it robustly detects the breaking of this symmetry via dynamical instability (Beetar et al., 2023).

6. Mathematical Summary and Key Formulas

The construction is summarized by:

• Krylov basis via Lanczos recursion:

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

• Time-evolved amplitudes:

$$|\psi(t)\rangle = \sum_{n=0}^{\infty} \phi_n(t) |K_n\rangle, \quad i \dot{\phi}_n(t) = a_n \phi_n + b_{n+1} \phi_{n+1} + b_n \phi_{n-1}$$

• Krylov complexity:

$K(t) = \sum_{n=0}^\infty n |\phi_n(t)|^2$

• Exact solution in the $su(1,1)$-algebraic regime:

$$K(t) = h(\omega^2+c^2) \frac{\sinh^2 (\omega t)}{\omega^2}$$

where $h$ is the Bargmann index determined by the representation (Beetar et al., 2023).

This framework and its analytic tractability in Lie-algebraic systems establish Krylov dynamical complexity as a central tool for probing operator growth, dynamical stability, phase transitions, and non-Hermitian symmetry breaking in quantum many-body and open systems (Beetar et al., 2023, Rabinovici et al., 8 Jul 2025).