Krylov complexity, path integrals, and instantons

Krylov complexity has emerged as an important tool in the description of quantum information and, in particular, quantum chaos. Here we formulate Krylov complexity $K(t)$ for quantum mechanical systems as a path integral, and argue that at large times, for classical chaotic systems with at least two minima of the potential, that have a plateau for $K(t)$, the value of the plateau is described by quantum mechanical instantons, as is the case for standard transition amplitudes. We explain and test these ideas in a simple toy model.

💡 Research Summary

The paper tackles a long‑standing gap in the analytical understanding of Krylov complexity (KC), a quantity that measures how far an initially simple operator or state spreads in the Lanczos‑Krylov chain under time evolution. While numerical studies have revealed a characteristic three‑stage behavior—bounded oscillations for integrable systems, unbounded growth for unbounded potentials, and a rapid rise followed by a long‑lived plateau for quantum chaotic models—no closed‑form description of the plateau has been available.

The authors first reformulate the Lanczos recursion in a continuous path‑integral language. By inserting complete sets of position eigenstates on both sides of the propagator ⟨x|e^{-iHt}|x’⟩, the amplitude ψₙ(t)=⟨Kₙ|e^{-iHt}|K₀⟩ is expressed as a double integral over initial and final positions weighted by the Krylov basis functions Kₙ(x) and the initial wavefunction ψ(x’). The propagator itself is written as a standard Feynman path integral, turning the whole expression for KC, K(t)=∑ₙ n|ψₙ(t)|², into the probability that a fictitious particle hops on a semi‑infinite auxiliary lattice whose “classical” potential V(x) is fixed by the original Hamiltonian in the Krylov basis. Physical time t now plays the role of Euclidean proper time, allowing a semiclassical saddle‑point analysis.

The central insight is that for chaotic systems the sum over real‑time classical trajectories becomes highly oscillatory at late times because tiny changes in the initial point lead to exponentially divergent trajectories (the hallmark of chaos). When one integrates over all initial and final positions, these oscillations cancel almost completely, effectively suppressing the contribution of real‑time paths to the late‑time KC. The dominant contribution therefore comes from Euclidean (imaginary‑time) saddle points—instantons—that connect distinct minima of the original potential V(x). In Euclidean language the particle moves in the inverted potential V_E(x)=−V(x); an instanton is a finite‑action trajectory that starts at one minimum of V(x) (a maximum of V_E) and ends at another, with vanishing velocity at both ends.

Carrying out the saddle‑point evaluation yields a compact formula for the plateau value: K_∞ = N e^{−2S_inst/ħ}, where S_inst is the Euclidean instanton action and N = ∑_{i≠j} |Kₙ^(x_i)ψ(x_j) + Kₙ^(x_j)ψ(x_i)|² is a purely kinematic factor built from the values of the Krylov wavefunctions at the turning points x_i, x_j (the minima of V). This expression shows that the plateau height is determined entirely by the instanton action and by the overlap of the Krylov basis with the initial state at the classical turning points.

To make the formalism practical, the authors introduce two auxiliary computational tools. The first is a “survival‑amplitude Lanczos algorithm” that bypasses the direct computation of Lanczos coefficients by iterating the survival amplitude itself; this regularizes models where the naive recursion diverges. The second is a position‑space implementation of the Lanczos chain that works directly with continuous variables (x₁, x₂) and avoids any artificial Fock‑space cutoff. Both methods are model‑independent and can be applied to a broad class of quantum systems.

The theoretical framework is tested on a concrete toy model: two harmonic oscillators coupled by a weak exponential interaction. Classically this system has a double‑well potential, guaranteeing the existence of instanton solutions. The authors perform a multi‑pronged analysis: (i) they compute Lanczos coefficients and Krylov wavefunctions numerically for low n; (ii) they simulate K(t) up to t·g≈30 and observe the formation of a plateau; (iii) they develop a perturbative early‑time expansion and demonstrate that contributions from real‑time classical paths cancel in a chaotic fashion; (iv) they evaluate the instanton action analytically (by solving the Euclidean equation of motion) and compare the resulting K_∞ from the formula above with the numerical plateau, finding excellent agreement for n up to 30. They also explore parameter regimes where the instanton does not exist (single‑well potentials) and confirm that in those cases K(t) either keeps growing or remains oscillatory, never forming a plateau.

The paper concludes that the late‑time plateau of Krylov complexity, previously observed only numerically, can be understood as a semiclassical instanton effect. This bridges the gap between quantum information diagnostics (Krylov complexity) and traditional field‑theoretic tools (instantons, Euclidean path integrals). The result suggests that any quantum chaotic system whose Krylov potential possesses multiple minima will exhibit a plateau whose height is set by the instanton action. This insight can be applied to SYK, Schwarzian, random‑matrix models, and potentially to experimental platforms such as superconducting qubits or cold‑atom simulators, where the instanton action might be inferred from measured plateau values.

Future directions outlined include studying multi‑instanton contributions (relevant for potentials with more than two minima), extending the analysis to higher‑dimensional systems, and exploring connections with out‑of‑time‑order correlators and spectral form factors. The authors also propose that the instanton‑based plateau formula could serve as a new quantitative diagnostic of quantum chaos, complementing existing measures like OTOCs and level statistics.