krylov-complexity-loschmidt-amplitude

name	krylov-complexity-loschmidt-amplitude
description	Relate Krylov complexity to the Loschmidt amplitude to diagnose quantum dynamics and quantum chaos. Use when analyzing operator growth, OTOCs, Lyapunov exponents, or Krylov-space methods in many-body quantum systems. Triggers: Krylov complexity, Loschmidt amplitude, operator growth, quantum chaos, Lyapunov exponent, Lanczos, eigenvalues of the Krylov operator.
category	quantum
source_arxiv	2607.10921

Krylov Complexity from Loschmidt Amplitude

What this skill is for

A compact, reusable recipe for connecting two diagnostics of quantum dynamics — Krylov complexity (operator growth / chaos) and the Loschmidt amplitude (return amplitude under perturbed evolution) — so that Krylov complexity can be expressed as a derivative of a Loschmidt amplitude. This gives an algebraic bridge between operator-growth diagnostics and the overlap-decay signatures of quantum chaos (e.g. Lyapunov-controlled decay).

Source paper: Krylov Complexity from Loschmidt Amplitude (arXiv:2607.10921, Debarghya Chakraborty, 2026-07-12).

Core idea

• Krylov complexity $C(t)$ measures the spread of a state/operator along the Krylov chain generated by the Lanczos (or Heisenberg) recursion. It is a leading probe of quantum chaos and operator growth.
• Loschmidt amplitude $G(t) = \langle \psi_0 | e^{iH_1 t} e^{-iH_0 t} | \psi_0 \rangle$ is the overlap of a state evolved under two slightly different Hamiltonians $H_0, H_1$. Its modulus decay is, in certain systems, controlled by the classical Lyapunov exponent $\lambda$.
• The key result: by exploiting the algebraic properties of the Krylov complexity operator, $C(t)$ can be written as the derivative of a Loschmidt amplitude (a generating-function identity). This unifies the operator-growth viewpoint with the return-amplitude viewpoint.

Standard computational pipeline

1. Build the Krylov basis via Lanczos on the operator $A(t)$ in the Heisenberg picture (or on a state under $H$):
   • Tri-diagonalize the Liouvillian / Hamiltonian to get the Krylov chain coefficients $a_n, b_n$ (Lanczos recurrence).
2. Compute Krylov complexity: $$ C(t) = \sum_n n, |c_n(t)|^2, \qquad |\psi(t)\rangle = \sum_n c_n(t), |K_n\rangle, $$ where $|K_n\rangle$ are Krylov basis vectors.
3. Compute the Loschmidt amplitude for two nearby Hamiltonians $H_0, H_1 = H_0 + \delta H$:
   • Evolve $|\psi_0\rangle$ forward under $H_1$ and backward under $H_0$ (or equivalently evaluate $G(t)=\langle\psi_0|e^{iH_1 t}e^{-iH_0t}|\psi_0\rangle$).
   • Monitor $|G(t)|^2$; exponential decay rate $\sim 2\lambda$ signals chaotic (Lyapunov) behavior.
4. Relate the two: use the Krylov-complexity operator algebra to express $C(t) = \partial_\lambda \ln G(t)$ (or the appropriate derivative identity from the paper) — verify numerically on a benchmark (e.g. kicked rotator, SYK, or transverse-field Ising).
5. Cross-check with the OTOC $F(t) = \langle W^\dagger(t) W(0)\rangle$: Krylov complexity and OTOC scrambling times should be consistent.

Pitfalls

• The derivative identity holds under the algebraic structure of the Krylov operator; do not assume a generic $C(t)=\partial_t G(t)$ without the operator relation — verify on a known solvable case first.
• Lanczos coefficients $b_n$ become noisy after ~$N$ iterations (ghost eigenvalues) for finite-dimensional numerics; restrict the Krylov order.
• Loschmidt amplitude decay being "Lyapunov-controlled" is system-dependent (it requires well-developed chaos); in integrable systems expect power-law / revivals instead.
• arXiv API is frequently rate-limited; fetch IDs individually with spacing, or use the browser tool for the abstract page.

Verification

• Reproduce $C(t)$ growth (linear then plateau) on a textbook chaotic model.
• Confirm $|G(t)|^2$ exponential decay rate matches the known Lyapunov exponent for that model.
• Confirm the Krylov/Loschmidt derivative identity holds to numerical precision on the benchmark.

Activation keywords

Krylov complexity, Loschmidt amplitude, operator growth, quantum chaos, Lyapunov exponent, Lanczos recursion, Krylov space, OTOC, Heisenberg evolution, quantum dynamics diagnostics, eigenvalues of Krylov operator.