Krylov Complexity from Loschmidt Amplitude

Technical Summary: Krylov Complexity from Loschmidt Amplitude

Problem Statement
Krylov complexity has emerged as a powerful diagnostic for quantum dynamics, offering a clear definition of wavefunction delocalization within the Hilbert space via the Lanczos recursion method. However, a fundamental limitation exists: Krylov space measures (like complexity) can only reconstruct the system up to the autocorrelation function. Two physically distinct systems with identical autocorrelation functions yield identical Krylov expansions. Consequently, translating Krylov basis features back into the language of local Hilbert space operators requires additional, model-specific input. Furthermore, while connections exist between Krylov complexity and other chaos measures (like the Out-of-Time-Ordered Correlator or OTOC), a unified framework linking the growth of Krylov complexity directly to the sensitivity of quantum states under Hamiltonian perturbations—specifically through the lens of Loschmidt amplitudes—remains to be fully developed. This paper addresses the need to express Krylov complexity and its time evolution as derivatives of Loschmidt amplitudes, thereby providing a classification scheme for complexity dynamics and establishing geometric bounds.

Methodology
The paper employs the formalism of the Lanczos algorithm to construct an orthonormal Krylov basis $\{|n\rangle\}$ from an initial state $|\psi(0)\rangle$ and a Hamiltonian $H$. The dynamics are encoded in the tridiagonal matrix $M$ (the representation of $H$ in the Krylov basis), characterized by Lanczos coefficients $a_n$ and $b_n$.

The core methodology involves introducing two auxiliary deformations to the Hamiltonian $M$:

1. Angular Deformation ($\phi$): A rotation generated by the number operator $\hat{n}$ (which counts the Krylov index). The deformed Hamiltonian is defined as $M(\phi) = e^{i\hat{n}\phi} M e^{-i\hat{n}\phi}$.
2. Euclidean Deformation ($\tau$): A deformation generated by the operator $F = i[B, M]$, where $B = i[\hat{n}, M]$ acts as a "momentum" operator. This corresponds to evolving the initial state via Euclidean time $e^{-\tau H}$.

The authors utilize these deformations to define Loschmidt amplitudes $G(t, \phi) = \langle 0 | e^{iMt} e^{-iM(\phi)t} | 0 \rangle$ and their $\tau$-dependent counterparts. By analyzing the spectral properties of these amplitudes and the algebraic structure of the operators $M$, $B$, and $\hat{n}$, the authors derive relations between the moments of Krylov complexity and the derivatives of these amplitudes. The analysis relies heavily on the spectral function $\Phi(E)$ (the Fourier transform of the autocorrelation function) and the properties of orthogonal polynomials associated with the Lanczos coefficients.

Key Contributions and Results

1. Krylov Complexity as a Loschmidt Derivative:
   The paper establishes that Krylov complexity $\langle \hat{n}(t) \rangle$ is the leading-order phase response of a Loschmidt amplitude. Specifically, the $k$-th moment of the complexity distribution is given by the $k$-th derivative of $G(t, \phi)$ with respect to $\phi$ at $\phi=0$:
   $$\langle \hat{n}^k(t) \rangle = (-i)^k \frac{\partial^k}{\partial \phi^k} G(t, \phi) \Big|_{\phi=0}$$
   This formulation defines a "spectral propagator" $K(E_1, E_2, \phi)$ that encodes the entire complexity distribution.

2. Time-Derivative and $\tau$-Classification:
   The time derivative of Krylov complexity is expressed as the derivative of a distinct Loschmidt amplitude involving a Euclidean time deformation $\tau$:
   $$\partial_t \langle \hat{n}(t) \rangle = i \partial_\tau \langle 0 | e^{iM(0)t} e^{-i(M(0) + \tau F)t} | 0 \rangle \Big|_{\tau=0}$$
   The operator $F = i[B, M]$ governs this perturbation. The paper proposes a classification scheme for Krylov complexity dynamics based on the growth and boundedness of the entries of $F$.

   • Bounded Perturbations: If $F$ is a bounded operator (e.g., when Lanczos coefficients grow slowly or saturate), the perturbation can be approximated by a finite-rank truncation. This allows the complexity dynamics to be captured by a measurable amplitude with a finite cutoff $n^*$.
   • Unbounded Perturbations: In chaotic systems where Lanczos coefficients grow rapidly (e.g., linearly), $F$ is unbounded, and finite-rank truncations fail to capture the dynamics accurately.

3. General Bounds on Complexity:
   Using the Cauchy-Schwarz inequality and the Frobenius norm of $F$, the authors derive a general upper bound for Krylov complexity in terms of the autocorrelation function $C(t)$:
   $$\langle \hat{n}(t) \rangle \leq \|F\| \sqrt{\int_0^t ds_1 \int_0^t ds_2 (t-s_1)(t-s_2) |C(s_1-s_2)|^2}$$
   This bound holds with minimal assumptions about the growth of Lanczos coefficients, provided $\|F\|$ is finite. For specific cases (like Random Matrix Theory with constant $b_n$), the paper derives the precise late-time asymptotic behavior of all complexity moments.

4. Complexity Geometry and Volume Bounds:
   The paper constructs a two-dimensional Riemannian geometry spanned by time $t$ and the deformation angle $\phi$. The metric is induced from the Fubini-Study metric on the Hilbert space.

   • Volume Bound: The authors demonstrate that Krylov complexity is upper-bounded by the volume of this geometry up to time $t$. Specifically, $\langle \hat{n}(t) \rangle \leq \frac{1}{\pi} \text{Vol}_t$.
   • Saturation: This bound is saturated when the algebra between $\hat{n}$, $M$, and $B$ closes (e.g., in $SL(2, \mathbb{R})$ or Heisenberg-Weyl algebras), leading to spaces of constant curvature.
   • Geodesic Deviation: The scalar curvature of this geometry is shown to be related to the convexity of the Krylov variance $\Delta n(t)$.

5. Statistical Relation between IPR and Variance:
   A surprising statistical property is identified: the Inverse Participation Ratio (IPR) of the Krylov wavefunctions is inversely proportional to the standard deviation of the complexity distribution ($\Delta n(t)$) for a wide class of systems:
   $$\text{IPR}_t \sim \frac{1}{\sqrt{\pi} \Delta n(t)}$$
   This relation holds for systems with constant Lanczos coefficients and those governed by complexity algebras, suggesting a robust statistical behavior of Krylov basis probabilities.

Significance and Claims
The paper claims to provide a unified analytical framework that treats Krylov complexity as a measure of sensitivity to Hamiltonian perturbations. By expressing complexity moments as derivatives of Loschmidt amplitudes, the work bridges the gap between abstract Krylov space measures and physical observables like the survival amplitude.

The significance lies in:

• Classification: Offering a new way to classify quantum dynamics based on the boundedness of the perturbation operator $F$, distinguishing between systems where complexity can be approximated by finite truncations and those requiring full spectral resolution.
• Geometric Interpretation: Establishing a rigorous geometric picture where complexity is bounded by the volume of a parameter space, linking quantum chaos concepts (Lyapunov exponents) to geometric curvature and trajectory separation.
• Predictive Power: Deriving general bounds and asymptotic behaviors directly from the autocorrelation function without requiring detailed knowledge of the full Lanczos sequence, particularly for systems with bounded spectral functions.

The authors maintain that while the connection to chaos exponents (like the Lyapunov exponent) is suggestive and analogous to classical definitions, establishing a precise physical identification of the Krylov exponent with chaos in general systems requires further consideration of the specific physical content of the autocorrelation function, which is beyond the scope of this work. The paper focuses on the structural and algebraic relations within the Krylov formalism.