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Geometry of Krylov Complexity
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We develop a geometric approach to operator growth and Krylov complexity in many-body quantum systems governed by symmetries. We start by showing a direct link between a unitary evolution with the Liouvillian and the displacement operator of appropriate generalized coherent states. This connection maps operator growth to a purely classical motion in phase space. The phase spaces are endowed with a natural information metric. We show that, in this geometry, operator growth is represented by geodesics and Krylov complexity is proportional to a volume. This geometric perspective also provides two novel avenues towards computation of Lanczos coefficients and sheds new light on the origin of their maximal growth. We describe the general idea and analyze it in explicit examples among which we reproduce known results from the Sachdev-Ye-Kitaev model, derive operator growth based on SU(2) and Heisenberg-Weyl symmetries, and generalize the discussion to conformal field theories. Finally, we use techniques from quantum optics to study operator evolution with quantum information tools such as entanglement and Renyi entropies, negativity, fidelity, relative entropy and capacity of entanglement.
Forward citations
Cited by 14 Pith papers
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Krylov Complexity from Loschmidt Amplitude
Krylov complexity is the ϕ-derivative of a Loschmidt amplitude and is upper-bounded by the volume of the induced (t,ϕ) Fubini-Study geometry.
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Krylov-Space Memory Cores
Anomalous initial states in otherwise thermalizing models leave compact, stationary low-depth Krylov-space cores—regions with persistent fluctuations, Gibbs mismatch, and current activity—while generic states do not.
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The Normalised Wigner Negativity Rate as a Second-Moment Probe of Infall in AdS$_3$
Normalized Krylov-Wigner negativity rate matches Krylov variance growth and equals tidal stretch rate R ∝ C P_ρ if and only if Δ=1 in AdS3.
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Krylov complexity has it all
Krylov complexity's Taylor coefficients recursively determine all Lanczos coefficients, making it a complete descriptor of operator dynamics, with caveats for spread complexity.
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Krylov Complexity, Confinement and Universality
Holographic calculations show the proper-momentum proxy for Krylov complexity oscillates in every confining geometry with a smooth infrared cap, with frequency set by the confinement scale.
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Attention in Krylov Space: Transformer-Based Extrapolation of Lanczos Coefficients
A transformer trained on short Lanczos-coefficient prefixes extrapolates coefficients and reconstructed observables more accurately than asymptotic fits, and transfers across system sizes in the two tested chaotic models.
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Grand Canonical vs Canonical Krylov Complexity in Double-Scaled Complex SYK Model
In double-scaled complex SYK, grand-canonical Krylov complexity is the charge-weighted sum of canonical complexities, saturating a conjectured inequality.
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Krylov Complexity of Supersymmetric SYK Models
In finite-size N=2 SYK, breaking supersymmetry with an irrelevant deformation pushes late-time Krylov complexity to roughly half the maximal Krylov-space bound, while a mass deformation leaves saturation complexity a ...
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Quantum Cosmology in Krylov Space: Complexity and Entropy
In a sharply peaked Gaussian state of a flat FLRW universe with a massless scalar clock, Krylov state complexity grows as σ²(φ−φ0)²/4 and operator complexity is exactly twice that, in both Wheeler-DeWitt and loop quan...
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Complexity of PXP scars revisited
In the PXP model, the arch in the Lanczos coefficients is traced to a linear sl(3) part of the Hamiltonian, and the arch width is proposed as a signal distinguishing scarred from thermalizing states.
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The Information Content of Krylov Observables: A Machine Learning Approach
Under chaos, the normalized Wigner negativity χ(t) carries information about the fine return dynamics that spread complexity C(t) cannot, with the asymmetry gap rising from +0.33 to +0.77 across the integrable-to-GUE ...
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Emergence of Krylov complexity through quantum walks: An exploration of the quantum origins of complexity
Reducing a graph walk to distance-layers reproduces Krylov/spread complexity, yielding analytic finite-q SYK Lanczos coefficients and hypercube complexity D sin²(t/D), with faster saturation than classical-walk circuits.
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Trotterization, Operator Scrambling, and Entanglement
Trotter simulation error for observables is bounded by operator scrambling, and sufficient entanglement reduces this error to a normalized Frobenius-norm scaling.
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Krylov Complexity and $c$-function along RG Flows
Along holographic RG flows, the acceleration of spread complexity and the covariant c-function are algebraically related: inversely in fixed-dimension domain walls and Dp-branes, co-monotonically in twisted compactifications.
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