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arXiv:2610.03883

On Information Propagation in Krylov Subspaces

Pablo Martinez-Azcona, Maximilian Meyer-Moelleringhof, Adolfo del Campo, Apollonas S. Matsoukas-Roubeas

quant-phcond-mat.stat-mechhep-thSubmitted Oct 2, 2026• Announced Oct 6, 2026• 0 citations
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Abstract

Krylov subspace methods provide an efficient tool for modeling any quantum initial value problem as a one dimensional chain. A time ordered path formulation, in which the dynamical generator successively splits probability between paths, yields a speed limit within such dynamical subspaces. For power law growing Lanczos coefficients $b_n \sim n^δ$ we find super ballistic lightcones, with an exponentially spreading lightcone for maximal growth $δ= 1$. Causality in the Krylov chain constrains the Lanczos coefficients to grow at most linearly. We illustrate the validity of the above in the $SL(2, R)$ and Heisenberg Weyl algebras, and in a many body skew field Ising model. Moreover, we extend the above bound to the family of $Q$ complexities. Lastly, mapping to a spin chain gives a Lieb Robinson bound analogue and opens the problem to standard condensed matter methods.

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Krylov subspace methods provide an efficient tool for modeling any quantum initial value problem as a one dimensional chain. A time ordered path formulation, in which the dynamical generator successively splits probability between paths, yields a speed limit within such dynamical subspaces. For power law growing Lanczos coefficients bn∼nδb_n \sim n^δbn​∼nδ we find super ballistic lightcones, with an exponentially spreading lightcone for maximal growth δ=1δ= 1δ=1. Causality in the Krylov chain constrains the Lanczos coefficients to grow at most linearly. We illustrate the validity of the above in the SL(2,R)SL(2, R)SL(2,R) and Heisenberg Weyl algebras, and in a many body skew field Ising model. Moreover, we extend the above bound to the family of QQQ complexities. Lastly, mapping to a spin chain gives a Lieb Robinson bound analogue and opens the problem to standard condensed matter methods.
Recently, the propagation of information through quantum many-body systems, developed to study quantum chaos, have found many applications from black holes to disordered spin systems. Among other quantitative tools, Krylov complexity has been explored as a diagnostic tool for information scrambling in quantum many-body systems. We introduce a universal limit to the growth of the Krylov complexity in dissipative open quantum systems by utilizing the uncertainty relation for non-Hermitian operators. We also present the analytical results of Krylov complexity for characteristic behavior of Lanczos coefficients in dissipative systems. The validity of these results is demonstrated by explicit study of transverse-field Ising model under dissipative effects.
We study the operator growth problem and its complexity in the many-body localization (MBL) system from the Lanczos algorithm perspective. Using the Krylov basis, the operator growth problem can be viewed as a single-particle hopping problem on a semi-infinite chain with the hopping amplitudes given by the Lanczos coefficients. We find that, in the MBL systems, the Lanczos coefficients scale as ∼ n/ ln(n) asymptotically, same as in the ergodic systems, but with an additional even-odd alteration and an effective randomness. We use a simple linear extrapolation scheme as an attempt to extrapolate the Lanczos coefficients to the thermodynamic limit. With the original and extrapolated Lanczos coefficients, we study the properties of the emergent single-particle hopping problem via its spectral function, integrals of motion, Krylov complexity, wavefunction profile and return probability. Our numerical results of the above quantities suggest that the emergent single-particle hopping problem in the MBL system is localized when initialized on the first site. We also study the operator growth in the MBL phenomenological model, whose Lanczos coefficients also have an even-odd alteration, but approach constants asymptotically. The Krylov complexity grows linearly in time in this case.
Krylov subspace methods quantify operator growth in quantum many-body systems through Lanczos coefficients that encode how operators spread under time evolution. Although these diagnostics were originally motivated by questions of chaos and integrability, quadratic fermionic Hamiltonians are often expected to exhibit trivial Lanczos structure. Here we show that, in the long-range Kitaev chain, Lanczos coefficients generated from local boundary operators sharply diagnose whether the lowest excitation gap is controlled by boundary-localized or bulk-extended modes. We introduce the KrylovKrylovKrylov staggeringstaggeringstaggering parameterparameterparameter for the Lanczos coefficients. In the short-range Kitaev chain with balanced hopping and pairing, we derive analytically for arbitrary system size (valid in the thermodynamic limit) and show that this quantity is exactly constant and its sign cleanly distinguishes the topological phase with Majorana edge modes from the trivial phase. Away from that limit, long-range couplings and pairing-hopping imbalance deform the simple flat structure and analytical control is lost, nevertheless, we show that the sign pattern of the diagnostic still tracks whether the lowest excitation gap is controlled by boundary modes or by bulk excitations. These results are enabled by an exact single-particle operator Lanczos algorithm, as derived in this work, which reduces the recursion from exponentially large operator space to a finite-dimensional linear problem and achieves machine precision for chains of hundreds of sites. Krylov diagnostics thus emerge as practical probes of boundary-versus-bulk low-energy physics in topological superconductors with broken U(1)U(1)U(1) symmetry and algebraically decaying couplings.
We study the operator growth problem and its complexity in the many-body localization (MBL) system from the Lanczos algorithm perspective. Using the Krylov basis, the operator growth problem can be viewed as a single-particle hopping problem on a semi-infinite chain with the hopping amplitudes given by the Lanczos coefficients. We find that, in the MBL systems, the Lanczos coefficients scale as \sim n/\ln(n) ∼ n / ln ( n ) asymptotically, same as in the ergodic systems, but with an additional even-odd alteration and an effective randomness. We use a simple linear extrapolation scheme as an attempt to extrapolate the Lanczos coefficients to the thermodynamic limit. With the original and extrapolated Lanczos coefficients, we study the properties of the emergent single-particle hopping problem via its spectral function, integrals of motion, Krylov complexity, wavefunction profile and return probability. Our numerical results of the above quantities suggest that the emergent single-particle hopping problem in the MBL system is localized when initialized on the first site. We also study the operator growth in the MBL phenomenological model, whose Lanczos coefficients also have an even-odd alteration, but approach constants asymptotically. The Krylov complexity grows linearly in time in this case.
Krylov state complexity, or spread complexity, has emerged as a sharp and versatile diagnostic of quantum chaos, information spreading, and many-body dynamics. Built from the Lanczos algorithm and grounded in the optimal-basis theorem, Krylov complexity thereby provides a robust spectroscopic window into quantum dynamics. A central theme is the characteristic overshoot observed in chaotic systems: a complexity peak in which chaotic evolution drives the state deeper into the Krylov chain than in integrable systems before relaxing to equilibrium. This behavior, tied to random-matrix universality classes of spectral statistics, is illustrated across a broad range of models, including quantum billiards, quantum spin chains, and variants of the SYK model. We also discuss proposed holographic descriptions of Krylov complexity in Einstein gravity, and conclude by outlining future directions and open problems, including time-dependent systems and quantum-field-theoretic formulations. A Mathematica notebook is provided for numerical exploration of Krylov complexity and spectral statistics across models.
Krylov and Lanczos approximations are used in quantum dynamics, quantum subspace methods, and Hamiltonian learning. A practical question is how long an mmm-dimensional Krylov truncation can be trusted. We argue that this time is fixed by causal propagation on the associated Jacobi chain. The relevant distance is not the Krylov index itself, but the inhomogeneous transport metric ρ(m,n)=∑j=min⁡(m,n)max⁡(m,n)−11/bjρ(m,n) = \sum_{j=\min(m,n)}^{\max(m,n)-1} 1/b_jρ(m,n)=∑j=min(m,n)max(m,n)−1​1/bj​, where bjb_jbj​ is the Lanczos hopping across the bond j↔j+1j \leftrightarrow j+1j↔j+1. We prove a Lieb--Robinson bound in this metric. Its small-weight limit gives the velocity vLR=2v_{\rm LR} = 2vLR​=2, meaning that propagation is exponentially suppressed outside the cone ρ(m,n)≃2∣t∣ρ(m,n) \simeq 2|t|ρ(m,n)≃2∣t∣. The error of a finite Krylov approximation to the return amplitude is a round-trip effect: information has to travel from the probe to the truncation boundary and back. Combining the Lieb--Robinson bound with Duhamel's formula yields a lower bound on the error of the truncated dynamics. For a fixed tolerance εεε, let the break time t∗(m;ε)t_\ast(m;ε)t∗​(m;ε) denote the longest time for which the mmm-dimensional truncation is guaranteed to reproduce the exact return amplitude within error εεε. We show that t∗(m;ε)≥τm[1−o(1)]t_\ast(m;ε) \ge τ_m[1-o(1)]t∗​(m;ε)≥τm​[1−o(1)], where τm=ρ(0,m)=∑j<m1/bjτ_m = ρ(0,m) = \sum_{j<m} 1/b_jτm​=ρ(0,m)=∑j<m​1/bj​. When the probe spreads along the chain, this lower bound is also tight, so t∗(m)≃τmt_\ast(m) \simeq τ_mt∗​(m)≃τm​. The situation is different when the probe excites only a localized part of the spectrum, or a part already resolved by the truncation. In this case, essentially no signal reaches the boundary. Beyond a state-dependent Krylov dimension m∗m_\astm∗​, the approximation can therefore remain accurate at all times, and the break time is effectively infinite. Numerical tests on spin chains and random Jacobi matrices support t∗(m)≃τmt_\ast(m) \simeq τ_mt∗​(m)≃τm​ in the transport-limited regime.