Quantum Dynamics in Krylov Space: Methods and Applications

1. Introduction

• Krylov methods identify the minimal subspace in which quantum dynamics unfolds, avoiding full Hamiltonian diagonalization and explicit state storage.They apply to Schrödinger and Heisenberg evolution and support correlation-function, response, and spectral analyses.Source: §1 · p. 4
• The review develops operator Krylov constructions, Lanczos algorithms, operator growth, and Krylov complexity as tools for characterizing many-body dynamics and quantum chaos.It also relates these ideas to universal operator growth, quantum Lyapunov bounds, and applications in quantum technologies.Source: §1 · pp. 4–6

2.1. Spectral statistics and random matrix theory

• RMT provides reference spectral statistics for quantum chaos, contrasting correlated chaotic spectra with the uncorrelated spectra expected for generic integrable systems.Gaussian ensembles are classified by orthogonal, unitary, and symplectic symmetries through the Dyson index.Source: §2.1 · pp. 6–7

2.2. Spectral Form Factor

• The SFF probes spectral correlations: its chaotic dip-ramp-plateau structure connects early energy-density decay, two-level correlations, and the late plateau.The plateau approaches Z(2β)/Z(β)^2 without degeneracies, while generalized filters extend the construction to different spectral windows and open dynamics.Source: §2.2 · pp. 7–8

2.3. Out-of-time order correlators (OTOCs) and scrambling

• OTOCs diagnose scrambling through exponential growth only within a parametrically separated window between dissipation and Ehrenfest times.The required small parameter commonly arises from semiclassical or large-N limits, restricting direct Lyapunov interpretations in generic systems.Source: §2.3 · p. 9
• A positive quantum Lyapunov exponent signals scrambling rather than necessarily quantum chaos, because integrable systems such as LMG can exhibit saddle-dominated scrambling.The MSS bound constrains finite-temperature Lyapunov growth, while at infinite temperature it is trivially satisfied.Source: §2.3 · pp. 9–10

2.4. Quantum complexity

• Quantum complexity measures resources for preparing states or implementing operations, but typical states and unitaries can require exponentially large resources.Circuit, geometric, field-theoretic, and holographic formulations address this problem with different choices of gates, generators, reference states, and regularization.Source: §2.4 · p. 11

3.1. Preliminaries: vectorization of operators

• Vectorization maps d × d operators to d^2-dimensional vectors, allowing Liouvillian and Lindbladian superoperators to represent operator or density-matrix evolution as matrix dynamics.The Hilbert–Schmidt inner product becomes the Euclidean inner product in the vectorized representation, with row-wise and column-wise conventions equivalent when used consistently.Source: §3.1 · p. 12

3.2. Krylov basis and operators: Lanczos algorithm

• Nested commutators generated by the Liouvillian span a generally smaller Krylov space, making operator evolution computationally more tractable and exposing support growth in k-local many-body systems.After n Liouvillian applications, an initially p-local operator can acquire support of order O(nk + p), while the number of terms grows and produces operator scrambling.Source: §3.2 · p. 13
• Lanczos construction requires a local, nonconserved initial operator and an inner product, but numerical orthogonality is lost through rounding errors unless full or partial re-orthogonalization is used.The Krylov dimension depends on the initial probe and distinct energy gaps; because interacting integrable systems can saturate its upper bound, it is not by itself a diagnostic of chaos.Source: §3.2 · pp. 13–15
• The Lanczos algorithm orthonormalizes Liouvillian-generated operators into a Krylov basis whose nonnegative coefficients define a tridiagonal Liouvillian and completely determine unitary operator dynamics.The resulting wavefunction obeys a nearest-neighbor hopping equation with rates b_n and b_n+1, while the autocorrelation function encodes all Lanczos coefficients.Source: §3.2 · pp. 14–16

3.3. Krylov complexity and cumulants

• Krylov complexity is the mean position of the operator distribution on the chain, while its variance and higher cumulants capture spread, skewness, kurtosis, and related features.Because the mean is insensitive to distributional spread, Krylov complexity is better suited to operator growth than scrambling; the full normalized distribution supplies the missing information.Source: §3.3 · pp. 16–18
• Krylov complexity depends on the Hamiltonian and initial operator, unlike a fixed computational basis, and yields simpler coefficient equations for operator evolution in the Krylov basis.Krylov complexity also equals the expectation value of a number-like Krylov operator, while other conventional complexity measures have proposed but potentially subtle connections to it.Source: §3.3 · p. 17

3.4. Krylov entropy

• For operators satisfying the Eigenstate Thermalization Hypothesis, Krylov complexity grows exponentially during scrambling, linearly until the Heisenberg time, and then saturates, while Krylov entropy grows logarithmically after scrambling.The late-time complexity–entropy relation is logarithmic, whereas at early times the two quantities follow a product-logarithmic relation instead.Source: §3.4 · p. 19
• Lanczos ascent, plateau, and descent organize the late-time dynamics: complexity reaches K ∼O(S) by t∗∼O(log S), grows linearly to K ∼O(e2S) by tH ∼O(eS), and remains there during descent.In chaotic systems, DK ∼e2S and the plateau wavefunction is uniformly distributed across Krylov space; the displayed regimes are schematic rather than numerically scaled.Source: Figure 4 · p. 18, §3.4 · p. 19

4. Lanczos algorithm: Monic version and orthogonal polynomials

• The monic Lanczos algorithm omits stepwise normalization, producing orthogonal rather than orthonormal operators while yielding the same Lanczos coefficients as the orthonormal construction.Its polynomial form satisfies a three-term recurrence and is orthogonal with respect to a measure by Favard’s theorem [211]; the monic version is especially useful for Toda-chain methods.Source: §4 · pp. 20–21

5.1. Statement of the hypothesis

• For generic chaotic many-body systems, the universal operator growth hypothesis predicts asymptotically linear Lanczos growth, with a one-dimensional logarithmic correction, while finite systems eventually terminate or saturate.The coefficients’ slope depends on the Hamiltonian and initial local operator, and the appropriate growth regime must precede finite-size effects.Source: §5.1 · p. 22

5.2. Does linear growth of Lanczos coe

• Linear Lanczos growth is necessary but not sufficient for chaos: the integrable LMG model exhibits linear coefficients and exponential Krylov complexity because of saddle-dominated scrambling.For LMG, the OTOC Lyapunov exponent saturates λOTOC = 2α, while the Krylov dimension remains below its saturation bound; classically chaotic examples can also show the expected linear growth.Source: §5.2 · pp. 23–24, Figure 5 · p. 24
• The growth hypothesis has limited direct applicability in QFT and CFT: free theories can show linear growth without a UV cutoff, while an IR scale can split coefficients into even and odd sequences.These continuum features challenge the smoothness assumption in the discrete-system hypothesis and motivate extensions to field theories.Source: §5.2 · p. 24

5.3.1. Lanczos coe

• The moment method derives Lanczos coefficients from spectral-function moments and autocorrelation derivatives, with Dyck and Motzkin paths providing combinatorial evaluations of moments and Liouvillian matrix elements.For Hermitian unitary evolution, odd moments and a_n vanish, reducing Motzkin paths to Dyck paths; non-Hermitian or non-unitary evolution generally retains both coefficient sets.Source: §5.3.1 · pp. 25, 27, 29
• Autocorrelation singularities constrain Lanczos growth: C(t) singularities require unbounded linear growth, while C(t)=sech(αt) gives b_n=αn, K(t)∼e^2αt, and exponential spectral decay.Sublinear b_n∼α√n corresponds to Gaussian spectral decay, whereas one-dimensional local systems can reach b_n∼n/log n because of slower high-frequency decay.Source: §5.3.1 · pp. 28–29
• Weighted Dyck-path saddle-point methods relate Lanczos-coefficient growth to asymptotic moments, including linear, constant, and logarithmically corrected cases.For b_n = αn^δ, the saddle-point action determines the moments; constant coefficients recover Cat(n) ∼ 4^n, while logarithmic corrections modify the action.Source: §5.3.1 · pp. 29–30
• Smooth moments do not ensure smooth Lanczos coefficients: a periodic autocorrelation can produce odd/even branches with sublinear and linear growth, respectively.For the mock autocorrelation, branch splitting begins near n* ≈ 30; non-decaying or power-law autocorrelation behavior is linked to unusual splitting.Source: §5.3.1 · pp. 31–32

5.3.2. An example: large q SYK model

• In large-q SYK, the analytic Lanczos coefficients grow as b_n ∼ αn with α = J, yielding exponential Krylov complexity with growth coefficient λ_K = 2J.The Krylov variance grows exponentially with coefficient 4J; these results use the leading O(1/q) autocorrelation at infinite temperature.Source: §5.3.2 · pp. 33–34
• Numerical SYK4 coefficients grow linearly before system-size-dependent saturation, while Krylov complexity shows transient exponential and linear regimes crossing at t* ∝ log N/q.The saturation value scales as b_n = 0.0465N + 0.0337; finite-size data do not directly match the large-q analytic result, and late-time complexity saturation requires the full coefficient profile.Source: Figure 9 · p. 34, §5.3.2 · p. 35

5.4. Revisiting Lanczos coe

• The Toda-chain method computes Lanczos coefficients from autocorrelation derivatives by forming Hankel determinants, satisfying Hirota and Toda equations and yielding parameter-dependent Flaschka variables.An appropriate Euclidean-time cutoff τ0, usually zero but potentially nonzero in CFT, gives the physical coefficients; polynomial Ansatzes reproduce the moment-method sequences [212].Source: §5.4 · pp. 35–36
• The moment and Toda-chain methods are especially useful when autocorrelation functions are known analytically, including SYK and conformal field theories.In QFT/CFT, exact autocorrelations can enable exact Lanczos coefficients even when infinite degrees of freedom prevent constructing an orthonormal or bi-orthonormal basis.Source: §5.4 · p. 37

6.1. Coherent states

• Coherent states generated by ladder-operator displacement operators describe Liouvillian evolution and operator growth for the closed SL(2, R), Heisenberg-Weyl, and SU(2) algebras.The construction uses symmetry-adapted reference states and displacement parameters to represent time-evolved operators in Krylov space.Source: §6.1 · p. 37

6.1.1. SL(2, R) algebra

• SL(2, R) symmetry governs operator evolution; as time progresses, higher Krylov-wavefunction modes dominate, the delocalization length ξ(t) grows exponentially, and Krylov complexity grows exponentially.The autocorrelation is φ0(t) = sech(αt) ≡ C1(t), while Krylov complexity is K(t) = sinh2(αt) ∼e2αt.Source: §6.1.1 · pp. 39–40

6.1.2. Heisenberg-Weyl (HW) algebra

• The Heisenberg-Weyl algebra produces coherent-state Krylov evolution whose complexity grows quadratically over time, exemplifying sub-exponential growth from sub-linear Lanczos coefficients.Its infinite-dimensional number basis is generated by creation and annihilation operators, and the corresponding wavefunctions are normalized coherent-state amplitudes.Source: §6.1.2 · p. 41

6.1.3. SU(2) algebra

• For SU(2), Krylov basis states coincide with finite-dimensional spin states, with dimension DK = 2j + 1 and Lanczos coefficients vanishing at the space boundary.The coefficients peak at n = j + 1/2, while the periodic complexity has average K = j, proportional to the spin.Source: §6.1.3 · pp. 43–44

6.2. Complexity algebra: the simplicity hypothesis

• The complexity algebra defines an anti-Liouvillian from the Krylov tridiagonal structure, makes Krylov complexity a norm, and yields a diagonal [L, M] whose nested commutator with K vanishes.The relation [K, [L, M]] = 0 follows because K is diagonal in the Krylov basis and the diagonal entries of [L, M] are determined by differences between squared Lanczos coefficients.Source: §6.2 · pp. 44–45
• The simplicity hypothesis redefines the Krylov complexity operator so that it closes an algebra with L and M, restricting Lanczos-coefficient growth to at most linear.The resulting nested-commutator relations determine differential equations that fully characterize Krylov-complexity dynamics.Source: §6.2 · pp. 45–46
• Linear Lanczos growth produces exponential Krylov-complexity growth associated with SL(2, R), while sublinear growth produces quadratic growth associated with the Heisenberg-Weyl algebra.The linear case allows arbitrary independent α and γ asymptotically, whereas the sublinear case has bn = α √n.Source: §6.2 · p. 46
• In finite dimensions, termination of the Lanczos sequence yields bn = ω √n(DK −n), periodic Krylov complexity, and an SU(2) algebra.The algebraic classification also enables analytic coherent-state descriptions of operator evolution, with more general complexity algebras proposed [281].Source: §6.2 · p. 47

6.3. Dispersion bound and quantum speed limits to the complexity growth rate

• The Robertson uncertainty relation in Krylov space yields a dispersion bound in which the first Lanczos coefficient b1 provides an upper bound on the speed of evolution.The bound is formulated for the Krylov complexity superoperator and parallels Mandelstam-Tamm time-energy limits.Source: §6.3 · pp. 47–48
• Dispersion-bound saturation is equivalent to closure of the complexity algebra, the associated Lanczos-coefficient form, and the Krylov-complexity differential equation [284].Linear Lanczos growth is sufficient but not necessary, so saturation is not generally tied to chaos.Source: §6.3 · p. 48
• Generic systems saturate the dispersion bound through O(t4) but deviate when O(t6) terms matter, while random-matrix Hamiltonians neither saturate the bound nor yield exponential Krylov-complexity growth [232] [292].These observations associate exponential Krylov-complexity growth more closely with scrambling than with quantum chaos; saturation also bounds Krylov entropy and matches geometric operator quantum-speed-limit saturation.Source: §6.3 · pp. 48–49

Large-q SYK derivation

• In large-q SYK, the diagrammatic construction yields Lanczos coefficients b1 = 1/q and bn = √n(n −1)/2, matching Eq. (82) to leading order in q.The resulting Krylov basis produces s = n(q −2) + 1, establishing operator size concentration.Source: §7 · pp. 49, 52

Operator size concentration

• Operator size concentration means the n-th Krylov basis element is a linear combination of Majorana strings with the same size s = n(q −2) + 1, so n counts nested-commutator generations.The first step gives s = q −1, and the basis index tracks the operator’s generation.Source: §7 · p. 50

Liouvillian action

• The Liouvillian splits into L+ and L−, which respectively increase and decrease operator size, providing the diagrammatic growth and reduction processes.Repeated L+ actions create child arcs within parent arcs, producing ordered diagrams for the Krylov basis.Source: §7 · pp. 50–51

Large-q SYK derivation

• The diagrammatic observables are formed after closing melon diagrams and disorder averaging, while subleading contributions are neglected because they have negligible effects on disorder-averaged observables.The leading-order diagrams themselves are not disorder-averaged.Source: §7 · pp. 50–51

Combinatorial derivation

• The action of L− removes a childless arc from a parent diagram; marked ordered diagrams are ordered diagrams with one marked child, whose removal produces an unmarked ordered diagram.Source: §7 · p. 51
• The identity governing marked-child removal leads to the Krylov basis and shows that the Liouvillian action is effectively similar to the action of L+.Special cases n = 0, 1 are evaluated independently, while the general construction applies for n ≥2.Source: §7 · pp. 51–52

Holographic interpretation

• The radial operator-graph direction has an intuitive but not rigorously established analogy with radial motion in a holographic bulk or infall toward a black hole.This interpretation is presented as a possible picture for operator growth and Krylov complexity, not as a demonstrated result.Source: §7 · p. 52

8.1. Finite temperature inner product and Lanczos algorithm

• Finite-temperature Krylov construction changes the inner product and orthogonality, while the Krylov subspace definition remains unchanged.The Lanczos basis and coefficients therefore acquire temperature dependence; standard and Wightman inner products are common choices.Source: §8.1 · pp. 53–54
• For chaotic systems in d > 1, finite-temperature Lanczos coefficients asymptotically grow linearly; at low temperature, this growth persists until n ≲ J/T.In the approximately continuous regime J/T ≫ 1, the slope is πT.Source: §8.1 · p. 54

8.2. SYK at finite temperature

• In SYK4, Lanczos coefficients grow linearly before a system-size-controlled plateau, while Krylov-complexity growth slows at lower temperature but approaches a temperature-independent late-time rate.The numerical results retain exponential-to-linear complexity growth despite temperature-dependent early-time rates.Source: Figure 14 · p. 55, §8.2 · pp. 56–57

8.3. Krylov exponent at finite temperature

• The refined finite-temperature OTOC bound remains applicable from low to infinite temperatures, while large-q SYK saturates the left bound at all temperatures.The right inequality is tight only at low temperature before the refinement extends its applicability across the full temperature range.Source: §8.3 · p. 57

9.1. Krylov space and spread complexity

• The pure-state Krylov basis is generated from powers of the Hamiltonian acting on the initial state and obtained by the Lanczos algorithm.The Hamiltonian becomes tridiagonal in this basis, although its diagonal coefficients are generally nonzero.Source: §9.1 · pp. 58–59
• Spread complexity is the mean position of the wavefunction on the Krylov chain, and the Krylov basis minimizes the associated spreading cost.This optimality holds over finite time and, for discrete-time unitary circuits, at all times.Source: §9.1 · p. 60

9.3. Spread complexity in RMT

• In GUE random matrices, numerical Lanczos coefficients match analytic expressions, while their ensemble means depend on the density of states rather than the initial state.The mean relation implicitly averages over initial states for a fixed Hamiltonian.Source: §9.3 · pp. 62–63, Figure 16 · p. 62

9.4. Spread complexity in the thermofield double state

• TFD spread complexity exhibits linear growth, a peak, a decrease, and a plateau; its peak is highest for GSE, followed by GUE and GOE.At fixed dimension, both the peak and saturation plateau decrease with temperature, while uncorrelated levels remove the peak.Source: §9.4 · pp. 64–65, Figure 18 · p. 65

9.5. A digression: Spectral Complexity

• Spectral and spread complexity share initial growth in GUE but have different late-time saturation, with spectral complexity reaching twice the spread-complexity saturation.Spectral-complexity saturation can reflect chaos, whereas spread-complexity saturation does not generically do so.Source: §9.5 · p. 66

10. Krylov Space of Density Operators

• Mixed-state Krylov space is generated by repeated Liouvillian action on an initial density operator and orthogonalized using the Hilbert-Schmidt inner product.The resulting Liouvillian is tridiagonal in the Krylov basis, and the amplitudes encode the density-matrix evolution relative to the initial state.Source: §10 · pp. 67–68
• The proposed Lanczos normalization preserves the initial density matrix’s unit trace, unlike a convention that normalizes it by its purity.The first Krylov element need not be normalized, while later elements satisfy standard orthonormality.Source: §10 · p. 68

10.1. Properties of the elements of the density matrix Krylov chain

• For a unit-trace initial state, Krylov-chain traces alternate between even and odd elements, so elements with n ≥ 1 are not themselves physical density matrices.The identity operator occupies the even Krylov sector, and in the absence of degeneracies the zero-eigenvalue subspace collapses to the maximally mixed state.Source: §10.1 · p. 69

10.2. Constraints on the evolution

• Trace preservation constrains only even amplitudes, while unitary evolution preserves purity; positivity further requires real amplitudes and nonnegative density-matrix eigenvalues.Determining the complete positivity constraints on Krylov amplitudes remains an open problem.Source: §10.2 · p. 70

11.1. An introduction to the Lindblad master equation

• Lindblad master equations describe Markovian dissipative evolution that preserves complete positivity and trace, with Hamiltonian and jump-operator contributions.The rates satisfy µ_k ≥ 0 for the Markovian case; negative rates generally correspond to non-Markovian evolution.Source: §11.1 · pp. 71–72
• The microscopic system–environment derivation requires weak or singular coupling together with Born, Markov, and rotating-wave approximations.These assumptions impose product system–environment states, separated timescales, and negligible cross-frequency terms.Source: §11.1 · pp. 71–72
• Continuous measurement yields Lindblad dynamics when jump operators equal measurement operators and all dissipation rates equal the measurement rate η(t) = µ ≥ 0.Increasing the measurement rate drives the dynamics farther from unitary evolution.Source: §11.1 · p. 73
• The adjoint Lindblad equation describes dissipative operator evolution in the Heisenberg picture, with unitary and dissipative superoperator parts.For fermionic jump operators and initial operators, the sign depends on their parity; the identity remains preserved under unital evolution.Source: §11.1 · pp. 73–74
• Vectorization converts the Lindbladian map on d × d matrices into a d^2 × d^2 operator acting on length-d^2 vectors, enabling spectrum computations.The construction uses the infinite-temperature inner product and rescales jump operators by √µ_k when convenient.Source: §11.1 · p. 74

12.1.1. Arnoldi iteration

• Arnoldi iteration builds an orthonormal Krylov basis from the full Lindbladian and transforms it into an upper Hessenberg form.Without dissipation, the construction reduces to the usual Lanczos algorithm and the matrix becomes tridiagonal with Lanczos coefficients b_n.Source: §12.1.1 · p. 75

12.1.2. Closed Krylov basis

• The closed Krylov basis uses the closed-system Liouvillian while retaining dissipative operator dynamics, producing a non-Hermitian tight-binding model.Numerics in interacting fermions and finite-fermion SYK find diagonal coefficients that grow linearly before finite saturation, although the behavior lacks analytical support.Source: §12.1.2 · p. 76

12.1.3. Bi-Lanczos algorithm

• Bi-Lanczos constructs mutually bi-orthonormal Krylov spaces and represents the Lindbladian tridiagonally, unlike Arnoldi’s upper Hessenberg representation.Both methods capture non-Hermiticity and reduce to ordinary Lanczos when dissipation is absent.Source: §12.1.3 · pp. 76–78

12.2. Structure of the Lindbladian

• Bi-Lanczos dynamics map to a non-Hermitian tight-binding chain where b_n and b_{n+1} govern neighboring hops and a_n gives local on-site dissipation.Because total probability is not conserved, normalized amplitudes define Krylov complexity as the particle’s average position.Source: Figure 21 · p. 78, §12.2 · pp. 79–80
• In the bi-Lanczos representation, physical Lindbladian spectra require real off-diagonal coefficients b_n and purely imaginary diagonal coefficients a_n = i|a_n|.This follows from eigenvalues being real or occurring in complex-conjugate pairs.Source: §12.2 · pp. 78–79

12.3. Constraints imposed by Trace Preservation

• Trace preservation appears as unitality in the Heisenberg picture and constrains the identity component in the bi-orthonormal Krylov basis.Choosing a traceless initial operator makes the identity the zero vector; complete-positivity constraints on the Lanczos coefficients remain open.Source: §12.3 · pp. 80–81

13.1. Dissipative SYK Model

• Dissipative SYK has linear and p-body nonlinear dissipators, with operator-size concentration making annihilation rates proportional to operator size and dissipation strength.The nonlinear class reduces to the linear dissipator at p = 1; numerical analysis uses p = 2 and a large-N, large-M limit with R = M/N fixed.Source: §13.1 · pp. 81–82

13.2. Analytical approach: moment method

• The moment method produces both even and odd autocorrelation moments, with dissipation-independent off-diagonal coefficients and imaginary diagonal coefficients linear in dissipation and n.The leading moments are ˜m1 = w/2, ˜m2 = 1, and ˜m3 = w, while the resulting coefficients grow linearly in n.Source: §13.2 · pp. 82–83

13.2.1. Arnoldi iteration

• In Arnoldi iteration, diagonal coefficients grow with dissipation and n before finite-size saturation, whereas primary off-diagonal elements remain close to closed-system values but are unequal.Additional smaller off-diagonal elements prevent a generally tridiagonal Lindbladian and complicate direct Krylov-complexity evaluation.Source: Figure 22 · p. 84, §13.2.1 · p. 84

13.3.1. Bi-Lanczos algorithm

• Bi-Lanczos coefficients show the same asymptotic linear diagonal growth, but equal upper and lower off-diagonal coefficients produce a purely tridiagonal Lindbladian.Both methods agree in slope and saturation after averaging 100 realizations, while small systems exhibit numerical instability and benefit from re-orthogonalization.Source: §13.3.1 · p. 85

13.4. Krylov complexity in chaotic open quantum systems

• The proposed operator-growth hypothesis has both Lanczos-coefficient sets asymptotically linear in n, with the dissipative parameter scaling as µ ∝λ for linear and µ ∝RV2 for p-body dissipation.The coefficient χ is independent of n and dissipation, and the hypothesis contains the unitary result as a special case.Source: §13.4 · p. 85

13.4.1. Continuum limit: large n result

• In the continuum large-n limit, Krylov complexity grows exponentially as K(t) ∼e2αt and saturates at a value inversely proportional to weak dissipation, Ksat = α/(χµ) + O(µ).The saturation time depends logarithmically on the inverse dissipation strength.Source: §13.4.1 · p. 86

13.4.2. Exact results

• The exact non-Hermitian tight-binding solution reproduces the continuum asymptotics, with a stationary exponential wavefunction tail, logarithmic dissipative timescale, and dissipation-dependent complexity plateau.At zero dissipation it recovers exponential unitary growth, while fixed positive dissipation gives a plateau independent of the initial growth parameter and system size.Source: §13.4.2 · pp. 87–88, Figure 23 · p. 88
• The plateau arises because dissipation strengthens with operator size and balances scrambling, while the Krylov-complexity distribution remains broad with standard deviation comparable to its mean.The same plateau behavior is reported for operator size and OTOC, and the mechanism is interpreted as continuous environment-induced measurement suppressing unitary exponential growth.Source: §13.4.2 · p. 89

13.5. Pole structure of autocorrelation and spectral density

• Under dissipation, pole distances govern Lanczos growth: vertical distance controls b_n, while horizontal displacement controls a_n, with generic dissipation producing a combined diagonal shift [136].In weak dissipation, the closest poles shift along the negative real axis while retaining their distance from the imaginary axis; the resulting coefficients are a_n = iµ(2n + 1) ∼ iµχn and b_n = αn.Source: §13.5 · pp. 90–91, Figure 24 · p. 91
• The pole-based autocorrelation model illustrates the universal growth hypothesis but does not provide exact generic-dissipation dynamics, which would require non-Markovian analysis.The derived Lanczos expressions hold for any dissipation strength, although the discussion focuses on small µ for continuity with Markovian dissipation.Source: §13.5 · pp. 90–91
• The spectral function remains exponentially decaying under weak dissipation, although its leading term is exponential and its subleading term has product-exponential high-frequency decay [136].The leading behavior is exp(−#|ω|), while the subleading behavior is ω exp(−#|ω|), assuming smooth Lanczos coefficients; a parallel large q SYK analysis appears in.Source: §13.5 · p. 92, Figure 25 · p. 92

14.1. Krylov complexity in Quantum Field Theories

• In free CFT, Lanczos coefficients grow linearly indefinitely and the Krylov growth rate saturates the MSS bound, λK = 2π/β = λMSS.This behavior results from the infinite UV cutoff and is altered by regulating the theory.Source: §14.1 · p. 94
• UV regulation changes indefinite Lanczos growth into a plateau, so coefficients beyond the cutoff are needed to assess quantum-field-theory chaos.For finite Λ, the saturation value is linearly proportional to the UV regulator.Source: §14.1 · pp. 94–95
• Infrared scales produce distinct even and odd Lanczos branches: massive theories show persistent staggering, while compact manifolds can have different linear slopes beyond the original universality.For massive theories both branches have slope α = πT; compact manifolds exhibit slopes different from πT.Source: §14.1 · p. 95

14.2. Krylov complexity in Holography

• Holographic Krylov complexity is computed from thermal Wightman two-point functions using methods analogous to the field-theory construction.The holographic calculation uses Euclidean continuation and the Lanczos coefficients derived from the thermal correlator.Source: §14.2 · p. 95
• Thermal AdS3 exhibits two-slope behavior and early trapping of Krylov complexity at infrared values, making its asymptotic value independent of the UV cutoff.This differs from computational or holographic complexities that are explicitly UV-cutoff dependent.Source: §14.2 · p. 96
• In DSSYK, non-perturbative Lanczos spectra support interpreting Krylov state complexity as a non-perturbative description of wormhole size, while finite Hilbert-space dimension causes saturation.The perturbative regime agrees with, but the wormhole interpretation breaks down in the non-perturbative Lanczos-descent regime.Source: §14.2 · pp. 96–97

15. Krylov complexity and Integrability

• Krylov complexity differentiates integrable and non-integrable systems across XXZ, Ising, Bose-Hubbard, and Floquet models, including weakly and strongly broken integrability.The review uses “non-integrable” for systems whose integrability is disrupted by an added term.Source: §15 · p. 97
• Lanczos coefficients grow sublinearly in the integrable case and linearly in the non-integrable case, while the Krylov dimension saturates its bound in both regimes.The Krylov dimension remains fixed across the two scenarios and is constrained by finite system size.Source: §15 · pp. 97–98
• Integrable limits exhibit greater Lanczos-spectrum disorder, quantified by logarithmic variance and associated with Krylov localization in an auxiliary Anderson model.This disorder difference persists when integrability is strongly broken.Source: §15 · p. 97
• Late-time Krylov-complexity saturation increases toward the chaotic phase but remains below DK/2, unlike the benchmark behavior of genuinely chaotic SYK4 dynamics.The trend is reported across XXZ, Ising, and Bose-Hubbard models, but the review notes exceptions in XXZ and Ising chains.Source: §15 · pp. 97–98
• The MFIM figures compare initial and full Lanczos spectra across integrable and non-integrable limits.The displayed parameters are (g, h) = (1, 0) and (g, h) = (−1.05, 0.5).Source: Figure 26 · p. 98, §15 · p. 98
• For the MFIM parameters shown, the non-integrable regime has higher Krylov complexity than the integrable regime because of Krylov localization.The figures use N = 6, periodic boundary conditions, and the initial operator σz_1.Source: §15 · p. 98
• Non-integrable MFIM dynamics follows GOE Wigner-Dyson statistics and shows a complexity peak followed by saturation, unlike the integrable regime.The saturation value is identical in both regimes and depends only on Hilbert-space dimension.Source: §15 · p. 98

16.1. Applications to Quantum Control

• Counterdiabatic driving provides a shortcut to adiabatic state preparation without requiring slow driving, addressing noise, errors, and environmental coupling.It is also called transitionless quantum driving.Source: §16.1 · p. 98
• Many-body counterdiabatic terms generally involve nonlocal multiple-body interactions, motivating variational approximations and digitized counterdiabatic algorithms.Truncated controls are desirable and can be implemented through digital quantum schemes.Source: §16.1 · p. 99
• The Krylov expansion fixes counterdiabatic coefficients through Lanczos coefficients, avoiding their approximate determination by a variational approach.The construction uses O = ∂λH(λ) and has dA = DK/2 or (DK − 1)/2 for even or odd Krylov dimension.Source: §16.1 · p. 100
• Krylov expansions relate counterdiabatic-driving features to operator growth and Lanczos coefficients, while the counterdiabatic norm quantifies protocol cost.The norm is also related to fidelity susceptibility and the quantum geometric tensor.Source: §16.1 · p. 100
• Krylov-expanded counterdiabatic constructions may extend to open systems, quantum metrology, and parallel transport in matrix-product-state algorithms.The relevant transported objects include Liouvillian eigenstates, natural orbitals, and parametric derivatives of evolution generators.Source: §16.1 · p. 100

16.2. Applications to Quantum Computing

• Krylov methods are costly for many-body quantum computers because the basis dimension scales exponentially with system size, while circuit models naturally implement unitaries rather than generator powers.These constraints motivate unitary-based variants.Source: §16.2 · pp. 100–101
• Unitary Krylov variants support approximate real- and imaginary-time evolution, ground- and thermal-state preparation, and hybrid quantum-classical algorithms.Other approaches use combined unitary evolutions or construct the Krylov basis exactly on a quantum computer.Source: §16.2 · p. 101
• Exact quantum construction of the Krylov basis can reduce exponential classical cost and be achieved in polynomial time and memory, while Lindbladian extensions remain an open prospect.Recent efforts reviewed here primarily address Hamiltonian-generated dynamics.Source: §16.2 · p. 101

17. Open problems

• Time-dependent dynamics can be treated using Floquet operators, Trotterized circuits, stepwise constant generators, or Magnus-operator powers, but broader formulations remain under development.Floquet-based Krylov dynamics corresponds to diffusion with inhomogeneous, time-dependent nearest-neighbor hopping.Source: §17 · p. 101
• Extending Krylov constructions from Markovian Lindblad evolution to non-Markovian, time-dependent-rate, stochastic, quantum-jump, and continuous-measurement dynamics is an open direction.A key question is whether relations among Lanczos coefficients, operator correlation functions, and state survival probabilities persist.Source: §17 · pp. 101–102
• Krylov methods have recently been applied to Wigner phase-space distributions, while analogous treatments of P and Q distributions remain possible extensions.These representations are relevant across quantum foundations, many-body physics, and quantum optics.Source: §17 · p. 102
• It remains unknown whether symmetry classes from Hermitian, non-Hermitian, and open-system classifications leave clear signatures in Lanczos coefficients or Krylov complexity.The review also asks whether integrable Krylov dynamics extends beyond isolated systems and connects to open-system notions of integrability.Source: §17 · p. 102
• Developing these extensions and assessing their usefulness in applications remain open prospects for future studies.The review presents these questions as formal and applied directions rather than settled results.Source: §17 · p. 102

• The review frames Krylov methods as tools for characterizing many-body quantum evolution, thermalization, quantum chaos, and complexity.Source: §18 · p. 102
• Lanczos constructions are presented for unitary evolution of pure and mixed states, while Arnoldi and bi-Lanczos generalizations address open quantum systems.Source: §18 · p. 102
• The moment method provides an alternative construction from two-point autocorrelation functions, supporting analyses of quantum chaos in quantum field theory and holography via AdS/CFT.Source: §18 · p. 103
• Analytical and numerical examples connect coherent states, geometric structure, quantum speed limits, RMT, spin chains, SYK models, quantum chaos bounds, integrability, and quantum control.Source: §18 · p. 103
• Experimental methods for probing Krylov complexity remain nascent, although digital quantum simulations of spectral form factors and OTOCs provide a foundation for implementation on quantum devices.Source: §18 · p. 103

• Notation covers Hilbert-space and thermal objects, including H, I, β, Z(β), d, and DK for Hamiltonian, identity, inverse temperature, partition function, dimension, and Krylov dimension.Source: List of Symbols and Acronyms · p. 104
• Dynamical notation includes the Liouvillian L, Lindbladian Lo, observables O(t), density matrices ρ(t), Lanczos or Arnoldi coefficients, Krylov amplitudes, and state or operator complexity measures.Source: List of Symbols and Acronyms · p. 104
• Acronyms identify Random Matrix Theory, spectral form factors, out-of-time-order correlators, Sachdev-Ye-Kitaev models, quantum field or conformal field theory, Anti-de Sitter space, open quantum systems, and counterdiabatic driving.Source: List of Symbols and Acronyms · p. 104

• Foundational topics include universal operator growth, Krylov complexity, Krylov spread complexity, entropy and cumulants, numerical algorithms, and applications to SYK and related models.Source: A Brief List of References of primary… · p. 105
• The references cover coherent states, quantum speed limits, integrability, localization, entanglement, modular Hamiltonians, density matrices, QFT/CFT, quantum mechanics, billiards, and Toda-chain methods.Source: A Brief List of References of primary… · p. 105
• Applications span open quantum systems, Arnoldi and bi-Lanczos algorithms, driven and Floquet systems, unitary circuits, spin chains, Nielsen and circuit complexity, scars, holography, control, phase space, and RMT.Source: A Brief List of References of primary… · p. 105