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Geometry of Krylov Complexity

Pawel Caputa, Javier M. Magan, Dimitrios Patramanis

arXiv:2109.03824v2hep-thcond-mat.stat-mechcond-mat.str-elquant-ph

TL;DR

The physical and operational meaning of Krylov complexity remains unclear, and computing it can be technically difficult. The paper develops a symmetry-based geometric approach that connects operator evolution with classical phase-space motion, interpreting Krylov complexity as a volume and simplifying Lanczos-coefficient computation.

  • Problem

    Krylov complexity has unresolved physical and operational meaning, and computing it requires numerics while its universal features remain difficult to understand.

    Source: §I · p. 2

  • Method

    The paper uses symmetries, generalized coherent states, and their information geometry to geometrize operator growth and Krylov complexity.

    Source: §I · p. 2

  • Results

    When the Liouvillian and Hamiltonian have the stated ladder-operator structure, operator dynamics is equivalent to classical phase-space motion, and Krylov complexity is the volume defined by that motion.

    Source: §IX · p. 18

  • Takeaways & Limitations

    The geometric identification gives Krylov complexity a physical interpretation and simplifies computation of Lanczos coefficients.

    Source: §IX · p. 18

Abstract

from arXiv · show

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.

I. INTRODUCTION

The paper takes up the conceptual and technical difficulties of operator complexity by developing a symmetry-based geometric approach to Krylov complexity. It reviews how the Lanczos construction represents operator evolution in a Krylov basis.

I. INTRODUCTION

  • The paper develops a geometric approach to Krylov complexity to clarify its physical meaning and simplify technical analysis of operator growth.The introduction identifies unresolved operational meaning and the difficulty of computing Krylov complexity as motivations.Source: §I · p. 2

Operator Growth

  • Operator growth quantifies how a simple Heisenberg operator mixes into increasingly complex operators, particularly under chaotic dynamics.Solving Heisenberg evolution is out of reach in many chaotic quantum systems, often leaving numerical analysis for specific models.Source: §I · p. 1, Operator Growth · p. 3

Lanczos Algorithm and Krylov Basis

  • The Lanczos algorithm orthogonalizes repeated Liouvillian actions into a Krylov basis, producing coefficients that encode the operator's evolution.The coefficients help determine the operator wavefunction and its amplitudes.Source: Lanczos Algorithm and Krylov Basis · pp. 3–4
  • In the Krylov basis, the Liouvillian is tridiagonal and the operator amplitudes obey a discrete Schrödinger equation.Given the Lanczos coefficients, the amplitudes can be solved from the initial condition to determine the operator wavefunction.Source: Lanczos Algorithm and Krylov Basis · p. 4

Krylov Complexity

The section presents Krylov complexity as an average position in the Krylov chain and uses the SYK model to illustrate its growth. It then connects symmetry-based operator evolution to coherent states and information geometry.

Krylov Complexity

  • Krylov complexity is the evolving operator's average position on a chain whose sites correspond to Krylov basis vectors.The paper notes that this minimal complexity definition is ambiguous, but gives it a geometric interpretation.Source: Krylov Complexity · p. 4

SYK example

  • In SYK, the negative-binomial probabilities yield exponentially growing Krylov complexity controlled by the maximal Lyapunov exponent λ = 2π/β.The exponential growth is described as more universal than the operator-dependent scrambling time.Source: SYK example · p. 5

III. LIOUVILLIAN AND SYMMETRY: GENERAL IDEA

  • For symmetry-governed dynamics, the Liouvillian acts on the Krylov basis as a sum of ladder operators that furnish symmetry-group representations.The proportionality factor α is not fixed by symmetry and depends on physical details, including the inner product.Source: §III · p. 6
  • Symmetry actions determine Lanczos coefficients directly; under specified conditions, quadratic equations for them constrain their growth to at most linear in n.This agrees with the maximal operator growth hypothesis in the examples discussed.Source: §III · p. 6
  • Liouvillian evolution becomes a generalized coherent-state displacement, so operator growth traces a trajectory through coherent-state Hilbert space.The displacement parameter is ξ = iαt, and expansion in an orthonormal basis gives the Krylov amplitudes.Source: §III · p. 6
  • The coherent-state information geometry relates Krylov complexity to volume and links the Liouvillian and complexity operator to isometry generators.The resulting complexity algebra is isomorphic to the algebra of geometric isometries.Source: §III · p. 6

IV. LIOUVILLIAN AND SYMMETRY: EXAMPLES

Section IV uses symmetries and generalized coherent states to recast operator evolution; in the SL(2,R) example, evolution follows a hyperbolic geodesic and Krylov complexity tracks enclosed volume.

Example I: SL(2,R)

  • The operator wavefunction is a Perelomov coherent state with ξ=iαt, mapping operator evolution to a classical trajectory in coherent-state phase space.In polar coordinates, the trajectory has ρ=2αt and φ=π/2.Source: Example I: SL(2,R) · p. 7
  • Identifying the Krylov basis with the SL(2,R) representation basis lets ladder-operator actions determine the Lanczos coefficients directly.Source: Example I: SL(2,R) · p. 7
  • Krylov complexity is proportional to highest weight h and grows exponentially with λ=2α, while the n=0 amplitude is the SYK autocorrelation with η=2h.Source: Example I: SL(2,R) · p. 7
  • The coherent-state information metric is the hyperbolic-disc metric; Krylov complexity generates translations in φ, an isometry of that geometry.The translation-generator interpretation follows from −i∂φ producing a factor n.Source: Example I: SL(2,R) · pp. 7–8
  • Krylov complexity is proportional to the information-geometric volume enclosed from the origin to the trajectory at ρ=2αt, φ=π/2.The trajectory is a geodesic on the hyperbolic disc.Source: Example I: SL(2,R) · p. 8, Figure 1 · p. 8
  • The geodesic length from the disc center grows linearly in time, whereas Krylov complexity grows exponentially in this example.For the trajectory, the geodesic length is L=ρf with ρf=αt.Source: Example I: SL(2,R) · pp. 7–8

Example II: SU(2)

The SU(2) example maps operator growth to spin coherent-state dynamics in a finite-dimensional Krylov space. Its complexity rises and falls along a geodesic on a positively curved sphere, with Krylov complexity proportional to geometric volume.

Example II: SU(2)

  • The SU(2) Krylov space has dimension 2j + 1, with basis states identified with spin states |j, −j + n⟩ for n = 0, ..., 2j.The initial state is the lowest-weight state |j, −j⟩.Source: Example II: SU(2) · p. 9
  • The Lanczos coefficients grow as α√(2j n) to a maximum and then decrease to their final value.Their rise and fall are shown in the sample distribution for j = 20.Source: Example II: SU(2) · p. 9, Figure 2 · p. 9
  • The SU(2) probabilities form a binomial distribution with λ = sin(αt), and the j = 5 wavefunctions peak at later αt symmetrically.The wavefunction symmetry reflects the symmetry of the Lanczos coefficients.Source: Example II: SU(2) · p. 9, Figure 3 · p. 9
  • For a free harmonic oscillator, the SU(2) result at j = 1/2 and α = ω is recovered up to operator normalization, while temperature changes normalization but not the operator wavefunction.The assignment of α is temperature independent in this class of models.Source: Example II: SU(2) · p. 10
  • Krylov complexity grows quadratically as KO ∼ 2jα²t² at early times, reaches Kmax = 2j at t = π/(2α), and returns to zero at t = π/α.There is no exponential growth in this SU(2) case.Source: Example II: SU(2) · p. 10
  • The information geometry is a constant-positive-curvature sphere, and operator growth follows a geodesic whose enclosed volume confirms the proportionality to Krylov complexity.The curvature decreases for large spin j.Source: Example II: SU(2) · p. 10

Example III: Heisenberg-Weyl

The Heisenberg-Weyl example yields an infinite-dimensional Krylov chain with coherent-state evolution, while the 2d CFT example uses two copies of SL(2,R). In both settings, the information geometry organizes operator growth as geodesic motion and relates volume to Krylov complexity.

Example III: Heisenberg-Weyl

  • For Heisenberg-Weyl symmetry, the infinite-dimensional Krylov basis has Lanczos coefficients bn = α√n.The Liouvillian is L = α(a† + a).Source: Example III: Heisenberg-Weyl · p. 10
  • The Heisenberg-Weyl coherent-state wavefunction has Poisson probabilities, while its Lanczos growth is non-maximal and its autocorrelation decays exponentially.Figure 4 plots the wavefunctions against n for αt = 1, 2, 4, 6, 8, and 10.Source: Example III: Heisenberg-Weyl · p. 11, Figure 4 · p. 11
  • Heisenberg-Weyl Krylov complexity grows quadratically at early times and remains quadratic for all times.Its information metric is the flat complex plane, the trajectory is a geodesic, and Krylov complexity is proportional to volume.Source: Example III: Heisenberg-Weyl · p. 11

Example IV: Conformal Field Theories in 2d

  • The 2d CFT construction uses two SL(2,R) displacement operators, producing a product of left and right wavefunctions and a consistent two-point function.The complexity is sensitive to operator spin when β+ ≠ β−.Source: Example IV: Conformal Field Theories… · pp. 11–12
  • For 2d CFTs, the information geometry is a product of two Euclidean Poincaré discs, with geodesic evolution and Krylov complexity proportional to volume.The complexity is the sum of position expectation values in the left and right chains.Source: Example IV: Conformal Field Theories… · p. 12
  • The symmetry-determined CFT generalization does not distinguish chaotic from non-chaotic CFTs; composite operators, thermal CFTs on a circle, or Virasoro dynamics require additional spectral information or further analysis.The considered Virasoro subalgebra example has complexity that depends on the central charge c.Source: Example IV: Conformal Field Theories… · p. 12

V. COMPLEXITY ALGEBRA AND GEOMETRY

The section proposes deriving Lanczos coefficients from closure of an algebra built from the Liouvillian and a generalized ladder operator. It connects the resulting algebra to complexity operators and information-geometry isometries, while showing that the relation varies across symmetry examples.

V. COMPLEXITY ALGEBRA AND GEOMETRY

  • Closing the algebra of the Liouvillian, operator B, and their commutator imposes a recurrence for Lanczos coefficients and bounds the commutator eigenvalues to grow at most linearly in n.Requiring b0 = 1 fixes the arbitrary constant in the resulting coefficient family; the family was also derived from the Toda chain approach [29].Source: §V · p. 13
  • For SL(2,R), the commutator operator is related to the Krylov complexity operator up to a constant and proportionality factor, and both grow exponentially with the same rate.This suggests that L0, or more generally energy in CFTs, may witness operator growth, though the Krylov operator definition is more robust for numerically analyzed generic systems.Source: §V · p. 13
  • The operators L, B, and ˜K correspond to information-metric isometries; ˜K generates translations in φ and the Krylov complexity operator does so as well.The absolute value of the expectation of B also grows exponentially during the stated operator dynamics.Source: §V · p. 13
  • For SU(2), ˜K and the Krylov complexity operator are related and generate rotations in the information metric.The section assigns L = α(J+ + J−), B = α(J+ − J−), and ˜K = −4α²J0.Source: §V · p. 13
  • For Heisenberg-Weyl symmetry, the commutator is proportional to the identity, making its relation to the Krylov complexity operator less natural.The relation is not merely a simple constant shift.Source: §V · p. 13
  • The proposed classification program would specify how many operators must be added to the Liouvillian and B to close the algebra, then analyze the resulting representations.The authors identify other Lie groups and their deformations as useful examples for this program.Source: §V · pp. 12, 14

VI. RELATION TO GEOMETRIC COMPLEXITY: PARTICLE ON A GROUP

The section connects operator dynamics to classical motion on generalized coherent-state phase spaces, then relates Krylov complexity to their information geometry. It also compares this volume measure with geodesic-length notions of circuit complexity.

VI. RELATION TO GEOMETRIC COMPLEXITY: PARTICLE ON A GROUP

  • Nielsen complexity treats circuits as paths through unitary transformations chosen by time-dependent Hamiltonians, but its geodesic length is not itself the Krylov measure described here.The section frames the connection as clarifying the operational meaning of the operator wavefunction and Krylov complexity.Source: §VI · pp. 14–15
  • A coherent-state transition amplitude yields a path-integral action whose classical approximation gives equations of motion for operator growth.The classical Hamiltonian is the symbol, or coherent-state expectation value, of the quantum Hamiltonian.Source: §VI · p. 14
  • The classical dynamics use group-specific Poisson brackets on hyperbolic SL(2,R), spherical SU(2), and flat Heisenberg–Weyl phase spaces.The section assumes coherence-preserving Hamiltonians from the relevant Lie algebra; the illustrated parameters are taken constant.Source: §VI · pp. 14–15
  • For SL(2,R), the classical Hamilton equations become complex Riccati equations, and the earlier operator-wavefunction trajectory solves them for a=0 and b=c=α.The solution corresponds to H=α(L−1 + L1); its phase-sign difference is attributed to forward versus backward evolution.Source: §VI · p. 15
  • The earlier trajectory also solves the SU(2) equations for a=0 and H=α(J+ + J−), while the Heisenberg–Weyl case gives H=α(a† + a).The latter is described as particle motion on the Heisenberg–Weyl phase space.Source: §VI · p. 15
  • Krylov complexity measures phase-space volume in the Fubini–Study metric, while the radial geodesic length grows only linearly in time.Angular displacements relate to Nielsen complexity with F1 cost functions and connect to the introduction’s discussion of classical chaos.Source: §VI · p. 15

VII. QUANTUM INFORMATION TOOLS FOR OPERATOR GROWTH

The section uses a two-mode coherent-state representation to derive a density matrix for the evolving operator and examine its quantum-information measures. In the SL(2,R) example, these measures show varied time dependence, and several connect to the Lyapunov exponent or geometric bounds.

VII. QUANTUM INFORMATION TOOLS FOR OPERATOR GROWTH

  • The authors focus on the chaotic SL(2,R) example, noting that most findings also hold for other coherent states and that non-chaotic examples are treated elsewhere.The section’s main message is that the quantum-information tools can be written in terms of ϕ_n(t) and are sensitive to the Lyapunov exponent 2α.Source: §VII · pp. 16–17
  • The two-mode representation yields a density matrix for the evolving operator whose eigenvalues are the probabilities in the Krylov basis.The coherent states become entangled two-mode squeezed states, with k excitations separating the modes.Source: §VII · p. 16
  • For k=0, the operator’s proper temperature grows exponentially at large times with the Lyapunov exponent, resembling the temperature or energy behavior of infalling black-hole perturbations.The proposed temperature comes from identifying the reduced state with a thermal harmonic-oscillator state.Source: §VII · p. 16
  • For k=0, the K-entropy is the von Neumann entropy of the operator density matrix and grows linearly at late times with a proportionality factor equal to the Lyapunov exponent.The paper compares this rate to the upper bound on Kolmogorov–Sinai entropy given by the sum of Lyapunov exponents.Source: §VII · pp. 16–17
  • For k=0, capacity of entanglement saturates to 1 exponentially fast as t→∞, with twice the Lyapunov exponent, after its early-time growth.The authors say the physical interpretation of this saturation as thermalisation is unclear.Source: §VII · p. 17
  • The section surveys entanglement negativity, relative entropy, and fidelity as additional probes of operator growth, with late-time fidelity growing linearly.Negativity resembles the growth of a geodesic length from the hyperbolic-disc origin; negativity and late-time relative entropy are bounded by the geodesic length up to ρ(t).Source: §VII · p. 17

VIII. DISCUSSION: CFT GENERALIZATIONS

The discussion extends the symmetry-based framework to CFT generalizations and concludes that Lanczos dynamics can be understood as classical motion on phase space. It identifies further CFT settings and initial states for study, while noting that the analyzed examples are constrained by universal symmetry kinematics.

VIII. DISCUSSION: CFT GENERALIZATIONS

  • Most analyzed examples are constrained by universal symmetry kinematics; genuine holographic applications require setups or operators sensitive to detailed spectra and dynamics.Source: §VIII · p. 17
  • Further CFT directions include extending the 2d symmetry analysis to Virasoro and higher-dimensional conformal algebras, including Wn and BMS examples.Source: §VIII · p. 17
  • For coincident products O1(t)O2(t), the lowest-dimension operator in their OPE describes the growth after normalization.The composite-operator autocorrelation involves a four-point function, and the lowest-dimension OPE contribution controls the coincident-point limit.Source: §VIII · p. 18
  • For non-coincident products O1(0)O2(t), the OPE yields blocks rather than a single operator, motivating block Krylov complexity and a possible relation to kinematic-space volumes.Source: §VIII · p. 18
  • The framework could also address general initial states, including boundary and TFD states, bulk-local-operator states, and high-energy SYK states.Some such settings involve return amplitudes related to spectral form factors.Source: §VIII · p. 18

IX. CONCLUSIONS

  • The Liouvillian/Hamiltonian in the Lanczos basis can be expressed through ladder operators, making the dynamics equivalent to classical evolution on the associated phase space.Krylov complexity is related to phase-space isometry-generating vector fields, and its expectation value gives the volume defined by classical motion.Source: §IX · p. 18
  • The authors identify photon-based quantum-optics experiments as a possible way to study the classical process representing SYK operator-growth wavefunctions.Source: §IX · p. 18
  • The relation between operator complexity and phase-space volume remains open for Liouvillians built from arbitrary combinations of algebra elements, with a possible connection to holographic complexity.The authors also suggest that relating classical operator motion to OTOC-based chaos measures may help derive the chaos bound more physically.Source: §IX · pp. 18–19

Appendix A: Lanczos Algorithm

The appendix carries out the Lanczos construction for Lie-algebra Liouvillians, starting from an initial operator or state and iteratively building Krylov basis vectors and coefficients. It describes SL(2,R) and two-mode representations, then extends the procedure to two-dimensional CFTs and more general initial states.

Appendix A: Lanczos Algorithm

  • The Lanczos algorithm iteratively applies the Liouvillian to an initial operator or state to construct Krylov basis vectors and Lanczos coefficients.In the SL(2,R) example, the initial operator/state is chosen as a highest-weight state.Source: Appendix A · p. 19
  • For the SL(2,R) construction, the resulting Krylov basis vectors are orthonormal, and the appendix also carries out the algorithm in a two-mode representation.Source: Appendix A · p. 20
  • In the two-mode representation, the algorithm uses a k-photon-added state and constructs an orthonormal Krylov basis with associated Lanczos coefficients.The two representations are related by 2h = k + 1.Source: Appendix A · pp. 20–21
  • For 2d CFTs, left and right Liouvillians define two sets of Lanczos coefficients that determine the operator wavefunction in the Krylov basis.The procedure can also be repeated for a more general initial state with a nonzero Liouvillian expectation value.Source: Appendix A · p. 21

Appendix B: SL(2, R) subalgebras of Virasoro

The appendix applies the symmetry-based construction to Virasoro modes that form an SL(2,R) subalgebra. It derives Lanczos coefficients and Krylov complexity, finding a large-n growth rate that depends on the central charge.

Appendix B: SL(2, R) subalgebras of Virasoro

  • The modes L0, Lk, and L−k for fixed integer k > 0 form an SL(2,R) algebra and define a displacement-operator construction.Source: Appendix B · p. 21
  • The appendix constructs an orthonormal basis and Lanczos coefficients for the Virasoro subalgebra using the displacement-operator approach.Source: Appendix B · pp. 21–22
  • The large-n Lanczos growth generalizes to α_k = kα, and the resulting growth rate is sensitive to the CFT central charge.Source: Appendix B · p. 22

Appendix C: CFT Correlators

The appendix reviews how conformal symmetry determines CFT two-point functions and how conformal maps produce correlators on a circle or at finite temperature. It obtains the Lorentzian correlator used in operator growth by analytic continuation of the Euclidean result.

Appendix C: CFT Correlators

  • Quasi-primary operators transform under chiral and anti-chiral conformal transformations, with their dimension and spin encoded by h and h̄.Source: Appendix C · p. 22
  • Global conformal symmetry fixes vacuum two-point functions of quasi-primary operators through invariance under infinitesimal transformations.Source: Appendix C · p. 22
  • Exponential conformal maps yield universal two-point correlators on a finite-size circle and on a line at finite temperature.Source: Appendix C · pp. 22–23
  • The Lorentzian correlator used in operator growth is obtained by analytically continuing the Euclidean correlator.Source: Appendix C · p. 23

Appendix D: Coherent States: Formulas

This appendix collects coherent-state formulas used in the main text for SU(1,1)/SL(2,R), SU(2), and Heisenberg-Weyl examples. It also records associated expectation values and phase-space quantities.

Appendix D: Coherent States: Formulas

  • The appendix identifies Perelomov, spin, and standard coherent states for SU(1,1)/SL(2,R), SU(2), and Heisenberg-Weyl, respectively; these states are over-complete and nonorthogonal.Source: Appendix D · p. 23
  • Along ρ = 2αt and φ = π/2, both ˜KO and the anti-Hermitian B grow exponentially while the Liouvillian vanishes.Source: Appendix D · p. 24
  • It gives coordinate expressions for Poisson brackets across the three symmetry examples and relates SL(2,R) symbols to hyperbolic-space embedding coordinates and an induced metric.Source: Appendix D · p. 24
  • The formulas use coherent-state expectation values of Lie-algebra generators, including ⟨z|a|z⟩ = a, ⟨z|a†|z⟩ = z̄, and ⟨z|n̂|z⟩ = |z|².Source: Appendix D · pp. 23–24
  • For trajectories ρ = 2αt, θ = 2αt, and r = αt, the F1 norms in the SL(2,R), SU(2), and Heisenberg-Weyl examples have a universal form.Source: Appendix D · p. 24
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