Complex-Time Krylov Spaces

• Complex-Time Krylov Spaces are defined as subspaces generated by evolution maps evolving in complex time, combining real-time dynamics with imaginary-time damping.
• They employ higher-order propagators that interpolate between conventional Lanczos methods and full complex-time evolution, improving the accuracy of Green’s function reconstructions.
• These spaces provide actionable frameworks that address challenges in orthonormalization, conditioning, and non-unitary evolution through contour-time and spectral filtering techniques.

Complex-time Krylov spaces are Krylov-type subspaces generated by evolution maps rather than only by monomials of a Hamiltonian. In the most explicit recent usage, they are subspaces of the form

$$\mathcal H_D^{\tau}=\mathrm{span}\Bigl\{|\Psi\rangle,\hat U(\tau)|\Psi\rangle,\dots,\hat U^{D-1}(\tau)|\Psi\rangle\Bigr\},$$

where the contour time is $\tau(t)=t(1-i\tan\alpha)$ and $\hat U(\tau)$ combines real-time evolution with imaginary-time damping. A broader, partly extrapolative usage treats bases built from snapshots $e^{-inHz}|\psi_0\rangle$ with $z\in\mathbb C$ as the natural complex-time analogue of real-time snapshot bases. In current literature, the subject is not a single unified formalism but a family of closely related constructions linking Lanczos/Krylov methods, Green’s-function reconstruction, higher-order propagator generators, contour-time dynamics, non-Hermitian evolution, Lie-algebraic time dependence, and polynomial spectral filtering (Paeckel, 2024, Čindrak et al., 7 Mar 2026, Takahashi et al., 22 Oct 2025, Murugan et al., 2 Feb 2026, Medina-Guerra et al., 11 Feb 2025, Grabarits et al., 6 May 2026, Chowdhury et al., 6 Jul 2026).

1. Standard Krylov construction and its evolution-adapted reinterpretation

In the standard state-Krylov setting, Schrödinger evolution

$$\partial_t \ket{\psi(t)}=-iH\ket{\psi(t)},\qquad \ket{\psi(0)}=\ket{\psi_0},$$

implies

$$\ket{\psi(t)}=e^{-iHt}\ket{\psi_0} =\sum_{k=0}^\infty \frac{(-iHt)^k}{k!}\ket{\psi_0},$$

so the dynamics lies in the Krylov subspace

$$\mathrm K_m=\mathrm{Span}\{\ket{\psi_0},H\ket{\psi_0},\ldots,H^{m-1}\ket{\psi_0}\}.$$

Orthogonalizing $\{\ket{\psi_0},H\ket{\psi_0},\ldots\}$ gives the usual Krylov basis $\mathcal B_K=\{\ket{k_n}\}_{n=0}^{m-1}$. For Hermitian $\tau(t)=t(1-i\tan\alpha)$0, the basis is generated by the Lanczos three-term recursion, and in this basis $\tau(t)=t(1-i\tan\alpha)$1 becomes tridiagonal. The spread complexity used in this framework is

$\tau(t)=t(1-i\tan\alpha)$2

so the amplitudes evolve as a wavefunction on a semi-infinite tight-binding chain with hopping amplitudes set by the Lanczos coefficients (Čindrak et al., 7 Mar 2026).

A central reinterpretation is that the usual Krylov basis is the basis induced by a first-order approximation to time evolution rather than a uniquely privileged optimal basis. For a short step $\tau(t)=t(1-i\tan\alpha)$3,

$\tau(t)=t(1-i\tan\alpha)$4

which motivates the first-order generator $\tau(t)=t(1-i\tan\alpha)$5. After orthonormalization, the identity contributes no new direction and the effective generator is just $\tau(t)=t(1-i\tan\alpha)$6. This immediately suggests a higher-order family,

$\tau(t)=t(1-i\tan\alpha)$7

interpolating between the conventional generator $\tau(t)=t(1-i\tan\alpha)$8 at $\tau(t)=t(1-i\tan\alpha)$9 and the exact finite-step propagator

$\hat U(\tau)$0

At $\hat U(\tau)$1, the raw generating set is a set of time snapshots,

$\hat U(\tau)$2

This is already much closer to generalized-time and contour-based Krylov constructions than to the monomial basis $\hat U(\tau)$3. A direct extrapolation is to replace $\hat U(\tau)$4 by $\hat U(\tau)$5 and consider

$\hat U(\tau)$6

that is, a basis built from complex-time snapshots. This extrapolation is formally immediate when $\hat U(\tau)$7 is well defined and the vectors remain linearly independent, but the detailed consequences for orthonormalization, conditioning, and non-unitarity are not analyzed in that work (Čindrak et al., 7 Mar 2026).

2. Higher-order generators, spread minimization, and snapshot bases

The higher-order-generator framework directly challenges the widely held assumption that the ordinary Lanczos basis minimizes spread complexity. For grade $\hat U(\tau)$8, the comparison between the usual basis $\hat U(\tau)$9 and the basis obtained by orthonormalizing

$e^{-inHz}|\psi_0\rangle$0

yields a rigorous counterexample: for any target time $e^{-inHz}|\psi_0\rangle$1, there exists $e^{-inHz}|\psi_0\rangle$2 such that

$e^{-inHz}|\psi_0\rangle$3

The proof sets $e^{-inHz}|\psi_0\rangle$4, so that the state at time $e^{-inHz}|\psi_0\rangle$5 lies entirely in the span of the first two basis vectors of the infinite-order basis, while the standard first-order basis generically has nonzero weight on higher levels even for arbitrarily short positive times. The result is an existence theorem for a $e^{-inHz}|\psi_0\rangle$6-adapted generator rather than a universal statement that one fixed higher-order basis dominates for all times simultaneously (Čindrak et al., 7 Mar 2026).

The same work introduces a natural time scale for choosing $e^{-inHz}|\psi_0\rangle$7 by comparing adjacent Dyson terms $e^{-inHz}|\psi_0\rangle$8. Defining scrambling to begin when

$e^{-inHz}|\psi_0\rangle$9

the paper arrives at

$z\in\mathbb C$0

Numerically, for a $z\in\mathbb C$1 GUE example with $z\in\mathbb C$2 and $z\in\mathbb C$3, the orders $z\in\mathbb C$4 all show initial growth, but higher-order complexities can become smaller than the first-order complexity after a crossover. For $z\in\mathbb C$5 GUE matrices with $z\in\mathbb C$6, $z\in\mathbb C$7, all higher-order generators $z\in\mathbb C$8 yield smaller spread complexity than the first-order basis over an early-time window $z\in\mathbb C$9, then may exceed it in a short intermediate region $$\partial_t \ket{\psi(t)}=-iH\ket{\psi(t)},\qquad \ket{\psi(0)}=\ket{\psi_0},$$0, after which all generators approach the same late-time average. As $$\partial_t \ket{\psi(t)}=-iH\ket{\psi(t)},\qquad \ket{\psi(0)}=\ket{\psi_0},$$1 increases, the matrix representation in the orthonormalized basis loses tridiagonality and develops broader, eventually dense upper support. A plausible implication is that generalized-time constructions based on finite-step propagators naturally favor non-tridiagonal effective recurrences rather than ordinary Lanczos chains (Čindrak et al., 7 Mar 2026).

3. Complex-time Krylov expansion for Green’s functions

The most explicit formulation of a complex-time Krylov space appears in work on dynamical Green’s functions in strongly correlated many-body systems. The starting point is the resolvent form

$$\partial_t \ket{\psi(t)}=-iH\ket{\psi(t)},\qquad \ket{\psi(0)}=\ket{\psi_0},$$2

together with the time-domain representation over a finite real-time window $$\partial_t \ket{\psi(t)}=-iH\ket{\psi(t)},\qquad \ket{\psi(0)}=\ket{\psi_0},$$3. In standard tensor-network calculations, the accessible time domain limits the frequency resolution to

$$\partial_t \ket{\psi(t)}=-iH\ket{\psi(t)},\qquad \ket{\psi(0)}=\ket{\psi_0},$$4

which is the Nyquist–Shannon constraint for truncated real-time Fourier reconstruction. The key reformulation partitions the time axis into intervals of length $$\partial_t \ket{\psi(t)}=-iH\ket{\psi(t)},\qquad \ket{\psi(0)}=\ket{\psi_0},$$5, introduces

$$\partial_t \ket{\psi(t)}=-iH\ket{\psi(t)},\qquad \ket{\psi(0)}=\ket{\psi_0},$$6

and decomposes the exact Green’s function as

$$\partial_t \ket{\psi(t)}=-iH\ket{\psi(t)},\qquad \ket{\psi(0)}=\ket{\psi_0},$$7

where the correction $$\partial_t \ket{\psi(t)}=-iH\ket{\psi(t)},\qquad \ket{\psi(0)}=\ket{\psi_0},$$8 encodes precisely the information lost by truncating the real-time domain (Paeckel, 2024).

The complex-time Krylov space is then defined by a tilted contour

$$\partial_t \ket{\psi(t)}=-iH\ket{\psi(t)},\qquad \ket{\psi(0)}=\ket{\psi_0},$$9

for which

$$\ket{\psi(t)}=e^{-iHt}\ket{\psi_0} =\sum_{k=0}^\infty \frac{(-iHt)^k}{k!}\ket{\psi_0},$$0

The resulting subspace

$$\ket{\psi(t)}=e^{-iHt}\ket{\psi_0} =\sum_{k=0}^\infty \frac{(-iHt)^k}{k!}\ket{\psi_0},$$1

is generated by complex-time evolved states rather than by powers of $$\ket{\psi(t)}=e^{-iHt}\ket{\psi_0} =\sum_{k=0}^\infty \frac{(-iHt)^k}{k!}\ket{\psi_0},$$2. The basis is nonorthogonal, so one constructs the Gram matrix $$\ket{\psi(t)}=e^{-iHt}\ket{\psi_0} =\sum_{k=0}^\infty \frac{(-iHt)^k}{k!}\ket{\psi_0},$$3, diagonalizes $$\ket{\psi(t)}=e^{-iHt}\ket{\psi_0} =\sum_{k=0}^\infty \frac{(-iHt)^k}{k!}\ket{\psi_0},$$4, forms $$\ket{\psi(t)}=e^{-iHt}\ket{\psi_0} =\sum_{k=0}^\infty \frac{(-iHt)^k}{k!}\ket{\psi_0},$$5, discards nonpositive eigenmodes, projects the Hamiltonian, and diagonalizes the effective Hamiltonian in the orthonormal basis. The missing tail $$\ket{\psi(t)}=e^{-iHt}\ket{\psi_0} =\sum_{k=0}^\infty \frac{(-iHt)^k}{k!}\ket{\psi_0},$$6 is then reconstructed directly on the real-frequency axis through the projected Hamiltonian and the overlaps $$\ket{\psi(t)}=e^{-iHt}\ket{\psi_0} =\sum_{k=0}^\infty \frac{(-iHt)^k}{k!}\ket{\psi_0},$$7. The method therefore does not perform analytic continuation in the conventional sense; it uses complex-time snapshots to build a low-energy subspace and evaluates the correction directly on the real axis.

Its central claim is that the Nyquist–Shannon limit is bypassed because the method does not rely solely on Fourier inversion of real-time data on $$\ket{\psi(t)}=e^{-iHt}\ket{\psi_0} =\sum_{k=0}^\infty \frac{(-iHt)^k}{k!}\ket{\psi_0},$$8. Instead it adds analytic structure through the exact decomposition $$\ket{\psi(t)}=e^{-iHt}\ket{\psi_0} =\sum_{k=0}^\infty \frac{(-iHt)^k}{k!}\ket{\psi_0},$$9, low-energy filtering through non-unitary complex-time evolution, and subspace reconstruction through a projected resolvent. In benchmarks, the critical $$\mathrm K_m=\mathrm{Span}\{\ket{\psi_0},H\ket{\psi_0},\ldots,H^{m-1}\ket{\psi_0}\}.$$0 Heisenberg chain with $$\mathrm K_m=\mathrm{Span}\{\ket{\psi_0},H\ket{\psi_0},\ldots,H^{m-1}\ket{\psi_0}\}.$$1, real-time evolution only to $$\mathrm K_m=\mathrm{Span}\{\ket{\psi_0},H\ket{\psi_0},\ldots,H^{m-1}\ket{\psi_0}\}.$$2, complex-time evolution to $$\mathrm K_m=\mathrm{Span}\{\ket{\psi_0},H\ket{\psi_0},\ldots,H^{m-1}\ket{\psi_0}\}.$$3, $$\mathrm K_m=\mathrm{Span}\{\ket{\psi_0},H\ket{\psi_0},\ldots,H^{m-1}\ket{\psi_0}\}.$$4 Krylov states, $$\mathrm K_m=\mathrm{Span}\{\ket{\psi_0},H\ket{\psi_0},\ldots,H^{m-1}\ket{\psi_0}\}.$$5, and $$\mathrm K_m=\mathrm{Span}\{\ket{\psi_0},H\ket{\psi_0},\ldots,H^{m-1}\ket{\psi_0}\}.$$6 showed that the corrected dynamical spin structure factor removes artificial oscillations and agrees extremely well with exact data around $$\mathrm K_m=\mathrm{Span}\{\ket{\psi_0},H\ket{\psi_0},\ldots,H^{m-1}\ket{\psi_0}\}.$$7. The same framework reconstructed local time-dependent Green’s functions out to times $$\mathrm K_m=\mathrm{Span}\{\ket{\psi_0},H\ket{\psi_0},\ldots,H^{m-1}\ket{\psi_0}\}.$$8 from data available only up to $$\mathrm K_m=\mathrm{Span}\{\ket{\psi_0},H\ket{\psi_0},\ldots,H^{m-1}\ket{\psi_0}\}.$$9, with absolute errors below $\{\ket{\psi_0},H\ket{\psi_0},\ldots\}$0. In the two-dimensional SSH model on an $\{\ket{\psi_0},H\ket{\psi_0},\ldots\}$1 square lattice, with real-time evolution again limited to $\{\ket{\psi_0},H\ket{\psi_0},\ldots\}$2 and complex-time evolution to $\{\ket{\psi_0},H\ket{\psi_0},\ldots\}$3, the corrected spectra resolved dressed two-electron structures and an $\{\ket{\psi_0},H\ket{\psi_0},\ldots\}$4-bipolaron signature absent in the uncorrected spectra (Paeckel, 2024).

4. Contour time, imaginary-time-like deformations, and geometric formulations

Not all work relevant to complex-time Krylov spaces studies a literal complex evolution parameter $\{\ket{\psi_0},H\ket{\psi_0},\ldots\}$5. One important line instead deforms the initial state by a real function of the Hamiltonian,

$\{\ket{\psi_0},H\ket{\psi_0},\ldots\}$6

while keeping the real-time generator equal to the original Hermitian $\{\ket{\psi_0},H\ket{\psi_0},\ldots\}$7. For $\{\ket{\psi_0},H\ket{\psi_0},\ldots\}$8, this gives coherent Gibbs states and the survival amplitude

$\{\ket{\psi_0},H\ket{\psi_0},\ldots\}$9

The paper’s main result is that such deformations leave the Krylov subspace unchanged but reorganize the Krylov basis and the Lanczos coefficients through an isospectral Lax flow

$\mathcal B_K=\{\ket{k_n}\}_{n=0}^{m-1}$0

leading to Toda equations for linear and quadratic deformations. This provides a controlled description of imaginary-time-like weighting inside a fixed Krylov subspace, but it is not a general theory of evolution in a single complex time variable (Takahashi et al., 22 Oct 2025).

A different but related contour notion appears in a Schwinger–Keldysh formulation of operator Krylov dynamics. There the relevant object is a real-time closed contour with forward and backward branches, and Krylov complexity is treated as an in-in observable generated by a Schwinger–Keldysh functional. In the semiclassical limit, the Lanczos coefficients define an effective Hamiltonian

$\mathcal B_K=\{\ket{k_n}\}_{n=0}^{m-1}$1

so asymptotically linear Lanczos growth produces hyperbolic trajectories and exponential complexity growth. This is contour-time Krylov dynamics rather than arbitrary complex-time evolution, but it identifies a natural path-integral structure for general contour extensions (Murugan et al., 2 Feb 2026).

Geometric work on operator Krylov complexity supplies another important bridge. When the Liouvillian takes the form $\mathcal B_K=\{\ket{k_n}\}_{n=0}^{m-1}$2, Liouvillian evolution is identified with a generalized coherent-state displacement operator with parameter $\mathcal B_K=\{\ket{k_n}\}_{n=0}^{m-1}$3. In $\mathcal B_K=\{\ket{k_n}\}_{n=0}^{m-1}$4, $\mathcal B_K=\{\ket{k_n}\}_{n=0}^{m-1}$5, Heisenberg–Weyl, and 2d CFT examples, operator growth is mapped to classical motion in coherent-state phase space, the trajectories are geodesics in the information metric, and Krylov complexity is proportional to an enclosed volume. This strongly suggests an analytic-continuation route $\mathcal B_K=\{\ket{k_n}\}_{n=0}^{m-1}$6, but that continuation is not carried out in the paper itself (Caputa et al., 2021).

5. Time-dependent and non-Hermitian generalizations

Time dependence introduces a separate generalization. For

$\mathcal B_K=\{\ket{k_n}\}_{n=0}^{m-1}$7

one proposal defines the time-dependent Krylov basis through the extended generator $\mathcal B_K=\{\ket{k_n}\}_{n=0}^{m-1}$8. In a broad Lie-algebraic class, however, the exact dynamics can be reduced in the interaction picture to

$\mathcal B_K=\{\ket{k_n}\}_{n=0}^{m-1}$9

where $\tau(t)=t(1-i\tan\alpha)$00 belong to an embedded rank-one algebra. The exact time-dependent Krylov amplitudes then satisfy

$\tau(t)=t(1-i\tan\alpha)$01

with representation-theoretic coefficients

$\tau(t)=t(1-i\tan\alpha)$02

The same framework admits Wei–Norman factorization, an exact single-exponential representation, and a quantum speed limit of the same functional form as in the time-independent case. The paper does not formulate a full $\tau(t)=t(1-i\tan\alpha)$03 theory, but it provides unusually explicit algebraic infrastructure for such a development (Grabarits et al., 6 May 2026).

Non-Hermitian evolution supplies another route to complex-amplitude dynamics. In the Ising chain with a complex-valued transverse magnetic field,

$\tau(t)=t(1-i\tan\alpha)$04

the paper studies normalized non-unitary evolution

$\tau(t)=t(1-i\tan\alpha)$05

expanded in an orthonormal Hilbert-space Krylov basis generated by powers of $\tau(t)=t(1-i\tan\alpha)$06. The spectrum

$\tau(t)=t(1-i\tan\alpha)$07

makes the long-time behavior depend on the imaginary spectral structure. When the imaginary part is gapped, the state asymptotically approaches the non-Hermitian Bogoliubov vacuum and the spread approaches a static value; in that gapped regime, the approach to stationarity splits into three dynamical phases distinguished by the asymptotic decay of the Krylov spread fidelity. When the imaginary spectrum is gapless, the time-averaged spread density rather than the pointwise late-time spread matches the static Bogoliubov-vacuum value. This is not a theory of complex time as a variable, but it demonstrates that ordinary Krylov constructions survive under normalized non-unitary evolution and that the relevant control parameter can be the imaginary spectral gap rather than only the real excitation structure (Medina-Guerra et al., 11 Feb 2025).

6. Polynomial filters, operator extensions, and conceptual status

A further development addresses the relative initial-state problem. For a fixed self-adjoint $\tau(t)=t(1-i\tan\alpha)$08 and reference seed $\tau(t)=t(1-i\tan\alpha)$09, a normalized polynomially filtered state

$\tau(t)=t(1-i\tan\alpha)$10

induces the transformed scalar measure

$\tau(t)=t(1-i\tan\alpha)$11

Orthogonality then yields an exact finite-band transfer from the reference Fourier–orthogonal-polynomial moments $\tau(t)=t(1-i\tan\alpha)$12 to the shifted amplitudes $\tau(t)=t(1-i\tan\alpha)$13, together with projected Christoffel–Darboux kernels for cumulative probabilities and spread complexity. The construction covers confluent roots, complex seed coefficients, support loss, and terminal quotients in finite dimension. It also extends to operator Krylov complexity after the replacement $\tau(t)=t(1-i\tan\alpha)$14 and $\tau(t)=t(1-i\tan\alpha)$15, so polynomial seeds become nested-commutator descendants $\tau(t)=t(1-i\tan\alpha)$16. This is exact for polynomial filters and provides an exact polynomial calculus for approximating filtered states such as $\tau(t)=t(1-i\tan\alpha)$17 or $\tau(t)=t(1-i\tan\alpha)$18 by truncated polynomial surrogates. The paper explicitly notes, however, that thermal filters such as $\tau(t)=t(1-i\tan\alpha)$19 generally produce infinite-band transformations rather than the finite-band connectors of the polynomial proposition (Chowdhury et al., 6 Jul 2026).

The conceptual status of complex-time Krylov spaces is therefore deliberately plural. Ordinary Lanczos should not automatically be treated as a generally optimal basis for spread minimization; complex-time Green’s-function reconstruction bypasses rather than violates the Nyquist–Shannon theorem; imaginary-time-like deformation, Schwinger–Keldysh contour time, non-Hermitian real-time propagation, and polynomially filtered seed dynamics are closely related but are not identical to a full theory of evolution at arbitrary $\tau(t)=t(1-i\tan\alpha)$20; and several issues repeatedly appear as undeveloped or future-work directions, including conditioning, orthogonality loss, non-normality, stability, contour choice, and the consequences of non-unitarity for probability-based complexity measures (Čindrak et al., 7 Mar 2026, Paeckel, 2024, Takahashi et al., 22 Oct 2025, Murugan et al., 2 Feb 2026, Medina-Guerra et al., 11 Feb 2025, Grabarits et al., 6 May 2026).