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Shortcuts to Adiabaticity for non-Hermitian systems in Krylov Space

T0 review · 1 major / 5 minor · reviewed 2026-07-10 · grok-4.5

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Pith's one-line read A Krylov-space expansion turns the adiabatic gauge potential of non-Hermitian systems into a sparse matrix equation, so counterdiabatic drives can be built without diagonalization and with controlled locality.

desk verdict Clean non-Hermitian generalization of Krylov AGP that works and is usable; main limit is the weak-non-Hermiticity regime they already flag. read the letter →

arxiv 2607.07802 v1 pith:W6LJG5YA submitted 2026-07-08 quant-ph cond-mat.othercond-mat.stat-mech

Ankit W. Shrestha , Budhaditya Bhattacharjee , Adolfo del Campo This is my paper · ORCID
classification quant-phcond-mat.othercond-mat.stat-mech
keywords shortcutstoadiabaticityadiabaticgaugepotentialnon-HermitiansystemsKrylovspacebi-LanczosalgorithmArnoldiiterationPTsymmetrycounterdiabaticdriving
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Shortcuts to adiabaticity let a quantum system follow an adiabatic path in finite time by adding a counterdiabatic drive. That drive is built from the adiabatic gauge potential, which is hard to compute for many-body systems and had been available for non-Hermitian Hamiltonians only in model-specific form. This paper supplies a general construction: an integral representation of the gauge potential is rewritten as a nested-commutator series, then projected onto the bi-Lanczos or Arnoldi Krylov basis generated from the driving term. The result is a sparse tridiagonal or upper-Hessenberg matrix equation whose solution is the exact or truncated gauge potential. On a decaying two-level atom the method recovers the known exact drive and diverges at the exceptional point; on the interacting Hatano–Nelson chain a few dozen Krylov terms already suppress excess energy; on a PT-symmetric Heisenberg chain the gauge-potential norm itself tracks the PT-breaking transition. Because only a small fraction of the full Krylov space is needed, the route is practical for large non-Hermitian many-body systems where full diagonalization is impossible.

What carries the argument

The bi-Lanczos/Arnoldi Krylov basis of the Liouvillian superoperator Lλ = [Hλ, ·] acting on ∂λH. Only the odd-indexed vectors appear, so the gauge potential is a short linear combination whose coefficients solve a sparse matrix equation (tridiagonal for bi-Lanczos, upper-Hessenberg for Arnoldi).

What would settle it

Apply a truncated Krylov counterdiabatic drive to the interacting Hatano–Nelson model (or the PT Heisenberg chain) and measure residual excess energy or final-state fidelity; if the residual fails to drop monotonically with Krylov order or remains large far from exceptional points, the claim that a small Krylov fraction already yields accurate control is falsified.

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Extended reading notes

Core claim

The adiabatic gauge potential of a non-Hermitian Hamiltonian admits an integral representation that expands into a nested-commutator series; when that series is written in the bi-Lanczos or Arnoldi Krylov basis generated from ∂λH, the exact or truncated gauge potential reduces to the solution of a sparse tridiagonal or upper-Hessenberg matrix equation that generalizes the Hermitian Krylov construction.

Load-bearing premise

The construction assumes weak non-Hermiticity: the imaginary parts of energy differences do not produce large exponential growth, so the biorthogonal counterdiabatic term remains valid.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

1 major / 5 minor

Summary. The manuscript develops a diagonalization-free framework for counterdiabatic shortcuts to adiabaticity in non-Hermitian systems by expressing the adiabatic gauge potential (AGP) in Krylov space. Starting from an integral representation of the AGP (Eq. 21, App. A), the authors recast it as a nested-commutator series and generate the basis via bi-Lanczos and Arnoldi algorithms, reducing the problem to sparse tridiagonal or upper-Hessenberg matrix equations (Eqs. 29, 32) that generalize the Hermitian construction. Truncated expansions are shown to converge rapidly. Demonstrations include recovery of the exact drive and exceptional-point divergence for a decaying two-level atom, suppression of excess energy in the interacting Hatano–Nelson model with few Krylov vectors, detection of the PT-breaking transition via AGP norm in a Heisenberg chain, and closed-form AGP results for the non-Hermitian transverse-field Ising model that capture both Ising and PT transitions.

Significance. If the results hold, the work supplies a practical, systematically improvable route to counterdiabatic control of many-body non-Hermitian systems, where exact diagonalization is prohibitive and prior STA constructions were largely model-specific. The reduction to sparse matrix equations, the controlled locality of nested commutators, and the rapid convergence with a small fraction of the Krylov space are concrete advances over spectral formulas. The AGP-norm diagnostic of PT transitions and exceptional points, the exact NH-TFIM solution, recovery of the known two-level result, and the publicly available simulation codes are particular strengths that make the contribution both theoretically clean and usable.

major comments (1)
  1. The central construction (biorthogonal CD term Eq. 18, integral representation Eq. 21, and the subsequent Krylov matrix equations) rests on the weak-non-Hermiticity condition Eq. 17 (§II.A). The paper is explicit that the usual adiabatic condition is insufficient for strongly complex spectra and that the AGP diverges at exceptional points (recovered for the two-level atom). All many-body examples stay inside the safe regime (real spectrum for open-boundary Hatano–Nelson; real spectrum in the PT-unbroken phase). This is a legitimate scope restriction rather than an internal inconsistency, but the abstract and conclusion currently advertise a general non-Hermitian framework without a crisp statement of the domain of validity. A short, prominent paragraph delimiting when the integral/Krylov equations guarantee transitionless driving (and what fails near strong non-Hermiticity or singular bi
minor comments (5)
  1. Figs. 1–2: axis labels and legends are readable, but the caption of Fig. 2 should state more clearly that the plotted points are the absolute excess energy evaluated at the oscillation maxima, and should quote the system parameters (N, J, U, h0) already given for Fig. 1.
  2. Notation: the same symbol L is used for the Liouvillian superoperator and for system size in the NH-TFIM section; a brief local redefinition or a different letter for one of them would reduce momentary confusion.
  3. Eq. (29) and the surrounding text: the statement that c_{2d_A} and b_{2d_A} “do not exist and can be taken as zero” for even K is correct but terse; a one-sentence reminder that the last row/column is simply truncated would help readers implementing the matrix.
  4. References: the recent experimental non-Hermitian STA demonstration in a superconducting qubit (Erdamar et al., PRX Quantum 2026) is already cited; a short sentence in the introduction noting that the present Krylov construction is complementary to that single-qubit experiment would improve context.
  5. Typographical: “Schr¨ odinger” and similar accented characters appear inconsistently rendered in a few places; a final pass for UTF-8 consistency is recommended.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: AGP is defined from the biorthogonal spectral/integral formula and re-expressed in Krylov space; matrix equations for α_k are solved from the constraint, not fitted to dynamics.

full rationale

The derivation chain is self-contained. The AGP begins from the standard biorthogonal definition (Eqs. 18–20) or the derived integral representation (Eq. 21, Appendix A), is rewritten as the nested-commutator series (Eq. 24) under a spectral-gap assumption, and is then expanded in the bi-Lanczos/Arnoldi basis generated from ∂_λH. The coefficients α_k are obtained by solving the sparse matrix equations (29) or (32) that enforce the constraint L[i∂_λH + L A] = 0; they are not fitted to any target trajectory or excess-energy data. Truncation (Eq. 33) and the AGP-norm diagnostic (Eqs. 58–60) are derived observables. Self-citations to the authors’ prior Hermitian Krylov constructions ([17,18]) supply the starting point that is generalized, but the non-Hermitian bi-orthogonal coefficients {c_n}, the integral representation, the variational consistency check, and the closed-form NH-TFIM solution are derived independently inside the paper. Numerical checks (excess-energy suppression, recovery of the two-level exact drive, detection of PT/EP features) are external validations, not inputs. The only mild self-reference is the natural extension of the Hermitian Krylov method; it is not load-bearing for the new claims. Score 1 reflects that minor, non-circular self-citation.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The construction rests on standard linear-algebra and non-Hermitian spectral theory plus the weak-non-Hermiticity assumption that keeps the biorthogonal CD term valid. No free parameters are fitted to data; truncation order M is a controlled approximation parameter, not a fit. No new physical entities are postulated.

free parameters (1)
  • Krylov truncation order M = model-dependent (e.g. 80, 50)
    Chosen by hand to demonstrate convergence (e.g., M=80 for Hatano–Nelson, d_A=50 for NH-TFIM); not fitted to data but controls the accuracy of the approximate AGP.
assumptions (4)
  • domain assumption Weak non-Hermiticity: exp(∫ Im(E_n−E_m) dt′) ≈ 1 so that the biorthogonal counterdiabatic term remains valid (Eq. 17).
    Invoked in §II.A to justify using the Ibáñez et al. form of H_CD; fails near exceptional points or for strong non-Hermiticity.
  • domain assumption Existence of a complete biorthogonal eigenbasis away from exceptional points (geometric = algebraic multiplicity).
    Standard non-Hermitian spectral theory; used throughout the integral representation and matrix-element formulae.
  • standard math Hilbert–Schmidt (or biorthogonal) inner product on operators and the associated Frobenius norm for the AGP.
    Defines the Krylov orthonormalization and the AGP-norm diagnostic (Eqs. 58–60).
  • standard math Baker–Campbell–Hausdorff expansion of the integral representation yields only odd nested commutators when a spectral gap is present.
    Used to justify the series (Eq. 24) and the restriction to odd Krylov vectors.

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Cite this review

Pith. "Pith review of Shortcuts to Adiabaticity for non-Hermitian systems in Krylov Space." pith.science (2026). https://pith.science/paper/W6LJG5YA

@misc{pith2026260707802,
  author       = {Pith},
  title        = {Pith review of: Shortcuts to Adiabaticity for non-Hermitian systems in Krylov Space},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/W6LJG5YA}},
  note         = {Machine review of arXiv:2607.07802}
}
read the original abstract

Shortcuts to adiabaticity (STA) reproduce adiabatic dynamics in finite time, but their counterdiabatic implementation relies on the adiabatic gauge potential (AGP), which is difficult to compute and implement in many-body systems and whose extension to open and non-Hermitian settings has remained largely model-specific. Here, we develop a general, diagonalization-free framework for engineering STA in non-Hermitian systems by representing the AGP in Krylov space. Starting from an integral representation of the counterdiabatic control, we recast the AGP as a nested-commutator series with controlled locality and generate the associated Krylov basis using the bi-Lanczos and Arnoldi algorithms. This reduces the exact or truncated AGP to a sparse tridiagonal or upper-Hessenberg matrix equation that generalizes the Hermitian construction. We demonstrate the method on a decaying two-level atom, where it recovers the exact drive and signals the exceptional point; on the interacting Hatano-Nelson model, where truncated controls rapidly suppress nonadiabatic excitations; and on a PT-symmetric Heisenberg chain, whose AGP norm detects the PT-symmetry-breaking transition. Throughout, the expansion converges with only a small fraction of the full Krylov space, offering a practical route to fast, accurate control of many-body non-Hermitian systems.

Figures

Figures reproduced from arXiv: 2607.07802 by the authors.

Figure 1
Figure 1. FIG. 1. Excess energy [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Absolute value of the excess energy [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. shows the AGP norm (Eq. (60)) for differ￾ent approximate AGPs constructed using first M odd￾indexed Krylov basis vectors. For the calculation, we fix χ ≈ 10−7 so that the Hamiltonian is non-Hermitian, but the spectrum is real up to the machine precision. We can see that it rapidly approaches the exact AGP norm in Eq. (58). Moreover, it also captures the phase transition at ξ = 1/2 [PITH_FULL_IMAGE:figures/full_fig_… view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Norm of the adiabatic gauge potential for the Non [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]

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Forward citations

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Reference graph

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