Heisenberg–Weyl Oscillator Scales Hermite in Krylov Construction

Abhishek Chowdhury and Ajit Prasad Mahapatra of the School of Basic Sciences at the Indian Institute of Technology Bhubaneswar report a new approach to understanding Krylov complexity, a property determined by both a Hamiltonian and its initial state. Their work demonstrates that changing the initial state defines a new Krylov problem, rather than fundamentally reshaping the calculations within the Krylov construction. The researchers evaluate the construction in three canonical families: the Heisenberg, Weyl/Charlier oscillator, the SU(2)/Krawtchouk spin, and the constant-coefficient tight-binding/Chebyshev chain. A Hermite central-limit scaling of Charlier serves as a continuous-spectrum check of this process, and is not a validation of their calculations. This organization allows for efficient computation because the same reference Krylov data can be reused to survey seed deformations without repeating Lanczos in the original Hilbert space.

Research into Krylov complexity has revealed a link between seemingly disparate physical systems, showing that changing the initial state defines a new Krylov problem. The team derives “exact finite sums for individual amplitudes and projected Christoffel–Darboux kernels,” enabling precise calculations of spread complexity even with modified initial states. This approach allows for a controlled modification of the filtered spectral measure, described as |Q(E)|^{2}\mathrm{d}\mu(E)/N_{Q}, and the construction is evaluated in three canonical Jacobi families: the Heisenberg, Weyl/Charlier oscillator, the compact S​U​(2)/Krawtchouk spin, and the constant-coefficient tight-binding/Chebyshev chain. This organization allows for efficient computation; the same reference Krylov data can be reused to survey seed deformations without repeating Lanczos in the original Hilbert space. The method extends to operator Krylov complexity, where polynomial seeds become avenues for exploring Liouville space dynamics and mixed state preparations. According to the researchers, this results in an exact relative calculus in which a solved cyclic problem generates a family of polynomially related initial-state dynamics without repeating Lanczos in the original Hilbert space.

Researchers report a new approach to mapping how alterations to the starting conditions reshape the computational landscape, moving beyond treating the Hamiltonian in isolation. This refined approach necessitates recognizing that changing the initial state at a fixed Hamiltonian defines a new Krylov problem.

Their work shows that changing the initial state while keeping the Hamiltonian fixed defines a new Krylov problem, moving beyond the assumption that complexity solely resides within the Hamiltonian itself. The researchers are developing an approach allowing a solved problem to generate a family of related initial-state dynamics without repeating Lanczos in the original Hilbert space. This approach isn’t simply solving individual problems, but leveraging a broader, interconnected framework. To check this process, they employed a Hermite central-limit scaling of Charlier as a continuous-spectrum check, ensuring accuracy as calculations approach a continuous limit. They emphasize the interconnectedness of Hamiltonian and initial state. This refined approach promises to accelerate complexity calculations, particularly in many-body systems, by allowing reuse of reference Krylov data to survey seed deformations.

The ability to efficiently calculate quantum complexity is increasingly vital as researchers seek to benchmark quantum computers and understand the limits of quantum simulations. The team derives “exact finite sums for individual amplitudes and projected Christoffel–Darboux kernels,” enabling precise calculations of spread complexity even with modified initial states. The filtered spectral measure is the positive polynomial modification |Q(E)|²dμ(E)/N_Q, and orthogonality turns this measure change into a finite-band transfer from reference Fourier, orthogonal-polynomial moments to shifted Krylov amplitudes.

Their approach centers on polynomial spectral filters, where a new seed state is derived from the original via a polynomial operation on the Hamiltonian, represented as. The formulae derive exact finite sums for individual amplitudes and projected Christoffel, Darboux kernels, enabling precise calculations of spread complexity even with modified initial states.

The interplay between seemingly disparate physical systems emerges as a key feature in advanced calculations of computational complexity. The team’s work centers on organizing finite seed families by a “matrix-valued parent measure whose scalar compressions recover the individual shifted problems,” allowing for efficient computation; the same reference Krylov data can be repurposed for multiple calculations. This organization isn’t merely an abstract mathematical exercise; it’s a practical method for ensuring the accuracy of computations used to model complex quantum phenomena. As the researchers state, existing data can be reused for multiple calculations.

The pursuit of efficient methods for calculating Krylov complexity, a measure of how quickly quantum systems evolve, has led researchers to explore novel mathematical techniques for validating their calculations, particularly as systems grow in size and approach a continuous limit. A key component of this validation is a Hermite central-limit scaling of Charlier as a continuous-spectrum check of this process. This suggests a unifying framework for understanding complexity across diverse physical scenarios. This relative initial-state problem is addressed by solving for normalized polynomial filters, allowing for a precise mapping of spectral changes and a finite-band transfer of information.

Their work centers on understanding how changing the initial state impacts the resulting complexity, recognizing that altering the initial state at fixed H generally changes the Lanczos coefficients and the ordered Krylov basis. This approach introduces polynomial spectral filters, effectively modifying the initial state via operations like Q(H)|K₀⟩, allowing for controlled exploration of the system’s evolution. Calculations show that these polynomial seeds act as spectral filters, influencing how the wavefunction spreads along the Krylov chain. This is particularly useful in scenarios where researchers want to isolate initial-state dependence from changes in the Hamiltonian or Hilbert space dimensions. Extending this framework to operator Krylov complexity, the researchers replace the Hamiltonian H with the Liouvillian ℒ = [H, ⋅], effectively shifting the problem into Liouville space.

Their work, reported in a preprint dated July 6, 2026, focuses on specific initial states used in Krylov subspace methods, and reveals how manipulating these seeds impacts the resulting complexity measures. The team’s approach centers on understanding that the initial state, combined with the Hamiltonian, defines the complexity, rather than the Hamiltonian alone. They establish that changing the initial state defines a new Krylov problem, rather than fundamentally reshaping the calculations. This filtered spectral measure is the positive polynomial modification |Q(E)|²dμ(E)/N_Q, enabling a controlled shift in the calculation without entirely restarting the process. This is particularly useful because this reuse of reference Krylov data allows for surveying seed deformations without repeating Lanczos in the original Hilbert space. This refined technique promises to accelerate complexity calculations and unlock deeper insights into quantum systems.