Researchers have developed an exact method to calculate Krylov complexity, also known as spread complexity, for quantum states that are polynomial transformations of an initial state. This complexity is a joint property of the system's Hamiltonian and the initial state, and its calculation typically requires generating a new Krylov basis for each state. The new approach allows for determining how complexity changes when the initial state is modified by a polynomial filter, without the need to repeat the costly Lanczos process in the original Hilbert space.

The method addresses the relative initial-state problem for normalized polynomial filters of the form $\ket{\psi_Q}=Q(H)\ket{K_0}/\sqrt{N_Q}$, where $Q(H)$ is a polynomial of the Hamiltonian $H$ and $\ket{K_0}$ is the initial state. The key lies in how polynomial filtering modifies the spectral measure, transforming the problem into a finite-band transfer from reference Fourier-orthogonal-polynomial moments to shifted Krylov amplitudes. They have derived exact finite sums for individual amplitudes and projected Christoffel-Darboux kernels for cumulative probabilities and spread complexity.

The developed formulas are robust, covering cases such as confluent roots, complex seed coefficients, support loss, and terminal quotients in finite dimensions. The team validated their construction in three canonical Jacobi families: the Heisenberg-Weyl/Charlier oscillator, the compact SU(2)/Krawtchouk spin, and the constant-coefficient tight-binding/Chebyshev chain. A Hermite central-limit scaling of Charlier was also included to check the Christoffel jump machinery in a continuous spectrum. This framework provides an exact relative calculus, generating a family of polynomially related initial-state dynamics from a single solved cyclic problem.

This advance is significant for the study of quantum dynamics and information in complex systems, including black holes and many-body models, where Krylov complexity is an important metric for characterizing information growth. The ability to efficiently calculate complexity for a family of related states simplifies the analysis of how perturbations or transformations affect the evolution of complexity, opening new avenues for understanding thermalization and black hole formation in the context of string theory and quantum gravity.