Quantum 2025 – wissenschaftliches Programm

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THU: Thursday Contributed Sessions

THU 7: Entanglement and Complexity: Contributed Session to Symposium II

THU 7.5: Vortrag

Donnerstag, 11. September 2025, 15:15–15:30, ZHG008

Measurable Krylov spaces and eigenenergy count in quantum state dynamics — •Saud Čindrak, Lina Jaurigue, and Kathy Lüdge — Technische Universität Ilmenau, Ilmenau, Germany

Krylov complexity is defined on the Krylov space, which consists of the powers of the Hamiltonian acting on the initial state. We prove that an equivalent space can be constructed by taking time-evolved states as a basis, which is also quantum-mechanically measurable. The Krylov complexities computed with respect to both spaces exhibit almost identical behavior, thus enabling the use of Krylov complexity for systems where the Hamiltonian is unknown or in experimental settings. This is particularly relevant for quantum machine learning, where the system is described by unitaries and the Hamiltonian is not explicitly known. We then use this newly defined Krylov space to introduce the effective dimension, which captures the extent to which the state has evolved in the Krylov basis. This measure is upper-bounded by the number of pairwise distinct eigenvalues of the Hamiltonian, thereby providing a method to experimentally determine the number of eigenenergies.

[1] S. Čindrak, L. Jaurigue, K.Lüdge, J. High Energ. Phys 2024, 83

Keywords: Krylov complexity; Quantum expressivity; Quantum machine learning