Towards Verifications of Krylov Complexity

• Published in: Progress of theoretical and experimental physics Vol. 2024; no. 6
• Main Author: Sasaki, Ryu
• Format: Journal Article
• Language: English
• Published: Oxford Oxford University Press 01.06.2024
• Subjects: Quantum physics
• ISSN: 2050-3911; 2050-3911
• DOI: 10.1093/ptep/ptae073

Krylov complexity is considered to provide a measure of the growth of operators evolving under Hamiltonian dynamics. The main strategy is the analysis of the structure of Krylov subspace $\mathcal {K}_M(\mathcal {H},\eta )$ spanned by the multiple applications of the Liouville operator $\mathcal {L}$ defined by the commutator in terms of a Hamiltonian $\mathcal {H}$, $\mathcal {L}:=[\mathcal {H},\cdot ]$ acting on an operator η, $\mathcal {K}_M(\mathcal {H},\eta )=\text{span}\lbrace \eta ,\mathcal {L}\eta ,\ldots ,\mathcal {L}^{M-1}\eta \rbrace$. For a given inner product (·, ·) of the operators, the orthonormal basis $\lbrace \mathcal {O}_n\rbrace$ is constructed from $\mathcal {O}_0=\eta /\sqrt{(\eta ,\eta )}$ by the Lanczos algorithm. The moments $\mu _m=(\mathcal {O}_0,\mathcal {L}^m\mathcal {O}_0)$ are closely related to the important data {bn}, called Lanczos coefficients. I present exact and explicit expressions of the moments {μm} for 16 quantum mechanical systems that are exactly solvable in both the Schrödinger and Heisenberg pictures. The operator η is the variable of the eigenpolynomials. Among them, six systems show a clear sign of “noncomplexity” with vanishing higher Lanczos coefficients bm = 0, m ≥ 3.