On Krylov complexity in open systems: an approach via bi-Lanczos algorithm

Abstract

Continuing the previous initiatives [1, 2], we pursue the exploration of operator growth and Krylov complexity in dissipative open quantum systems. In this paper, we resort to the bi-Lanczos algorithm generating two bi-orthogonal Krylov spaces, which individually generate non-orthogonal subspaces. Unlike the previously studied Arnoldi iteration, this algorithm renders the Lindbladian into a purely tridiagonal form, thus opening up a possibility to study a wide class of dissipative integrable and non-integrable systems by computing Krylov complexity at late times. Our study relies on two specific systems, the dissipative transverse-field Ising model (TFIM) and the dissipative interacting XXZ chain. We find that, for the weak coupling, initial Lanczos coefficients can efficiently distinguish integrable and chaotic evolution before the dissipative effect sets in, which results in more fluctuations in higher Lanczos coefficients. This results in the equal saturation of late-time complexity for both integrable and chaotic cases, making the notion of late-time chaos dubious.

Mixed Riccati–Lyapunov Balanced Truncation for Order Reduction of Electrical Circuit Systems

“…Positive-real balanced truncation (PRBT), also known as positive-real Riccati balance truncation, is specifically tailored for passive systems, as outlined in Note 1. In this approach, the controllability Gramian matrix R c and the observability Gramian matrix R o are calculated using two positive-real Riccati Equations ( 6) and (7) or satisfy Lemma 1 (Lemma KYP) [32,33]. The sequence of steps to implement the PRBT algorithm is presented in detail in [18,25,26].…”

“…The Gramians RL c and RL o are diagonal, symmetric matrices that are positive definite and satisfy Equations ( 12), (7), and Lemma 1. Therefore, the MRLBS demonstrates stability and passivity.…”

“…Since M 1 is a diagonal, positive definite, symmetric matrix, satisfying Equations ( 12), (7), and Lemma 1, the reduced-order system in the output of Algorithm 2 preserves the stability and passivity of the original LTI-CTD system (1). □ Corollary 2.…”

“…To reduce the order of electrical and electronic circuit systems, there are several groups of methods and algorithms to mention, such as the Krylov subspace methods group [5]; techniques related to the iterative rational Krylov algorithm (IRKA) such as Asymptotic Waveform Evaluation (AWE) [6], Lanczos and the Arnold-algorithm [7], Padé approximation [8], etc. ; the Proper Orthogonal Decomposition (POD) [9] group of methods and similar algorithms such as Principal Component Analysis (PCA) [10], etc.…”

Krylov complexity in a natural basis for the Schrödinger algebra

“…We believe that there are two main reasons. First, Krylov complexity can be applied to any quantum system making it computationally available, at least in principle, for a plethora of different cases including but not limited to condensed matter and many-body systems [13][14][15][16][17], quantum and conformal field theories [3][4][5][18][19][20], open systems [21][22][23][24][25], topological phases of matter [26,27] and many other topics related to aspects of the above and not only [28][29][30][31]. Second, it is related by its construction to inherent properties and characteristic parameters of the system, namely the Hamiltonian and the Hilbert space that it defines.…”