Operator dynamics in Lindbladian SYK: a Krylov complexity perspective

Using spread complexity and spread entropy, we study non-unitary quantum dynamics. For non-hermitian Hamiltonians, we extend the bi-Lanczos construction for the Krylov basis to the Schrödinger picture. Moreover, we implement an algorithm adapted to complex symmetric Hamiltonians. This reduces the computational memory requirements by half compared to the bi-Lanczos construction. We apply this construction to the one-dimensional tight-binding Hamiltonian subject to repeated measurements at fixed small time intervals, resulting in effective non-unitary dynamics. We find that the spread complexity initially grows with time, followed by an extended decay period and saturation. The choice of initial state determines the saturation value of complexity and entropy. In analogy to measurement-induced phase transitions, we consider a quench between hermitian and non-hermitian Hamiltonian evolution induced by turning on regular measurements at different frequencies. We find that as a function of the measurement frequency, the time at which the spread complexity starts growing increases. This time asymptotes to infinity when the time gap between measurements is taken to zero, indicating the onset of the quantum Zeno effect, according to which measurements impede time evolution.

Spread complexity for measurement-induced non-unitary dynamics and Zeno effect

Bhattacharya,

Das,

Dey

et al. 2024

Inflationary Krylov complexity

Li,

Liu

2024

In this work, we have systematically investigated the Krylov complexity of curvature perturbation for the modified dispersion relation in inflation, using the algorithm in closed system and open system. Since many quantum gravitational frameworks could lead to this kind of modified dispersion relation, our analysis could be applied to the string cosmology, loop gravity, e.t.c. Following the Lanczos algorithm, we find the very early universe is an infinite, many-body, and maximal chaotic system. Our numerics shows that the Lanczos coefficient and Lyapunov index of the standard dispersion relation are mainly determined by the scale factor. As for the modified case, it is nearly determined by the momentum. In a method of the closed system, we discover that the Krylov complexity will show irregular oscillation before the horizon exits. The modified case will present faster growth after the horizon exists. Since the whole universe is an open system, the approach of an open system is more realistic and reliable. Then, we construct the exact wave function which is very robust only requiring the Lanczos coefficient proportional to n (main quantum number). Based on it, we find the Krylov complexity and Krylov entropy could nicely recover in the case of a closed system under the weak dissipative approximation, in which our analysis shows that the evolution of Krylov complexity will not be the same with the original situation. We also find the inflationary period is a strong dissipative system. Meanwhile, our numerics clearly shows the Krylov complexity will grow during the whole inflationary period. But for the small scales, there will be a peak after the horizon exits. Our analysis reveals that the dramatic change in background (inflation) will significantly impact the evolution of Krylov complexity. Since the curvature perturbation will transit from the quantum level to the classical level. We could expect that the decoherence will highly impact the Krylov complexity during inflation.

Spread complexity in saddle-dominated scrambling

Huh,

Jeong,

Pedraza

2024

Recently, the concept of spread complexity, Krylov complexity for states, has been introduced as a measure of the complexity and chaoticity of quantum systems. In this paper, we study the spread complexity of the thermofield double state within integrable systems that exhibit saddle-dominated scrambling. Specifically, we focus on the Lipkin-Meshkov-Glick model and the inverted harmonic oscillator as representative examples of quantum mechanical systems featuring saddle-dominated scrambling. Applying the Lanczos algorithm, our numerical investigation reveals that the spread complexity in these systems exhibits features reminiscent of chaotic systems, displaying a distinctive ramp-peak-slope-plateau pattern. Our results indicate that, although spread complexity serves as a valuable probe, accurately diagnosing true quantum chaos generally necessitates additional physical input. We also explore the relationship between spread complexity, the spectral form factor, and the transition probability within the Krylov space. We provide analytical confirmation of our numerical results, validating the Ehrenfest theorem of complexity and identifying a distinct quadratic behavior in the early-time regime of spread complexity.