Stochastic Krylov Dynamics: Revisiting Operator Growth in Open Quantum Systems
Arpan Bhattacharyya, S. Shajidul Haque, Jeff Murugan, Mpho Tladi, Hendrik J. R. Van Zyl
TLDR
This paper shows operator growth in open quantum systems becomes stochastic, driven by environmental coupling and diffusion, altering Krylov complexity's geometric picture.
Key contributions
- Extends Krylov complexity's geometric description to open quantum systems.
- Derives an effective action for operator growth under Lindblad dynamics using Schwinger-Keldysh.
- Demonstrates environmental coupling introduces diffusion, making operator growth stochastic.
- Identifies dissipation as a relevant perturbation to the chaotic Krylov fixed point.
Why it matters
This work fundamentally redefines operator growth in open quantum systems, showing it as a stochastic process rather than deterministic. It's crucial for understanding complexity in realistic quantum systems interacting with their environment, impacting quantum computing and black hole physics.
Original Abstract
In closed quantum systems, Krylov complexity admits a geometric description; operator growth is equivalent to Hamiltonian flow in an emergent phase space whose structure is fixed by the Lanczos coefficients. We show that this picture survives, albeit in a fundamentally altered form, once the system is coupled to an environment.Using a Schwinger-Keldysh formulation of the full counting statistics of the Krylov position, we derive an effective action for operator growth under Lindblad dynamics. Even for the minimal case of dephasing, the phase-space dynamics ceases to be Hamiltonian; environmental coupling generates diffusion in the variable conjugate to Krylov depth, converting deterministic trajectories in to stochastic ones. The hyperbolic mechanism underlying exponential complexity growth is therefore broadened and, beyond a parametrically controlled scale, destroyed.This identifies dissipation as a relevant perturbation of the chaotic Krylov fixed point and reveals operator growth in open systems as a problem of stochastic dynamics in an emergent phase space.
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