Exact Model Reduction for Continuous-Tim

Exact Model Reduction for Continuous-Time Open Quantum Dynamics – Quantum
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Exact Model Reduction for Continuous-Time Open Quantum Dynamics
Abstract
We consider finite-dimensional many-body quantum systems described by time-independent Hamiltonians and Markovian master equations, and present a systematic method for constructing smaller-dimensional, reduced models that $exactly$ reproduce the time evolution of a set of initial conditions or observables of interest. Our approach exploits Krylov operator spaces and their extension to operator algebras, and may be used to obtain reduced linear models of minimal dimension, well-suited for simulation on classical computers, or reduced quantum models that preserve the structural constraints of physically admissible quantum dynamics, as required for simulation on quantum computers. Notably, we prove that the reduced quantum-dynamical generator is still in Lindblad form. By introducing a new type of $\textit{observable-dependent symmetries}$, we show that our method provides a non-trivial generalization of techniques that leverage symmetries, unlocking new reduction opportunities. We quantitatively benchmark our method on paradigmatic open many-body systems of relevance to condensed-matter and quantum-information physics. In particular, we demonstrate how our reduced models can quantitatively describe decoherence dynamics in central-spin systems coupled to structured environments, magnetization transport in boundary-driven dissipative spin chains, and unwanted error dynamics on information encoded in a noiseless quantum code.
Popular summary
Because simulating the dynamics of quantum systems is notoriously hard, it is imperative that the available resources are used as efficiently as possible, by focusing on computing only quantities of interest. Here, we focus on Markovian open quantum systems evolving under a Lindblad master equation, and propose a model reduction procedure for finding a smaller model that reproduces exactly the expectation values of a set of observables, for a given set of initial conditions. Our procedure guarantees that, when such a reduced model exists, it retains the fundamental quantum requirements of complete positivity and trace preservation.
Our core idea is to construct operator subspaces that contain the trajectory of states (evolved in Schrodinger picture), or observables (evolved in Heisenberg picture). These spaces allow us to find the minimal amount of resources needed to reproduce the trajectories exactly. We then enlarge these operator subspaces to associative algebras, the natural structure for defining a quantum probability space, and prove that the resulting reduced model is still in Lindblad form. We show that our approach genuinely extends techniques that exploit symmetries, by introducing a novel notion of observable-dependent symmetry. We further test our methodology on prototypical models motivated by condensed-matter and quantum-information physics.
In many practical cases, exact model reduction may be too stringent a requirement. Our work represents a stepping stone towards approximate quantum model reduction that preserves the properties of complete positivity and conservation of total probability. Other future directions include extensions to parametrized families of Lindblad models and infinite-dimensional settings.
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