Multiple-scale integro-differential perturbation method for generic non-Markovian environments

TL;DR

This paper introduces a multiple-scale perturbation method for integro-differential equations to analyze non-Markovian effects in quantum systems, demonstrating how memory effects can accelerate quantum dynamics and aid in reservoir engineering.

Contribution

The paper presents a novel multiple-scale perturbation approach for non-Markovian quantum dynamics, enabling controllable approximations and insights into time-scale emergence and quantum speed-up.

Findings

Existence of up to two long-term and two short-term modes in regular networks.

Non-Markovianity can significantly accelerate quantum walk dynamics.

Reservoir engineering enhances quantum algorithm performance.

Abstract

Non-Markovianity may significantly speed up quantum dynamics when the system interacts strongly with an infinite large reservoir, of which the coupling spectrum should be fine-tuned. The potential benefits are evident in many dynamics schemes, especially the continuous-time quantum walk. Difficulty exists, however, in producing closed-form solutions with controllable accuracy against the complexity of memory kernels. Here, we introduce a new multiple-scale perturbation method that works on integro-differential equations for general study of memory effects in dynamical systems. We propose an open-system model in which a continuous-time quantum walk is enclosed in a non-Markovian reservoir, that naturally corresponds to an error correction algorithm scheme. By applying the multiple-scale method we show how emergence of different time scales is related to transition of system dynamics into the non-Markovian regime. We find that up to two long-term modes and two short-term modes exist in regular networks, limited by their intrinsic symmetries. In addition to the effective approximation by our perturbation method on general forms of reservoirs, the speed-up of quantum walks assisted by non-Markovianity is also confirmed, revealing the advantage of reservoir engineering in designing time-sensitive quantum algorithms.