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arXiv:2204.06936

Description and complexity of Non-markovian open quantum dynamics

Rahul Trivedi

Published Apr 14, 2022• 0 citations
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Abstract

Understanding and simulating non-Markovian quantum dynamics remains an important challenge in open quantum system theory. A key advance in this endeavour would be to develop a unified mathematical description of non-Markovian dynamics, and classify its complexity in the many-body setting. In this paper, we identify a general class of non-Markovian memory kernels, described by complex-valued radon measures, and define their dynamics through a regularization procedure constructing the corresponding system-environment unitary groups. Building on this definition, we then consider $k-$local many-body non-Markovian systems with physically motivated assumptions on the total variation and smoothness of the memory kernels. We establish that their dynamics can be efficiently approximated on quantum computers, thus providing a rigorous verification of the Extended Church-Turing thesis for this general class of non-Markovian open quantum systems.

DOI
10.48550/arxiv.2204.06936

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Understanding and simulating non-Markovian quantum dynamics remains an important challenge in open quantum system theory. A key advance in this endeavour would be to develop a unified mathematical description of non-Markovian dynamics, and classify its complexity in the many-body setting. In this paper, we identify a general class of non-Markovian memory kernels, described by complex-valued radon measures, and define their dynamics through a regularization procedure constructing the corresponding system-environment unitary groups. Building on this definition, we then consider k−k-k−local many-body non-Markovian systems with physically motivated assumptions on the total variation and smoothness of the memory kernels. We establish that their dynamics can be efficiently approximated on quantum computers, thus providing a rigorous verification of the Extended Church-Turing thesis for this general class of non-Markovian open quantum systems.
The basic features of the dynamics of open quantum systems, such as the dissipation of energy, the decay of coherences, the relaxation to an equilibrium or non-equilibrium stationary state, and the transport of excitations in complex structures are of central importance in many applications of quantum mechanics. The theoretical description, analysis and control of non-Markovian quantum processes play an important role in this context. While in a Markovian process an open system irretrievably loses information to its surroundings, non-Markovian processes feature a flow of information from the environment back to the open system, which implies the presence of memory effects and represents the key property of non-Markovian quantum behavior. Here, we review recent ideas developing a general mathematical definition for non-Markoviantiy in the quantum regime and a measure for the degree of memory effects in the dynamics of open systems which are based on the exchange of information between system and environment. We further study the dynamical effects induced by the presence of system-environment correlations in the total initial state and design suitable methods to detect such correlations through local measurements on the open system.
Markovian approximation is a widely-employed idea in the descriptions of the dynamics of open quantum systems. Although it is usually claimed to be a concept inspired by classical Markovianity, the physical meaning of this concept, however, is vague. In this thesis, I compare the descriptions of classical stochastic processes and open quantum systems, and show that those inherent differences lead to the non-trivial problem of characterizing quantum non-Markovianity. With various examples, I argue that the current most often used definitions of quantumMarkovianity in the literature do not fully capture the memoryless property of OQSs. To address this issue, I take a different approach by studying a host of Markov-related concepts in the quantum regime. Some of these concepts have long been used in quantum theory, while others are first proposed in this thesis. I define all these concepts under a unified framework, which allows one to rigorously build hierarchy relations among them. In this sense, quantum non-Markovianity is highly context-dependent. The results in this thesis, summarized as a hierarchy figure, are expected to bring clarity to the nature of quantumnon-Markovianity.
The reduced description of the quantum dynamic processes in the condensed phase environment leads to the equation of motion with a memory kernel. Such a memory effect, termed non-Markovianity, presents more complex dynamics compared to its memoryless or Markovian counterpart, and many chemical systems have been demonstrated through numerical simulations to exhibit non-Markovian quantum dynamics. Explicitly how the memory impacts the dynamic process remains largely unexplored. In this work, we focus on ways to separate the non-Markovian contributions from the dynamics and study the non-Markovian effects. Specifically, we developed a rigorous procedure for mapping the exact non-Markovian quantum propagator to the Lindblad form. Consequently, it allows us to extract the negative decay rate from the Lindbladian that is the signature of the non-Markovianity. By including or excluding the negative rate in the time evolution, we can decisively pinpoint the influence of non-Markovianity on the system's properties such as coherence, entanglement, and equilibrium state distribution. The understanding of such memory effects on the dynamic process suggests the possibility of leveraging non-Markovianity for quantum control.
Existing quantum algorithms for simulating dynamical systems -- from Hamiltonian simulation to linear and nonlinear differential equations solvers -- simulate Markovian dynamics, in which the system's future evolution depends solely on its current state. We turn our attention to developing quantum algorithms for non-Markovian dynamical systems where the system's future evolution depends on its past history and thus has memory. Specifically, we develop efficient algorithms for linear Volterra integro-differential equations (VIDEs) with a convolution memory kernel that output a quantum state encoding the state description over a time interval or at a particular time. Given efficient circuits for the problem inputs, our algorithms achieve an exponential speedup in system size over existing classical algorithms. We develop an algorithm for general kernels assuming that M<1\textsf{M} < 1M<1, where M\textsf{M}M characterizes the strength of the memory term relative to the dissipation of the Markovian part of the dynamics. We complement this with lower bounds for general-kernel VIDEs when M≥1\textsf{M} \geq 1M≥1, showing that the problem becomes intractable for a family of systems. However, by specializing to structured kernels which admit concise decompositions over exponentials, we develop efficient quantum algorithms even when M≥1\textsf M \geq 1M≥1 by converting the VIDE into a larger set of ODEs, a procedure which we call Markovianization. As an application of the overall framework, we discuss the Mori-Zwanzig formalism used in open quantum systems and fluid dynamics. Overall, our results expand the range of dynamical systems that quantum computers can simulate efficiently.
A non-Markovian process is one that retains `memory' of its past. A systematic understanding of these processes is necessary to fully describe and harness a vast range of complex phenomena; however, no such general characterisation currently exists. This long-standing problem has hindered advances in understanding physical, chemical and biological processes, where often dubious theoretical assumptions are made to render a dynamical description tractable. Moreover, the methods currently available to treat non-Markovian quantum dynamics are plagued with unphysical results, like non-positive dynamics. Here we develop an operational framework to characterise arbitrary non-Markovian quantum processes. We demonstrate the universality of our framework and how the characterisation can be rendered efficient, before formulating a necessary and sufficient condition for quantum Markov processes. Finally, we stress how our framework enables the actual systematic analysis of non-Markovian processes, the understanding of their typicality, and the development of new master equations for the effective description of memory-bearing open-system evolution.
In the first part of this thesis we extensively review the general theory of open quantum systems. We introduce new approaches to describe, in a hopefully more transparent way, some aspects which are usually misleading in a broad range of the literature. In addition, we revise new concepts and methods developed in recent years that complement the general theory mainly developed in the 1970s. In the second part, we focus our attention on the so-called Markovian processes which have been described in the previous part. We study the validity of typical equations used in simple systems such as harmonic oscillators, and propose and check different strategies to describe Markovian dynamics in more complicated interacting systems. Finally we propose a counter-intuitive situation where some level of noise can be actually useful for information tasks. In the third and last part, we approach the problem of the non-Markovian description of open quantum systems. To quantify how non-Markovian an open quantum system is we introduce a computable measure and apply it to some relevant situations. We also propose a way of probing the complexity of an environment by visualizing properties of a pair of qubits embedded in it. In addition, we develop a technique based on orthogonal polynomial theory to transform a wide range of environments into simple chains of particles which can be efficiently simulated with numerical techniques.