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arXiv:1805.05111 · Physical Review A

Information-theoretic meaning of quantum information flow and its applications to amplitude amplification algorithms

Sudipto Singha Roy, Joonwoo Bae

Published Sep 3, 2019• 3 citations
arXivHTMLPDF

Abstract

The advantages of quantum information processing are in many cases obtained as consequences of quantum interactions, especially for computational tasks where two-qubit interactions are essential. In this work, we establish the framework of analyzing and quantifying loss or gain of information on a quantum system when the system interacts with its environment. We show that the information flow, the theoretical method of characterizing (non-)Markovianity of quantum dynamics, corresponds to the rate of the minimum uncertainty about the system given quantum side information. Thereafter, we analyze the information exchange among subsystems that are under the performance of quantum algorithms, in particular, the amplitude amplification algorithms where the computational process relies fully on quantum evolution. Different realizations of the algorithm are considered, such as (i) quantum circuits, (ii) analog computation, and (iii) adiabatic computation. It is shown that, in all the cases, our formalism provides insight into the process of amplifying the amplitude from the information flow or leakage on the subsystems.

DOI
10.1103/physreva.100.032303

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The advantages of quantum information processing are in many cases obtained as consequences of quantum interactions, especially for computational tasks where two-qubit interactions are essential. In this work, we establish the framework of analyzing and quantifying loss or gain of information on a quantum system when the system interacts with its environment. We show that the information flow, the theoretical method of characterizing (non-)Markovianity of quantum dynamics, corresponds to the rate of the minimum uncertainty about the system given quantum side information. Thereafter, we analyze the information exchange among subsystems that are under the performance of quantum algorithms, in particular, the amplitude amplification algorithms where the computational process relies fully on quantum evolution. Different realizations of the algorithm are considered, such as (i) quantum circuits, (ii) analog computation, and (iii) adiabatic computation. It is shown that, in all the cases, our formalism provides insight into the process of amplifying the amplitude from the information flow or leakage on the subsystems.
Exchange of information between a quantum system and its surrounding environment plays a fundamental role in the study of the dynamics of open quantum systems. Here we discuss the role of the information exchange in the non-Markovian behavior of dynamical quantum processes following the decoherence approach, where we consider a quantum system that is initially correlated with its measurement apparatus, which in turn interacts with the environment. We introduce a way of looking at the information exchange between the system and environment using the quantum loss, which is shown to be closely related to the measure of non-Markovianity based on the quantum mutual information. We also extend the results of Fanchini et al. [Phys. Rev. Lett. 112, 210402 (2014)] in several directions, providing a more detailed investigation of the use of the accessible information for quantifying the backflow of information from the environment to the system. Moreover, we reveal a clear conceptual relation between the entanglement- and mutual-information-based measures of non-Markovianity in terms of the quantum loss and accessible information. We compare different ways of studying the information flow in two theoretical examples. We also present experimental results on the investigation of the quantum loss and accessible information for a two-level system undergoing a zero temperature amplitude damping process. We use an optical approach that allows full access to the state of the environment.
This paper explores the application of information theory to analyze the computational complexity of quantum algorithms. Traditional approaches to complexity analysis often rely on circuit models and gate complexity, which can be insufficient for fully characterizing the resource requirements of quantum computation. We propose utilizing information-theoretic concepts, specifically quantum entropy and mutual information, to provide a more nuanced understanding of how quantum circuits process information. This approach allows us to quantify the information-theoretic limits of quantum algorithms, providing insights into the number of qubits and gates necessary for efficient execution. We examine several key aspects, including the role of entanglement, the impact of measurement, and the relationship between quantum entropy and algorithmic complexity. The goal is to establish a framework for assessing quantum algorithms based on their fundamental information-processing capabilities, moving beyond purely circuit-based evaluations. This work contributes to a deeper theoretical understanding of quantum computation and its potential limitations.
This paper investigates the application of quantum information entropy as a novel approach to analyzing algorithm complexity. Traditional methods often rely on classical computational models and may not fully capture the resource demands inherent in quantum algorithms. We propose a framework where algorithms are translated into quantum information processing steps, and quantum information entropy is utilized to quantify the complexity of these processes. The core claim is that this approach offers a more accurate assessment of computational resource requirements, ultimately guiding the design and optimization of algorithms. We present a theoretical framework and explore potential applications, highlighting the unique insights gained through the lens of quantum information entropy. The analysis focuses on identifying key metrics related to entanglement, coherence, and quantum state manipulation, which directly influence the entropy values. We demonstrate how variations in these metrics can significantly impact the overall complexity of an algorithm. Furthermore, we discuss the limitations of this approach and suggest future research directions.
Quantum memory effects can be qualitatively understood as a consequence of an environment-to-system backflow of information. Here, we analyze and compare how this concept is interpreted and implemented in different approaches to quantum non-Markovianity. We study a nonoperational approach, defined by the distinguishability between two system states characterized by different initial conditions, and an operational approach, which is defined by the correlation between different outcomes associated to successive measurement processes performed over the system of interest. The differences, limitations, and vantages of each approach are characterized in detail by considering diverse system-environment models and dynamics. As a specific example, we study a non-Markovian depolarizing map induced by the interaction of the system of interest with an environment characterized by incoherent and coherent self-dynamics.
Mixing dynamical maps describing open quantum systems can lead from Markovian to non-Markovian processes. Being surprising and counter-intuitive, this result has been used as argument against characterization of non-Markovianity in terms of information exchange. Here, we demonstrate that, quite the contrary, mixing can be understood in a natural way which is fully consistent with existing theories of memory effects. In particular, we show how mixing-induced non-Markovianity can be interpreted in terms of the distinguishability of quantum states, system-environment correlations and the information flow between system and environment.
Quantum information science has altered the concept of the method of processing and transporting information through the laws of quantum mechanics. Quantum entanglement is one of its basic properties, which is instrumental to non-classical correlations between spatially separate particles. New quantum communication networks and quantum computing systems distributed are based on these interrelations. In this paper, quantum information flow dynamics of multi-particle entangled networks with particular emphasis to the processes governing information transmission, redistribution and maintenance of coupled quantum nodes are discussed. The paper discloses the way in which the entangled structures comprising multiple entities can help in facilitating the sharing of information beyond the capabilities of the classical communication systems. Complex entangled structures are analyzed in terms of the channels through which the flow of quantum information is analyzed in respect to the theoretical models of multi-qubit systems and networked quantum channels. The way the communication between the particles influences the coherence, strength of correlation, as well as the stability of the whole network in general, is given particular attention. The effect of environmental noise and decoherence which can cause a break of entanglement and the loss of efficiency in quantum information exchange is also considered in the analysis. Moreover, the article discusses the application of network topology in order to establish the behaviour of entangled systems. The impact of the structural changes on information transfer reliability and speed is determined by the evaluation of the various forms of quantum nodes and connections. These findings suggest that entangled networks structured appropriately will be able to enhance stability of quantum communication and assist in distributing quantum resources more efficiently. Overall, the present research can be deemed as part of the bigger initiative of scaling up quantum communication infrastructures. The research builds knowledge in the sense that it provides an insight into the behavior of information in multi-particle entangled networks with important considerations being made as to the advancement of robust quantum networks which can be used in future to support the realization of future applications in the area of secure communication, distributed computing and the future developments in information processing technologies.