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arXiv:1007.1089

Quantum memories and Landauer's principle

Robert Alicki

Published Oct 1, 2010• 1 citation
arXivHTMLPDF

Abstract

Two types of arguments concerning (im)possibility of constructing a scalable, exponentially stable quantum memory equipped with Hamiltonian controls are discussed. The first type concerns ergodic properties of open Kitaev models which are considered as promising candidates for such memories. It is shown that, although the 4D Kitaev model provides stable qubit observables, the Hamiltonian control is not possible. The thermodynamical approach leads to the new proposal of the revised version of Landauer's principle and suggests that the existence of quantum memory implies the existence of the perpetuum mobile of the second kind. Finally, a discussion of the stability property of information and its implications is presented.

DOI
10.1142/9789814317443_0001

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Two types of arguments concerning (im)possibility of constructing a scalable, exponentially stable quantum memory equipped with Hamiltonian controls are discussed. The first type concerns ergodic properties of open Kitaev models which are considered as promising candidates for such memories. It is shown that, although the 4D Kitaev model provides stable qubit observables, the Hamiltonian control is not possible. The thermodynamical approach leads to the new proposal of the revised version of Landauer's principle and suggests that the existence of quantum memory implies the existence of the perpetuum mobile of the second kind. Finally, a discussion of the stability property of information and its implications is presented.
It is argued using a Gedankenexperiment that a scalable quantum memory could be used as a perpetuum mobile of the second kind and hence cannot be realized in nature. The reasoning is based on the assumption that the Landauer's principle for measurements is a consequence of the second law of thermodynamics and not an independent postulate. This implies a modification of the Landauer's principle when applied for discrimination of equilibrium (metastable) states. While identification of the metastable state can be done at infinitesimally low cost, any change of such state involves dissipation of energy proportional to its stability factor.
We prove a no-go theorem for storing quantum information in equilibrium systems. Namely, quantum information cannot be stored in a system with time-independent Hamiltonian interacting with heat bath of temperature T>0T>0T>0 during time that grows with the number of used qubits. We prove it by showing, that storing quantum information for macroscopic time would imply existence of perpetuum mobile of the second kind. The general results are illustrated by the Kitaev model of quantum memory. In contrast, classical information can be stored in equilibrium states for arbitrary long times. We show how it is possible via phase-transition type phenomena. Our result shows that there is a fundamental difference between quantum and classical information in {\it physical} terms.
We analyse stability of the four-dimensional Kitaev model — a candidate for scalable quantum memory — in finite temperature within the weak coupling Markovian limit. It is shown that, below a critical temperature, certain topological qubit observables X and Z possess relaxation times exponentially long in the size of the system. Their construction involves polynomial in system size algorithm which uses as an input the results of measurements performed on all individual spins. We also discuss the drawbacks of such candidate for quantum memory and mention the implications of the stability of qubit for statistical mechanics.
This thesis is devoted to the study of coherent storage of quantum information as well as its potential applications.Quantum memories are crucial to harnessing the potential of quantum physics for information processing tasks.They are required for almost all quantum computation proposals.However, despite the large arsenal of theoretical techniques and proposals dedicated to their implementation, the realization of long-lived quantum memories remains an elusive task.Encoding information in quantum states associated to many-body topological phases of matter and protecting them by means of a static Hamiltonian is one of the leading proposals to achieve quantum memories.While many genuine and well publicized virtues have been demonstrated for this approach, equally real limitations were widely disregarded.In the first two projects of this thesis, we study limitations of passive Hamiltonian protection of quantum information under two different noise models.Chapter 2 deals with arbitrary passive Hamiltonian protection for a many body system under the effect of local depolarizing noise.It is shown that for both constant and time dependent Hamiltonians, the optimal enhancement over the natural single-particle memory time is logarithmic in the number of particles composing the system.The main argument involves a monotonic increase of entropy against which a Hamiltonian can provide little protection.Chapter 3 considers the recoverability of quantum information when it is encoded in a many-body state and evolved under a Hamiltonian composed of known geometrically local interactions and a weak yet unknown Hamiltonian perturbation.We obtain some generic criteria which must be fulfilled by the encoding of information.For specific proposals of protecting Hamiltonian and encodings such as Kitaev's toric code and a subsystem code proposed by Bacon, we additionally provide example perturbations capable of destroying the memory which imply upper bounds for the provable memory times.vi
The natural framework to discuss thermodynamics at the quantum level is the theory of open quantum systems. Memory effects arising from strong system-environment correlations may lead to information back-flow, that is non-Markovian behaviour. The relation between non-Markovianity and quantum thermodynamics has been until now largely unexplored. Here we show by means of Landauer's principle that memory effects control the amount of work extraction by erasure in presence of realistic environments.
The paper is motivated by the discussion of feasibility of large scale quantum computations which should incorporate both unitarity of quantum dynamics for information bearing degrees of freedom and stability with respect to environmental noise. The minimal thermodynamic cost of a single CNOT gate, which is equivalent to the minimal cost of a quantum measurement of a binary observable is analyzed using a generic quantum model of one bit memory. For this model stability of memory with respect to thermal and quantum noise and the error of readout can be quantified. One obtains the relations between the minimal work which is invested in a measurement or CNOT gate, the error and the stability factor. The basic formula differs from the standard Landauer one and seems to be much more realistic. The results show the fundamental conflict between stability and irreversibility of information processing. This explains the feasibility of classical stable and scalable information processing performed by irreversible gates and suggests impossibility of large scale quantum computations based on unitary gates.