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arXiv:quant-ph/0603260

Can one build a quantum hard drive? A no-go theorem for storing quantum information in equilibrium systems

Robert Alicki, Michał Horodecki

Published Mar 28, 2006• 8 citations
arXivHTMLPDF

Abstract

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 $T>0$ 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.

DOI
10.48550/arxiv.quant-ph/0603260

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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.
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.
The results of this thesis concern the real-world realization of quantum computers, specifically how to build their "hard drives" or quantum memories. These are many-body quantum systems, and their building blocks are qubits, the same way bits are the building blocks of classical computers. Quantum memories need to be robust against thermal noise, noise that would otherwise destroy the encoded information, similar to how strong magnetic field corrupts data classically stored in magnetic many-body systems (e.g., in hard drives). In this work I focus on a subset of many-body models, called quantum doubles, which, in addition to storing the information, could be used to perform the steps of the quantum computation, i.e., work as a "quantum processor". In the first part of my thesis, I investigate how long a subset of quantum doubles (qudit surface codes) can retain the quantum information stored in them, referred to as their memory time. I prove an upper bound for this memory time, restricting the maximum possible performance of qudit surface codes. Then, I analyze the structure of quantum doubles, and find two interesting properties. First, that the high-level description of doubles, utilizing only their quasi-particles to describe their states, disregards key components of their microscopic properties. In short, quasi-particles (anyons) of quantum doubles are not in a one-to-one correspondence with the energy eigenstates of their Hamiltonian. Second, by investigating phase transitions of a simple quantum double, D(S3), I map its phase diagram, and interpret the physical processes the theory undergoes through terms borrowed from the Landau theory of phase transitions.
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.
The ability to protect quantum information from the effect of noise is one of the major goals of quantum information processing. In this article, we study limitations on the asymptotic stability of quantum information stored in passive NNN-qubit systems. We consider the effect of small imperfections in the implementation of the protecting Hamiltonian in the form of perturbations or weak coupling to a ground state environment. We thus depart from the usual Markovian approximation for a thermal bath by conce ntrating on models for which part of the evolution can be calculated exactly. We prove that, regardless of the protecting Hamiltonian, there exists a perturbed evolution that necessitates a final error correcting step for the state of the memory to be read. Such an error correction step is shown to require a finite error threshold, the lack thereof being exemplified by the 3D XZ-compass model \cite{bacon-2006}. We go on to present explicit weak Hamiltonian perturbations which destroy the logical information stored in the 2D toric code in a time O(log⁡(N))O(\log(N))O(log(N)).
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 ability to store quantum information without recourse to constant feedback processes would yield a significant advantage for future implementations of quantum information processing. In this Letter, limitations of the prototypical model, the Toric code in two dimensions, are elucidated along with a sufficient condition for overcoming these limitations. Specifically, the interplay between Hamiltonian perturbations and dynamically occurring noise is considered as a system in its ground state is brought into contact with a thermal reservoir. This proves that when utilizing the Toric code on N2 qubits in a 2D lattice as a quantum memory, the information cannot be stored for a time O(N). In contrast, the 2D Ising model protects classical information against the described noise model for exponentially long times. The results also have implications for the robustness of braiding operations in topological quantum computation.