39 research outputs found

    Quantum Error-Correcting Hybrid Codes

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    Remarkable contributions made in the field of quantum algorithms and theory since 1994 have paved the way for quantum information and quantum computing. Their substantial speed-up over classical algorithms encouraged further developments in quantum information theory that enable information transmission in a reliable and fault-tolerant manner. A huge family of error-correcting codes have been developed since then with improved parameters and code-generating methods to process quantum information in the presence of noise and imperfect quantum gates. Stabilizer codes are one of the important classes of quantum error correcting codes. Their simple structure makes these codes easier to implement in a fault-tolerant manner. Promising work in the domain of hybrid quantum error-correcting codes has further shown their advantages over general quantum error correction. In this thesis, we show various techniques for constructing error-correcting quantum codes, especially hybrid codes that transmit quantum-classical information over a single channel. A hybrid code can simultaneously transmit m bits of classical information and k bits of quantum information by building a collection of m quantum codes where each quantum message is associated with a classical message. Such codes have been shown to have better code parameters than the best known quantum codes using the same number of physical qubits. The first model is based on the use of codeword stabilized codes and union stabilizer codes while the second model uses subsystem codes by encoding the classical information in the gauge subsystem of the code. We also discuss various examples of good hybrid code constructions using these models and introduce the notion of using the framework of graph codes to encode and transmit both quantum and classical information since they allow for simpler fault-tolerant procedures. We finally propose various future directions to continue the work

    Avoiding coherent errors with rotated concatenated stabilizer codes

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    Coherent errors, which arise from collective couplings, are a dominant form of noise in many realistic quantum systems, and are more damaging than oft considered stochastic errors. Here, we propose integrating stabilizer codes with constant-excitation codes by code concatenation. Namely, by concatenating an [[n, k, d]] stabilizer outer code with dual-rail inner codes, we obtain a [[2n, k, d]] constant-excitation code immune from coherent phase errors and also equivalent to a Pauli-rotated stabilizer code. When the stabilizer outer code is fault-tolerant, the constant-excitation code has a positive fault-tolerant threshold against stochastic errors. Setting the outer code as a four-qubit amplitude damping code yields an eight-qubit constant-excitation code that corrects a single amplitude damping error, and we analyze this code’s potential as a quantum memory

    Fault-tolerant quantum computer architectures using hierarchies of quantum error-correcting codes

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    Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2008.Includes bibliographical references (p. 221-238).Quantum computers have been shown to efficiently solve a class of problems for which no efficient solution is otherwise known. Physical systems can implement quantum computation, but devising realistic schemes is an extremely challenging problem largely due to the effect of noise. A quantum computer that is capable of correctly solving problems more rapidly than modern digital computers requires some use of so-called fault-tolerant components. Code-based fault-tolerance using quantum error-correcting codes is one of the most promising and versatile of the known routes for fault-tolerant quantum computation. This dissertation presents three main, new results about code-based fault-tolerant quantum computer architectures. The first result is a large new family of quantum codes that go beyond stabilizer codes, the most well-studied family of quantum codes. Our new family of codeword stabilized codes contains all known codes with optimal parameters. Furthermore, we show how to systematically find, construct, and understand such codes as a pair of codes: an additive quantum code and a classical (nonlinear) code. Second, we resolve an open question about universality of so-called transversal gates acting on stabilizer codes. Such gates are universal for classical fault-tolerant computation, but they were conjectured to be insufficient for universal fault-tolerant quantum computation. We show that transversal gates have a restricted form and prove that some important families of them cannot be quantum universal. This is strong evidence that so-called quantum software is necessary to achieve universality, and, therefore, fault-tolerant quantum computer architecture is fundamentally different from classical computer architecture. Finally, we partition the fault-tolerant design problem into levels of a hierarchy of concatenated codes and present methods, compatible with rigorous threshold theorems, for numerically evaluating these codes.(cont.) The methods are applied to measure inner error-correcting code performance, as a first step toward elucidation of an effective fault-tolerant quantum computer architecture that uses no more than a physical, inner, and outer level of coding. Of the inner codes, the Golay code gives the highest pseudothreshold of 2 x 10-3. A comparison of logical error rate and overhead shows that the Bacon-Shor codes are competitive with Knill's C₄/C₆ scheme at a base error rate of 10⁻⁴.by Andrew W. Cross.Ph.D

    Contrôle et optimisation de systèmes physiques : application à la mécanique quantique et au confinement magnétique dans les stellarators.

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    This PhD manuscript deals with the optimization and control of several physical systems. It is divided into three parts.The first part is devoted to stellarators. This type of nuclear fusion reactor poses many challenges related to optimization. We focus on an inverse problem well known to physicists, modeling the optimal design of superconducting coils generating a given magnetic field. We conduct both a theoretical and a numerical study of an extension of this problem, involving shape optimization. Then, we develop a new method to prove the existence of optimal shapes in the case of hypersurface optimization problems. Finally, we study and optimize the Laplace forces acting on a current surface density. The second part of this manuscript deals with the control of finite dimensional quantum systems. We rigorously study the combination of the rotating wave approximation with the adiabatic approximation. First, we obtain the robustness of a population transfer method on qubits. The latter then allows to extend results of Li and Khaneja on the ensemble control of qubits by restricting to the use of a single control. We also present a second contribution, devoted to the analysis of a chattering phenomenon for an optimal control problem of a quantum system. Finally, the third part is dedicated to the proof of a small-time global null controllability result for generalized Burgers' equations using a boundary layer.Cette thèse porte sur l’optimisation et le contrôle de plusieurs systèmes physiques : elle est composée de trois parties.La première partie est consacrée aux stellarators. Ce type de réacteur à fusion nucléaire pose de nombreux défis reliés à l’optimisation. Nous nous sommes concentrés sur un problème inverse bien connu des physiciens, modélisant la conception optimale de bobines supraconductrices générant un champ magnétique donné. Nous avons conduit une étude théorique et numérique d’une extension de ce problème, portant sur une optimisation de forme. Nous avons ensuite développé une nouvelle méthode afin de prouver l’existence de formes optimales dans le cas de problèmes d’optimisation d’hypersurfaces. Nous avons enfin effectué l’étude et l’optimisation des forces de Laplace s’exerçant sur une densité surfaciquede courant.La deuxième partie porte ensuite sur l’étude du contrôle de systèmes quantiques de dimension finie. Nous avons étudié rigoureusement la combinaison de l’approximation de l’onde tournante avec l’approximation adiabatique. Dans un premier temps, nous avons obtenu la robustesse des méthodes de transfert de population sur les qubits. Cette dernière permet alors d’étendre des résultats de Li et Khaneja sur le contrôle d’ensemble des qubits en se restreignant à l’utilisation d’un seul contrôle. Nous présentons égallement une seconde contribution, consacrée à l’analyse d’un phénomène de chattering pour un problème de contrôle optimal d’un système quantique.Enfin, la troisième partie est dédiée à la preuve d’un résultat de contrôlabilité à zéro en temps petit pour des équations de Burgers généralisées grâce à l’utilisation d’une couche limite
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