Holographic quantum error-correcting codes: Toy models for the bulk/boundary correspondence

We propose a family of exactly solvable toy models for the AdS/CFT correspondence based on a novel construction of quantum error-correcting codes with a tensor network structure. Our building block is a special type of tensor with maximal entanglement along any bipartition, which gives rise to an isometry from the bulk Hilbert space to the boundary Hilbert space. The entire tensor network is an encoder for a quantum error-correcting code, where the bulk and boundary degrees of freedom may be identified as logical and physical degrees of freedom respectively. These models capture key features of entanglement in the AdS/CFT correspondence; in particular, the Ryu-Takayanagi formula and the negativity of tripartite information are obeyed exactly in many cases. That bulk logical operators can be represented on multiple boundary regions mimics the Rindler-wedge reconstruction of boundary operators from bulk operators, realizing explicitly the quantum error-correcting features of AdS/CFT recently proposed by Almheiri et. al in arXiv:1411.7041.Comment: 40 Pages + 25 Pages of Appendices. 38 figures. Typos and bibliographic amendments and minor correction

Resource optimization for fault-tolerant quantum computing

In this thesis we examine a variety of techniques for reducing the resources required for fault-tolerant quantum computation. First, we show how to simplify universal encoded computation by using only transversal gates and standard error correction procedures, circumventing existing no-go theorems. We then show how to simplify ancilla preparation, reducing the cost of error correction by more than a factor of four. Using this optimized ancilla preparation, we develop improved techniques for proving rigorous lower bounds on the noise threshold. Additional overhead can be incurred because quantum algorithms must be translated into sequences of gates that are actually available in the quantum computer. In particular, arbitrary single-qubit rotations must be decomposed into a discrete set of fault-tolerant gates. We find that by using a special class of non-deterministic circuits, the cost of decomposition can be reduced by as much as a factor of four over state-of-the-art techniques, which typically use deterministic circuits. Finally, we examine global optimization of fault-tolerant quantum circuits under physical connectivity constraints. We adapt techniques from VLSI in order to minimize time and space usage for computations in the surface code, and we develop a software prototype to demonstrate the potential savings.Comment: 231 pages, Ph.D. thesis, University of Waterlo

Measurement-based quantum computation with cluster states

In this thesis we describe the one-way quantum computer (QCc), a scheme of universal quantum computation that consists entirely of one-qubit measurements on a highly entangled multi-particle state, the cluster state. We prove universality of the QCc, describe the underlying computational model and demonstrate that the QCc can be operated fault-tolerantly. In Chapter 2 we show that the QCc can be regarded as a simulator of quantum logic networks. In this way, we give the universality proof and establish the link to the network model, the common model of quantum computation. We also indicate that the description of the QCc as a network simulator is not adequate in every respect. In Chapter 3 we derive the computational model underlying the one-way quantum computer, which is very different from the quantum logic network model. The QCc has no quantum input, no quantum output and no quantum register, and the unitary gates from some universal set are not the elementary building blocks of QCc-quantum algorithms. Further, all information that is processed with the QCc are the outcomes of one-qubit measurements and thus processing of information exists only at the classical level. The QCc is nevertheless quantum mechanical as it uses a highly entangled cluster state as the central physical resource. In Chapter 4 we show that there exist nonzero error thresholds for fault-tolerant quantum computation with the QCc. Further, we outline the concept of checksums in the context of the QCc which may become an element in future practicable and adequate methods for fault-tolerant QCc-computation.In dieser Dissertation beschreiben wir den Einweg-Quantenrechner (QCc), ein Schema zum universellen Quantenrechnen, das allein aus Einteilchenmessungen an einem hochgradig verschraenkten Vielteilchenzustand, dem Clusterzustand, besteht. Wir beweisen die Universalitaet des QCc, beschreiben das zugrunde liegende Rechnermodell und zeigen, dass der QCc fehlertolerantes Quantenrechnen erlaubt. In Kapitel 2 zeigen wir, dass der QCc als ein Simulator quantenlogischer Netzwerke aufgefasst werden kann. Damit beweisen wir dessen Universalitaet und stellen den Zusammenhang zum Netzwerkmodel her, welches das verbreitete Model eines Quantenrechners darstellt. Wir weisen auch darauf hin, dass die Beschreibung des QCc als Netzwerksimulator nicht in jeder Hinsicht passend ist. In Kapitel 3 leiten wir das dem Einweg-Quantenrechner zugrunde liegende Rechnermodell her. Es ist sehr verschieden vom Netzwerkmodell des Quantenrechners. Der QCc besitzt keinen Quanten-Input, keinen Quanten-Output und kein Quantenregister. Unitaere Quantengatter aus einem universellen Satz sind nicht die elementaren Bestandteile von QCc-Quantenalgorithmen. Darueber hinaus sind die Messergebnisse aus den Einteilchenmessungen die einzige Information, die vom QCc verarbeitet wird, und somit existiert Informationsverarbeitung beim QCc nur auf klassischem Niveau. Dennoch arbeitet der QCc fundamental quantenmechanisch, da er den hochverschraenkten Clusterzustand als zentrale physikalische Resource nutzt. In Kapitel 4 zeigen wir, dass positive Fehlerschranken fuer das fehlertolerante Quantenrechnen mit dem QCc existieren. Desweiteren skizzieren wir das Konzept der Pr{"u}fsummen im Zusammenhang mit dem QC, das ein Element zukuenfitiger praktikabler und zweckmaessiger Methoden fuer fehlertolerantes QCc-Quantenrechnen werden kann

Design of tch-type sequences for communications

This thesis deals with the design of a class of cyclic codes inspired by TCH codewords. Since TCH codes are linked to finite fields the fundamental concepts and facts about abstract algebra, namely group theory and number theory, constitute the first part of the thesis. By exploring group geometric properties and identifying an equivalence between some operations on codes and the symmetries of the dihedral group we were able to simplify the generation of codewords thus saving on the necessary number of computations. Moreover, we also presented an algebraic method to obtain binary generalized TCH codewords of length N = 2k, k = 1,2, . . . , 16. By exploring Zech logarithm’s properties as well as a group theoretic isomorphism we developed a method that is both faster and less complex than what was proposed before. In addition, it is valid for all relevant cases relating the codeword length N and not only those resulting from N = p

The Telecommunications and Data Acquisition Report

This quarterly publication provides archival reports on developments in programs managed by JPL's Telecommunications and Mission Operations Directorate (TMOD), which now includes the former Telecommunications and Data Acquisition (TDA) Office. In space communications, radio navigation, radio science, and ground-based radio and radar astronomy, it reports on activities of the Deep Space Network (DSN) in planning, supporting research and technology, implementation, and operations. Also included are standards activity at JPL for space data and information systems and reimbursable DSN work performed for other space agencies through NASA. The preceding work is all performed for NASA's Office of Space Communications (OSC)

Blueprint for a Scalable Photonic Fault-Tolerant Quantum Computer

Photonics is the platform of choice to build a modular, easy-to-network quantum computer operating at room temperature. However, no concrete architecture has been presented so far that exploits both the advantages of qubits encoded into states of light and the modern tools for their generation. Here we propose such a design for a scalable and fault-tolerant photonic quantum computer informed by the latest developments in theory and technology. Central to our architecture is the generation and manipulation of three-dimensional hybrid resource states comprising both bosonic qubits and squeezed vacuum states. The proposal enables exploiting state-of-the-art procedures for the non-deterministic generation of bosonic qubits combined with the strengths of continuous-variable quantum computation, namely the implementation of Clifford gates using easy-to-generate squeezed states. Moreover, the architecture is based on two-dimensional integrated photonic chips used to produce a qubit cluster state in one temporal and two spatial dimensions. By reducing the experimental challenges as compared to existing architectures and by enabling room-temperature quantum computation, our design opens the door to scalable fabrication and operation, which may allow photonics to leap-frog other platforms on the path to a quantum computer with millions of qubits.Comment: 38 pages, many figures. Comments welcom