Quantum Error Correction with the Toric-GKP Code

We examine the performance of the single-mode GKP code and its concatenation with the toric code for a noise model of Gaussian shifts, or displacement errors. We show how one can optimize the tracking of errors in repeated noisy error correction for the GKP code. We do this by examining the maximum-likelihood problem for this setting and its mapping onto a 1D Euclidean path-integral modeling a particle in a random cosine potential. We demonstrate the efficiency of a minimum-energy decoding strategy as a proxy for the path integral evaluation. In the second part of this paper, we analyze and numerically assess the concatenation of the GKP code with the toric code. When toric code measurements and GKP error correction measurements are perfect, we find that by using GKP error information the toric code threshold improves from $10\%$ to $14\%$. When only the GKP error correction measurements are perfect we observe a threshold at $6\%$. In the more realistic setting when all error information is noisy, we show how to represent the maximum likelihood decoding problem for the toric-GKP code as a 3D compact QED model in the presence of a quenched random gauge field, an extension of the random-plaquette gauge model for the toric code. We present a new decoder for this problem which shows the existence of a noise threshold at shift-error standard deviation $\sigma_0 \approx 0.243$ for toric code measurements, data errors and GKP ancilla errors. If the errors only come from having imperfect GKP states, this corresponds to states with just 4 photons or more. Our last result is a no-go result for linear oscillator codes, encoding oscillators into oscillators. For the Gaussian displacement error model, we prove that encoding corresponds to squeezing the shift errors. This shows that linear oscillator codes are useless for quantum information protection against Gaussian shift errors.Comment: 50 pages, 14 figure

A Computation in a Cellular Automaton Collider Rule 110

A cellular automaton collider is a finite state machine build of rings of one-dimensional cellular automata. We show how a computation can be performed on the collider by exploiting interactions between gliders (particles, localisations). The constructions proposed are based on universality of elementary cellular automaton rule 110, cyclic tag systems, supercolliders, and computing on rings.Comment: 39 pages, 32 figures, 3 table

Array languages and the N-body problem

This paper is a description of the contributions to the SICSA multicore challenge on many body planetary simulation made by a compiler group at the University of Glasgow. Our group is part of the Computer Vision and Graphics research group and we have for some years been developing array compilers because we think these are a good tool both for expressing graphics algorithms and for exploiting the parallelism that computer vision applications require. We shall describe experiments using two languages on two different platforms and we shall compare the performance of these with reference C implementations running on the same platforms. Finally we shall draw conclusions both about the viability of the array language approach as compared to other approaches used in the challenge and also about the strengths and weaknesses of the two, very different, processor architectures we used

Implementing and characterizing precise multi-qubit measurements

There are two general requirements to harness the computational power of quantum mechanics: the ability to manipulate the evolution of an isolated system and the ability to faithfully extract information from it. Quantum error correction and simulation often make a more exacting demand: the ability to perform non-destructive measurements of specific correlations within that system. We realize such measurements by employing a protocol adapted from [S. Nigg and S. M. Girvin, Phys. Rev. Lett. 110, 243604 (2013)], enabling real-time selection of arbitrary register-wide Pauli operators. Our implementation consists of a simple circuit quantum electrodynamics (cQED) module of four highly-coherent 3D transmon qubits, collectively coupled to a high-Q superconducting microwave cavity. As a demonstration, we enact all seven nontrivial subset-parity measurements on our three-qubit register. For each we fully characterize the realized measurement by analyzing the detector (observable operators) via quantum detector tomography and by analyzing the quantum back-action via conditioned process tomography. No single quantity completely encapsulates the performance of a measurement, and standard figures of merit have not yet emerged. Accordingly, we consider several new fidelity measures for both the detector and the complete measurement process. We measure all of these quantities and report high fidelities, indicating that we are measuring the desired quantities precisely and that the measurements are highly non-demolition. We further show that both results are improved significantly by an additional error-heralding measurement. The analyses presented here form a useful basis for the future characterization and validation of quantum measurements, anticipating the demands of emerging quantum technologies.Comment: 10 pages, 5 figures, plus supplemen

Enhancements to the STAGS computer code

The power of the STAGS family of programs was greatly enhanced. Members of the family include STAGS-C1 and RRSYS. As a result of improvements implemented, it is now possible to address the full collapse of a structural system, up to and beyond critical points where its resistance to the applied loads vanishes or suddenly changes. This also includes the important class of problems where a multiplicity of solutions exists at a given point (bifurcation), and where until now no solution could be obtained along any alternate (secondary) load path with any standard production finite element code

Bondi-Sachs Energy-Momentum for the CMC Initial Value Problem

The constraints on the asymptotic behavior of the conformal factor and conformal extrinsic curvature imposed by the initial value equations of general relativity on constant mean extrinsic curvature (CMC) hypersurfaces are analyzed in detail. We derive explicit formulas for the Bondi-Sachs energy and momentum in terms of coefficients of asymptotic expansions on CMC hypersurfaces near future null infinity. Precise numerical results for the Bondi-Sachs energy, momentum, and angular momentum are used to interpret physically Bowen-York solutions of the initial value equations on conformally flat CMC hypersurfaces of the type obtained earlier by Buchman et al. [Phys. Rev. D 80:084024 (2009)].Comment: version to be published in Phys. Rev.

Multi-Architecture Monte-Carlo (MC) Simulation of Soft Coarse-Grained Polymeric Materials: SOft coarse grained Monte-carlo Acceleration (SOMA)

Multi-component polymer systems are important for the development of new materials because of their ability to phase-separate or self-assemble into nano-structures. The Single-Chain-in-Mean-Field (SCMF) algorithm in conjunction with a soft, coarse-grained polymer model is an established technique to investigate these soft-matter systems. Here we present an im- plementation of this method: SOft coarse grained Monte-carlo Accelera- tion (SOMA). It is suitable to simulate large system sizes with up to billions of particles, yet versatile enough to study properties of different kinds of molecular architectures and interactions. We achieve efficiency of the simulations commissioning accelerators like GPUs on both workstations as well as supercomputers. The implementa- tion remains flexible and maintainable because of the implementation of the scientific programming language enhanced by OpenACC pragmas for the accelerators. We present implementation details and features of the program package, investigate the scalability of our implementation SOMA, and discuss two applications, which cover system sizes that are difficult to reach with other, common particle-based simulation methods

Towards universal quantum computation in continuous-variable systems

In this thesis we explore the possibility of creating continuousvariable quantum systems that are capable of supporting universal quantum computation. We begin by examining the measurement-based model, which employs sequences of measurements on highly entangled resource states, known as a cluster states. We suggest a method for the construction of Gaussian cluster states based on ensembles of atoms and quantum non-demolition interactions. We then go on to expand our model to allow for the inclusion of light modes as part of the cluster. This yields a new class of states, the composite cluster states. This leads us to propose a new architecture for the measurement-based model that uses these composite clusters to increase resource e ciency and reduce computational errors. The second part of this thesis concerns topological quantum computation. In states exhibiting topological degrees of freedom, quantum information can be stored as a non-local property of the physical system and manipulated by braiding quasiparticles known as anyons. Here we show how these ideas can be extended to continuous variables. We establish a continuous variable analogue of the Kitaev toric code, show that excitations correspond to continuous versions of Abelian anyons and investigate their behaviour under the condition of nite squeezing of the resource state. Finally, we expand our continuous variable topological model to include non-abelian excitations by constructing superpositions of CV toric code anyons. We derive the fusion and braiding behaviour of these non-abelian excitations and nd that they correspond to a CV analog of Ising anyons. Using these resources, we go on to suggest a computational scheme that encodes qubits within the fusion spaces of the CV Ising anyons and derive one- and two-qubit quantum gates operations that are implemented in a topological manner