23 research outputs found
The Many Worlds of Uncertainty
The status of the uncertainty relations varies between the different
interpretations of quantum mechanics. The aim of the current paper is to
explore their meanings within a certain neo-Everettian many worlds
interpretation. We will also look at questions that have been linked with the
uncertainty relations since Heisenberg's uncertainty principle: those of joint
and repeated measurement of non-commuting (or otherwise `incompatible')
observables. This will have implications beyond the uncertainty relations, as
we will see the fundamentally different way in which statistical statements are
interpreted in the neo-Everett theory that we use.Comment: 9 page
Quantum Computation and Many Worlds
An Everett (`Many Worlds') interpretation of quantum mechanics due to
Saunders and Zurek is presented in detail. This is used to give a physical
description of the process of a quantum computation. Objections to such an
understanding are discussed.Comment: This paper has been superceded by arXiv:0802.2504v1 [quant-ph
An introduction to many worlds in quantum computation
The interpretation of quantum mechanics is an area of increasing interest to
many working physicists. In particular, interest has come from those involved
in quantum computing and information theory, as there has always been a strong
foundational element in this field. This paper introduces one interpretation of
quantum mechanics, a modern `many-worlds' theory, from the perspective of
quantum computation. Reasons for seeking to interpret quantum mechanics are
discussed, then the specific `neo-Everettian' theory is introduced and its
claim as the best available interpretation defended. The main objections to the
interpretation, including the so-called ``problem of probability'' are shown to
fail. The local nature of the interpretation is demonstrated, and the
implications of this both for the interpretation and for quantum mechanics more
generally are discussed. Finally, the consequences of the theory for quantum
computation are investigated, and common objections to using many worlds to
describe quantum computing are answered. We find that using this particular
many-worlds theory as a physical foundation for quantum computation gives
several distinct advantages over other interpretations, and over not
interpreting quantum theory at all.Comment: Published version. This supercedes quant-ph/0210204. Comments welcom
Optimising the Solovay-Kitaev algorithm
The Solovay-Kitaev algorithm is the standard method used for approximating
arbitrary single-qubit gates for fault-tolerant quantum computation. In this
paper we introduce a technique called "search space expansion", which modifies
the initial stage of the Solovay-Kitaev algorithm, increasing the length of the
possible approximating sequences but without requiring an exhaustive search
over all possible sequences. We show that our technique, combined with a GNAT
geometric tree search outputs gate sequences that are almost an order of
magnitude smaller for the same level of accuracy. This therefore significantly
reduces the error correction requirements for quantum algorithms on encoded
fault-tolerant hardware.Comment: 9 page
Layer by layer generation of cluster states
Cluster states can be used to perform measurement-based quantum computation.
The cluster state is a useful resource, because once it has been generated only
local operations and measurements are needed to perform universal quantum
computation. In this paper, we explore techniques for quickly and
deterministically building a cluster state. In particular we consider
generating cluster states on a qubus quantum computer, a computational
architecture which uses a continuous variable ancilla to generate interactions
between qubits. We explore several techniques for building the cluster, with
the number of operations required depending on whether we allow the ability to
destroy previously created controlled-phase links between qubits. In the case
where we can not destroy these links, we show how to create an n x m cluster
using just 3nm -2n -3m/2 + 3 operations. This gives more than a factor of 2
saving over a naive method. Further savings can be obtained if we include the
ability to destroy links, in which case we only need (8nm-4n-4m-8)/3
operations. Unfortunately the latter scheme is more complicated so choosing the
correct order to interact the qubits is considerably more difficult. A half way
scheme, that keeps a modular generation but saves additional operations over
never destroying links requires only 3nm-2n-2m+4 operations. The first scheme
and the last scheme are the most practical for building a cluster state because
they split up the generation into the repetition of simple sections.Comment: 16 pages, 11 figure
Matrix multiplication is determined by orthogonality and trace
Any associative bilinear multiplication on the set of n-by-n matrices over
some field of characteristic not two, that makes the same vectors orthogonal
and has the same trace as ordinary matrix multiplication, must be ordinary
matrix multiplication or its opposite.Comment: 4 page
Generalised Compositional Theories and Diagrammatic Reasoning
This chapter provides an introduction to the use of diagrammatic language, or
perhaps more accurately, diagrammatic calculus, in quantum information and
quantum foundations. We illustrate the use of diagrammatic calculus in one
particular case, namely the study of complementarity and non-locality, two
fundamental concepts of quantum theory whose relationship we explore in later
part of this chapter.
The diagrammatic calculus that we are concerned with here is not merely an
illustrative tool, but it has both (i) a conceptual physical backbone, which
allows it to act as a foundation for diverse physical theories, and (ii) a
genuine mathematical underpinning, permitting one to relate it to standard
mathematical structures.Comment: To appear as a Springer book chapter chapter, edited by G.
Chirabella, R. Spekken
Quantum picturalism for topological cluster-state computing
Topological quantum computing is a way of allowing precise quantum
computations to run on noisy and imperfect hardware. One implementation uses
surface codes created by forming defects in a highly-entangled cluster state.
Such a method of computing is a leading candidate for large-scale quantum
computing. However, there has been a lack of sufficiently powerful high-level
languages to describe computing in this form without resorting to single-qubit
operations, which quickly become prohibitively complex as the system size
increases. In this paper we apply the category-theoretic work of Abramsky and
Coecke to the topological cluster-state model of quantum computing to give a
high-level graphical language that enables direct translation between quantum
processes and physical patterns of measurement in a computer - a "compiler
language". We give the equivalence between the graphical and topological
information flows, and show the applicable rewrite algebra for this computing
model. We show that this gives us a native graphical language for the design
and analysis of topological quantum algorithms, and finish by discussing the
possibilities for automating this process on a large scale.Comment: 18 pages, 21 figures. Published in New J. Phys. special issue on
topological quantum computin
Recursive quantum repeater networks
Internet-scale quantum repeater networks will be heterogeneous in physical
technology, repeater functionality, and management. The classical control
necessary to use the network will therefore face similar issues as Internet
data transmission. Many scalability and management problems that arose during
the development of the Internet might have been solved in a more uniform
fashion, improving flexibility and reducing redundant engineering effort.
Quantum repeater network development is currently at the stage where we risk
similar duplication when separate systems are combined. We propose a unifying
framework that can be used with all existing repeater designs. We introduce the
notion of a Quantum Recursive Network Architecture, developed from the emerging
classical concept of 'recursive networks', extending recursive mechanisms from
a focus on data forwarding to a more general distributed computing request
framework. Recursion abstracts independent transit networks as single relay
nodes, unifies software layering, and virtualizes the addresses of resources to
improve information hiding and resource management. Our architecture is useful
for building arbitrary distributed states, including fundamental distributed
states such as Bell pairs and GHZ, W, and cluster states.Comment: 14 page
Surface code quantum computing by lattice surgery
In recent years, surface codes have become a leading method for quantum error
correction in theoretical large scale computational and communications
architecture designs. Their comparatively high fault-tolerant thresholds and
their natural 2-dimensional nearest neighbour (2DNN) structure make them an
obvious choice for large scale designs in experimentally realistic systems.
While fundamentally based on the toric code of Kitaev, there are many variants,
two of which are the planar- and defect- based codes. Planar codes require
fewer qubits to implement (for the same strength of error correction), but are
restricted to encoding a single qubit of information. Interactions between
encoded qubits are achieved via transversal operations, thus destroying the
inherent 2DNN nature of the code. In this paper we introduce a new technique
enabling the coupling of two planar codes without transversal operations,
maintaining the 2DNN of the encoded computer. Our lattice surgery technique
comprises splitting and merging planar code surfaces, and enables us to perform
universal quantum computation (including magic state injection) while removing
the need for braided logic in a strictly 2DNN design, and hence reduces the
overall qubit resources for logic operations. Those resources are further
reduced by the use of a rotated lattice for the planar encoding. We show how
lattice surgery allows us to distribute encoded GHZ states in a more direct
(and overhead friendly) manner, and how a demonstration of an encoded CNOT
between two distance 3 logical states is possible with 53 physical qubits, half
of that required in any other known construction in 2D.Comment: Published version. 29 pages, 18 figure