24 research outputs found
Finding flows in the one-way measurement model
The one-way measurement model is a framework for universal quantum
computation, in which algorithms are partially described by a graph G of
entanglement relations on a collection of qubits. A sufficient condition for an
algorithm to perform a unitary embedding between two Hilbert spaces is for the
graph G, together with input/output vertices I, O \subset V(G), to have a flow
in the sense introduced by Danos and Kashefi [quant-ph/0506062]. For the
special case of |I| = |O|, using a graph-theoretic characterization, I show
that such flows are unique when they exist. This leads to an efficient
algorithm for finding flows, by a reduction to solved problems in graph theory.Comment: 8 pages, 3 figures: somewhat condensed and updated version, to appear
in PR
Theory of measurement-based quantum computing
In the study of quantum computation, data is represented in terms of linear
operators which form a generalized model of probability, and computations are
most commonly described as products of unitary transformations, which are the
transformations which preserve the quality of the data in a precise sense. This
naturally leads to "unitary circuit models", which are models of computation in
which unitary operators are expressed as a product of "elementary" unitary
transformations. However, unitary transformations can also be effected as a
composition of operations which are not all unitary themselves: the "one-way
measurement model" is one such model of quantum computation.
In this thesis, we examine the relationship between representations of
unitary operators and decompositions of those operators in the one-way
measurement model. In particular, we consider different circumstances under
which a procedure in the one-way measurement model can be described as
simulating a unitary circuit, by considering the combinatorial structures which
are common to unitary circuits and two simple constructions of one-way based
procedures. These structures lead to a characterization of the one-way
measurement patterns which arise from these constructions, which can then be
related to efficiently testable properties of graphs. We also consider how
these characterizations provide automatic techniques for obtaining complete
measurement-based decompositions, from unitary transformations which are
specified by operator expressions bearing a formal resemblance to path
integrals. These techniques are presented as a possible means to devise new
algorithms in the one-way measurement model, independently of algorithms in the
unitary circuit model.Comment: Ph.D. thesis in Combinatorics and Optimization. 199 pages main text,
26 PDF figures. Official electronic version available at
http://hdl.handle.net/10012/413
Quadratic Form Expansions for Unitaries
We introduce techniques to analyze unitary operations in terms of quadratic
form expansions, a form similar to a sum over paths in the computational basis
when the phase contributed by each path is described by a quadratic form over
. We show how to relate such a form to an entangled resource akin to
that of the one-way measurement model of quantum computing. Using this, we
describe various conditions under which it is possible to efficiently implement
a unitary operation U, either when provided a quadratic form expansion for U as
input, or by finding a quadratic form expansion for U from other input data.Comment: 20 pages, 3 figures; (extended version of) accepted submission to TQC
200
Finding Optimal Flows Efficiently
Among the models of quantum computation, the One-way Quantum Computer is one
of the most promising proposals of physical realization, and opens new
perspectives for parallelization by taking advantage of quantum entanglement.
Since a one-way quantum computation is based on quantum measurement, which is a
fundamentally nondeterministic evolution, a sufficient condition of global
determinism has been introduced as the existence of a causal flow in a graph
that underlies the computation. A O(n^3)-algorithm has been introduced for
finding such a causal flow when the numbers of output and input vertices in the
graph are equal, otherwise no polynomial time algorithm was known for deciding
whether a graph has a causal flow or not. Our main contribution is to introduce
a O(n^2)-algorithm for finding a causal flow, if any, whatever the numbers of
input and output vertices are. This answers the open question stated by Danos
and Kashefi and by de Beaudrap. Moreover, we prove that our algorithm produces
an optimal flow (flow of minimal depth.)
Whereas the existence of a causal flow is a sufficient condition for
determinism, it is not a necessary condition. A weaker version of the causal
flow, called gflow (generalized flow) has been introduced and has been proved
to be a necessary and sufficient condition for a family of deterministic
computations. Moreover the depth of the quantum computation is upper bounded by
the depth of the gflow. However, the existence of a polynomial time algorithm
that finds a gflow has been stated as an open question. In this paper we answer
this positively with a polynomial time algorithm that outputs an optimal gflow
of a given graph and thus finds an optimal correction strategy to the
nondeterministic evolution due to measurements.Comment: 10 pages, 3 figure
Generalized Flow and Determinism in Measurement-based Quantum Computation
We extend the notion of quantum information flow defined by Danos and Kashefi
for the one-way model and present a necessary and sufficient condition for the
deterministic computation in this model. The generalized flow also applied in
the extended model with measurements in the X-Y, X-Z and Y-Z planes. We apply
both measurement calculus and the stabiliser formalism to derive our main
theorem which for the first time gives a full characterization of the
deterministic computation in the one-way model. We present several examples to
show how our result improves over the traditional notion of flow, such as
geometries (entanglement graph with input and output) with no flow but having
generalized flow and we discuss how they lead to an optimal implementation of
the unitaries. More importantly one can also obtain a better quantum
computation depth with the generalized flow rather than with flow. We believe
our characterization result is particularly essential for the study of the
algorithms and complexity in the one-way model.Comment: 16 pages, 10 figure
Generalized Flow and Determinism in Measurement-based Quantum Computation
We extend the notion of quantum information flow defined by Danos and Kashefi
for the one-way model and present a necessary and sufficient condition for the
deterministic computation in this model. The generalized flow also applied in
the extended model with measurements in the X-Y, X-Z and Y-Z planes. We apply
both measurement calculus and the stabiliser formalism to derive our main
theorem which for the first time gives a full characterization of the
deterministic computation in the one-way model. We present several examples to
show how our result improves over the traditional notion of flow, such as
geometries (entanglement graph with input and output) with no flow but having
generalized flow and we discuss how they lead to an optimal implementation of
the unitaries. More importantly one can also obtain a better quantum
computation depth with the generalized flow rather than with flow. We believe
our characterization result is particularly essential for the study of the
algorithms and complexity in the one-way model.Comment: 16 pages, 10 figure
Determinism in the one-way model
We introduce a flow condition on open graph states (graph states with inputs
and outputs) which guarantees globally deterministic behavior of a class of
measurement patterns defined over them. Dependent Pauli corrections are derived
for all such patterns, which equalize all computation branches, and only depend
on the underlying entanglement graph and its choice of inputs and outputs.
The class of patterns having flow is stable under composition and
tensorization, and has unitary embeddings as realizations. The restricted class
of patterns having both flow and reverse flow, supports an operation of
adjunction, and has all and only unitaries as realizations.Comment: 8 figures, keywords: measurement based quantum computing,
deterministic computing; Published version, including a new section on
circuit decompositio