560 research outputs found
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
All unitaries having operator Schmidt rank 2 are controlled unitaries
We prove that every unitary acting on any multipartite system and having
operator Schmidt rank equal to 2 can be diagonalized by local unitaries. This
then implies that every such multipartite unitary is locally equivalent to a
controlled unitary with every party but one controlling a set of unitaries on
the last party. We also prove that any bipartite unitary of Schmidt rank 2 is
locally equivalent to a controlled unitary where either party can be chosen as
the control, and at least one party can control with two terms, which implies
that each such unitary can be implemented using local operations and classical
communication (LOCC) and a maximally entangled state on two qubits. These
results hold regardless of the dimensions of the systems on which the unitary
acts.Comment: Comments welcom
Universal Uhrig dynamical decoupling for bosonic systems
We construct efficient deterministic dynamical decoupling schemes protecting
continuous variable degrees of freedom. Our schemes target decoherence induced
by quadratic system-bath interactions with analytic time-dependence. We show
how to suppress such interactions to -th order using only pulses.
Furthermore, we show to homogenize a -mode bosonic system using only
pulses, yielding - up to -th order - an effective evolution
described by non-interacting harmonic oscillators with identical frequencies.
The decoupled and homogenized system provides natural decoherence-free
subspaces for encoding quantum information. Our schemes only require pulses
which are tensor products of single-mode passive Gaussian unitaries and SWAP
gates between pairs of modes.Comment: 17 pages, 2 figures
Entanglement and the Power of One Qubit
The "Power of One Qubit" refers to a computational model that has access to
only one pure bit of quantum information, along with n qubits in the totally
mixed state. This model, though not as powerful as a pure-state quantum
computer, is capable of performing some computational tasks exponentially
faster than any known classical algorithm. One such task is to estimate with
fixed accuracy the normalized trace of a unitary operator that can be
implemented efficiently in a quantum circuit. We show that circuits of this
type generally lead to entangled states, and we investigate the amount of
entanglement possible in such circuits, as measured by the multiplicative
negativity. We show that the multiplicative negativity is bounded by a
constant, independent of n, for all bipartite divisions of the n+1 qubits, and
so becomes, when n is large, a vanishingly small fraction of the maximum
possible multiplicative negativity for roughly equal divisions. This suggests
that the global nature of entanglement is a more important resource for quantum
computation than the magnitude of the entanglement.Comment: 22 pages, 4 figure
Stationary Phase Method in Discrete Wigner Functions and Classical Simulation of Quantum Circuits
One of the lowest-order corrections to Gaussian quantum mechanics in
infinite-dimensional Hilbert spaces are Airy functions: a uniformization of the
stationary phase method applied in the path integral perspective. We introduce
a ``periodized stationary phase method'' to discrete Wigner functions of
systems with odd prime dimension and show that the gate is the
discrete analog of the Airy function. We then establish a relationship between
the stabilizer rank of states and the number of quadratic Gauss sums necessary
in the periodized stationary phase method. This allows us to develop a
classical strong simulation of a single qutrit marginal on qutrit
gates that are followed by Clifford evolution, and show that
this only requires quadratic Gauss sums. This outperforms
the best alternative qutrit algorithm (based on Wigner negativity and scaling
as for precision) for any number of
gates to full precision
Noncommutative ball maps
In this paper, we analyze problems involving matrix variables for which we
use a noncommutative algebra setting. To be more specific, we use a class of
functions (called NC analytic functions) defined by power series in
noncommuting variables and evaluate these functions on sets of matrices of all
dimensions; we call such situations dimension-free. These types of functions
have recently been used in the study of dimension-free linear system
engineering problems.
In this paper we characterize NC analytic maps that send dimension-free
matrix balls to dimension-free matrix balls and carry the boundary to the
boundary; such maps we call "NC ball maps". We find that up to normalization,
an NC ball map is the direct sum of the identity map with an NC analytic map of
the ball into the ball. That is, "NC ball maps" are very simple, in contrast to
the classical result of D'Angelo on such analytic maps over C. Another
mathematically natural class of maps carries a variant of the noncommutative
distinguished boundary to the boundary, but on these our results are limited.
We shall be interested in several types of noncommutative balls, conventional
ones, but also balls defined by constraints called Linear Matrix Inequalities
(LMI). What we do here is a small piece of the bigger puzzle of understanding
how LMIs behave with respect to noncommutative change of variables.Comment: 46 page
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
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