423 research outputs found
Quantum Searching via Entanglement and Partial Diffusion
In this paper, we will define a quantum operator that performs the inversion
about the mean only on a subspace of the system (Partial Diffusion Operator).
This operator is used in a quantum search algorithm that runs in O(sqrt{N/M})
for searching an unstructured list of size N with M matches such that 1<= M<=N.
We will show that the performance of the algorithm is more reliable than known
{fixed operators quantum search algorithms} especially for multiple matches
where we can get a solution after a single iteration with probability over 90%
if the number of matches is approximately more than one-third of the search
space. We will show that the algorithm will be able to handle the case where
the number of matches M is unknown in advance such that 1<=M<=N in
O(sqrt{N/M}). A performance comparison with Grover's algorithm will be
provided.Comment: 19 pages. Submitted to IJQI. Please forward comments/enquires for the
first author to [email protected]
Quantum Nonlocal Boxes Exhibit Stronger Distillability
The hypothetical nonlocal box (\textsf{NLB}) proposed by Popescu and Rohrlich
allows two spatially separated parties, Alice and Bob, to exhibit stronger than
quantum correlations. If the generated correlations are weak, they can
sometimes be distilled into a stronger correlation by repeated applications of
the \textsf{NLB}. Motivated by the limited distillability of \textsf{NLB}s, we
initiate here a study of the distillation of correlations for nonlocal boxes
that output quantum states rather than classical bits (\textsf{qNLB}s). We
propose a new protocol for distillation and show that it asymptotically
distills a class of correlated quantum nonlocal boxes to the value , whereas in contrast, the optimal non-adaptive
parity protocol for classical nonlocal boxes asymptotically distills only to
the value 3.0. We show that our protocol is an optimal non-adaptive protocol
for 1, 2 and 3 \textsf{qNLB} copies by constructing a matching dual solution
for the associated primal semidefinite program (SDP). We conclude that
\textsf{qNLB}s are a stronger resource for nonlocality than \textsf{NLB}s. The
main premise that develops from this conclusion is that the \textsf{NLB} model
is not the strongest resource to investigate the fundamental principles that
limit quantum nonlocality. As such, our work provides strong motivation to
reconsider the status quo of the principles that are known to limit nonlocal
correlations under the framework of \textsf{qNLB}s rather than \textsf{NLB}s.Comment: 25 pages, 7 figure
Higher Order Decompositions of Ordered Operator Exponentials
We present a decomposition scheme based on Lie-Trotter-Suzuki product
formulae to represent an ordered operator exponential as a product of ordinary
operator exponentials. We provide a rigorous proof that does not use a
time-displacement superoperator, and can be applied to non-analytic functions.
Our proof provides explicit bounds on the error and includes cases where the
functions are not infinitely differentiable. We show that Lie-Trotter-Suzuki
product formulae can still be used for functions that are not infinitely
differentiable, but that arbitrary order scaling may not be achieved.Comment: 16 pages, 1 figur
Quantum rejection sampling
Rejection sampling is a well-known method to sample from a target
distribution, given the ability to sample from a given distribution. The method
has been first formalized by von Neumann (1951) and has many applications in
classical computing. We define a quantum analogue of rejection sampling: given
a black box producing a coherent superposition of (possibly unknown) quantum
states with some amplitudes, the problem is to prepare a coherent superposition
of the same states, albeit with different target amplitudes. The main result of
this paper is a tight characterization of the query complexity of this quantum
state generation problem. We exhibit an algorithm, which we call quantum
rejection sampling, and analyze its cost using semidefinite programming. Our
proof of a matching lower bound is based on the automorphism principle which
allows to symmetrize any algorithm over the automorphism group of the problem.
Our main technical innovation is an extension of the automorphism principle to
continuous groups that arise for quantum state generation problems where the
oracle encodes unknown quantum states, instead of just classical data.
Furthermore, we illustrate how quantum rejection sampling may be used as a
primitive in designing quantum algorithms, by providing three different
applications. We first show that it was implicitly used in the quantum
algorithm for linear systems of equations by Harrow, Hassidim and Lloyd.
Secondly, we show that it can be used to speed up the main step in the quantum
Metropolis sampling algorithm by Temme et al.. Finally, we derive a new quantum
algorithm for the hidden shift problem of an arbitrary Boolean function and
relate its query complexity to "water-filling" of the Fourier spectrum.Comment: 19 pages, 5 figures, minor changes and a more compact style (to
appear in proceedings of ITCS 2012
An Algorithmic Argument for Nonadaptive Query Complexity Lower Bounds on Advised Quantum Computation
This paper employs a powerful argument, called an algorithmic argument, to
prove lower bounds of the quantum query complexity of a multiple-block ordered
search problem in which, given a block number i, we are to find a location of a
target keyword in an ordered list of the i-th block. Apart from much studied
polynomial and adversary methods for quantum query complexity lower bounds, our
argument shows that the multiple-block ordered search needs a large number of
nonadaptive oracle queries on a black-box model of quantum computation that is
also supplemented with advice. Our argument is also applied to the notions of
computational complexity theory: quantum truth-table reducibility and quantum
truth-table autoreducibility.Comment: 16 pages. An extended abstract will appear in the Proceedings of the
29th International Symposium on Mathematical Foundations of Computer Science,
Lecture Notes in Computer Science, Springer-Verlag, Prague, August 22-27,
200
On the Minimum Degree up to Local Complementation: Bounds and Complexity
The local minimum degree of a graph is the minimum degree reached by means of
a series of local complementations. In this paper, we investigate on this
quantity which plays an important role in quantum computation and quantum error
correcting codes. First, we show that the local minimum degree of the Paley
graph of order p is greater than sqrt{p} - 3/2, which is, up to our knowledge,
the highest known bound on an explicit family of graphs. Probabilistic methods
allows us to derive the existence of an infinite number of graphs whose local
minimum degree is linear in their order with constant 0.189 for graphs in
general and 0.110 for bipartite graphs. As regards the computational complexity
of the decision problem associated with the local minimum degree, we show that
it is NP-complete and that there exists no k-approximation algorithm for this
problem for any constant k unless P = NP.Comment: 11 page
Necessary Condition for the Quantum Adiabatic Approximation
A gapped quantum system that is adiabatically perturbed remains approximately
in its eigenstate after the evolution. We prove that, for constant gap, general
quantum processes that approximately prepare the final eigenstate require a
minimum time proportional to the ratio of the length of the eigenstate path to
the gap. Thus, no rigorous adiabatic condition can yield a smaller cost. We
also give a necessary condition for the adiabatic approximation that depends on
local properties of the path, which is appropriate when the gap varies.Comment: 5 pages, 1 figur
Multipartite Nonlocal Quantum Correlations Resistant to Imperfections
We use techniques for lower bounds on communication to derive necessary
conditions in terms of detector efficiency or amount of super-luminal
communication for being able to reproduce with classical local hidden-variable
theories the quantum correlations occurring in EPR-type experiments in the
presence of noise. We apply our method to an example involving n parties
sharing a GHZ-type state on which they carry out measurements and show that for
local-hidden variable theories, the amount of super-luminal classical
communication c and the detector efficiency eta are constrained by eta 2^(-c/n)
= O(n^(-1/6)) even for constant general error probability epsilon = O(1)
Quantum Network Coding
Since quantum information is continuous, its handling is sometimes
surprisingly harder than the classical counterpart. A typical example is
cloning; making a copy of digital information is straightforward but it is not
possible exactly for quantum information. The question in this paper is whether
or not quantum network coding is possible. Its classical counterpart is another
good example to show that digital information flow can be done much more
efficiently than conventional (say, liquid) flow.
Our answer to the question is similar to the case of cloning, namely, it is
shown that quantum network coding is possible if approximation is allowed, by
using a simple network model called Butterfly. In this network, there are two
flow paths, s_1 to t_1 and s_2 to t_2, which shares a single bottleneck channel
of capacity one. In the classical case, we can send two bits simultaneously,
one for each path, in spite of the bottleneck. Our results for quantum network
coding include: (i) We can send any quantum state |psi_1> from s_1 to t_1 and
|psi_2> from s_2 to t_2 simultaneously with a fidelity strictly greater than
1/2. (ii) If one of |psi_1> and |psi_2> is classical, then the fidelity can be
improved to 2/3. (iii) Similar improvement is also possible if |psi_1> and
|psi_2> are restricted to only a finite number of (previously known) states.
(iv) Several impossibility results including the general upper bound of the
fidelity are also given.Comment: 27pages, 11figures. The 12page version will appear in 24th
International Symposium on Theoretical Aspects of Computer Science (STACS
2007
Fetching marked items from an unsorted database in NMR ensemble computing
Searching a marked item or several marked items from an unsorted database is
a very difficult mathematical problem. Using classical computer, it requires
steps to find the target. Using a quantum computer, Grover's
algorithm uses steps. In NMR ensemble computing,
Brushweiler's algorithm uses steps. In this Letter, we propose an
algorithm that fetches marked items in an unsorted database directly. It
requires only a single query. It can find a single marked item or multiple
number of items.Comment: 4 pages and 1 figur
- …