346 research outputs found
A universal adiabatic quantum query algorithm
Quantum query complexity is known to be characterized by the so-called
quantum adversary bound. While this result has been proved in the standard
discrete-time model of quantum computation, it also holds for continuous-time
(or Hamiltonian-based) quantum computation, due to a known equivalence between
these two query complexity models. In this work, we revisit this result by
providing a direct proof in the continuous-time model. One originality of our
proof is that it draws new connections between the adversary bound, a modern
technique of theoretical computer science, and early theorems of quantum
mechanics. Indeed, the proof of the lower bound is based on Ehrenfest's
theorem, while the upper bound relies on the adiabatic theorem, as it goes by
constructing a universal adiabatic quantum query algorithm. Another originality
is that we use for the first time in the context of quantum computation a
version of the adiabatic theorem that does not require a spectral gap.Comment: 22 pages, compared to v1, includes a rigorous proof of the
correctness of the algorithm based on a version of the adiabatic theorem that
does not require a spectral ga
Adiabatic quantum search algorithm for structured problems
The study of quantum computation has been motivated by the hope of finding
efficient quantum algorithms for solving classically hard problems. In this
context, quantum algorithms by local adiabatic evolution have been shown to
solve an unstructured search problem with a quadratic speed-up over a classical
search, just as Grover's algorithm. In this paper, we study how the structure
of the search problem may be exploited to further improve the efficiency of
these quantum adiabatic algorithms. We show that by nesting a partial search
over a reduced set of variables into a global search, it is possible to devise
quantum adiabatic algorithms with a complexity that, although still
exponential, grows with a reduced order in the problem size.Comment: 7 pages, 0 figur
Information-theoretic lower bounds for quantum sorting
We analyze the quantum query complexity of sorting under partial information.
In this problem, we are given a partially ordered set and are asked to
identify a linear extension of using pairwise comparisons. For the standard
sorting problem, in which is empty, it is known that the quantum query
complexity is not asymptotically smaller than the classical
information-theoretic lower bound. We prove that this holds for a wide class of
partially ordered sets, thereby improving on a result from Yao (STOC'04)
Finding a marked node on any graph by continuous-time quantum walk
Spatial search by discrete-time quantum walk can find a marked node on any
ergodic, reversible Markov chain quadratically faster than its classical
counterpart, i.e.\ in a time that is in the square root of the hitting time of
. However, in the framework of continuous-time quantum walks, it was
previously unknown whether such general speed-up is possible. In fact, in this
framework, the widely used quantum algorithm by Childs and Goldstone fails to
achieve such a speedup. Furthermore, it is not clear how to apply this
algorithm for searching any Markov chain . In this article, we aim to
reconcile the apparent differences between the running times of spatial search
algorithms in these two frameworks. We first present a modified version of the
Childs and Goldstone algorithm which can search for a marked element for any
ergodic, reversible by performing a quantum walk on its edges. Although
this approach improves the algorithmic running time for several instances, it
cannot provide a generic quadratic speedup for any . Secondly, using the
framework of interpolated Markov chains, we provide a new spatial search
algorithm by continuous-time quantum walk which can find a marked node on any
in the square root of the classical hitting time. In the scenario where
multiple nodes are marked, the algorithmic running time scales as the square
root of a quantity known as the extended hitting time. Our results establish a
novel connection between discrete-time and continuous-time quantum walks and
can be used to develop a number of Markov chain-based quantum algorithms.Comment: This version deals only with new algorithms for spatial search by
continuous-time quantum walk (CTQW) on ergodic, reversible Markov chains.
Please see arXiv:2004.12686 for results on the necessary and sufficient
conditions for the optimality of the Childs and Goldstone algorithm for
spatial search by CTQ
Quantum circuit implementation of the Hamiltonian versions of Grover's algorithm
We analyze three different quantum search algorithms, the traditional
Grover's algorithm, its continuous-time analogue by Hamiltonian evolution, and
finally the quantum search by local adiabatic evolution. We show that they are
closely related algorithms in the sense that they all perform a rotation, at a
constant angular velocity, from a uniform superposition of all states to the
solution state. This make it possible to implement the last two algorithms by
Hamiltonian evolution on a conventional quantum circuit, while keeping the
quadratic speedup of Grover's original algorithm.Comment: 5 pages, 3 figure
Search via Quantum Walk
We propose a new method for designing quantum search algorithms for finding a
"marked" element in the state space of a classical Markov chain. The algorithm
is based on a quantum walk \'a la Szegedy (2004) that is defined in terms of
the Markov chain. The main new idea is to apply quantum phase estimation to the
quantum walk in order to implement an approximate reflection operator. This
operator is then used in an amplitude amplification scheme. As a result we
considerably expand the scope of the previous approaches of Ambainis (2004) and
Szegedy (2004). Our algorithm combines the benefits of these approaches in
terms of being able to find marked elements, incurring the smaller cost of the
two, and being applicable to a larger class of Markov chains. In addition, it
is conceptually simple and avoids some technical difficulties in the previous
analyses of several algorithms based on quantum walk.Comment: 21 pages. Various modifications and improvements, especially in
Section
Symmetry-assisted adversaries for quantum state generation
We introduce a new quantum adversary method to prove lower bounds on the
query complexity of the quantum state generation problem. This problem
encompasses both, the computation of partial or total functions and the
preparation of target quantum states. There has been hope for quite some time
that quantum state generation might be a route to tackle the {\sc Graph
Isomorphism} problem. We show that for the related problem of {\sc Index
Erasure} our method leads to a lower bound of which matches
an upper bound obtained via reduction to quantum search on elements. This
closes an open problem first raised by Shi [FOCS'02].
Our approach is based on two ideas: (i) on the one hand we generalize the
known additive and multiplicative adversary methods to the case of quantum
state generation, (ii) on the other hand we show how the symmetries of the
underlying problem can be leveraged for the design of optimal adversary
matrices and dramatically simplify the computation of adversary bounds. Taken
together, these two ideas give the new result for {\sc Index Erasure} by using
the representation theory of the symmetric group. Also, the method can lead to
lower bounds even for small success probability, contrary to the standard
adversary method. Furthermore, we answer an open question due to \v{S}palek
[CCC'08] by showing that the multiplicative version of the adversary method is
stronger than the additive one for any problem. Finally, we prove that the
multiplicative bound satisfies a strong direct product theorem, extending a
result by \v{S}palek to quantum state generation problems.Comment: 35 pages, 5 figure
Information transmission via entangled quantum states in Gaussian channels with memory
Gaussian quantum channels have recently attracted a growing interest, since
they may lead to a tractable approach to the generally hard problem of
evaluating quantum channel capacities. However, the analysis performed so far
has always been restricted to memoryless channels. Here, we consider the case
of a bosonic Gaussian channel with memory, and show that the classical capacity
can be significantly enhanced by employing entangled input symbols instead of
product symbols.Comment: 13 pages, 5 figures, Workshop on Quantum entanglement in physical and
information sciences, Pisa, December 14-18, 200
Simulating quantum correlations as a distributed sampling problem
It is known that quantum correlations exhibited by a maximally entangled
qubit pair can be simulated with the help of shared randomness, supplemented
with additional resources, such as communication, post-selection or non-local
boxes. For instance, in the case of projective measurements, it is possible to
solve this problem with protocols using one bit of communication or making one
use of a non-local box. We show that this problem reduces to a distributed
sampling problem. We give a new method to obtain samples from a biased
distribution, starting with shared random variables following a uniform
distribution, and use it to build distributed sampling protocols. This approach
allows us to derive, in a simpler and unified way, many existing protocols for
projective measurements, and extend them to positive operator value
measurements. Moreover, this approach naturally leads to a local hidden
variable model for Werner states.Comment: 13 pages, 2 figure
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