261 research outputs found
A Quantum Random Walk Search Algorithm
Quantum random walks on graphs have been shown to display many interesting
properties, including exponentially fast hitting times when compared with their
classical counterparts. However, it is still unclear how to use these novel
properties to gain an algorithmic speed-up over classical algorithms. In this
paper, we present a quantum search algorithm based on the quantum random walk
architecture that provides such a speed-up. It will be shown that this
algorithm performs an oracle search on a database of items with
calls to the oracle, yielding a speed-up similar to other quantum
search algorithms. It appears that the quantum random walk formulation has
considerable flexibility, presenting interesting opportunities for development
of other, possibly novel quantum algorithms.Comment: 13 pages, 3 figure
Efficient estimation of nearly sparse many-body quantum Hamiltonians
We develop an efficient and robust approach to Hamiltonian identification for
multipartite quantum systems based on the method of compressed sensing. This
work demonstrates that with only O(s log(d)) experimental configurations,
consisting of random local preparations and measurements, one can estimate the
Hamiltonian of a d-dimensional system, provided that the Hamiltonian is nearly
s-sparse in a known basis. We numerically simulate the performance of this
algorithm for three- and four-body interactions in spin-coupled quantum dots
and atoms in optical lattices. Furthermore, we apply the algorithm to
characterize Hamiltonian fine structure and unknown system-bath interactions.Comment: 8 pages, 2 figures. Title is changed. Detailed error analysis is
added. Figures are updated with additional clarifying discussion
Quantum computation with devices whose contents are never read
In classical computation, a "write-only memory" (WOM) is little more than an
oxymoron, and the addition of WOM to a (deterministic or probabilistic)
classical computer brings no advantage. We prove that quantum computers that
are augmented with WOM can solve problems that neither a classical computer
with WOM nor a quantum computer without WOM can solve, when all other resource
bounds are equal. We focus on realtime quantum finite automata, and examine the
increase in their power effected by the addition of WOMs with different access
modes and capacities. Some problems that are unsolvable by two-way
probabilistic Turing machines using sublogarithmic amounts of read/write memory
are shown to be solvable by these enhanced automata.Comment: 32 pages, a preliminary version of this work was presented in the 9th
International Conference on Unconventional Computation (UC2010
Green function approach for scattering quantum walks
In this work a Green function approach for scattering quantum walks is
developed. The exact formula has the form of a sum over paths and always can be
cast into a closed analytic expression for arbitrary topologies and position
dependent quantum amplitudes. By introducing the step and path operators, it is
shown how to extract any information about the system from the Green function.
The method relevant features are demonstrated by discussing in details an
example, a general diamond-shaped graph.Comment: 13 pages, 6 figures, this article was selected by APS for Virtual
Journal of Quantum Information, Vol 11, Iss 11 (2011
Quantum fingerprinting
Classical fingerprinting associates with each string a shorter string (its
fingerprint), such that, with high probability, any two distinct strings can be
distinguished by comparing their fingerprints alone. The fingerprints can be
exponentially smaller than the original strings if the parties preparing the
fingerprints share a random key, but not if they only have access to
uncorrelated random sources. In this paper we show that fingerprints consisting
of quantum information can be made exponentially smaller than the original
strings without any correlations or entanglement between the parties: we give a
scheme where the quantum fingerprints are exponentially shorter than the
original strings and we give a test that distinguishes any two unknown quantum
fingerprints with high probability. Our scheme implies an exponential
quantum/classical gap for the equality problem in the simultaneous message
passing model of communication complexity. We optimize several aspects of our
scheme.Comment: 8 pages, LaTeX, one figur
Random Oracles in a Quantum World
The interest in post-quantum cryptography - classical systems that remain
secure in the presence of a quantum adversary - has generated elegant proposals
for new cryptosystems. Some of these systems are set in the random oracle model
and are proven secure relative to adversaries that have classical access to the
random oracle. We argue that to prove post-quantum security one needs to prove
security in the quantum-accessible random oracle model where the adversary can
query the random oracle with quantum states.
We begin by separating the classical and quantum-accessible random oracle
models by presenting a scheme that is secure when the adversary is given
classical access to the random oracle, but is insecure when the adversary can
make quantum oracle queries. We then set out to develop generic conditions
under which a classical random oracle proof implies security in the
quantum-accessible random oracle model. We introduce the concept of a
history-free reduction which is a category of classical random oracle
reductions that basically determine oracle answers independently of the history
of previous queries, and we prove that such reductions imply security in the
quantum model. We then show that certain post-quantum proposals, including ones
based on lattices, can be proven secure using history-free reductions and are
therefore post-quantum secure. We conclude with a rich set of open problems in
this area.Comment: 38 pages, v2: many substantial changes and extensions, merged with a
related paper by Boneh and Zhandr
Recurrence of biased quantum walks on a line
The Polya number of a classical random walk on a regular lattice is known to
depend solely on the dimension of the lattice. For one and two dimensions it
equals one, meaning unit probability to return to the origin. This result is
extremely sensitive to the directional symmetry, any deviation from the equal
probability to travel in each direction results in a change of the character of
the walk from recurrent to transient. Applying our definition of the Polya
number to quantum walks on a line we show that the recurrence character of
quantum walks is more stable against bias. We determine the range of parameters
for which biased quantum walks remain recurrent. We find that there exist
genuine biased quantum walks which are recurrent.Comment: Journal reference added, minor corrections in the tex
Almost uniform sampling via quantum walks
Many classical randomized algorithms (e.g., approximation algorithms for
#P-complete problems) utilize the following random walk algorithm for {\em
almost uniform sampling} from a state space of cardinality : run a
symmetric ergodic Markov chain on for long enough to obtain a random
state from within total variation distance of the uniform
distribution over . The running time of this algorithm, the so-called {\em
mixing time} of , is , where
is the spectral gap of .
We present a natural quantum version of this algorithm based on repeated
measurements of the {\em quantum walk} . We show that it
samples almost uniformly from with logarithmic dependence on
just as the classical walk does; previously, no such
quantum walk algorithm was known. We then outline a framework for analyzing its
running time and formulate two plausible conjectures which together would imply
that it runs in time when is
the standard transition matrix of a constant-degree graph. We prove each
conjecture for a subclass of Cayley graphs.Comment: 13 pages; v2 added NSF grant info; v3 incorporated feedbac
Ground states of unfrustrated spin Hamiltonians satisfy an area law
We show that ground states of unfrustrated quantum spin-1/2 systems on
general lattices satisfy an entanglement area law, provided that the
Hamiltonian can be decomposed into nearest-neighbor interaction terms which
have entangled excited states. The ground state manifold can be efficiently
described as the image of a low-dimensional subspace of low Schmidt measure,
under an efficiently contractible tree-tensor network. This structure gives
rise to the possibility of efficiently simulating the complete ground space
(which is in general degenerate). We briefly discuss "non-generic" cases,
including highly degenerate interactions with product eigenbases, using a
relationship to percolation theory. We finally assess the possibility of using
such tree tensor networks to simulate almost frustration-free spin models.Comment: 14 pages, 4 figures, small corrections, added a referenc
Quantum random walks with history dependence
We introduce a multi-coin discrete quantum random walk where the amplitude
for a coin flip depends upon previous tosses. Although the corresponding
classical random walk is unbiased, a bias can be introduced into the quantum
walk by varying the history dependence. By mixing the biased random walk with
an unbiased one, the direction of the bias can be reversed leading to a new
quantum version of Parrondo's paradox.Comment: 8 pages, 6 figures, RevTe
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