73,210 research outputs found
Quantum Walk Search through Potential Barriers
An ideal quantum walk transitions from one vertex to another with perfect
fidelity, but in physical systems, the particle may be hindered by potential
energy barriers. Then the particle has some amplitude of tunneling through the
barriers, and some amplitude of staying put. We investigate the algorithmic
consequence of such barriers for the quantum walk formulation of Grover's
algorithm. We prove that the failure amplitude must scale as
for search to retain its quantum runtime; otherwise, it searches
in classical time. Thus searching larger "databases" requires
increasingly reliable hop operations or error correction. This condition holds
for both discrete- and continuous-time quantum walks.Comment: 13 pages, 7 figure
Faster Quantum Walk Search on a Weighted Graph
A randomly walking quantum particle evolving by Schr\"odinger's equation
searches for a unique marked vertex on the "simplex of complete graphs" in time
. In this paper, we give a weighted version of this graph that
preserves vertex-transitivity, and we show that the time to search on it can be
reduced to nearly . To prove this, we introduce two novel
extensions to degenerate perturbation theory: an adjustment that distinguishes
the weights of the edges, and a method to determine how precisely the jumping
rate of the quantum walk must be chosen.Comment: 8 pages, 5 figure
Faster Search by Lackadaisical Quantum Walk
In the typical model, a discrete-time coined quantum walk searching the 2D
grid for a marked vertex achieves a success probability of in
steps, which with amplitude amplification yields an
overall runtime of . We show that making the quantum walk
lackadaisical or lazy by adding a self-loop of weight to each vertex
speeds up the search, causing the success probability to reach a constant near
in steps, thus yielding an
improvement over the typical, loopless algorithm. This improved runtime matches
the best known quantum algorithms for this search problem. Our results are
based on numerical simulations since the algorithm is not an instance of the
abstract search algorithm.Comment: 9 pages, 4 figure
Coined Quantum Walks on Weighted Graphs
We define a discrete-time, coined quantum walk on weighted graphs that is
inspired by Szegedy's quantum walk. Using this, we prove that many
lackadaisical quantum walks, where each vertex has integer self-loops, can
be generalized to a quantum walk where each vertex has a single self-loop of
real-valued weight . We apply this real-valued lackadaisical quantum walk to
two problems. First, we analyze it on the line or one-dimensional lattice,
showing that it is exactly equivalent to a continuous deformation of the
three-state Grover walk with faster ballistic dispersion. Second, we generalize
Grover's algorithm, or search on the complete graph, to have a weighted
self-loop at each vertex, yielding an improved success probability when .Comment: 14 pages, 5 figure
Grover Search with Lackadaisical Quantum Walks
The lazy random walk, where the walker has some probability of staying put,
is a useful tool in classical algorithms. We propose a quantum analogue, the
lackadaisical quantum walk, where each vertex is given self-loops, and we
investigate its effects on Grover's algorithm when formulated as search for a
marked vertex on the complete graph of vertices. For the discrete-time
quantum walk using the phase flip coin, adding a self-loop to each vertex
boosts the success probability from 1/2 to 1. Additional self-loops, however,
decrease the success probability. Using instead the Ambainis, Kempe, and Rivosh
(2005) coin, adding self-loops simply slows down the search. These coins also
differ in that the first is faster than classical when scales less than
, while the second requires that scale less than . Finally,
continuous-time quantum walks differ from both of these discrete-time
examples---the self-loops make no difference at all. These behaviors generalize
to multiple marked vertices.Comment: 16 pages, 7 figures; additional 2-page corrigendu
Quantum Walk Search on Johnson Graphs
The Johnson graph is defined by symbols, where vertices are
-element subsets of the symbols, and vertices are adjacent if they differ in
exactly one symbol. In particular, is the complete graph , and
is the strongly regular triangular graph , both of which are
known to support fast spatial search by continuous-time quantum walk. In this
paper, we prove that , which is the -tetrahedral graph, also
supports fast search. In the process, we show that a change of basis is needed
for degenerate perturbation theory to accurately describe the dynamics. This
method can also be applied to general Johnson graphs with fixed .Comment: 17 pages, 9 figure
Quantum Walk Search with Time-Reversal Symmetry Breaking
We formulate Grover's unstructured search algorithm as a chiral quantum walk,
where transitioning in one direction has a phase conjugate to transitioning in
the opposite direction. For small phases, this breaking of time-reversal
symmetry is too small to significantly affect the evolution: the system still
approximately evolves in its ground and first excited states, rotating to the
marked vertex in time . Increasing the phase does not change
the runtime, but rather changes the support for the 2D subspace, so the system
evolves in its first and second excited states, or its second and third excited
states, and so forth. Apart from the critical phases corresponding to these
transitions in the support, which become more frequent as the phase grows, this
reveals that our model of quantum search is robust against time-reversal
symmetry breaking.Comment: 14 pages, 8 figure
Quantum Walk on the Line through Potential Barriers
Quantum walks are well-known for their ballistic dispersion, traveling
away in steps, which is quadratically faster than a classical
random walk's diffusive spreading. In physical implementations of the walk,
however, the particle may need to tunnel through a potential barrier to hop,
and a naive calculation suggests this could eliminate the ballistic transport.
We show by explicit calculation, however, that such a loss does not occur.
Rather, the dispersion is retained, with only the coefficient
changing, which additionally gives a way to detect and quantify the hopping
errors in experiments.Comment: 14 pages, 6 figure
Engineering the Success of Quantum Walk Search Using Weighted Graphs
Continuous-time quantum walks are natural tools for spatial search, where one
searches for a marked vertex in a graph. Sometimes, the structure of the graph
causes the walker to get trapped, such that the probability of finding the
marked vertex is limited. We give an example with two linked cliques, proving
that the captive probability can be liberated by increasing the weights of the
links. This allows the search to succeed with probability 1 without increasing
the energy scaling of the algorithm. Further increasing the weights, however,
slows the runtime, so the optimal search requires weights that are neither too
weak nor too strong.Comment: 11 pages, 8 figure
Quantum Search with Multiple Walk Steps per Oracle Query
We identify a key difference between quantum search by discrete- and
continuous-time quantum walks: a discrete-time walk typically performs one walk
step per oracle query, whereas a continuous-time walk can effectively perform
multiple walk steps per query while only counting query time. As a result, we
show that continuous-time quantum walks can outperform their discrete-time
counterparts, even though both achieve quadratic speedups over their
corresponding classical random walks. To provide greater equity, we allow the
discrete-time quantum walk to also take multiple walk steps per oracle query
while only counting queries. Then it matches the continuous-time algorithm's
runtime, but such that it is a cubic speedup over its corresponding classical
random walk. This yields the first example of a greater-than-quadratic speedup
for quantum search over its corresponding classical random walk.Comment: 10 pages, 5 figure
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