11,037 research outputs found
Applications of the Quantum Algorithm for st-Connectivity
We present quantum algorithms for various problems related to graph connectivity. We give simple and query-optimal algorithms for cycle detection and odd-length cycle detection (bipartiteness) using a reduction to st-connectivity. Furthermore, we show that our algorithm for cycle detection has improved performance under the promise of large circuit rank or a small number of edges. We also provide algorithms for detecting even-length cycles and for estimating the circuit rank of a graph. All of our algorithms have logarithmic space complexity
Span programs and quantum algorithms for st-connectivity and claw detection
We introduce a span program that decides st-connectivity, and generalize the
span program to develop quantum algorithms for several graph problems. First,
we give an algorithm for st-connectivity that uses O(n d^{1/2}) quantum queries
to the n x n adjacency matrix to decide if vertices s and t are connected,
under the promise that they either are connected by a path of length at most d,
or are disconnected. We also show that if T is a path, a star with two
subdivided legs, or a subdivision of a claw, its presence as a subgraph in the
input graph G can be detected with O(n) quantum queries to the adjacency
matrix. Under the promise that G either contains T as a subgraph or does not
contain T as a minor, we give O(n)-query quantum algorithms for detecting T
either a triangle or a subdivision of a star. All these algorithms can be
implemented time efficiently and, except for the triangle-detection algorithm,
in logarithmic space. One of the main techniques is to modify the
st-connectivity span program to drop along the way "breadcrumbs," which must be
retrieved before the path from s is allowed to enter t.Comment: 18 pages, 4 figure
Approximate Span Programs
Span programs are a model of computation that have been used to design
quantum algorithms, mainly in the query model. For any decision problem, there
exists a span program that leads to an algorithm with optimal quantum query
complexity, but finding such an algorithm is generally challenging.
We consider new ways of designing quantum algorithms using span programs. We
show how any span program that decides a problem can also be used to decide
"property testing" versions of , or more generally, approximate the span
program witness size, a property of the input related to . For example,
using our techniques, the span program for OR, which can be used to design an
optimal algorithm for the OR function, can also be used to design optimal
algorithms for: threshold functions, in which we want to decide if the Hamming
weight of a string is above a threshold or far below, given the promise that
one of these is true; and approximate counting, in which we want to estimate
the Hamming weight of the input. We achieve these results by relaxing the
requirement that 1-inputs hit some target exactly in the span program, which
could make design of span programs easier.
We also give an exposition of span program structure, which increases the
understanding of this important model. One implication is alternative
algorithms for estimating the witness size when the phase gap of a certain
unitary can be lower bounded. We show how to lower bound this phase gap in some
cases.
As applications, we give the first upper bounds in the adjacency query model
on the quantum time complexity of estimating the effective resistance between
and , , of , and, when is a lower
bound on , by our phase gap lower bound, we can obtain , both using space
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