23,650 research outputs found
Quantum Algorithms for the Triangle Problem
We present two new quantum algorithms that either find a triangle (a copy of
) in an undirected graph on nodes, or reject if is triangle
free. The first algorithm uses combinatorial ideas with Grover Search and makes
queries. The second algorithm uses
queries, and it is based on a design concept of Ambainis~\cite{amb04} that
incorporates the benefits of quantum walks into Grover search~\cite{gro96}. The
first algorithm uses only qubits in its quantum subroutines,
whereas the second one uses O(n) qubits. The Triangle Problem was first treated
in~\cite{bdhhmsw01}, where an algorithm with query complexity
was presented, where is the number of edges of .Comment: Several typos are fixed, and full proofs are included. Full version
of the paper accepted to SODA'0
Derandomization of quantum algorithm for triangle finding
Derandomization is the process of taking a randomized algorithm and turning
it into a deterministic algorithm, which has attracted great attention in
classical computing. In quantum computing, it is challenging and intriguing to
derandomize quantum algorithms, due to the inherent randomness of quantum
mechanics. The significance of derandomizing quantum algorithms lies not only
in theoretically proving that the success probability can essentially be 1
without sacrificing quantum speedups, but also in experimentally improving the
success rate when the algorithm is implemented on a real quantum computer.
In this paper, we focus on derandomizing quanmtum algorithms for the triangle
sum problem (including the famous triangle finding problem as a special case),
which asks to find a triangle in an edge-weighted graph with vertices, such
that its edges sum up to a given weight.We show that when the graph is promised
to contain at most one target triangle, there exists a deterministic quantum
algorithm that either finds the triangle if it exists or outputs ``no
triangle'' if none exists. It makes queries to the edge weight
matrix oracle, and thus has the same complexity with the state-of-art
bounded-error quantum algorithm. To achieve this derandomization, we make full
use several techniques:nested quantum walks with quantum data structure,
deterministic quantum search with adjustable parameters, and dimensional
reduction of quantum walk search on Johnson graph
On the Power of Non-Adaptive Learning Graphs
We introduce a notion of the quantum query complexity of a certificate
structure. This is a formalisation of a well-known observation that many
quantum query algorithms only require the knowledge of the disposition of
possible certificates in the input string, not the precise values therein.
Next, we derive a dual formulation of the complexity of a non-adaptive
learning graph, and use it to show that non-adaptive learning graphs are tight
for all certificate structures. By this, we mean that there exists a function
possessing the certificate structure and such that a learning graph gives an
optimal quantum query algorithm for it.
For a special case of certificate structures generated by certificates of
bounded size, we construct a relatively general class of functions having this
property. The construction is based on orthogonal arrays, and generalizes the
quantum query lower bound for the -sum problem derived recently in
arXiv:1206.6528.
Finally, we use these results to show that the learning graph for the
triangle problem from arXiv:1210.1014 is almost optimal in these settings. This
also gives a quantum query lower bound for the triangle-sum problem.Comment: 16 pages, 1.5 figures v2: the main result generalised for all
certificate structures, a bug in the proof of Proposition 17 fixe
Matching Triangles and Triangle Collection: Hardness based on a Weak Quantum Conjecture
Classically, for many computational problems one can conclude time lower
bounds conditioned on the hardness of one or more of key problems: k-SAT, 3SUM
and APSP. More recently, similar results have been derived in the quantum
setting conditioned on the hardness of k-SAT and 3SUM. This is done using
fine-grained reductions, where the approach is to (1) select a key problem
that, for some function , is conjectured to not be solvable by any
time algorithm for any constant (in a
fixed model of computation), and (2) reduce in a fine-grained way to these
computational problems, thus giving (mostly) tight conditional time lower
bounds for them.
Interestingly, for Delta-Matching Triangles and Triangle Collection,
classical hardness results have been derived conditioned on hardness of all
three mentioned key problems. More precisely, it is proven that an
time classical algorithm for either of these two graph
problems would imply faster classical algorithms for k-SAT, 3SUM and APSP,
which makes Delta-Matching Triangles and Triangle Collection worthwhile to
study.
In this paper, we show that an time quantum algorithm for
either of these two graph problems would imply faster quantum algorithms for
k-SAT, 3SUM, and APSP. We first formulate a quantum hardness conjecture for
APSP and then present quantum reductions from k-SAT, 3SUM, and APSP to
Delta-Matching Triangles and Triangle Collection. Additionally, based on the
quantum APSP conjecture, we are also able to prove quantum lower bounds for a
matrix problem and many graph problems. The matching upper bounds follow
trivially for most of them, except for Delta-Matching Triangles and Triangle
Collection for which we present quantum algorithms that require careful use of
data structures and Ambainis' variable time search
Quantum algorithms for subset finding
Recently, Ambainis gave an O(N^(2/3))-query quantum walk algorithm for
element distinctness, and more generally, an O(N^(L/(L+1)))-query algorithm for
finding L equal numbers. We point out that this algorithm actually solves a
much more general problem, the problem of finding a subset of size L that
satisfies any given property. We review the algorithm and give a considerably
simplified analysis of its query complexity. We present several applications,
including two algorithms for the problem of finding an L-clique in an N-vertex
graph. One of these algorithms uses O(N^(2L/(L+1))) edge queries, and the other
uses \tilde{O}(N^((5L-2)/(2L+4))), which is an improvement for L <= 5. The
latter algorithm generalizes a recent result of Magniez, Santha, and Szegedy,
who considered the case L=3 (finding a triangle). We also pose two open
problems regarding continuous time quantum walk and lower bounds.Comment: 7 pages; note added on related results in quant-ph/031013
On the Power of Non-adaptive Learning Graphs
We introduce a notion of the quantum query complexity of a certificate structure. This is a formalization of a well-known observation that many quantum query algorithms only require the knowledge of the position of possible certificates in the input string, not the precise values therein.
Next, we derive a dual formulation of the complexity of a non-adaptive learning graph and use it to show that non-adaptive learning graphs are tight for all certificate structures. By this, we mean that there exists a function possessing the certificate structure such that a learning graph gives an optimal quantum query algorithm for it.
For a special case of certificate structures generated by certificates of bounded size, we construct a relatively general class of functions having this property. The construction is based on orthogonal arrays and generalizes the quantum query lower bound for the k-sum problem derived recently by Belovs and Špalek (Proceeding of 4th ACM ITCS, 323–328, 2013).
Finally, we use these results to show that the learning graph for the triangle problem by Lee et al. (Proceeding of 24th ACM-SIAM SODA, 1486–1502, 2013) is almost optimal in the above settings. This also gives a quantum query lower bound for the triangle sum problem.National Science Foundation (U.S.) (Scott Aaronson’s Alan T. Waterman Award
An Improved Approximation Algorithm for Quantum Max-Cut
We give an approximation algorithm for Quantum Max-Cut which works by
rounding an SDP relaxation to an entangled quantum state. The SDP is used to
choose the parameters of a variational quantum circuit. The entangled state is
then represented as the quantum circuit applied to a product state. It achieves
an approximation ratio of 0.582 on triangle-free graphs. The previous best
algorithms of Anshu, Gosset, Morenz, and Parekh, Thompson achieved
approximation ratios of 0.531 and 0.533 respectively. In addition, we study the
EPR Hamiltonian, which we argue is a natural intermediate problem which
isolates some key quantum features of local Hamiltonian problems. For the EPR
Hamiltonian, we give an approximation algorithm with approximation ratio on all graphs
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