119 research outputs found
Quantum walk based search algorithms
In this survey paper we give an intuitive treatment of the discrete time
quantization of classical Markov chains. Grover search and the quantum walk
based search algorithms of Ambainis, Szegedy and Magniez et al. will be stated
as quantum analogues of classical search procedures. We present a rather
detailed description of a somewhat simplified version of the MNRS algorithm.
Finally, in the query complexity model, we show how quantum walks can be
applied to the following search problems: Element Distinctness, Matrix Product
Verification, Restricted Range Associativity, Triangle, and Group
Commutativity.Comment: 16 pages, survey pape
Efficient quantum algorithms for some instances of the non-Abelian hidden subgroup problem
In this paper we show that certain special cases of the hidden subgroup
problem can be solved in polynomial time by a quantum algorithm. These special
cases involve finding hidden normal subgroups of solvable groups and
permutation groups, finding hidden subgroups of groups with small commutator
subgroup and of groups admitting an elementary Abelian normal 2-subgroup of
small index or with cyclic factor group.Comment: 10 page
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
Separating decision tree complexity from subcube partition complexity
The subcube partition model of computation is at least as powerful as
decision trees but no separation between these models was known. We show that
there exists a function whose deterministic subcube partition complexity is
asymptotically smaller than its randomized decision tree complexity, resolving
an open problem of Friedgut, Kahn, and Wigderson (2002). Our lower bound is
based on the information-theoretic techniques first introduced to lower bound
the randomized decision tree complexity of the recursive majority function.
We also show that the public-coin partition bound, the best known lower bound
method for randomized decision tree complexity subsuming other general
techniques such as block sensitivity, approximate degree, randomized
certificate complexity, and the classical adversary bound, also lower bounds
randomized subcube partition complexity. This shows that all these lower bound
techniques cannot prove optimal lower bounds for randomized decision tree
complexity, which answers an open question of Jain and Klauck (2010) and Jain,
Lee, and Vishnoi (2014).Comment: 16 pages, 1 figur
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
Linear time algorithm for quantum 2SAT
A canonical result about satisfiability theory is that the 2-SAT problem can
be solved in linear time, despite the NP-hardness of the 3-SAT problem. In the
quantum 2-SAT problem, we are given a family of 2-qubit projectors
on a system of qubits, and the task is to decide whether the Hamiltonian
has a 0-eigenvalue, or it is larger than for
some . The problem is not only a natural extension of the
classical 2-SAT problem to the quantum case, but is also equivalent to the
problem of finding the ground state of 2-local frustration-free Hamiltonians of
spin , a well-studied model believed to capture certain key
properties in modern condensed matter physics. While Bravyi has shown that the
quantum 2-SAT problem has a classical polynomial-time algorithm, the running
time of his algorithm is . In this paper we give a classical algorithm
with linear running time in the number of local projectors, therefore achieving
the best possible complexity.Comment: 20 page
On the black-box complexity of Sperner's Lemma
We present several results on the complexity of various forms of Sperner's
Lemma in the black-box model of computing. We give a deterministic algorithm
for Sperner problems over pseudo-manifolds of arbitrary dimension. The query
complexity of our algorithm is linear in the separation number of the skeleton
graph of the manifold and the size of its boundary. As a corollary we get an
deterministic query algorithm for the black-box version of the
problem {\bf 2D-SPERNER}, a well studied member of Papadimitriou's complexity
class PPAD. This upper bound matches the deterministic lower
bound of Crescenzi and Silvestri. The tightness of this bound was not known
before. In another result we prove for the same problem an
lower bound for its probabilistic, and an
lower bound for its quantum query complexity, showing
that all these measures are polynomially related.Comment: 16 pages with 1 figur
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