2,811 research outputs found
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
Improving Quantum Query Complexity of Boolean Matrix Multiplication Using Graph Collision
The quantum query complexity of Boolean matrix multiplication is typically
studied as a function of the matrix dimension, n, as well as the number of 1s
in the output, \ell. We prove an upper bound of O (n\sqrt{\ell}) for all values
of \ell. This is an improvement over previous algorithms for all values of
\ell. On the other hand, we show that for any \eps < 1 and any \ell <= \eps
n^2, there is an \Omega(n\sqrt{\ell}) lower bound for this problem, showing
that our algorithm is essentially tight.
We first reduce Boolean matrix multiplication to several instances of graph
collision. We then provide an algorithm that takes advantage of the fact that
the underlying graph in all of our instances is very dense to find all graph
collisions efficiently
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 Ć palek (Proceeding of 4th ACM ITCS, 323â328, 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, 1486â1502, 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 Aaronsonâs Alan T. Waterman Award
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
Applications of the Adversary Method in Quantum Query Algorithms
In the thesis, we use a recently developed tight characterisation of quantum
query complexity, the adversary bound, to develop new quantum algorithms and
lower bounds. Our results are as follows:
* We develop a new technique for the construction of quantum algorithms:
learning graphs.
* We use learning graphs to improve quantum query complexity of the triangle
detection and the -distinctness problems.
* We prove tight lower bounds for the -sum and the triangle sum problems.
* We construct quantum algorithms for some subgraph-finding problems that are
optimal in terms of query, time and space complexities.
* We develop a generalisation of quantum walks that connects electrical
properties of a graph and its quantum hitting time. We use it to construct a
time-efficient quantum algorithm for 3-distinctness.Comment: PhD Thesis, 169 page
Quantum query complexity of state conversion
State conversion generalizes query complexity to the problem of converting
between two input-dependent quantum states by making queries to the input. We
characterize the complexity of this problem by introducing a natural
information-theoretic norm that extends the Schur product operator norm. The
complexity of converting between two systems of states is given by the distance
between them, as measured by this norm.
In the special case of function evaluation, the norm is closely related to
the general adversary bound, a semi-definite program that lower-bounds the
number of input queries needed by a quantum algorithm to evaluate a function.
We thus obtain that the general adversary bound characterizes the quantum query
complexity of any function whatsoever. This generalizes and simplifies the
proof of the same result in the case of boolean input and output. Also in the
case of function evaluation, we show that our norm satisfies a remarkable
composition property, implying that the quantum query complexity of the
composition of two functions is at most the product of the query complexities
of the functions, up to a constant. Finally, our result implies that discrete
and continuous-time query models are equivalent in the bounded-error setting,
even for the general state-conversion problem.Comment: 19 pages, 2 figures; heavily revised with new results and simpler
proof
Quantum Algorithms for Classical Probability Distributions
We study quantum algorithms working on classical probability distributions. We formulate four different models for accessing a classical probability distribution on a quantum computer, which are derived from previous work on the topic, and study their mutual relationships.
Additionally, we prove that quantum query complexity of distinguishing two probability distributions is given by their inverse Hellinger distance, which gives a quadratic improvement over classical query complexity for any pair of distributions.
The results are obtained by using the adversary method for state-generating input oracles and for distinguishing probability distributions on input strings
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