58,094 research outputs found
Semiclassical Expansions, the Strong Quantum Limit, and Duality
We show how to complement Feynman's exponential of the action so that it
exhibits a Z_2 duality symmetry. The latter illustrates a relativity principle
for the notion of quantum versus classical.Comment: 5 pages, references adde
On the Quantum Complexity of Closest Pair and Related Problems
The closest pair problem is a fundamental problem of computational geometry:
given a set of points in a -dimensional space, find a pair with the
smallest distance. A classical algorithm taught in introductory courses solves
this problem in time in constant dimensions (i.e., when ).
This paper asks and answers the question of the problem's quantum time
complexity. Specifically, we give an algorithm in constant
dimensions, which is optimal up to a polylogarithmic factor by the lower bound
on the quantum query complexity of element distinctness. The key to our
algorithm is an efficient history-independent data structure that supports
quantum interference.
In dimensions, no known quantum algorithms perform
better than brute force search, with a quadratic speedup provided by Grover's
algorithm. To give evidence that the quadratic speedup is nearly optimal, we
initiate the study of quantum fine-grained complexity and introduce the Quantum
Strong Exponential Time Hypothesis (QSETH), which is based on the assumption
that Grover's algorithm is optimal for CNF-SAT when the clause width is large.
We show that the na\"{i}ve Grover approach to closest pair in higher dimensions
is optimal up to an factor unless QSETH is false. We also study the
bichromatic closest pair problem and the orthogonal vectors problem, with
broadly similar results.Comment: 46 pages, 3 figures, presentation improve
Why Philosophers Should Care About Computational Complexity
One might think that, once we know something is computable, how efficiently
it can be computed is a practical question with little further philosophical
importance. In this essay, I offer a detailed case that one would be wrong. In
particular, I argue that computational complexity theory---the field that
studies the resources (such as time, space, and randomness) needed to solve
computational problems---leads to new perspectives on the nature of
mathematical knowledge, the strong AI debate, computationalism, the problem of
logical omniscience, Hume's problem of induction, Goodman's grue riddle, the
foundations of quantum mechanics, economic rationality, closed timelike curves,
and several other topics of philosophical interest. I end by discussing aspects
of complexity theory itself that could benefit from philosophical analysis.Comment: 58 pages, to appear in "Computability: G\"odel, Turing, Church, and
beyond," MIT Press, 2012. Some minor clarifications and corrections; new
references adde
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