2,567 research outputs found
Computational Complexity for Physicists
These lecture notes are an informal introduction to the theory of
computational complexity and its links to quantum computing and statistical
mechanics.Comment: references updated, reprint available from
http://itp.nat.uni-magdeburg.de/~mertens/papers/complexity.shtm
Exponential algorithmic speedup by quantum walk
We construct an oracular (i.e., black box) problem that can be solved
exponentially faster on a quantum computer than on a classical computer. The
quantum algorithm is based on a continuous time quantum walk, and thus employs
a different technique from previous quantum algorithms based on quantum Fourier
transforms. We show how to implement the quantum walk efficiently in our
oracular setting. We then show how this quantum walk can be used to solve our
problem by rapidly traversing a graph. Finally, we prove that no classical
algorithm can solve this problem with high probability in subexponential time.Comment: 24 pages, 7 figures; minor corrections and clarification
Quantum walks on general graphs
Quantum walks, both discrete (coined) and continuous time, on a general graph
of N vertices with undirected edges are reviewed in some detail. The resource
requirements for implementing a quantum walk as a program on a quantum computer
are compared and found to be very similar for both discrete and continuous time
walks. The role of the oracle, and how it changes if more prior information
about the graph is available, is also discussed.Comment: 8 pages, v2: substantial rewrite improves clarity, corrects errors
and omissions; v3: removes major error in final section and integrates
remainder into other sections, figures remove
The Quantum Adiabatic Algorithm applied to random optimization problems: the quantum spin glass perspective
Among various algorithms designed to exploit the specific properties of
quantum computers with respect to classical ones, the quantum adiabatic
algorithm is a versatile proposition to find the minimal value of an arbitrary
cost function (ground state energy). Random optimization problems provide a
natural testbed to compare its efficiency with that of classical algorithms.
These problems correspond to mean field spin glasses that have been extensively
studied in the classical case. This paper reviews recent analytical works that
extended these studies to incorporate the effect of quantum fluctuations, and
presents also some original results in this direction.Comment: 151 pages, 21 figure
Quiet Planting in the Locked Constraint Satisfaction Problems
We study the planted ensemble of locked constraint satisfaction problems. We
describe the connection between the random and planted ensembles. The use of
the cavity method is combined with arguments from reconstruction on trees and
first and second moment considerations; in particular the connection with the
reconstruction on trees appears to be crucial. Our main result is the location
of the hard region in the planted ensemble. In a part of that hard region
instances have with high probability a single satisfying assignment.Comment: 21 pages, revised versio
Three Puzzles on Mathematics, Computation, and Games
In this lecture I will talk about three mathematical puzzles involving
mathematics and computation that have preoccupied me over the years. The first
puzzle is to understand the amazing success of the simplex algorithm for linear
programming. The second puzzle is about errors made when votes are counted
during elections. The third puzzle is: are quantum computers possible?Comment: ICM 2018 plenary lecture, Rio de Janeiro, 36 pages, 7 Figure
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