20,517 research outputs found
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
Ring Learning With Errors: A crossroads between postquantum cryptography, machine learning and number theory
The present survey reports on the state of the art of the different
cryptographic functionalities built upon the ring learning with errors problem
and its interplay with several classical problems in algebraic number theory.
The survey is based to a certain extent on an invited course given by the
author at the Basque Center for Applied Mathematics in September 2018.Comment: arXiv admin note: text overlap with arXiv:1508.01375 by other
authors/ comment of the author: quotation has been added to Theorem 5.
Quantum Computation by Adiabatic Evolution
We give a quantum algorithm for solving instances of the satisfiability
problem, based on adiabatic evolution. The evolution of the quantum state is
governed by a time-dependent Hamiltonian that interpolates between an initial
Hamiltonian, whose ground state is easy to construct, and a final Hamiltonian,
whose ground state encodes the satisfying assignment. To ensure that the system
evolves to the desired final ground state, the evolution time must be big
enough. The time required depends on the minimum energy difference between the
two lowest states of the interpolating Hamiltonian. We are unable to estimate
this gap in general. We give some special symmetric cases of the satisfiability
problem where the symmetry allows us to estimate the gap and we show that, in
these cases, our algorithm runs in polynomial time.Comment: 24 pages, 12 figures, LaTeX, amssymb,amsmath, BoxedEPS packages;
email to [email protected]
On solving systems of random linear disequations
An important subcase of the hidden subgroup problem is equivalent to the
shift problem over abelian groups. An efficient solution to the latter problem
would serve as a building block of quantum hidden subgroup algorithms over
solvable groups. The main idea of a promising approach to the shift problem is
reduction to solving systems of certain random disequations in finite abelian
groups. The random disequations are actually generalizations of linear
functions distributed nearly uniformly over those not containing a specific
group element in the kernel. In this paper we give an algorithm which finds the
solutions of a system of N random linear disequations in an abelian p-group A
in time polynomial in N, where N=(log|A|)^{O(q)}, and q is the exponent of A.Comment: 13 page
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