8,858 research outputs found
Prime Factorization in the Duality Computer
We give algorithms to factorize large integers in the duality computer. We
provide three duality algorithms for factorization based on a naive
factorization method, the Shor algorithm in quantum computing, and the Fermat's
method in classical computing. All these algorithms are polynomial in the input
size.Comment: 4 page
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
Quantum simulation of partially distinguishable boson sampling
Boson Sampling is the problem of sampling from the same output probability
distribution as a collection of indistinguishable single photons input into a
linear interferometer. It has been shown that, subject to certain computational
complexity conjectures, in general the problem is difficult to solve
classically, motivating optical experiments aimed at demonstrating quantum
computational "supremacy". There are a number of challenges faced by such
experiments, including the generation of indistinguishable single photons. We
provide a quantum circuit that simulates bosonic sampling with arbitrarily
distinguishable particles. This makes clear how distinguishabililty leads to
decoherence in the standard quantum circuit model, allowing insight to be
gained. At the heart of the circuit is the quantum Schur transform, which
follows from a representation theoretic approach to the physics of
distinguishable particles in first quantisation. The techniques are quite
general and have application beyond boson sampling.Comment: 25 pages, 4 figures, 2 algorithms, comments welcom
Approximating Turaev-Viro 3-manifold invariants is universal for quantum computation
The Turaev-Viro invariants are scalar topological invariants of compact,
orientable 3-manifolds. We give a quantum algorithm for additively
approximating Turaev-Viro invariants of a manifold presented by a Heegaard
splitting. The algorithm is motivated by the relationship between topological
quantum computers and (2+1)-D topological quantum field theories. Its accuracy
is shown to be nontrivial, as the same algorithm, after efficient classical
preprocessing, can solve any problem efficiently decidable by a quantum
computer. Thus approximating certain Turaev-Viro invariants of manifolds
presented by Heegaard splittings is a universal problem for quantum
computation. This establishes a novel relation between the task of
distinguishing non-homeomorphic 3-manifolds and the power of a general quantum
computer.Comment: 4 pages, 3 figure
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