15,174 research outputs found
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
Classical Ising model test for quantum circuits
We exploit a recently constructed mapping between quantum circuits and graphs
in order to prove that circuits corresponding to certain planar graphs can be
efficiently simulated classically. The proof uses an expression for the Ising
model partition function in terms of quadratically signed weight enumerators
(QWGTs), which are polynomials that arise naturally in an expansion of quantum
circuits in terms of rotations involving Pauli matrices. We combine this
expression with a known efficient classical algorithm for the Ising partition
function of any planar graph in the absence of an external magnetic field, and
the Robertson-Seymour theorem from graph theory. We give as an example a set of
quantum circuits with a small number of non-nearest neighbor gates which admit
an efficient classical simulation.Comment: 17 pages, 2 figures. v2: main result strengthened by removing
oracular settin
Majority-Vote Cellular Automata, Ising Dynamics, and P-Completeness
We study cellular automata where the state at each site is decided by a
majority vote of the sites in its neighborhood. These are equivalent, for a
restricted set of initial conditions, to non-zero probability transitions in
single spin-flip dynamics of the Ising model at zero temperature.
We show that in three or more dimensions these systems can simulate Boolean
circuits of AND and OR gates, and are therefore P-complete. That is, predicting
their state t time-steps in the future is at least as hard as any other problem
that takes polynomial time on a serial computer.
Therefore, unless a widely believed conjecture in computer science is false,
it is impossible even with parallel computation to predict majority-vote
cellular automata, or zero-temperature single spin-flip Ising dynamics,
qualitatively faster than by explicit simulation.Comment: 10 pages with figure
Fault Tolerance in Cellular Automata at High Fault Rates
A commonly used model for fault-tolerant computation is that of cellular
automata. The essential difficulty of fault-tolerant computation is present in
the special case of simply remembering a bit in the presence of faults, and
that is the case we treat in this paper. We are concerned with the degree (the
number of neighboring cells on which the state transition function depends)
needed to achieve fault tolerance when the fault rate is high (nearly 1/2). We
consider both the traditional transient fault model (where faults occur
independently in time and space) and a recently introduced combined fault model
which also includes manufacturing faults (which occur independently in space,
but which affect cells for all time). We also consider both a purely
probabilistic fault model (in which the states of cells are perturbed at
exactly the fault rate) and an adversarial model (in which the occurrence of a
fault gives control of the state to an omniscient adversary). We show that
there are cellular automata that can tolerate a fault rate (with
) with degree , even with adversarial combined
faults. The simplest such automata are based on infinite regular trees, but our
results also apply to other structures (such as hyperbolic tessellations) that
contain infinite regular trees. We also obtain a lower bound of
, even with purely probabilistic transient faults only
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