2,359 research outputs found
Quantum Knitting
We analyze the connections between the mathematical theory of knots and
quantum physics by addressing a number of algorithmic questions related to both
knots and braid groups.
Knots can be distinguished by means of `knot invariants', among which the
Jones polynomial plays a prominent role, since it can be associated with
observables in topological quantum field theory.
Although the problem of computing the Jones polynomial is intractable in the
framework of classical complexity theory, it has been recently recognized that
a quantum computer is capable of approximating it in an efficient way. The
quantum algorithms discussed here represent a breakthrough for quantum
computation, since approximating the Jones polynomial is actually a `universal
problem', namely the hardest problem that a quantum computer can efficiently
handle.Comment: 29 pages, 5 figures; to appear in Laser Journa
A 3-Stranded Quantum Algorithm for the Jones Polynomial
Let K be a 3-stranded knot (or link), and let L denote the number of
crossings in K. Let and be two positive real
numbers such that is less than or equal to 1.
In this paper, we create two algorithms for computing the value of the Jones
polynomial of K at all points of the unit circle in the complex
plane such that the absolute value of is less than or equal to .
The first algorithm, called the classical 3-stranded braid (3-SB) algorithm,
is a classical deterministic algorithm that has time complexity O(L). The
second, called the quantum 3-SB algorithm, is a quantum algorithm that computes
an estimate of the Jones polynomial of K at within a precision of
with a probability of success bounded below by $1-\epsilon_{2}%.
The execution time complexity of this algorithm is O(nL), where n is the
ceiling function of (ln(4/\epsilon_{2}))/(2(\epsilon_{2})^2). The compilation
time complexity, i.e., an asymptotic measure of the amount of time to assemble
the hardware that executes the algorithm, is O(L).Comment: 19 pages, 10 figures, to appear in Proc. SPIE, 6573-29, (2007
On the Complexity of Random Quantum Computations and the Jones Polynomial
There is a natural relationship between Jones polynomials and quantum
computation. We use this relationship to show that the complexity of evaluating
relative-error approximations of Jones polynomials can be used to bound the
classical complexity of approximately simulating random quantum computations.
We prove that random quantum computations cannot be classically simulated up to
a constant total variation distance, under the assumption that (1) the
Polynomial Hierarchy does not collapse and (2) the average-case complexity of
relative-error approximations of the Jones polynomial matches the worst-case
complexity over a constant fraction of random links. Our results provide a
straightforward relationship between the approximation of Jones polynomials and
the complexity of random quantum computations.Comment: 8 pages, 4 figure
On the Quantum Computational Complexity of the Ising Spin Glass Partition Function and of Knot Invariants
It is shown that the canonical problem of classical statistical
thermodynamics, the computation of the partition function, is in the case of
+/-J Ising spin glasses a particular instance of certain simple sums known as
quadratically signed weight enumerators (QWGTs). On the other hand it is known
that quantum computing is polynomially equivalent to classical probabilistic
computing with an oracle for estimating QWGTs. This suggests a connection
between the partition function estimation problem for spin glasses and quantum
computation. This connection extends to knots and graph theory via the
equivalence of the Kauffman polynomial and the partition function for the Potts
model.Comment: 8 pages, incl. 2 figures. v2: Substantially rewritte
Post Quantum Cryptography from Mutant Prime Knots
By resorting to basic features of topological knot theory we propose a
(classical) cryptographic protocol based on the `difficulty' of decomposing
complex knots generated as connected sums of prime knots and their mutants. The
scheme combines an asymmetric public key protocol with symmetric private ones
and is intrinsecally secure against quantum eavesdropper attacks.Comment: 14 pages, 5 figure
- …