Download Communications In Mathematical Physics - Volume 287 by M. Aizenman (Chief Editor) PDF

By M. Aizenman (Chief Editor)

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Existence of a Mating Fix a Yoccoz’ polynomial f c which is not critically finite, non-renormalizable, and such that c does not belong to the 1/2-limb of the Mandelbrot set. 11, there exists a parameter value a such that Ra has the same combinatorics of the puzzle as f c for all d ∈ N. 1. Consider any z ∈ J (Ra ) which is not a preimage of αa or pa . Then the nested sequence of puzzle pieces Pd (z) shrinks to z : Pd (z) = {z}. Proof. Assume first that there exists some N > 0 such that the orbit of z is disjoint from PN (−1).

The proof of the QMA-completeness result requires an additional idea beyond the usual techniques in the area: Not all illegal configurations can be ruled out by local checks, so instead we rule out such illegal configurations because they would, in the future, evolve into a state which can be seen locally to be illegal. Our construction implies (assuming the quantum Church-Turing thesis and that quantum computers cannot efficiently solve QMA-complete problems) that there are one-dimensional systems which take an exponential time to relax to their ground states at any temperature, making them candidates for being one-dimensional spin glasses.

The quantum adiabatic theorem guarantees that if the Hamiltonian is changed slowly enough, the system stays close The Power of Quantum Systems on a Line 43 to its ground state; the time required to safely move from the initial Hamiltonian to the final Hamiltonian is polynomial in the minimal spectral gap between the ground state and the first excited state over the course of the computation. The first adiabatic quantum algorithm was introduced by Farhi et al. [FGGS00, FGG+ 01], though the idea of encoding the solution to an optimization problem in the ground state of a Hamiltonian appeared as early as 1988 [ACdf90,ACdf88].

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