36,064 research outputs found
Qutrit Dichromatic Calculus and Its Universality
We introduce a dichromatic calculus (RG) for qutrit systems. We show that the
decomposition of the qutrit Hadamard gate is non-unique and not derivable from
the dichromatic calculus. As an application of the dichromatic calculus, we
depict a quantum algorithm with a single qutrit. Since it is not easy to
decompose an arbitrary d by d unitary matrix into Z and X phase gates when d >
2, the proof of the universality of qudit ZX calculus for quantum mechanics is
far from trivial. We construct a counterexample to Ranchin's universality
proof, and give another proof by Lie theory that the qudit ZX calculus contains
all single qudit unitary transformations, which implies that qudit ZX calculus,
with qutrit dichromatic calculus as a special case, is universal for quantum
mechanics.Comment: In Proceedings QPL 2014, arXiv:1412.810
Optimal convergence rates for the three-dimensional turbulent flow equations
In this paper we are concerned with the convergence rate of solutions to the
three-dimensional turbulent flow equations. By combining the -
estimates for the linearized equations and an elaborate energy method, the
convergence rates are obtained in various norms for the solution to the
equilibrium state in the whole space, when the initial perturbation of the
equilibrium state is small in -framework. More precisely, the optimal
convergence rates of the solutions and its first order derivatives in
-norm are obtained when the -norm of the perturbation is bounded for
some .Comment: 16 page
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