99 research outputs found
Regularity of degenerate Hessian equations
We show a second order a priori estimate for solutions to the complex k-Hessian equation on a compact Kähler manifold provided the (k−1)st root of the right hand side is C1,1. This improves an estimate of Hou-Ma-Wu (Math Res Lett 17:547-561, 2010). An example is provided to show that the exponent is sharp
Carleman estimates for elliptic operators with jumps at an interface: Anisotropic case and sharp geometric conditions
We consider a second-order selfadjoint elliptic operator with an anisotropic
diffusion matrix having a jump across a smooth hypersurface. We prove the
existence of a weight-function such that a Carleman estimate holds true. We
moreover prove that the conditions imposed on the weight function are
necessary.Comment: 43 page
Introducing Structure to Expedite Quantum Search
We present a novel quantum algorithm for solving the unstructured search
problem with one marked element. Our algorithm allows generating quantum
circuits that use asymptotically fewer additional quantum gates than the famous
Grover's algorithm and may be successfully executed on NISQ devices. We prove
that our algorithm is optimal in the total number of elementary gates up to a
multiplicative constant. As many NP-hard problems are not in fact unstructured,
we also describe the \emph{partial uncompute} technique which exploits the
oracle structure and allows a significant reduction in the number of elementary
gates required to find the solution. Combining these results allows us to use
asymptotically smaller number of elementary gates than the Grover's algorithm
in various applications, keeping the number of queries to the oracle
essentially the same. We show how the results can be applied to solve hard
combinatorial problems, for example Unique k-SAT. Additionally, we show how to
asymptotically reduce the number of elementary gates required to solve the
unstructured search problem with multiple marked elements.Comment: 22 pages, 7 figure
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