8,371 research outputs found
Hamiltonian Simulation Using Linear Combinations of Unitary Operations
We present a new approach to simulating Hamiltonian dynamics based on
implementing linear combinations of unitary operations rather than products of
unitary operations. The resulting algorithm has superior performance to
existing simulation algorithms based on product formulas and, most notably,
scales better with the simulation error than any known Hamiltonian simulation
technique. Our main tool is a general method to nearly deterministically
implement linear combinations of nearby unitary operations, which we show is
optimal among a large class of methods.Comment: 18 pages, 3 figure
Stochastic Optimization of PCA with Capped MSG
We study PCA as a stochastic optimization problem and propose a novel
stochastic approximation algorithm which we refer to as "Matrix Stochastic
Gradient" (MSG), as well as a practical variant, Capped MSG. We study the
method both theoretically and empirically
The high-dimensional cohomology of the moduli space of curves with level structures II: punctures and boundary
We give two proofs that appropriately defined congruence subgroups of the
mapping class group of a surface with punctures/boundary have enormous amounts
of rational cohomology in their virtual cohomological dimension. In particular
we give bounds that are super-exponential in each of three variables: number of
punctures, number of boundary components, and genus, generalizing work of
Fullarton-Putman. Along the way, we give a simplified account of a theorem of
Harer explaining how to relate the homotopy type of the curve complex of a
multiply-punctured surface to the curve complex of a once-punctured surface
through a process that can be viewed as an analogue of a Birman exact sequence
for curve complexes.
As an application, we prove upper and lower bounds on the coherent
cohomological dimension of the moduli space of curves with marked points. For
, we compute this coherent cohomological dimension for any number of
marked points. In contrast to our bounds on cohomology, when the surface has marked points, these bounds turn out to be independent of , and
depend only on the genus.Comment: 29 pages, 3 figures; some small correction
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