3,094 research outputs found
Cusps of lattices in rank 1 Lie groups over local fields
Let G be the group of rational points of a semisimple algebraic group of rank
1 over a nonarchimedean local field. We improve upon Lubotzky's analysis of
graphs of groups describing the action of lattices in G on its Bruhat-Tits tree
assuming a condition on unipotents in G. The condition holds for all but a few
types of rank 1 groups. A fairly straightforward simplification of Lubotzky's
definition of a cusp of a lattice is the key step to our results. We take the
opportunity to reprove Lubotzky's part in the analysis from this foundation.Comment: to appear in Geometriae Dedicat
Courant algebroids and Poisson Geometry
Given a manifold M with an action of a quadratic Lie algebra d, such that all
stabilizer algebras are co-isotropic in d, we show that the product M\times d
becomes a Courant algebroid over M. If the bilinear form on d is split, the
choice of transverse Lagrangian subspaces g_1, g_2 of d defines a bivector
field on M, which is Poisson if (d,g_1,g_2) is a Manin triple. In this way, we
recover the Poisson structures of Lu-Yakimov, and in particular the Evens-Lu
Poisson structures on the variety of Lagrangian Grassmannians and on the de
Concini-Procesi compactifications. Various Poisson maps between such examples
are interpreted in terms of the behaviour of Lagrangian splittings under
Courant morphisms
Codes and Protocols for Distilling , controlled-, and Toffoli Gates
We present several different codes and protocols to distill ,
controlled-, and Toffoli (or ) gates. One construction is based on
codes that generalize the triorthogonal codes, allowing any of these gates to
be induced at the logical level by transversal . We present a randomized
construction of generalized triorthogonal codes obtaining an asymptotic
distillation efficiency . We also present a Reed-Muller
based construction of these codes which obtains a worse but performs
well at small sizes. Additionally, we present protocols based on checking the
stabilizers of magic states at the logical level by transversal gates
applied to codes; these protocols generalize the protocols of 1703.07847.
Several examples, including a Reed-Muller code for -to-Toffoli distillation,
punctured Reed-Muller codes for -gate distillation, and some of the check
based protocols, require a lower ratio of input gates to output gates than
other known protocols at the given order of error correction for the given code
size. In particular, we find a T-gate to Toffoli gate code with
distance as well as triorthogonal codes with parameters
with very low prefactors in front of
the leading order error terms in those codes.Comment: 28 pages. (v2) fixed a part of the proof on random triorthogonal
codes, added comments on Clifford circuits for Reed-Muller states (v3) minor
chang
Homological Product Codes
Quantum codes with low-weight stabilizers known as LDPC codes have been
actively studied recently due to their simple syndrome readout circuits and
potential applications in fault-tolerant quantum computing. However, all
families of quantum LDPC codes known to this date suffer from a poor distance
scaling limited by the square-root of the code length. This is in a sharp
contrast with the classical case where good families of LDPC codes are known
that combine constant encoding rate and linear distance. Here we propose the
first family of good quantum codes with low-weight stabilizers. The new codes
have a constant encoding rate, linear distance, and stabilizers acting on at
most qubits, where is the code length. For comparison, all
previously known families of good quantum codes have stabilizers of linear
weight. Our proof combines two techniques: randomized constructions of good
quantum codes and the homological product operation from algebraic topology. We
conjecture that similar methods can produce good stabilizer codes with
stabilizer weight for any . Finally, we apply the homological
product to construct new small codes with low-weight stabilizers.Comment: 49 page
On Deformation Quantization of Poisson-Lie Groups and Moduli Spaces of Flat Connections
We give simple explicit formulas for deformation quantization of Poisson-Lie
groups and of similar Poisson manifolds which can be represented as moduli
spaces of flat connections on surfaces. The star products depend on a choice of
Drinfe\v{l}d associator and are obtained by applying certain monoidal functors
(fusion and reduction) to commutative algebras in Drinfe\v{l}d categories. From
a geometric point of view this construction can be understood as a quantization
of the quasi-Poisson structures on moduli spaces of flat connections.Comment: 11 page
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