4 research outputs found
Local Quantum Codes from Subdivided Manifolds
For , we demonstrate the existence of quantum codes which are local
in dimension with qubits, distance , and dimension
, up to a factor. The distance is optimal up to
the polylog factor. The dimension is also optimal for this distance up to the
polylog factor. The proof combines the existence of asymptotically good quantum
codes, a procedure to build a manifold from a code by Freedman-Hastings, and a
quantitative embedding theorem by Gromov-Guth
A Generalized Isoperimetric Inequality via Thick Embeddings of Graphs
We prove a generalized isoperimetric inequality for a domain diffeomorphic to
a sphere that replaces filling volume with -dilation. Suppose is an open
set in diffeomorphic to a Euclidean -ball. We show that in
dimensions at least 4 there is a map from a standard Euclidean ball of radius
about to , with degree 1 on the boundary, and
-dilation bounded by some constant only depending on . We also give
an example in dimension 3 of an open set where no such map with small
-dilation can be found. The generalized isoperimetric inequality is
reduced to a theorem about thick embeddings of graphs which is proved using the
Kolmogorov-Barzdin theorem and the max-flow min-cut theorem. The proof of the
counterexample in dimension 3 relies on the coarea inequality and a short
winding number computation
On Freedman's link packings
Recently, Freedman [arXiv:2301.00295] introduced the idea of packing a
maximal number of links into a bounded region subject to geometric constraints,
and produced upper bounds on the packing number in some cases, while commenting
that these bounds seemed far too large. We show that the smallest of these
"extravagantly large" bounds is in fact sharp by constructing, for any link, a
packing of exponentially many copies as a function of the available volume. We
also produce improved and generalized upper bounds.Comment: 16 pages, 3 figure
Non-Clifford and parallelizable fault-tolerant logical gates on constant and almost-constant rate homological quantum LDPC codes via higher symmetries
We study parallel fault-tolerant quantum computing for families of
homological quantum low-density parity-check (LDPC) codes defined on
3-manifolds with constant or almost-constant encoding rate. We derive generic
formula for a transversal gate of color codes on general 3-manifolds, which
acts as collective non-Clifford logical CCZ gates on any triplet of logical
qubits with their logical- membranes having a triple
intersection at a single point. The triple intersection number is a topological
invariant, which also arises in the path integral of the emergent higher
symmetry operator in a topological quantum field theory: the
gauge theory. Moreover, the transversal gate of the color code corresponds
to a higher-form symmetry supported on a codimension-1 submanifold, giving rise
to exponentially many addressable and parallelizable logical CZ gates. We have
developed a generic formalism to compute the triple intersection invariants for
3-manifolds and also study the scaling of the Betti number and systoles with
volume for various 3-manifolds, which translates to the encoding rate and
distance. We further develop three types of LDPC codes supporting such logical
gates: (1) A quasi-hyperbolic code from the product of 2D hyperbolic surface
and a circle, with almost-constant rate and
distance; (2) A homological fibre bundle code with
rate and distance; (3) A specific family of 3D
hyperbolic codes: the Torelli mapping torus code, constructed from mapping tori
of a pseudo-Anosov element in the Torelli subgroup, which has constant rate
while the distance scaling is currently unknown. We then show a generic
constant-overhead scheme for applying a parallelizable universal gate set with
the aid of logical- measurements.Comment: 40 pages, 31 figure