hyperbolic

Single-Shot Decoding of Linear Rate LDPC Quantum Codes with High Performance

We construct and analyze a family of low-density parity check (LDPC) quantum codes with a linear encoding rate, polynomial scaling distance and efficient decoding schemes. The code family is based on tessellations of closed, four-dimensional, …

The Critical Properties of the Ising Model in Hyperbolic Space

The Ising model exhibits qualitatively different properties in hyperbolic space in comparison to its flat space counterpart. Due to the negative curvature, a finite fraction of the total number of spins reside at the boundary of a volume in …

Single-Shot-Decoding with High Thresholds in LDPC Quantum Codes with Constant Encoding Rate

SLIDES It is believed that active quantum error correction will be an essential ingredient to build a scalable quantum computer. The currently favored scheme is the surface code due to its high decoding threshold and efficient decoding algorithm. However, it suffers from large overheads which are even more severe when parity check measurements are subject to errors and have to be repeated. Furthermore, the number of encoded qubits in the surface code does not grow with system size, leading to a sub-optimal use of the physical qubits.

The Ising Model in Curved Geometries

SLIDES The study of statistical mechanics in curved geometries has recently gained an increasing amount of attention. Particularly the Ising model in negatively curved spaces has been studied as a mean to understand exotic crystals, soft-matter and field theories in Anti-de Sitter spaces. We analyze the Ising model on in the hyperbolic plane as well as 2+1-Anti-de Sitter (AdS) space using high temperature series-expansion and Monte-Carlo simulations. While series expansions have been performed on the hyperbolic plane before, we study a wider class of hyperbolic lattices and go to much higher order, allowing us to analyze the dependency of critical phenomena on the magnitude of curvature.

The small stellated dodecahedron code and friends

We explore a distance-3 homological CSS quantum code, namely the small stellated dodecahedron code, for dense storage of quantum information and we compare its performance with the distance-3 surface code. The data and ancilla qubits of the small …

Hyperbolic and semi-hyperbolic surface codes for quantum storage

We show how a hyperbolic surface code could be used for overhead-efficient quantum storage. We give numerical evidence for a noise threshold of for the -hyperbolic surface code in a phenomenological noise model (as compared with for the toric …

Constructions and Noise Threshold of Hyperbolic Surface Codes

We show how to obtain concrete constructions of homological quantum codes based on tilings of 2D surfaces with constant negative curvature (hyperbolic surfaces). This construction results in two-dimensional quantum codes whose tradeoff of encoding …