19 research outputs found
Expander Graphs and Coding Theory
Expander graphs are highly connected sparse graphs which lie at the interface of many diļ¬erent ļ¬elds of study. For example, they play important roles in prime sieves, cryptography, compressive sensing, metric embedding, and coding theory to name a few. This thesis focuses on the connections between sparse graphs and coding theory. It is a major challenge to explicitly construct sparse graphs with good expansion properties, for example Ramanujan graphs. Nevertheless, explicit constructions do exist, and in this thesis, we survey many of these constructions up to this point including a new construction which slightly improves on an earlier edge expansion bound. The edge expansion of a graph is crucial in applications, and it is well-known that computing the edge expansion of an arbitrary graph is NP-hard. We present a simple algo-rithm for approximating the edge expansion of a graph using linear programming techniques. While Andersen and Lang (2008) proved similar results, our analysis attacks the problem from a diļ¬erent vantage point and was discovered independently. The main contribution in the thesis is a new result in fast decoding for expander codes. Current algorithms in the literature can decode a constant fraction of errors in linear time but require that the underlying graphs have vertex expansion at least 1/2. We present a fast decoding algorithm that can decode a constant fraction of errors in linear time given any vertex expansion (even if it is much smaller than 1/2) by using a stronger local code, and the fraction of errors corrected almost doubles that of Viderman (2013)
Tailoring three-dimensional topological codes for biased noise
Tailored topological stabilizer codes in two dimensions have been shown to
exhibit high storage threshold error rates and improved subthreshold
performance under biased Pauli noise. Three-dimensional (3D) topological codes
can allow for several advantages including a transversal implementation of
non-Clifford logical gates, single-shot decoding strategies, parallelized
decoding in the case of fracton codes as well as construction of fractal
lattice codes. Motivated by this, we tailor 3D topological codes for enhanced
storage performance under biased Pauli noise. We present Clifford deformations
of various 3D topological codes, such that they exhibit a threshold error rate
of under infinitely biased Pauli noise. Our examples include the 3D
surface code on the cubic lattice, the 3D surface code on a checkerboard
lattice that lends itself to a subsystem code with a single-shot decoder, the
3D color code, as well as fracton models such as the X-cube model, the
Sierpinski model and the Haah code. We use the belief propagation with ordered
statistics decoder (BP-OSD) to study threshold error rates at finite bias. We
also present a rotated layout for the 3D surface code, which uses roughly half
the number of physical qubits for the same code distance under appropriate
boundary conditions. Imposing coprime periodic dimensions on this rotated
layout leads to logical operators of weight at infinite bias and a
corresponding subthreshold scaling of the logical failure rate,
where is the number of physical qubits in the code. Even though this
scaling is unstable due to the existence of logical representations with
low-rate Pauli errors, the number of such representations scales only
polynomially for the Clifford-deformed code, leading to an enhanced effective
distance.Comment: 51 pages, 34 figure