Quantum Error Correction, Systolic Geometry, and Probabilistic Embeddings

A CSS quantum code C=(W1,W2) is a pair of orthogonal subspaces in 𝔽n2. The distance of C is the smallest hamming weight of a vector in W⊥1−W2 or W⊥2−W1. A large distance roughly means that the quantum code can correct many errors that affect the information stored in it.

In this talk we show how to construct a CSS quantum code that can be implemented in a 3-dimensional lattice and has near optimal distance. Along the way we will discuss a connection between quantum codes and systolic geometry made by Freedman and Hastings, as well as a probabilistic embedding result of Gromov and Guth.