Abstract and physical constraints on quantum LDPC codes

Abstract: DATE: 28 October, 2021 TIME: 11:00 am – 12:00 pm AEDT (Local Sydney time) TITLE: Abstract and physical constraints on quantum LDPC codes TOPIC: Quantum error correction

SPEAKER: Dr Anirudh Krishna AFFILIATION: Stanford University, California, USA

ABSTRACT: Seminal results by Bravyi, Poulin and Terhal have shown that quantum codes are limited by locality. As a consequence, all topological codes witness sharp tradeoffs between their rate and distance. Quantum LDPC codes can be viewed as a generalization of topological codes constructed using spatially-nonlocal connections. It is unclear what, if any, fundamental constraints these codes obey. The state-of-the-art code parameters are far from what their classical counterparts can achieve. We explore this question and present no-go results that shed some light on what is not possible.

We approach this question in two ways, using abstract and physical constraints. First, we use a graph-theoretic representation of a quantum code to show that the connectivity of this representation allows us to understand limitations of the associated code. We obtain generalizations of the Bravyi-Poulin-Terhal and Bravyi-Koenig bounds. We then study the complementary problem of embedding a code in D Euclidean dimensions. We ask how many long-range interactions we need to obtain a target code dimension k and distance d. Focusing on 2 dimensions (and ignoring polylogarthmic corrections), we find that a code with distance d requires Ω(d) interactions of length Ω(d/√n). Furthermore, a constant-rate code distance d requires Ω(n) interactions of length Ω(√d).

This is joint work with Nouédyn Baspin. It is based on the papers arXiv: 2106.00765 (https://arxiv.org/abs/2106.00765) and arXiv: 2109.10982 (https://arxiv.org/abs/2109.10982)

quantum computing and information

Audience: researchers in the topic

( paper )

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