Twisted Product Constructions for LDPC Quantum Codes

Simons Institute Workshop Quantum Wave in Computing Reunion 2021



The construction of low-density parity check (LDPC) quantum codes has some unique challenges in comparison to classical LDPC codes. While there are classical codes with constant encoding rate and linear distance, so-called good codes, no equivalent statement is known for LDPC quantum codes. Even more severely, no LDPC quantum codes with distance larger than polylog(N)sqrt(N) were known to exist for a long time.

Hastings-Haah-O’Donnell recently showed that this apparent polylog(N)sqrt(N) distance barrier can be broken [1]. They introduced “fiber bundle codes” which are inspired by a construction from differential geometry. Using a randomized procedure they obtain a family of codes which encode K in Theta(N^(3/5)) qubits and have distance D in Omega(N^(3/5)/polylog(N)) with high probability.

A similar, randomized construction, called “lifted product”, was introduced by Panteleev-Kalachev [2]. They further improved the distance scaling to be almost linear.

Breuckmann-Eberhardt showed how these codes can be constructed explicitly / without randomness via a third construction called the “balanced product” [3]. Further, balanced products are in some sense more symmetric, leading to a concrete construction of LDPC quantum codes that is conjectured to have optimal scaling code parameters.

We will give an introduction to these novel fiber bundle / lifted / balanced product constructions and their relationship.

[1] Hastings, Haah, and O’Donnell, “Fiber bundle codes”, arXiv:2009.03921, 2020 [2] Panteleev and Kalachev, “Quantum LDPC codes with almost linear minimum distance”, arXiv:2012.04068, 2020 [3] Breuckmann and Eberhardt, “Balanced Product Quantum Codes”, arXiv:2012.09271, 2020