Talks
Spring 2020

Twisted Product Constructions for LDPC Quantum Codes

Friday, Jul. 16, 2021 1:15 pm2:00 pm

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Speaker: 

Nikolas Breuckmann (University College London)

Location: 

Zoom

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

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