QCDA Seminar

The joint QCDA seminar is a bi-weekly meeting of the QCDA groups where an internal or external speaker presents new results in quantum fault-tolerance or quantum information.

 

Joining the Meetings

The QCDA meeting is open to non-QCDA members. In order to join the QCDA meeting please send an email to Nikolas Breuckmann (n.breuckmann@ucl.ac.uk) to receive the Zoom invitations and links to the recorded talks.

 

Schedule

18th January 2021 (15:00 London time): Anthony Leverrier

Title: Quantum XYZ Product Codes

Abstract: We study a three-fold variant of the hypergraph product code construction, differing from the standard homological product of three classical codes. When instantiated with 3 classical LDPC codes, this “XYZ product” yields a non CSS quantum LDPC code which might display a large minimum distance. The simplest instance of this construction, corresponding to the product of 3 repetition codes, is a non CSS variant of the 3-dimensional toric code known as the Chamon code. The general construction was introduced in Maurice’s PhD thesis, but has remained poorly understood so far. The reason is that while hypergraph product codes can be analyzed with combinatorial tools, the XYZ product codes depend crucially on the algebraic properties of the parity-check matrices of the three classical codes, making their analysis much more involved.
Our main motivation for studying XYZ product codes is that the natural representatives of logical operators are two-dimensional objects. This contrasts with standard hypergraph product codes in 3 dimensions which always admit one-dimensional logical operators. In particular, specific instances of XYZ product codes might display a minimum distance as large as Θ(N^2/3) which would beat the current record for the minimum distance of quantum LDPC codes held by fiber bundle codes. While we do not prove this result here, we obtain the dimension of a large class of XYZ product codes, and when restricting to codes with dimension 1, we reduce the problem of computing the minimum distance to a more elementary combinatorial problem involving binary 3-tensors. We also discuss in detail some families of XYZ product codes in three dimensions with local interaction. Some of these codes seem to share properties with Haah’s cubic codes and might be interesting candidates for self-correcting quantum memories with a logarithmic energy barrier.

https://arxiv.org/abs/2011.09746

 

8th February 2021 (16:00 London time): Pavel Panteleev

Title: Quantum LDPC Codes with Almost Linear Minimum Distance

Abstract: We give a construction of quantum LDPC codes of dimension \Theta(\log N) and distance \Theta(N/\log N) as the code length N\to\infty. Using a product of chain complexes this construction also provides a family of quantum LDPC codes of distance \Omega(N^{1-\alpha/2}/\log N) and dimension \Omega(N^\alpha \log N), where 0 \le \alpha < 1. We also introduce and study a new operation called lifted product, which naturally generalizes the product operations for quantum codes and chain complexes. Moreover, as a simple byproduct of our results on quantum codes, we obtain a new result on classical codes. We show that for any fixed R < 1 there exists an asymptotically good family of classical quasi-cyclic LDPC codes of rate at least R with, in some sense, optimal circulant size \Omega(N/\log N) as the code length N\to\infty.

https://arxiv.org/abs/2012.04068