From Möbius Strips to Twisted Toric Codes

Abstract

In the past few years, the search for good quantum low density parity check (qLDPC) codes suddenly took flight, and many different constructions of these codes have since been presented, including many product constructions. As these code constructions have a natural interpretation in the language of homology, this thesis studies the interplay between homological algebra and various recent product constructions of qLDPC codes.

First, we provide an overview of the theory of singular homology, cellular homology, and homological algebra over vector spaces, and use this theory to analyse two product constructions: the hypergraph product construction and the distance balancing procedure. These constructions can be interpreted as tensor products of chain complexes over vector spaces. We present new proofs for results from the literature regarding the distance and the number of encoded qubits of these product codes.

Secondly, we survey the theory of homology over ring modules, and use this theory to interpret and analyse another product code construction (the lifted product code construction), which is a generalisation of the hypergraph product construction. We prove a Künneth theorem for these codes, and use this theorem to prove a formula for the number of qubits such codes encode.

Thirdly, we investigate the homology of fibre bundles. To this end, we provide an overview of the theory of covering spaces and of fibre bundles, as well as an overview of the theory of homology with local coefficients and of spectral sequences. We then calculate the homology of a specific class of fibre bundles using three different methods. After this discussion, we consider two product code constructions: the fibre bundle product construction and the balanced product construction. We explicate their mathematical foundations, and use these insights to prove two new results for the number of qubits that these product codes encode. Finally, we explain under which conditions these two code constructions and the lifted product construction coincide.

Lastly, we consider a specific example of fibre bundle product codes: the twisted toric code. We determine an analytic expression for the distance of such a code and verify this expression using numerical simulations. Furthermore, we perform an extensive numerical study on these codes to determine how performing a twist alters the scaling of the logical error rate of such codes. We present a new analytic method to explain the scaling of these codes on a small domain, and verify the validity of these calculations by comparing them with the results of our numerical simulations.