Hypergraph product codes: a bridge to scalable quantum computers

A physical machine for storage and manipulation of information, being physical, will always be subject to noise and failure. For this reason, the design of fault-tolerant architectures is of prime importance for building a working quantum computer. Quantum error correction codes offer a possible elegant framework for fault-tolerance when provided with methods to operate qubits without corrupting the information stored therein. This work specialises in hypergraph product (HGP) codes and seeks to lay the groundwork for a quantum computer architecture based on them.
The leading approach to fault-tolerant quantum computation is, today, based on the planar code. A planar-code-based quantum computer, however, would require dramatic qubit overhead and we believe that good low-density parity-check (LDPC) codes are necessary to attain the full potential of quantum computing. The HGP codes, of which the planar code is an instance, are not, strictly speaking, good LDPC codes. Still, they are an efficient alternative. On the one hand, the best HGP codes improve upon the planar code as they can store multiple logical qubits. On the other, they are not considered good because their noise robustness is sub-optimal. Nonetheless, we see the design of a HGP-based quantum computer as a bridge between the currently-favoured planar code design and the gold standard of good LDPC codes. A HGP-based architecture would inform our knowledge on how to design fault-tolerant protocols when a code stores multiple logical qubits, which is, to a large extent, still an open question.
Our first original contribution is a decoding algorithm for all families of two-fold HGP codes. Second, we exhibit a constructive method to implement some logical encoded operations, given HGP codes with particular symmetries. Last, we propose the concept of confinement as an essential characteristic for a code family to be robust against syndrome measurement errors. Importantly, we show that both expander and three-dimensional HGP codes have the desired confinement property.