Towards Practical Quantum Error Correction and Fault Tolerance

Abstract

This thesis will explore quantum error correction from several different perspectives. We will examine both the tasks of how to implement quantum error correcting codes in hardware and how to develop error correcting codes with good properties. Firstly we will examine a specific, hardware-motivated computational scheme in which a class of bosonic code called rotation symmetric bosonic codes is concatenated with a well-known topological error correcting code, the surface code. We study this scheme subject to realistic noisy measurement and noise due to photon loss. We envision realising this scheme with superconducting qubit technology. The overhead in terms of number of qubits and time required for this scheme is significant however, motivating an alternative approach of investigating better error correcting codes. With this is mind, we then examine error correction approach that features condensed matter models at its core with the aim of finding more resource-efficient codes. We begin by studying a method of code construction called fractalization, which we apply to the Raussendorf model. Our novel contribution is to fractalize the code in both space and time and demonstrate the existence, for specific values of the generating parameters, of fracton excitations that are immobile in both space and time. Finally, we further pursue new code constructions by exploring the self-correction properties of classical hyperbolic codes. We study a family of classical hyperbolic codes for which we find numerical evidence consistent with a polynomial energy barrier. This would indicate that they are likely to display self-correcting behaviour. By studying such a simple classical model we not only gain intuition about hyperbolic codes in general, but also suggest that classical hyperbolic codes are likely to be good candidates for the construction of quantum codes via a product construction.