Error Analysis of Bosonic Codes

Abstract

Quantum computers have been shown to be able to perform computations that are thus far thought to be infeasible through classical computation. This is achieved by taking advantage of certain properties and phenomena unique to quantum systems. Unfortunately, engineering particular quantum systems that are useful for computation is difficult and there is still a long road towards building a robust, large-scale quantum computer. A key approach to building quantum computers that are robust to noise is the development of Quantum Error Correcting codes. In recent years, there have been a surge of progress in the development and realisation of bosonic codes, which are quantum error correcting codes that encode quantum information into the infinite-dimensional Hilbert space of an oscillator. Due to its large Hilbert space, bosonic codes are able to protect quantum information using a single, or a few physical systems. This thesis will take two different but complementary approaches to the development of bosonic codes and bosonic code qubits. The first will be a numerical study of errors affecting bosonic codes within a fault-tolerant system. Specifically, we introduce the concatenated Bacon-Shor rotation-symmetric bosonic code and evaluate the entanglement fidelity of a teleportation based error correction scheme against pure loss errors. In the second approach, we study the recently proposed gyrator qubit by Rymarz et al., which realises a bosonic code known as the Gottesman-Kitaev-Preskill (GKP) code in its low energy eigenspace. We support this study with a Wentzel-Kramers-Brillouin (WKB) analysis of the Zak transformed Hamiltonian of the qubit. This allows us to derive the low-energy spectrum and eigenstates of the qubit and derive matrix elements corresponding to transitions between qubit states when the system is weakly coupled to a thermal environment.