Encoding a Qubit into an Oscillator with Near-Term Experimental Devices

A universal, large-scale quantum computer would be a powerful tool with applications of high value to mankind. For example, such a computer could significantly speed up the search for new medications or materials. However, the error rates of current qubit designs are simply too large to enable interesting computations. Therefore, both error correction and improved designs of qubits are needed. In 2001, Gottesman, Kitaev and Preskill proposed an encoding (GKP code) where a qubit is stored in a harmonic oscillator — a system that can be controlled and manufactured with high precision, and therefore have comparatively high coherence times. Moreover, the code offers good protection against losses, a simple gate set, and error correction circuits that are comparatively easy to implement. The drawback is that encoding a qubit into a GKP code state is a challenging task. In this thesis, we develop efficient schemes to encode a GKP qubit. Bosonic codes, where a qubit is stored in an oscillator, and in particular the GKP code are still relatively unknown. Therefore, we will start the thesis with an overview of the field, and provide the reader with the tools to analyze a GKP code, as these are quite different from standard error correcting codes. A tool which is important to understand, and that describes a protocol that encodes a GKP qubit is the so-called phase estimation algorithm. This algorithm allows to measure the eigenvalue of any unitary operation, and is one of the cornerstones of quantum information. We will show how phase estimation can be applied to encode a GKP qubit, and what the requirements for
an experiment attempting to do so are. A major advantage of the GKP code over other encodings is that it can tolerate significant photon loss before the encoded information is lost. In addition, states that are closely related to the GKP qubit can be used to violate Bell’s inequalities (i. e. prove the presence of entanglement), even in the presence of large noise. Both these applications make the code particularly interesting in the optical regime, where error correction usually cannot be done while the signal is travelling. In this thesis, we will analyze an encoding protocol originally proposed by H.M. Vasconcelos, L. Sanz, and S. Glancy, Optics Letters 35, 3261 (2010) that relied on post-selection, and show that any output state can be used as a GKP code state with a simple change of frame, providing an exponential speedup. In 2019, two separate experiments generated a GKP code state for the first time: C.Flühmann et al., Nature 566, 513 (2019) realized a GKP qubit in the motional mode of a trapped ion, while P. Campagne-Ibarcq et al., Nature 584, 368 (2020) realized it with a transmon qubit coupled to a microwave cavity. However, both these experiments employ phase estimation, which is slow because it requires many measurements in sequence. We propose a circuit that allows a single-shot measurement of the GKP stabilizers, and analyze the performance of such a measurement as well as the impact of noise.