The fabulous world of GKP codes

• URL: http://arxiv.org/abs/2412.02442v1
• Date: Tue, 03 Dec 2024 13:27:37 GMT
• Title: The fabulous world of GKP codes
• Authors: Jonathan Conrad,
• Abstract summary: Quantum error correction is an essential ingredient in the development of quantum technologies. A quintessential tool in quantum error correction is the stabilizer formalism, which tames quantum systems by enforcing symmetries. A Gottesman-Kitaev-Preskill code is a stabilizer code that creates a logical subspace within an infinite dimensional Hilbert space by endowing it with translational symmetries.
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• Abstract: Quantum error correction is an essential ingredient in the development of quantum technologies. Its subject is to investigate ways to embed quantum Hilbert spaces into a physical system such that this subspace is robust against small imperfections in the physical systems. This task is exceedingly complex: for one, this is due to the vast diversity of possible physical systems with different structure to use. For another, every physical setting also comes with its own imperfections that need to be protected against. Bred by the complexity of a technological ambition, research on quantum error correction has developed into a large field of research that ranges from engineering of small systems with a single photon to the creation of macroscopic topological phases of matter and models of complex emergent physics. A quintessential tool in quantum error correction is the stabilizer formalism, which tames quantum systems by enforcing symmetries. A Gottesman-Kitaev-Preskill (GKP) code is a stabilizer code that creates a logical subspace within an infinite dimensional Hilbert space by endowing it with translational symmetries. While in practice the infinitude of the Hilbert space, as well as the infinitude of the translational symmetry group are considered as obstacles for implementation, in theory these are precisely the features that make the theory of GKP codes particularly rich, well behaved and well-connected to fascinating topics in mathematics. The purpose of this thesis is to explore these connections: to understand the coding theoretic and practical properties of GKP codes, utilizing its rich mathematical foundation, and to provide a foundation for future research. Along this journey we discover -- through the looking glass of GKP codes -- how quantum error correction fits into a fabulous mathematical world and formulate a series of dreams about possible directions of research.

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