Stabilizer subsystem decompositions for single- and multi-mode Gottesman-Kitaev-Preskill codes

Abstract

The Gottesman-Kitaev-Preskill (GKP) error correcting code encodes a finite dimensional logical space in one or more bosonic modes, and has recently been demonstrated in trapped ions and superconducting microwave cavities. In this work we introduce a new subsystem decomposition for GKP codes that we call the stabilizer subsystem decomposition, analogous to the usual approach to quantum stabilizer codes. The decomposition has the defining property that a partial trace over the non-logical stabilizer subsystem is equivalent to an ideal decoding of the logical state. We describe how to decompose arbitrary states across the subsystem decomposition using a set of transformations that move between the decompositions of different GKP codes. Besides providing a convenient theoretical view on GKP codes, such a decomposition is also of practical use. We use the stabilizer subsystem decomposition to efficiently simulate noise acting on single-mode GKP codes, and in contrast to more conventional Fock basis simulations, we are able to to consider essentially arbitrarily large photon numbers for realistic noise channels such as loss and dephasing.

arXiv comment34 pages, 12 figures, 1 table; referee feedback implemented

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• Two-qubit operations for finite-energy Gottesman-Kitaev-Preskill encodings
• Passive environment-assisted quantum communication with GKP states
• Bosonic quantum error correction using squeezed Fock states
• Logical channel for heralded and pure loss with the Gottesman-Kitaev-Preskill code
• Bosonic Pauli+: Efficient Simulation of Concatenated Gottesman-Kitaev-Preskill Codes
• Classical simulation of circuits with realistic odd-dimensional Gottesman-Kitaev-Preskill states
• Using a Kerr interaction for GKP magic state preparation
• Achievable rates for concatenated square Gottesman-Kitaev-Preskill codes
• Logical channels in approximate Gottesman-Kitaev-Preskill error correction
• Protected phase gate for the $0$-$\pi$ qubit using its internal modes
• Fault-Tolerant Non-Clifford GKP Gates using Polynomial Phase Gates and On-Demand Noise Biasing