# Gottesman-Kitaev-Preskill codes: A lattice perspective

Jonathan Conrad1,2, Jens Eisert1,2, and Francesco Arzani1

1Dahlem Center for Complex Quantum Systems, Physics Department, Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany
2Helmholtz-Zentrum Berlin für Materialien und Energie, Hahn-Meitner-Platz 1, 14109 Berlin, Germany

### Abstract

We examine general Gottesman-Kitaev-Preskill (GKP) codes for continuous-variable quantum error correction, including concatenated GKP codes, through the lens of lattice theory, in order to better understand the structure of this class of stabilizer codes. We derive formal bounds on code parameters, show how different decoding strategies are precisely related, propose new ways to obtain GKP codes by means of glued lattices and the tensor product of lattices and point to natural resource savings that have remained hidden in recent approaches. We present general results that we illustrate through examples taken from different classes of codes, including scaled self-dual GKP codes and the concatenated surface-GKP code.

Error correction is extremely important in quantum computers because quantum systems are very shy: as soon as someone observes them, they stop displaying their most peculiar features. If the state was supposed to encode some information to transmit or process, uncontrolled interactions of the system with its surroundings may result in logical errors. To overcome this, some form of error correction must be used: instead of being directly printed onto a quantum system, information is embedded into some larger space in such a way as to be screened from the typical interactions that occur with the environment. The embedding is called error correcting code. Bosonic codes embed finite systems (such as qubits) into infinite-dimensional ones. Here we are interested in so-called Gottesman-Kitaev-Preskill (GKP) codes, named after their inventors. They rely on discrete translation symmetries in the phase space of an ensemble of harmonic oscillators. Such symmetries are represented by lattices, which are mathematical objects studied in many branches of pure and applied mathematics and information science. This article elaborates on general methods to analyse and construct GKP codes grounded on results from lattice theory.

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### Cited by

[1] Ching-Yi Lai and Hao-Chung Cheng, "Learning Quantum Circuits of Some T Gates", IEEE Transactions on Information Theory 68 6, 3951 (2022).

[2] Frank Schmidt and Peter van Loock, "Quantum error correction with higher Gottesman-Kitaev-Preskill codes: Minimal measurements and linear optics", Physical Review A 105 4, 042427 (2022).

[3] Baptiste Royer, Shraddha Singh, and S.M. Girvin, "Encoding Qubits in Multimode Grid States", PRX Quantum 3 1, 010335 (2022).

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