# Square-lattice GKP code[1]

## Description

Single-mode GKP qudit-into-oscillator code based on the rectangular lattice. Its stabilizer generators are oscillator displacement operators \(\hat{S}_q(2\alpha)=e^{-2i\alpha \hat{p}}\) and \(\hat{S}_p(2\beta)=e^{2i\beta \hat{x}}\). To ensure \(\hat{S}_q(2\alpha)\) and \(\hat{S}_p(2\beta)\) generate a stabilizer group that is Abelian, there is a constraint that \(\alpha\beta=2q\pi\) where \(q\) is an integer denoting the logical dimension.

Codewords can be expressed as equal weight superpositions of coherent states on a rectangular lattice in phase space with spatial period \(2\sqrt{\pi}\). The exact GKP state is non-normalizable, so approximate constructs have to be considered.

The \(q=1\) trivial encoding is spanned by the canonical GKP state or grid state, \begin{align} |GKP\rangle=\sum_{n\in\mathbb{Z}}|x=n\sqrt{2\pi}\rangle~, \tag*{(1)}\end{align} where \(|x\rangle\) are single-mode position states.

## Protection

## Encoding

## Gates

## Decoding

## Fault Tolerance

## Realizations

## Notes

## Parents

## Cousins

- Approximate quantum error-correcting code (AQECC) — Square-lattice GKP codes approximately protect against photon loss [2,3,39].
- Rotor code — Because square-lattice GKP error states are parameterized by two modular (i.e., periodic) variables of position and momentum, measuring one of the GKP stabilizers constrains the oscillator Hilbert space into that of a rotor.
- \(\mathbb{Z}^n\) hypercubic lattice code — GKP codewords, when written in terms of coherent states, form a square lattice in phase space.
- Kitaev current-mirror qubit code — Current-mirror code phase gates utilize ancillary osillators in square-lattice GKP states [40,41].
- Zero-pi qubit code — Zero-pi code phase gates utilize ancillary osillators in square-lattice GKP states [40,41].
- Rotor GKP code — GKP (rotor GKP) codes protect against shifts in linear (angular) degrees of freedom.
- Number-phase code — Square-lattice GKP codes utilize translational symmetry in phase space, while number-phase codes utilize rotational symmetry. The two are related via a mapping [42].
- Asymmetric quantum code — GKP code parameters against position and momentum displacements can be tuned by the choice of lattice (e.g., square vs rectangular).
- Modular-qudit GKP code — The square-lattice GKP code can be obtained from the modular-qudit code by taking the physical qudit dimension to be infinite [1; Sec. II].
- Spin GKP code — Spin-GKP code constructions utilize the Holstein-Primakoff mapping [43] (see also [44]) to convert various expressions for square-lattice GKP states into codes for spin systems.

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## Page edit log

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- Victor V. Albert (2022-03-22)
- Victor V. Albert (2021-12-15)
- Yijia Xu (2021-12-14)

## Cite as:

“Square-lattice GKP code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/gkp