Table 1 Examples and comparisons of generalized NP codes

From: Generalized number-phase lattice encoding of a bosonic mode for quantum error correction

Lattice

s

\(f=\frac{p}{q}\)

{θn}

syndromes for \(\hat{D}({{{{\bf{n}}}}}_{e})\)

Rect.

> 1

0

\(\sqrt{{2}^{-K}\left(\begin{array}{l}K\\ n\end{array}\right)}\)

(k, ϕe)

Obl.

1

≠ 0

\(\left\langle \right.n\left\vert \alpha,r\right\rangle\)

(0, mfπ + ϕe)

Diam.

> 1

\(\frac{1}{2}\)

\(\left\langle \right.n\left\vert \alpha,r\right\rangle\)

\((k,\frac{mf\pi }{s}+{\phi }_{e})\)

  1. s, f, {θn} are the parameters of generalized NP codes, which are defined in Eq. (8). The first and second numbers of real array \((k,\bar{\phi })\) represent error syndromes read out by number parity measurement and phase measurement, respectively.
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