Bosonic \(q\)-ary expansion[1] 

A one-to-one mapping between basis states on \(n\) prime-dimensional qudits (of dimension \(q=p\)) and the subspace of the first \(p^n\) single-mode Fock states. While this mapping offers a way to map qudits into a single mode, noise models for the two code families induce different notions of locality and thus qualitatively different physical interpretations [2].

The mapping allows one to think of the Fock subspace as a tensor product of \(n\) \(p\)-dimensional qudits by performing a \(p\)-ary expansion of the Fock-state labels and treating each digit as an index of a qudit basis state. The Fock state integer label \(N\geq 0\) is expanded in the \(p\)-ary expansion as \begin{align} N=\sum_{\mu=0}^{\infty}b_{\mu}p^{\mu}~, \tag*{(1)}\end{align} with each \(p\)-ary string \(b_{\mu}\in\mathbb{Z}_p\) corresponding to the basis-state label of qudit \(\mu\).

In the binary case, the first qubit's \(Z\)-operator is the parity operator \(Z_0=(-1)^{\hat{n}}\), while the second qubit's \(Z\)-operator is the two-photon parity \(Z_1=(-1)^{\frac{1}{2}\hat{n}(\hat{n}-1)}\) [3,4]. These satisfy \(Z_{1}aZ_{1}=aZ_{0}\).

Pauli operators for the constituent qudits can be expressed in terms of bosonic raising and lowering operators. The modular-qudit Pauli-\(Z\) operator for qudit \(\mu\) is the Fock-space rotation \begin{align} Z_{\mu}=\exp\left[i\frac{2\pi}{p}{\hat{n} \choose p^{\mu}}\right]~, \tag*{(2)}\end{align} where \(\hat n\) is the mode's occupation number operator. This can be proven by Lucas's theorem.

The Pauli-\(X\) operator is expressed as \begin{align} X_{\mu}=\frac{1-P_{\mu}^{(d-1)}}{\sqrt{\left(\hat{n}+p^{\mu}\right)_{p^{\mu}}}}a^{p^{\mu}}+\frac{P_{\mu}^{(p-1)}}{\sqrt{\left(\hat{n}\right)_{p^{\mu}(p-1)}}}a^{\dagger p^{\mu}\left(p-1\right)}~, \tag*{(3)}\end{align} where \(\left(a\right)_{b}\) is the falling factorial, and where the qudit projector is \begin{align} P_{\mu}^{(k)}=\frac{1}{p}\sum_{l\in\mathbb{Z}_{p}}Z_{\mu}^{l}e^{-i\frac{2\pi}{p}kl}~. \tag*{(4)}\end{align}