Bosonic rotation code[1]

Description

A single-mode Fock-state bosonic code whose codespace is preserved by a phase-space rotation by a multiple of \(2\pi/N\) for some \(N\). The rotation symmetry ensures that encoded states have support only on every \(N^{\textrm{th}}\) Fock state. For example, single-mode Fock-state codes for \(N=2\) encoding a qubit admit basis states that are, respectively, supported on Fock state sets \(\{|0\rangle,|4\rangle,|8\rangle,\cdots\}\) and \(\{|2\rangle,|6\rangle,|10\rangle,\cdots\}\).

Encoding of a \(q\)-dimensional logical qudit admit a basis whose elements are eigenstates of the rotation operator \(\exp\left(\mathrm{i} 2\pi \hat{n}/qN \right)\), where \(\hat{n}\) is the number operator diagonal in the Fock basis. Basis elements are of the form \(\sum_{j=0}^\infty c_j |(kq+j)N \rangle\) for some coefficients \(c_j\) and \(k\). This is because the rotation acting on a general Fock-state superposition \(\sum_n a_n |n\rangle\) yields \(\sum_n a_n \exp\left(\mathrm{i} 2\pi n / qN \right) |n\rangle\). In order for a codeword to be an eigenvector of this operation, \(a_n\) must be zero whenever \(n \neq (kq+j)N\) for some \(k\).

Codewords can be uniquely specified by choosing a primitive state \(|\Theta\rangle\). To ensure valid (orthogonal and nonzero) codewords, \(|\Theta\rangle\) must satisfy the following requirement: for each \(j \in \mathbb{Z}_q\), \(|\Theta\rangle\) must have support on the Fock state \(|(k_j q+j)N\rangle\) for at least one \(k_j \in \mathbb{N}_0\). From such a primitive, the codewords \(\{|\overline{j}\rangle \mid j \in \mathbb{Z}_q\}\) are constructed as \begin{align} |\overline{j}\rangle \propto \sum_{m=0}^{2N-1} \mathrm{e}^{2\pi \mathrm{i} m (\hat n / N + j) / q} |\Theta\rangle~. \tag*{(1)}\end{align}