\(((6,2,3))\) transversal-\(\mathbb{Z}_{10}\) code [1]

Description

Six-qubit code that realizes gates from the group \(\mathbb{Z}_{10}\) transversally [1]. It has the minimum length among codes encoding a qubit with distance at least three whose single-block transversal gates are not all Clifford in any one logical basis [2; Sec. IV.B.4]. Up to local unitaries and qubit permutations, it is the only \(((6,2,3))\) code with a single-block transversal gate whose logical action has order five up to a phase [2; Cor. IV.7].

With the qubits split into the first four and the last two, the codewords are [2; Eqs. (122-123)] \begin{align} \begin{split} |\overline{0}\rangle &= \tfrac{1}{\sqrt{10}}\left(|0000\rangle(|00\rangle+|11\rangle)+|S_3\rangle|10\rangle\right)+\tfrac{1}{\sqrt{15}}|E\rangle|01\rangle\\ |\overline{1}\rangle &= \tfrac{1}{\sqrt{10}}\left(|1111\rangle(|00\rangle+|11\rangle)-|S_1\rangle|01\rangle\right)-\tfrac{1}{\sqrt{15}}|E^{*}\rangle|10\rangle~, \end{split}\tag*{$(1)$} \end{align}where \(|S_j\rangle\) is the sum of all four-qubit basis states of weight \(j\). The state \(|E^{*}\rangle\) is the complex conjugate of \(|E\rangle=|1100\rangle+|0011\rangle+\omega(|1010\rangle+|0101\rangle)+\omega^{2}(|1001\rangle+|0110\rangle)\) with \(\omega=e^{2\pi i/3}\). See Ref. [1] for another presentation of the codewords.