Complete-hypergraph CSS code[1,2]

Description

Family of qubit CSS codes, one for each rank \(t\geq2\) and each number \(k\geq t\) of logical qubits, whose physical qubits store the parities of all subsets of at most \(t\) logical qubits. A layer of physical single-qubit diagonal gates at level \(t\) of the Clifford hierarchy realizes every addressable diagonal logical gate at that level, at the smallest block length for which this is possible [2].

One physical qubit is placed on each nonempty subset \(A\subseteq\{1,\dots,k\}\) of size at most \(t\), so that \(n=\sum_{s=1}^{t}\binom{k}{s}\). For fixed \(t\), this is of order \(\Theta(k^t)\) [2]. There are no \(X\)-type stabilizer generators. The \(Z\)-type generators are the weight-three checks \(Z_A Z_{A\setminus\{i\}} Z_{\{i\}}\) for \(|A|\geq2\) and a chosen \(i\in A\), which together enforce \(z_A=\bigoplus_{i\in A}z_{\{i\}}\) [2]. Logical operators are \(\overline{Z}_i=Z_{\{i\}}\) on a singleton qubit and \(\overline{X}_i=\prod_{A\ni i}X_A\). The \(X\)-logical subspace is the classical simplex code punctured to the coordinates labelled by subsets of size at most \(t\). Physical qubits carrying the parity of more than two logical qubits were introduced as the higher-order parity encoding [1]. The rank-\(t\) family is that encoding with a completely symmetric choice of codespace, taking every subset of size at most \(t\).

At rank \(t=k\) the puncturing is trivial. There are \(n=2^k-1\) physical qubits, one per nonzero vector of \(\mathbb{F}_2^k\), the \(Z\)-type stabilizer space is the classical Hamming code, and the \(X\)-logical subspace is the full simplex code. This member is the simplex phantom code [3] which attains the phantom-code bound \(n\geq2^k-1\) [4].