Bipartite cyclic cluster (BCC) code[1]
Description
Cyclic CSS code constructed from a bipartite cluster state with cyclic invariance, emphasizing simplicity of state preparation over simplicity of stabilizers.
A BCC code encodes \(k\) logical qubits into \(n = k n_0\) physical qubits. Qubits are labeled by pairs \((m,j)\) with \(m\) defined modulo \(n_0\) and \(1 \leq j \leq k\), partitioned into sets \(A\) (indices \(j \in \mathcal{A}\)) and \(B\) (indices \(j \notin \mathcal{A}\)). The code is defined by a bipartite graph \(G\) on the qubits, with edges only between \(A\) and \(B\), that is invariant under cyclic shifts \((m,j) \to (m+1,j)\). Code states are prepared by initializing \(A\)-qubits in \(|\pm\rangle\) and \(B\)-qubits in \(|0/1\rangle\), then applying CNOT gates (source in \(A\), target in \(B\)) along the edges of \(G\). The \(2^k\) resulting basis states confirm that \(k\) logical qubits are encoded. After a Hadamard on each \(B\)-qubit, this gives a CSS code.
The \(X\)-type stabilizers arise from conjugating \(X_{m,j} X_{m+1,j}\) (for \(j \in \mathcal{A}\)) by the CNOT circuit, and the \(Z\)-type stabilizers from \(Z_{m,j} Z_{m+1,j}\) (for \(j \in \mathcal{B}\)).
For \(k=2\), the graph \(G\) is parameterized by a set \(\mathcal{S}\) of odd integers modulo \(n = 2n_0\): even qubit \(m\) connects to odd qubit \(m'\) iff \((m'-m) \bmod n \in \mathcal{S}\).
A related non-CSS construction, the cyclic cluster code, drops the bipartiteness requirement on \(G\). For \(d=3\), this construction applied to the \([[10,2,3]]\) BCC code yields the \([[5,1,3]]\) five-qubit perfect code [1].
Protection
The distance satisfies \(d \leq |\mathcal{S}| + 1\), since \(U^\dagger X_{m,j} U\) for \(j \in \mathcal{A}\) gives a logical operator of weight \(|\mathcal{S}|+1\) [1]. A weight-four GB code of odd distance \(d\) satisfies \(n \geq 1 + d^2\) [2], so the \([[d^2+1,2,d]]\) BCC codes are optimal among weight-four BCC codes.Encoding
Initialize \(A\)-qubits in \(|+\rangle\) and \(B\)-qubits in \(|0\rangle\), then apply CNOT gates along edges of \(G\), followed by Hadamard on each \(B\)-qubit.Transversal Gates
For \(k=2\), all BCC codes admit a transversal Hadamard-SWAP logical gate via qubit permutation \((m,1) \mapsto (-m,2)\) followed by Hadamard on all qubits [1].The \([[d^2+1,2,d]]\) BCC codes also admit a logical Hadamard without SWAP via \(m \mapsto md \pmod{n}\) followed by Hadamard [1].Fault Tolerance
For distance-3 BCC codes (\(|\mathcal{S}|=2\)), a single error produces at most weight-1 logical error, so fault tolerance is preserved [1].Cousins
- Cluster-state code— BCC codes are obtained by applying Hadamard on the \(B\)-sublattice of a bipartite cluster state, converting CZ-gate preparation into CNOT-gate preparation [1].
- Twisted XZZX toric code— The \([[d^2+1,2,d]]\) twisted XZZX toric codes (parameters \(a=1,b=d\)) are Clifford-equivalent to BCC codes for odd \(d\) [1].
Primary Hierarchy
References
- [1]
- M. B. Hastings, “A Class of Cyclic Quantum Codes”, (2025) arXiv:2509.06865
- [2]
- R. Wang and L. P. Pryadko, “Distance bounds for generalized bicycle codes”, (2022) arXiv:2203.17216
Page edit log
- Victor V. Albert (2026-05-26) — most recent
Cite as:
“Bipartite cyclic cluster (BCC) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/bipartite_cyclic_cluster