Subsystem Hypergraph Product Simplex (SHYPS) code[1]
Description
Family of quantum LDPC codes obtained by combining the subsystem hypgergraph product code construction with classical simplex codes. The result are CSS subsystem codes with weight-three gauge generators and code parameters \([n=(2^r − 1)^2, k=r^2, d=2^{r-1}]\) for \(r \geq 3\).
Due to their symmetric structure, SHYPS codes inherit the large authomorphism group of the underlying classical simplex codes. More precisely, \(|Aut(SHYPS(r))| \geq |GL_{r}(\mathbb{F}_2)|^2\), which is exponential in the number of logical qubits. This large authomorphism group can be leveraged to obtain a depth-one fault-tolerant implementation for a large set of logical Clifford operators. This set of depth-one Clifford generators is sufficiently large to allow for efficient compilation, i.e., any \(m\)-qubit Clifford operator can be executed in depth of order \(O(m)\). Due to single-shot properties of SHYPS codes, only order \(O(m)\) rounds of syndrome extraction are required to fault-tolerantly execute any logical \(m\)-qubit Clifford circuit.
Protection
Simulations of logical Clifford gates on two code blocks of the \([[49, 9, 4]]\) SHYPS code show a \(0.018\%\) pseudo-threshold under circuit-level noise with a BP-OSD decoder [1]. Memory simulations of the \([[49, 9, 4]]\) and \([[225, 16, 8]]\) SHYPS codes under circuit-level noise with a sliding window decoder using BPOSD were also performed in [1].Transversal Gates
Cross-block transversal CNOT gates \(\prod_{i=1}^n CNOT_{i, n+(\sigma_1\otimes\sigma_2)(i)}\) for \(\sigma_1, \sigma_2\) authomorphisms of the classical simplex code. These operators implement a generating set of cross-block logical CNOT gates.In-block phase-type fold-transversal gates with \(SP_{2k}(\mathbb{F}_2)\) representation \(\begin{bmatrix} I & (\sigma \otimes \sigma^T)\tau \\ 0 & 1\end{bmatrix}\) for \(\sigma\) an authomorphism of the classical simplex code, and \(\tau\) a self-inverse permutation which acts like \(\tau (e_i \otimes e_j) = e_j \otimes e_i\) for the canonical basis \(\{e_i\}\) of \(GF(2)^\sqrt{n}\). These operators implement a generating set of logical in-block diagonal gates [1].Cross-block phase-type fold-transversal gates \(\prod_{i=1}^n CZ_{i, n+(\sigma_1\otimes\sigma_2)\tau(i)}\) for \(\sigma_1, \sigma_2\) authomorphisms of the classical simplex code and \(\tau\) as above. These operators implement a generating set of logical cross-block diagonal gates [1].Fold-transversal Hadamard gate \(H^{\otimes n} \tau \), with \(\tau\) as above. Implements logical Hadamard-SWAP operator \(H^{\otimes k} \tau_k \), with \(\tau_k\) defined analogously to \(\tau\) [1].Gates
Roughly \(4m\) cycles of depth-one physical Clifford operations followed by syndrome extraction yield arbitrary \(m\)-qubit logical Clifford gates [1].Worst-case logical Clifford operation on \(b\) blocks can be implemented fault-tolerantly in depth roughly \(4bk\) using at most \(b\) auxiliary code blocks [1].Fault Tolerance
Logical Clifford operation on \(b\) blocks can be implemented fault-tolerantly in depth roughly \(4b r^2\) [1].Cousins
- \([2^m-1,m,2^{m-1}]\) simplex code— SHYPS code gauge generator matrices are constructed from hypergraph products of simplex codes [1].
- Single-shot code— Due to single-shot properties of SHYPS codes, only order \(O(m)\) rounds of syndrome extraction are required to fault-tolerantly execute any logical \(m\)-qubit Clifford circuit.
Primary Hierarchy
References
- [1]
- A. J. Malcolm et al., “Computing Efficiently in QLDPC Codes”, (2025) arXiv:2502.07150
Page edit log
- Alexis Schotte (2025-02-26) — most recent
- Victor V. Albert (2025-02-26)
- Victor V. Albert (2025-02-14)
Cite as:
“Subsystem Hypergraph Product Simplex (SHYPS) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2025. https://errorcorrectionzoo.org/c/shyps