Subsystem Hypergraph Product Simplex (SHYPS) code[1]
Description
Family of quantum LDPC codes obtained by combining the subsystem hypergraph product code construction with classical simplex codes. The results 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 automorphism 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 automorphism 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 \(O(m)\). SHYPS codes exhibit practical single-shot features, so only order \(O(m)\) rounds of syndrome extraction are required to fault-tolerantly execute any logical \(m\)-qubit Clifford circuit.
Protection
Memory simulations of the \([[49, 9, 4]]\) and \([[225, 16, 8]]\) SHYPS codes under circuit-level noise, using a sliding-window BPLSD decoder, yield pseudo-thresholds of approximately \(0.32\%\) and \(0.35\%\), respectively [1]. Depth-126 logical Clifford-circuit simulations on two blocks of the \([[49, 9, 4]]\) SHYPS code were also performed in [1].Rate
The exact encoding rate is \(k/n = r^2/(2^r-1)^2\), i.e., asymptotically as \(\Theta( (\log n)^2/n )\) [1].Transversal and Permutation-Based Gates
Cross-block transversal CNOT gates \(\prod_{i=1}^n CNOT_{i, n+(\sigma_1\otimes\sigma_2)(i)}\) for \(\sigma_1, \sigma_2\) automorphisms 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 \(\begin{pmatrix} I & (\sigma \otimes \sigma^T)\tau \\ 0 & 1\end{pmatrix}\) for \(\sigma\) an automorphism 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 \(\mathbb{F}_2^{\sqrt{n}}\). 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 \(\sigma_1, \sigma_2\) automorphisms 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
Arbitrary \(m\)-qubit logical Clifford gates can be implemented in \(4m( 1+o(1) )\) logical cycles, each consisting of a depth-one physical Clifford layer followed by a depth-six syndrome-extraction round [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].Decoding
BPLSD decoder [1].Fault Tolerance
Logical Clifford operation on \(b\) blocks can be implemented fault-tolerantly in depth \(4br^2( 1+o(1) )\) while remaining compatible with one syndrome-extraction round between logical generators [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— SHYPS codes exhibit practical single-shot signatures, including logical error-rate stability under small-window sliding-window decoding and constant single-shot distance \(d_{\mathrm{ss}}=3\), which supports using one syndrome-extraction round between logical generators [1].
Primary Hierarchy
References
- [1]
- A. J. Malcolm et al., “Computing Efficiently in QLDPC Codes”, (2025) arXiv:2502.07150
Page edit log
- Victor V. Albert (2026-06-08) — most recent
- Alexis Schotte (2025-02-26)
- 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.), 2026. https://errorcorrectionzoo.org/c/shyps, arXiv:2606.11484