Canonical LP code [1]
Description
Binary LP code over a cyclic group \(\mathbb{Z}_{\ell}\) that admits a canonical logical basis, whose operators are base-code codewords laid along the rows and columns of a grid of \(\ell\)-qubit fibers [1; Def. 3.4]. This basis is that of HGP codes with each logical qubit replaced by a cyclic orbit of \(\ell\) logical qubits [1].
For base matrices \(A\in R^{m_A\times n_A}\) and \(B\in R^{m_B\times n_B}\) over \(R=\mathbb{F}_2[x]/(x^{\ell}+1)\), the left block of qubits is an \(n_A\times m_B\) grid of fibers. Each canonical logical \(Z\)-type (\(X\)-type) operator is a codeword of \(\ker A\) (\(\ker B^{*}\)) placed along one column (row) of this grid, and conjugate operators meet on one qubit [1; Def. 3.4, Prop. 3.5]. The \(k_c=r_A r_B^*\ell\) canonical logical qubits \(\bar{q}_{i,j,m}\) form an \(r_A\times r_B^*\) grid of logical fibers, with \(r_A=n_A-m_A\) and \(r_B^*=m_B-n_B\), and any further logical qubits are residual [1; Thm. 3]. When the lifts of \(A\) and \(B^{*}\) have full row rank, the basis exists iff the code is a canonical-product code, with \(k=k_c\) [1; Prop. 3.7, Appx. B.5]. Up to column permutation, \(A\) then consists of \(r_A\) information columns and a square block \(A'\) with invertible lift, and likewise for \(B^{*}\) [1; Exam. 5.5].
Protection
A canonical-product code has distance at most \(1+\gamma_A\,\omega_{\mathrm{col}}(A)\) [1; Exam. 5.5]. Here, \(\gamma_A\) is the maximum column weight of the inverse of the lift of \(A'\), and \(\omega_{\mathrm{col}}(A)\) is the maximum column weight of \(A\). Constant \(\gamma_A\) and LDPC base matrices therefore cap the distance at a constant [1; Appx. D.5]. The symmetric \([[952,112,17]]\) code with \(\gamma_A=4\) and \(\omega_{\mathrm{col}}(A)=4\) attains the bound [1; Appx. D.5].Transversal and Permutation-Based Gates
Cyclically shifting every fiber by \(t\) positions is a qubit permutation that implements the logical cyclic shift \(\bar{q}_{i,j,m}\to\bar{q}_{i,j,m+t}\) on all logical fibers in parallel [1; Prop. 3.8].For symmetric codes \(LP(A,A^*)\), transversal Hadamard followed by transposing the grid of fibers implements logical Hadamard on every canonical logical qubit followed by the fold permutation \(\bar{q}_{i,j,m}\leftrightarrow\bar{q}_{j,i,m}\) [1; Prop. 3.8]. A fold-transversal circuit of \(S\), \(S^{\dagger}\), and \(CZ\) gates implements logical \(S\) on the diagonal logical fibers and logical \(CZ\) between \(\bar{q}_{i,j,m}\) and \(\bar{q}_{j,i,m}\) for \(i\neq j\) [1; Prop. 3.8].Gates
Graph surgery on symmetric codes measures any Pauli product of canonical logical qubits by bridging copies of only \(r_A\) seed surgery gadgets, independently of \(\ell\) [1; Thm. 4]. A fixed canonical extractor instead measures any Pauli product of canonical logical qubits by masking its connections to the data code [1; Thm. 5]. Its ancilla graph is a cyclic lift closing each check of \(A\) into a cycle through its bits [1; Def. 4.9]. Parallel intra-column hypergraph surgery measures any set of logically disjoint \(Z\)-type (\(X\)-type) Pauli products within a column (row) of logical fibers in one logical cycle [1; Def. 5.4]. Its skeleton is the lift of \(A\), punctured and augmented with weight-two rows to select the measured operators [1; Def. 5.4]. For canonical-product codes, a thickening of \(\gamma_A\) layers instead of \(d\) suffices [1; Exam. 5.5]. Bridging a column gadget with a row gadget extends this to mixed-type and \(Y\)-type products [1; Thm. 7]. Parallel inter-column surgery measures \(\bar{Z}_{i,j,m}\bar{Z}_{i,j',m}\) for all \(i,m\) at once, and the merged code is \(LP(A,B')\) with \(B'\) being \(B\) plus one weight-two column [1; Def. 5.6]. Column-parallel magic-state injection teleports \(r_A\ell\) \(|T\rangle\) states from \(r_A\) stacks of \(\ell\) distance-\(d_s\) surface codes into one column of logical fibers [1; Def. 5.7]. Each stack is itself the LP code \(LP(D,D^*)\) of the repetition-code check matrix \(D\), and the transfer passes through a transistor code \(LP(A,D^*)\) by two rounds of inter-column surgery [1; Def. 5.7]. Repeating the protocol for each of the \(r_B^*\) columns injects all \(k_c\) magic states in time \(O(r_B^* d_s)\) [1; Table I].Fault Tolerance
A canonical extractor whose ancilla graph has small-set soundness at least \(r_A\) preserves the code distance [1; Thm. 5]. Column-parallel magic-state injection has phenomenological fault distance at least \(\min(d,d_s)\) [1; Thm. 8].Notes
See Ref. [1; Tables II–V] for surgery and injection gadgets on explicit canonical LP codes.Cousins
- Hypergraph product (HGP) code— The canonical logical basis of canonical LP codes is that of HGP codes with full-rank base matrices, with each logical qubit replaced by a fiber of \(\ell\) logical qubits [1; Sec. III]. The fold-transversal Hadamard-SWAP and \(S\)-\(CZ\) gates of symmetric HGP codes carry over to symmetric canonical LP codes [1; Prop. 3.8].
- Permutationally self-dual (PSD) CSS code— Symmetric canonical LP codes \(LP(A,A^*)\) are permutationally self-dual. Transposing the grid of fibers on both blocks of qubits is an involutive \(XZ\)-duality [1; Prop. 3.5]. Composed with transversal Hadamard, it yields a Hadamard-SWAP logical gate [1; Prop. 3.8].
Primary Hierarchy
References
- [1]
H. Zheng, G. Zheng, L. Jiang, and Q. Xu, “Logical computation with canonical lifted product codes”, (2026) arXiv:2607.28605
Page edit log
- Victor V. Albert (2026-10-26) — most recent
Cite as:
“Canonical LP code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/canonical_lifted_product, arXiv:2606.11484