Binary Triorthogonal and CSS-T Codes for Quantum Error Correction
Eduardo Camps-Moreno, Hiram H. López, Gretchen L. Matthews, Diego Ruano, Rodrigo San-José, Ivan Soprunov
TL;DR
This work addresses the design and analysis of binary triorthogonal codes and their CSS-T generalizations to enable transversal $T$-gate implementations in quantum error correction. It develops a propagation rule: if a code $C$ yields a triorthogonal $[[n,k,d]]$ QECC and a vector $v\in (C^{\star 2})^{\perp}\setminus C$ with odd weight, then $C+\langle v\rangle$ yields a triorthogonal $[[n,k+1,d]]$ QECC, while also characterizing the poset of triorthogonal codes. A key result is that a binary triorthogonal code uniquely determines the parameters of its associated triorthogonal quantum code, with the quantum code defined by the pair $(C,\mathrm{Hull}(C))$ and invariant under equivalent triorthogonal generators; a matrix-form theorem further restricts allowable basis changes. These findings connect triorthogonal codes to CSS-T codes, clarifying when transversal $T$-gates induce logical operations and how code parameters are preserved under equivalence. The work provides tools for constructing robust QECCs with stable parameters, aiding magic-state distillation and fault-tolerant quantum computation through structured code design and poset analysis.
Abstract
In this paper, we study binary triorthogonal codes and their relation to CSS-T quantum codes. We characterize the binary triorthogonal codes that are minimal or maximal with respect to the CSS-T poset, and we also study how to derive new triorthogonal matrices from existing ones. Given a binary triorthogonal matrix, we characterize which of its equivalent matrices are also triorthogonal. As a consequence, we show that a binary triorthogonal matrix uniquely determines the parameters of the corresponding triorthogonal quantum code, meaning that any other equivalent matrix that is also triorthogonal gives rise to a triorthogonal quantum code with the same parameters.