Alessio Baldelli

Alessio Baldelli, Davide Orsucci, Francisco Lázaro et al.

Stabilizer codes are often constructed within the Calderbank--Shor--Steane (CSS) framework, where two mutually orthogonal binary classical codes define $X$ and $Z$-type stabilizer generators. While this structure is algebraically convenient, additional non-CSS constraints may help suppress low-weight logical operators and improve decoding performance in the finite-length regime. We thus introduce quantum XYZ stabilizer codes, whose parity-check matrix (PCM) is built from three pairwise orthogonal binary PCMs associated with $X$-, $Y$-, and $Z$-type stabilizer generators. A nontrivial point is that an XYZ code instance is not automatically genuinely non-CSS: the same stabilizer group may admit a CSS generating set. We characterize this collapse, obtaining algebraic and rank conditions for deciding when the $Y$-type checks are redundant and when they define genuinely non-CSS stabilizer constraints. We also derive upper and lower bounds on the quantum minimum distance, including bounds for mixed Pauli logical operators. The novel framework includes a known non-CSS topological code, namely the XYZ$^2$ hexagonal code, and yields also sparse finite-length quantum low-density parity-check (qLDPC) constructions from intersecting-subset and quasi-dyadic code families. Simulations under depolarizing code-capacity noise and quaternary belief propagation decoding show that the proposed XYZ qLDPC instances can outperform representative CSS qLDPC instances with similar finite-length parameters.

Alessio Baldelli, Olai Å. Mostad, Hsuan-Yin Lin et al.

A triorthogonal code is a binary quantum Calderbank-Shor-Steane (CSS) code defined by a triorthogonal matrix. Triorthogonal codes are a key ingredient in magic-state distillation, since they allow for transversal $\mathsf{T}$ gates, a non-Clifford logical operation useful for achieving universal fault-tolerant quantum computation. Their construction is challenging because it must satisfy simultaneous pairwise and triple-wise overlap constraints, as well as row-weight requirements. In this work, we study the construction and decoding of triorthogonal codes with prescribed dual-distance properties. We derive an existence criterion for even-weight triorthogonal generator matrices with a target dual minimum distance. The criterion combines triorthogonality constraints with MacWilliams identities via Krawtchouk-polynomial conditions on the dual weight distribution, yielding an integer linear programming formulation for the construction problem. We find new nontrivial triorthogonal codes that are not necessarily generated by classical triply-even codes. The decoding performance of high-distance triorthogonal codes obtained via the doubling construction is then evaluated over the dephasing channel. We compare bounded-distance decoding, belief propagation plus ordered-statistics post-processing, and a GRAND-based decoder adapted to the quantum setting, which turns out to be a promising option.

Alessio Baldelli, Marco Baldi, Massimo Battaglioni et al.

Building scalable quantum computers requires quantum error-correcting codes that enable reliable operations in the presence of noise. Motivated by such need, this paper introduces two constructions of high-rate, quantum dual-containing (DC) Calderbank-Shor-Steane (CSS) low-density parity-check (LDPC) codes based on quasi-dyadic matrices. Their DC structure enables the transversal implementation of the Hadamard gate, and, jointly with the sparsity of their parity-check matrices enable low-complexity decoding via a standard binary belief-propagation algorithm. We provide several theoretical results concerning the cycle properties of these CSS codes. We also investigate their automorphism groups as well as their minimum distance. Furthermore, through numerical simulations, we show that the quantum CSS LDPC codes obtained through these constructions achieve better finite-length error rate performance than existing DC codes across different block lengths and code rates.

Alessio Baldelli, Marco Baldi, Davide De Zuane et al.

Bit-Flipping (BF) decoders are a family of decoders widely employed in post-quantum cryptographic schemes based on Quasi-Cyclic Moderate-Density Parity-Check (QC-MDPC) codes, such as BIKE. BF decoders suffer from trapping sets, corresponding to low-weight error patterns that likely lead to decoding failures. For QC-MDPC codes, the most relevant family of trapping sets is that of near-codewords, which are error patterns associated to low-weight syndromes. Indeed, recent works show that error patterns having a large overlap with near-codewords are the main culprits for decoding failures at very low Decoding Failure Rate (DFR) values. In this paper, we show that any BF decoder can be tweaked and made somehow aware of near-codewords, which means being able to recognize, and recover from, bad configurations due to near-codewords. We show that this modification results in minimal computational overhead. Through intensive numerical simulations, we evaluate the effectiveness of this approach on several BF decoders, considering both toy code parameters and BIKE parameters for NIST security category 1. Our results show drastic reductions in the DFR. We also find that, with this modification, a recently proposed BF variant called BF-Max outperforms the two decoders used by BIKE within the NIST competition.