Qudit low-density parity-check codes

Qudits offer significant advantages over qubit-based architectures, including more efficient gate compilation, reduced resource requirements, improved error-correction primitives, and enhanced capabilities for quantum communication and cryptography. Yet, one of the most promising families of quantum error correction codes, namely quantum low-density parity-check (LDPC) codes, have so far been mostly restricted to qubits. Here, we generalize recent advancements in LDPC codes from qubits to qudits. We introduce a general framework for finding qudit LDPC codes and apply our formalism to several promising types of LDPC codes. We generalize bivariate bicycle codes, including their coprime variant; hypergraph product codes, including the recently proposed La-cross codes; subsystem hypergraph product (SHYPS) codes; high-dimensional expander codes, which make use of Ramanujan complexes; and fiber bundle codes. Using the qudit generalization formalism, we then numerically search for and decode several novel qudit codes compatible with near-term hardware. Our results highlight the potential of qudit LDPC codes as a versatile and hardware-compatible pathway toward scalable quantum error correction.

💡 Research Summary

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This paper presents a comprehensive theory and practical methodology for extending quantum low‑density parity‑check (LDPC) codes—previously developed almost exclusively for qubit (two‑level) systems—to qudit (d‑level) architectures. The authors begin by motivating the transition to qudits, highlighting their advantages in gate synthesis, resource overhead, magic‑state distillation, and quantum communication/cryptography. They then acknowledge that the most promising LDPC families (bicycle, hypergraph product, subsystem hypergraph product, high‑dimensional expander, and fiber‑bundle codes) have largely remained confined to the binary field (F_2).

Mathematical Foundations (Section 2).
The paper reviews finite‑field theory, ring and ideal concepts, and primitive polynomials, emphasizing the distinction between prime‑order fields (e.g., (F_5)) and extension fields (e.g., (F_{5^2})). It introduces chain complexes and homological algebra, showing how CSS stabilizer codes can be expressed as a sequence of vector spaces over (F_q) connected by boundary maps (\partial) with the property (\partial^2=0). This formalism guarantees that X‑type and Z‑type stabilizers commute, a prerequisite for any valid quantum code.

Qudit‑ization Framework (Section 2.4).
The central contribution is a systematic “quditization” procedure that lifts binary parity‑check matrices (H_X, H_Z) to matrices over an arbitrary finite field (F_q) while preserving the CSS orthogonality condition (H_X H_Z^{\top}=0). Two key techniques are employed: (i) replacing binary entries with powers of a primitive field element (\omega) to embed multi‑level phase information, and (ii) adjusting row/column weights to satisfy a coprime condition that ensures commutation even when the field characteristic is not 2. The authors prove that sparsity (LDPC property), code dimension, and distance can be retained under this mapping.

Generalized Code Families.

1. Bivariate Bicycle Codes (Section 3).
   The authors map the classical bicycle construction to the polynomial ring (F_q