Abstract

Fault-tolerant quantum computing (FTQC) is undergoing a paradigm shift from conventional 2-level qubits toward higher-dimensional d-level qudits (d = q = p m) and asymptotically efficient quantum Low-Density ParityCheck (qLDPC) codes. Non-binary qudits dramatically increase physical quantum information density and reduce cross-talk, while qLDPC codes achieve macroscopic code rates and distance with orders-of-magnitude fewer physical qubits than surface codes. Concurrently, faulttolerant quantum cryptanalysis (such as Shor’s algorithm for discrete logarithms over binary elliptic curves) demands massive reversible modular arithmetic over Galois fields F2m. However, both quantum error correction (QEC) decoding and reversible arithmetic face a severe classical computational bottleneck: processing high-throughput algebraic operations over finite fields (Galois vector additions, multiplications, matrix-vector syndrome checks, and polynomial root-finding) within the microsecond-scale physical coherence window of quantum processors (tcycle ≤ 1 µs). In this paper, we present Galois-Q, a unified, synthesizable domain-specific microarchitecture designed specifically for finite-field quantum error correction, syndrome decoding, and fault-tolerant reversible field arithmetic. Tracing its lineage to the foundational algebraic research by Tummala, Alapati, and Awtar (2003)—which explored finite field constructions, binary Hamming error-correcting codes over Z2, and residue-based secret codes over Z26 and base-100 systems—Galois-Q elevates these classical mathematical foundations into a premier quantum computer architecture. Galois-Q incorporates four core architectural innovations: (1) a Reconfigurable Galois Field Vector Execution Unit (GF-VU) executing SIMD arithmetic across F2, F2m (m ∈ {2, 3, 4, 8, 13}), and modular rings Zq; (2) a Quantum Syndrome Evaluation & Inversion Pipeline (Q-SEIP) accelerating non-binary [[n, k, d]]q Quantum Reed-Solomon and quantum Hamming/BCH codes via parallel Chien search and Euclidean solvers; (3) a Bivariate Polynomial Matrix Engine (BPME) accelerating syndrome extraction and beliefpropagation/ordered-statistics decoding (BP-OSD) for bivariate bicycle qLDPC codes over F2[x, y]/⟨x ℓ −1, ym −1⟩; and (4) a Reversible Quantum Finite-Field Multiplier (RQFM) that optimizes Clifford+T quantum arithmetic circuits for cryptanalytic coprocessors. We synthesize, prototype, and evaluate Galois-Q on FPGA and 28 nm CMOS technology. Experimental results show that Galois-Q achieves a deterministic syndrome decoding latency of 142 ns for [[8, 2, 4]]4 qudit codes and 284 ns for [[72, 12, 6]] bivariate bicycle qLDPC codes, staying well within the 1 µs coherence deadline and achieving up to 28.6× speedup over software decoders. Furthermore, RQFM slashes reversible T-depth by 64.8% and ancilla overhead by 42.1%, demonstrating that finite-field microarchitectures are an indispensable pillar of scalable quantum computing systems.

Creative Commons License

Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 License.

Download

Share

COinS