This paper proposes a novel quantum error correction scheme that uses prime phases derived from the modulo-30 orbital structure to encode logical qubits. By mapping prime residues to distinct phase shifts, we construct a set of approximately orthogonal quantum states that are robust against phase noise. Using quantum phase estimation (QPE), we demonstrate that a logical qubit encoded in the phases of primes 7 and 31 can tolerate phase errors up to 0.6 rad with near-zero logical error rate. Numerical simulations verify the theoretical error threshold and show the performance decay beyond the tolerance limit. Comparison with the 3-qubit repetition code shows that our scheme has unique advantages in continuous noise handling, resource overhead, and cascadability. We also extend the noise model to test uniform noise, 1/f noise, and bit-flip noise, verifying the robustness of the scheme. Finally, using four primes to encode two logical bits, we achieve a decoding accuracy of up to 98.4% under 0.3 rad noise, and prove that the performance de- pends only on the modulo-30 residues of the primes, not on their magnitude. This work establishes a direct link between number theory and quantum error correction, offering a lightweight, hardware-friendly alternative to mainstream codes.
Huang Feiyue (Thu,) studied this question.