Quantum Topological Error Correction Codes: The Classical-to-Quantum Isomorphism Perspective

We conceive and investigate the family of classical topological error correction codes (TECCs), which have the bits of a codeword arranged in a lattice structure. We then present the classical-to-quantum isomorphism to pave the way for constructing their quantum dual pairs, namely, the quantum TECCs (QTECCs). Finally, we characterize the performance of QTECCs in the face of the quantum depolarizing channel in terms of both the quantum-bit error rate (QBER) and fidelity. Specifically, from our simulation results, the threshold probability of the QBER curves for the color codes, rotated-surface codes, surface codes, and toric codes are given by <inline-formula> <tex-math notation="LaTeX">$1.8 \times 10^{-2}$ </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">$1.3 \times 10^{-2}$ </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">$6.3 \times 10^{-2}$ </tex-math></inline-formula>, and <inline-formula> <tex-math notation="LaTeX">$6.8 \times 10^{-2}$ </tex-math></inline-formula>, respectively. Furthermore, we also demonstrate that we can achieve the benefit of fidelity improvement at the minimum fidelity of 0.94, 0.97, and 0.99 by employing the 1/7-rate color code, the 1/9-rate rotated-surface code, and 1/13-rate surface code, respectively.