Quantum Topological Error Correction Codes are Capable of Improving the Performance of Clifford Gates

The employment of quantum error correction codes (QECCs) within quantum computers potentially offers a reliability improvement for both quantum computation and communications tasks. However, incorporating quantum gates for performing error correction potentially introduces more sources of quantum decoherence into the quantum computers. In this scenario, the primary challenge is to find the sufficient condition required by each of the quantum gates for beneficially employing QECCs in order to yield reliability improvements given that the quantum gates utilized by the QECCs also introduce quantum decoherence. In this treatise, we approach this problem by firstly presenting the general framework of protecting quantum gates by the amalgamation of the transversal configuration of quantum gates and quantum stabilizer codes (QSCs), which can be viewed as syndrome-based QECCs. Secondly, we provide examples of the advocated framework by invoking quantum topological error correction codes (QTECCs) for protecting both transversal Hadamard gates and CNOT gates. The simulation and analytical results explicitly show that by utilizing QTECCs, the fidelity of the quantum gates can be beneficially improved, provided that quantum gates satisfying a certain minimum depolarization fidelity threshold <inline-formula> <tex-math notation="LaTeX">$(F_{th})$ </tex-math></inline-formula> are available. For instance, for protecting transversal Hadamard gates, the minimum fidelity values required for each of the gates in order to attain fidelity improvements are 99.74%, 99.73%, 99.87%, and 99.86%, when they are protected by colour, rotated-surface, surface, and toric codes, respectively. These specific <inline-formula> <tex-math notation="LaTeX">$F_{th}$ </tex-math></inline-formula> values are obtained for a very large number of physical qubits <inline-formula> <tex-math notation="LaTeX">$(n \rightarrow \infty)$ </tex-math></inline-formula>, when the quantum coding rate of the QTECCs approaches zero <inline-formula> <tex-math notation="LaTeX">$(r_{Q} \rightarrow 0)$ </tex-math></inline-formula>. Ultimately, the framework advocated can be beneficially exploited for employing QSCs to protect large-scale quantum computers.