Abstract
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 (F th ) 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 F th values are obtained for a very large number of physical qubits (n → ∞), when the quantum coding rate of the QTECCs approaches zero (r Q → 0). Ultimately, the framework advocated can be beneficially exploited for employing QSCs to protect large-scale quantum computers.
Highlights
Quantum error correction codes (QECCs) have the potential of offering a promising reliability improvement in quantum computers
Since we considered imperfect quantum gates and imperfect stabilizer measurements in our scheme, we invoked the quantum topological error correction codes (QTECCs) for correcting the errorneous physical qubits
The additional benefit of employing QTECCs is that the number of stabilizer measurements experienced by each of the physical qubits remains constant, as we increase the number of physical qubits
Summary
Quantum error correction codes (QECCs) have the potential of offering a promising reliability improvement in quantum computers. A popular member of the QECC family is constituted by quantum stabilizer codes (QSCs) [1]–[7], which can be viewed as the class of syndrome-based QECCs. The field of QECCs benefitted from a rapid pace. The associate editor coordinating the review of this manuscript and approving it for publication was Jin Sha. development because under certain conditions we can transform various classes of powerful classical error correction codes into their quantum counterparts, such as Quantum Turbo Codes (QTCs) [8], [9], Quantum Low-Density ParityCheck (QLDPC) codes [10], [11], and Quantum Polar Codes (QPCs) [12], [13].
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