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

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. INDEX TERMS Quantum error correction codes, quantum stabilizer codes, fault-tolerant, quantum topological codes, quantum gates.

Gate-Error-Resilient Quantum Steane Codes

“…Replacing an ancilla qubit by an ancilla state means that the CNOT gates in the stabilizer can be applied to the ancilla relying on the same circuit construction as a transversal CNOT gate. This inherits the fault-tolerant properties of the transversal CNOT gates, see [3], [15]. The transversal construction ensures that there are no scenarios whereby a single qubit error can result in increased-weight errors at the circuit's output.…”

“…However, the prerequisite for using this scheme is that it needs clean all-zero ancilla qubits in order to achieve fault-tolerance. In addition, this scheme relies on a full stabilizer measurement, which is costly compared to the non-fault tolerant unitary encoding circuit in terms of qubit overheads [3], [15]. Nevertheless, if the architectural assumptions of the stabilizer circuit are met by the processor, the encoderless scheme imposes no further connectivity constraints on the device.…”

Mitigation of Decoherence-Induced Quantum-Bit Errors and Quantum-Gate Errors Using Steane’s Code

“…In this section the specific version that corrects a single qubit bit-flip error is described. However, the 2 Commuting operators satisfy results for the phase-flip error are equivalent. The full circuit of implementing the repetition code is shown in Figure 4.…”

“…A CNOT gate error in the two qubit depolarizing channel is the same as that in Eq. 33except that 4 2 − 1 combinations of the J = 4 operators {I , X , Y , Z } are applied, each with probability…”

“…A fault tolerant implementation of the CNOT gate relies on a so-called transversal architecture, as seen in Figure 1 [2]. The CNOT gate will be discussed in Section II-A and Transversal Gates in Section V. To elaborate, the left hand side of Figure 1 shows the uncoded circuit, whilst the right hand side portrays the fault tolerantly encoded circuitry, where the represents a NOT gate and S is a syndrome decoder.…”