Fault-tolerant quantum stabilizer codes for improving the fidelity of transversal CNOT gates

Abstract

In support of large-scale practical quantum algorithms Quantum-Error-Correction-Codes (QECC) are designed for mitigating the component errors inherent in quantum circuits. A QECC attaches carefully selected redundancy to quantum information in such a way that the individual qubit errors can be corrected without corrupting the logical qubit state, where the encoding and decoding circuits are built by individual quantum gates. If these components are error-prone, they increase the qubit error probability, hence leading to an even more grave corruption of the data qubits. Therefore constructing QECCs reliant on fault tolerant circuitry is crucial for creating quantum solutions. Fault tolerant QECCs are capable of providing error rate improvements in quantum processors as long as components operate below a certain gate error probability. We start by quantifying the depolarization probability bound, below which the family of transversal QECCs give a better error probability than an uncoded gate. Both a low-complexity repetition code and Steane’s 7-bit QECC are characterized. In this context it is observed that the Frame-Error-Rates attained are lower-bounded according to the gate error probability occurring in the non-fault tolerant encoding circuits. We address this problem by proposing the ‘encoderless’ quantum code, which replaces the encoder circuit by a fault-tolerant single-qubit gate arrangement. As a further benefit, in contrast to state preparation techniques, our encoderless scheme requires no prior knowledge of the input information, therefore realistic unknown states can be encoded fault-tolerantly. Our encoderless quantum code delivers a frame error rate that is three orders of magnitude lower than that of the corresponding scheme relying on a non-fault-tolerant encoder, when the gate error probability is as high as 10−3. Next, we consider two practical applications of fault-tolerant QECCs, in quantum communication protocols; Quantum teleportation allows an unknown quantum state to be transmitted between two separated locations. To achieve this the system requires both classical and quantum channel, for communicating a pair of classical bits and an entangled quantum bit from the transmitter to the receiver. It is commonly assumed in the literature that both channels are error free, even though under realistic conditions this is unlikely to be the case. Hence we propose and investigate a secure and reliable quantum teleportation scheme, when both the classical and quantum channels exhibit errors. It is found that both the security and reliability of the teleportation may be improved, when powerful turbo codes are employed. Finally, we quantify the fault-tolerance improvements attained by a [4, 2, 2] error detection code in IBM’s open-access devices. Up to 100 logical gates are activated in the Ibmq Bogota and Ibmq Santiago devices and we found that a [4, 2, 2] code’s logical gate set may be deemed fault-tolerant for gate sequences larger than 10 gates. However, certain circuits did not satisfy the fault tolerance criterion. In some cases the encoded-gate sequences show a high error rate that is lower bounded at ≈ 0.1, whereby the error inherent in these circuits cannot be mitigated by classical post-selection. A comparison of the experimental results to a simple error model reveal that the dominant gate errors cannot be readily represented by the popular Pauli error model. Finally, it is most accurate to assess the fault tolerance criterion when the circuits tested are restricted to those that give rise to an output state with a low dimension.

More information

Identifiers

ORCID for Lajos Hanzo: orcid.org/0000-0002-2636-5214

Catalogue record

Export record