Design of quantum error correcting code for biased error on heavy-hexagon structure

Abstract

Surface code is an error-correcting method that can be applied to the implementation of a usable quantum computer. At present, a promising candidate for a usable quantum computer is based on superconductor-specifically transmon. Because errors in transmon-based quantum computers appear biasedly as Z type errors, tailored surface and XZZX codes have been developed to deal with the type errors. Even though these surface codes have been suggested for lattice structures, since transmons-based quantum computers, developed by IBM, have a heavy-hexagon structure, it is natural to ask how tailored surface code and XZZX code can be implemented on the heavy-hexagon structure. In this study, we provide a method for implementing tailored surface code and XZZX code on a heavy-hexagon structure. Even when there is no bias, we obtain $0.231 \%$ as the threshold of the tailored surface code, which is much better than $0.21 \%$ and $0.209 \%$ as the thresholds of the surface code and XZZX code, respectively. Furthermore, we can see that even though a decoder, which is not the best of the syndromes, is used, the thresholds of the tailored surface code and XZZX code increase as the bias of the Z error increases. Finally, we show that in the case of infinite bias, the threshold of the surface code is $0.264\%$, but the thresholds of the tailored surface code and XZZX code are $0.296 \%$ and $0.328 \%$ respectively.

Appendix A: Measurement circuit of Z stabilizer on heavy-hexagon structure

Fig. 11

Z Stabilizer using six flag qubits in the heavy-hexagon structure. a Connection among qubits. b Measurement circuit. The blue dots denote data qubits, and the red dots denote flag qubits. H is a syndrome qubit. For a measurement of the Z syndrome, the quantum states of the syndrome qubits (flag qubits) are prepared as $|{+}\rangle$($|{0}\rangle$)

To construct a surface code in a heavy-hexagon structure, the measurement circuits of the X and Z stabilizers should be built. Because the measurement circuits of the X stabilizer are discussed in the main text, in the Appendix, we explain the measurement circuits of the Z stabilizer.

Figure 11 shows the assignment of qubits and gates to construct a measurement circuit for the Z syndrome in the heavy-hexagon structure. In the measurement circuit of the Z stabilizer, the X error is passed to the flag qubit through CNOT operator to evaluate the parity of the X error of the data qubit.

The parity of the X error, which passed through the flag qubit to the syndrome qubit, is checked in the syndrome qubit. It should be noted that the quantum state of the flag qubits should be prepared as $|{0}\rangle$ and applied Hadamard operators, in order for the X error of the data qubit not to affect a flag qubit. The quantum state of the syndrome qubit is prepared as $|{+}\rangle$, to check the parity of the X error. In addition, in the measurement circuit of the Z stabilizer the CNOT operator is constructed using the flag qubit adjacent to a data qubit as the target qubit and the flag qubit adjacent to the syndrome qubit as the control qubit. The CNOT gate between a data qubit and a flag qubit is built using the data qubit as the control qubit and the flag qubit as the target qubit.