Fault-tolerant double-circular connectivity pattern for quantum stabilizer codes

Recently, the circular connectivity pattern has been presented for a class of stabilizer quantum error correction codes. The circular connectivity pattern for such a class of stabilizer codes can be implemented in a resource-efficient manner using a single ancilla and native two-qubit Controlled-Not-Swap gates (CNS) gates, which may be interesting for demonstrating error-correction codes with superconducting quantum processors. However, one concern is that this scheme is not fault-tolerant. And it might not apply to the Calderbank-Shor-Steane (CSS) codes. In this paper, we present a fault-tolerant version of the circular connectivity pattern, named the double-circular connectivity pattern. This pattern is an implementation for syndrome-measurement circuits with a flagged error correction scheme for stabilizer codes. We illustrate that this pattern is available for Steane code (a CSS code), Laflamme’s five-qubit code, and Shor’s nine-qubit code. For Laflamme’s five-qubit code and Shor’s nine-qubit code, the pattern has the property that it uses only native two-qubit CNS gates, which are more efficient in the superconducting quantum platform.