Asymmetry of CNOT gate operation in superconducting transmon quantum processors using cross-resonance entangling

Controlled-NOT (CNOT) gates are commonly included in the standard gate set of quantum processors and provide an important way to entangle qubits. For fixed-frequency qubits using the cross-resonance entangling technique, using the higher-frequency qubit to control the lower-frequency qubit enables much shorter entangling times than using the lower-frequency qubit as the control. Consequently, when implementing a CNOT gate where logical control by the lower-frequency qubit is needed, compilers may implement this functionality by using an equivalent circuit such as placing Hadamard gates on both qubits before and after a CNOT gate controlled by the higher-frequency qubit. However, since the implementation is different depending on which qubit is the control, a natural question arises regarding the relative performance of the implementations. We have explored this using quantum processors on the IBM Q network. The basic circuit used consisted of operations to create a Bell State, followed by the inverse operations so as to return the qubits to their initial state in the absence of errors (Hadamard + CNOT + barrier + CNOT + Hadamard). The circuit depth was varied using multiples of this basic circuit. An asymmetry in the error of the final state was observed that increased with the circuit depth. The strength and direction of the asymmetry was unique but repeatable for each pair of coupled qubits tested. This observation suggests that the asymmetry in CNOT implementation should be characterized for the qubits of interest and incorporated into circuit transpilation to obtain the best accuracy for a particular computation.