Fig. 4: Parity measurement and disturbance effects.

a Parity measurement circuit for the JPM scheme (left) and the CNOT scheme (right) in weight-2 (upper panel) and weight-4 (bottom panel) parity measurement. Here, the circuit, including the solid-box framed Hadamard gates, represents the measure-Z procedure, and the dotted-box framed Hadamard gates will be involved in the measure-X procedure. In particular, the circuit of measure-X procedure in the CNOT scheme can be further simplified (see Supplementary Materials). b Weight-2 and c weight-4 parity measurement in the iSWAP-based JPM scheme and the CZ-based CNOT scheme. Left panel: measure-Z procedure. Right panel: measure-X procedure. The average parity detection fidelity of the JPM scheme [b measure-Z: 95.3%; measure-X: 95.1%; total: 95.2% and c measure-Z: 90.4%; measure-X: 89.5%; total: 90.0%] is on par with the CNOT scheme [b measure-Z: 93.3%; measure-X: 92.5%; total: 92.9% and c measure-Z: 88.6%; measure-X: 87.2%; total: 87.9%] in the experiment. Notice that here the CZ gate time is set to be close to the gate time of the 2-qubit JPM unitary gate for accurate comparison. d Circuit diagrams for verifying disturbance effects on data-qubit subspace based on the JPM scheme with the reference group (experimental group) shown in left (right) panel. The Dynamical Decoupling (DD) pulses are utilized with optimized pulse intervals to mitigate dephasing effect during the measurement of syndrome qubit. e Measured expectation values of multiqubit Pauli operators for the reference group (top panel) and the experimental group conditioned on the syndrome qubit in \(\left\vert 0\right\rangle\) state and \(\left\vert 1\right\rangle\) state (bottom panel). Here, the numbers on the horizontal axis represent the order of the Pauli set in terms of I,  X,  Y,  Z for each qubit. (f) The corresponding density matrix of Bell states \(\left| {\Phi }_{+}\right\rangle=\left(\left\vert 01\right\rangle+\left\vert 10\right\rangle \right)/\sqrt{2}\) and \(\left| {\Psi }_{+}\right\rangle=\left(\left\vert 00\right\rangle+\left\vert 11\right\rangle \right)/\sqrt{2}\) measured in the experimental group conditioned on the syndrome qubit in \(\left\vert 0\right\rangle\) state and \(\left\vert 1\right\rangle\) state.