Experimental demonstration of a high-fidelity virtual two-qubit gate

We experimentally demonstrate a virtual two-qubit gate and characterize it using quantum process tomography (QPT). The virtual two-qubit gate decomposes an actual two-qubit gate into single-qubit unitary gates and projection gates in quantum circuits for expectation-value estimation. We implement projection gates via midcircuit measurements. The deterministic sampling scheme reduces the number of experimental circuit evaluations required for decomposing a virtual two-qubit gate. We also apply quantum error mitigation to suppress the effect of measurement errors and improve the average gate fidelity of a virtual controlled-$Z$ (CZ) gate to $f_{\mathrm{av}}=0.9938\pm 0.0002$. Our results highlight a practical approach to implement virtual two-qubit gates with high fidelities, which are useful for simulating quantum circuits using fewer qubits and implementing two-qubit gates on a distant pair of qubits.

Figure 1

Decomposition of a CZ gate and quantum error mitigation. (a) Decomposition of a simple two-qubit quantum circuit with a single CZ gate and other one-qubit gates. See text for the explanation. (b) Calculation of the final expectation value using the expectation values ${\nu }_j^i$ obtained in the respective decomposition circuits in (a). (c) Measurement error mitigation for a single-qubit unitary operation. (d) Quantum error mitigation for a projection gate and a measurement.

Figure 2

Quantum process tomography results for a CZ gate. (a) Ideal CZ gate in the Pauli transfer matrix representation. (b) Experimentally-implemented virtual CZ gate without quantum error mitigation. (c) Experimentally-implemented virtual CZ gate with quantum error mitigation.

Figure 3

Gram matrix and quasiprobability vectors. (a) Gram matrix for a single-qubit measurement. (b)–(d) Quasiprobability vectors for measurement observables $X, Y$, and $Z$, respectively.

Figure 4

Quasiprobability vectors for projection gates. (a) Quasiprobability vector $\gamma$ corresponding to projection gate ${\tilde{\mathrm{\Pi }}}_{+Z}$. (b) Quasiprobability vector $\delta$ corresponding to projection gate ${\tilde{\mathrm{\Pi }}}_{-Z}$.