  We propose a gate optimization method, which we call variational quantum gate
optimization (VQGO). VQGO is a method to construct a target multi-qubit gate by
optimizing a parametrized quantum circuit which consists of tunable
single-qubit gates with high fidelities and fixed multi-qubit gates with
limited controlabilities. As an example, we apply the proposed scheme to the
models relevant to superconducting qubit systems. We show in numerical
simulations that the high-fidelity CNOT gate can be constructed with VQGO using
cross-resonance gates with finite crosstalk. We also demonstrate that fast and
a high-fidelity four-qubit syndrome extraction can be implemented with
simultaneous cross-resonance drives even in the presence of non-commutative
crosstalk. VQGO gives a pathway for designing efficient gate operations for
quantum computers.
