Error-Tolerant Geometric Quantum Control for Logical Qubits with Minimal Resources

Geometric quantum computation offers a practical strategy toward robust quantum computation due to its inherent error tolerance. However, the rigorous geometric conditions lead to complex and/or error-disturbed quantum controls, especially for logical qubits that involve more physical qubits, whose error tolerance is effective in principle, but their experimental demonstration is still demanding. Thus, how to best simplify the needed control and manifest its full advantage has become the key to widespread appli-cations of geometric quantum computation. Here we propose a fast and robust geometric scheme, with decoherence-free subspace encoding, and present its physical implementation on superconducting quan-tum circuits, where we only utilize the experimentally demonstrated parametrically tunable coupling to achieve high-fidelity geometric control over logical qubits. Numerical simulation verifies that it can effi-ciently combine the error tolerance from both the geometric phase and logical-qubit encoding, displaying our gate-performance superiority over the conventional dynamical one without encoding, in terms of both gate fidelity and robustness. Therefore, our scheme can consolidate both error suppression methods for logical-qubit control, which sheds light on the future large-scale quantum computation.