Phase-modulated entangling gates robust against static and time-varying errors

In a prominent class of entangling gates, encompassing the Mølmer-Sørensen gate in trapped ions and the resonator-induced phase gate in superconducting circuits, qubits are entangled via shared coupling to bosonic oscillator modes. A major source of error in these entangling gates is residual qubit-oscillator coupling at the conclusion of an operation. We present a technique that employs discrete phase shifts in the mediating field driving the gate to ensure all modes are de-excited at arbitrary user-defined times, increasing the gate fidelity and scalability. We demonstrate its use across a range of parameters with a pair of ¹⁷¹Yb⁺ ions and observe a significant reduction in gate error under non-ideal conditions, saturating measurement-fidelity limits (~97%) in cases where an unmodulated "primitive" gate would only achieve ~50% fidelity. The technique provides a unified framework to achieve robustness against both static and time-varying error sources captured in the filter-function formalism. Experiments agree well with theoretical predictions for gate robustness as a function of static detuning and time-dependent laser amplitude and trap-frequency error processes.