  High-fidelity entangling gates are essential for quantum computation.
Currently, most approaches to designing such gates are based either on simple,
analytical pulse waveforms or on ones obtained from numerical optimization
techniques. In both cases, it is typically not possible to obtain a global
understanding of the space of waveforms that generate a target gate operation,
making it challenging to design globally optimal gates. Here, we show that in
the case of weakly coupled qubits, it is possible to find all pulses that
implement a target entangling gate. We do this by mapping quantum evolution
onto geometric space curves. We derive the minimal conditions these curves must
satisfy in order to guarantee a gate with a desired entangling power is
implemented. Pulse waveforms are extracted from the curvatures of these curves.
We illustrate our method by designing fast, CNOT-equivalent entangling gates
for silicon quantum dot spin qubits with fidelities exceeding 99%. We show that
fidelities can be further improved while maintaining low bandwidth requirements
by using geometrically derived pulses as initial guesses in numerical
optimization routines.
