  The speed of elementary quantum gates ultimately sets the limit on the speed
at which quantum circuits can operate. For a fixed physical interaction
strength between two qubits, the speed of any two-qubit gate is limited even
with arbitrarily fast single-qubit gates. In this work, we explore the
possibilities of speeding up two-qubit gates beyond such a limit by expanding
our computational space outside the qubit subspace, which is experimentally
relevant for qubits encoded in multi-level atoms or anharmonic oscillators. We
identify an optimal theoretical bound for the speed limit of a two-qubit gate
achieved using two qudits with a bounded interaction strength and arbitrarily
fast single-qudit gates. In addition, we find an experimentally feasible
protocol using two parametrically coupled superconducting transmons that
achieves this theoretical speed limit in a non-trivial way. We also consider
practical scenarios with limited single-qudit drive strengths and off-resonant
transitions. For such scenarios, we develop an open-source, machine learning
assisted, quantum optimal control algorithm that can achieve a speedup close to
the theoretical limit with near-perfect gate fidelity. This work opens up a new
avenue to speed up two-qubit gates when the physical interaction strength
between qubits cannot be easily increased while extra states outside the qubit
subspace can be well controlled.
