Exact requirements for battery-assisted qubit gates

We consider the implementation of a unitary gate on a qubit system S via a global energy-preserving operation acting on S and an auxiliary system B that can be seen as a battery. We derive a simple, asymptotically exact expression for the implementation error as a function of the battery state, which we refer to as the it Unitary Defect. Remarkably, this quantity is independent of the specific gate being implemented, highlighting a universal property of the battery itself. We show that minimizing the unitary defect, under given physical constraints on the battery state, is mathematically equivalent to solving a Lagrangian optimization problem, often corresponding to finding the ground state of a one-dimensional quantum system. Using this mapping, we identify optimal battery states that achieve the highest precision under constraints on energy, squared energy, number of levels and Quantum Fisher Information. Overall, our results provide an efficient method for establishing bounds on the physical requirements needed to implement a unitary gate via energy-preserving operations and for determining the corresponding optimal protocols.