  We consider the problem of quantum multi-parameter estimation with
experimental constraints and formulate the solution in terms of a convex
optimization. Specifically, we outline an efficient method to identify the
optimal strategy for estimating multiple unknown parameters of a quantum
process and apply this method to a realistic example. The example is two
electron spin qubits coupled through the dipole and exchange interactions with
unknown coupling parameters -- explicitly, the position vector relating the two
qubits and the magnitude of the exchange interaction are unknown. This coupling
Hamiltonian generates a unitary evolution which, when combined with arbitrary
single-qubit operations, produces a universal set of quantum gates. However,
the unknown parameters must be known precisely to generate high-fidelity gates.
We use the Cram\'er-Rao bound on the variance of a point estimator to construct
the optimal series of experiments to estimate these free parameters, and
present a complete analysis of the optimal experimental configuration. Our
method of transforming the constrained optimal parameter estimation problem
into a convex optimization is powerful and widely applicable to other systems.
