Chemistry background: In magnetic molecules, it is sometimes the case that one can adjust the time-independent Hamiltonian by chemical design. This means there is freedom to adjust parameters in the time-independent Hamiltonian during the design phase, but when the device is prepared these parameters cannot be adjusted further. An example would be molecules containing magnetic ions with a well-defined spin anisotropy and which communicate with each other via dipolar interactions: if it is chemically possible to position the spins in 3D space and to orient their magnetization axes, one has a certain control over the final form of the time-independent Hamiltonian. (For a state-of-the-art example, see A modular design of molecular qubits to implement universal quantum gates).
Operational problem: In a given experimental setup aiming to implement quantum computing, there will be a collection of physical operations (described by a time-dependent Hamiltonian) which in principle allow for arbitrary quantum logical operations. In practice, the number of physical operations needed to implement a certain quantum algorithm (or even an elementary logical quantum gate) also depends on the details of the time-independent Hamiltonian. (See for example Spin qubits with electrically gated polyoxometalate molecules, where sqrt(SWAP) is spontaneously "implemented" by a simple waiting time).
Goal + question: Whenever one aims for a certain quantum logical manipulation (such as a Quantum Fourier Transform), it seems obvious that there will be some time-independent Hamiltonians that will require a smaller number of physical operations than others. Our goal would be to find design criteria; to find the parameters that we can adjust by chemical design that makes the number of operations small, not in a particular case but for a typical collection of quantum transformations. In the form of a question: is there a reasonable set of quantum logical operations that could be used in this context to benchmark the typical efficiency of time-dependent spin Hamiltonians?