# What set of quantum logical operations can one use to benchmark spin Hamiltonians?

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?

• This doesn't answer your question about a good collection of gates, but you may be interested in some works designing time independent Hamiltonians to achieve specific tasks, such as perfect state transfer, making GHZ states and making W-states. The GHZ case may well not work for the sort of Hamiltonian you're looking at, but the others would. – DaftWullie Apr 9 '18 at 7:27

Typically, error correcting codes require Clifford gates. For the surface codes, CNOTs (both ways around), and preparation and measurement in the $Z$ and $X$ basis are usually used. But your exact choice of code, and how to implement that code, will affect the precise details of the gate set. But it will typically be a subset of the multi qubit Clifford group, made using single and two qubit generators.