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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?

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    $\begingroup$ 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. $\endgroup$ – DaftWullie Apr 9 '18 at 7:27
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The answer arguably depends on the problem you wish to solve with your computation. More specifically, are you wanting to optimize near-term applications in the NISQ era, or are you wanting to build a fully scalable, fault-tolerant and universe quantum computer?

For the latter, you need to think about error correction. Pretty much everything that will happen in a fault-tolerant quantum computer will be part of error correction. So whether we want to run Shor's algorithm or simulate a quantum system, we nevertheless mostly need to optimize for error correction.

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.

So for fault-tolerant QC, the best option is to figure out good ways to implement the Clifford group.

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