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Suppose we have a 1D spin chain evolving in time according to some Hamiltonian $H_t(p_0, p_1, p_2 \ldots)$, where $p_i$ are classical parameters ``set by the lab equipment". Divide time into discrete intervals of some length $\Delta t$: we are allowed to change the $p_i$ every step. In addition, assume the ability to choose whether or not to perform a projective measurement in the spin-z basis at each site at each timestep.

What is an example of a nearest-neighbour spin-chain Hamiltonian that can implement quantum computation in this way?

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  • $\begingroup$ Sounds like homework? $\endgroup$ Commented Oct 14, 2019 at 20:27
  • $\begingroup$ It's a research problem. I'm well past my homework days, I'm afraid. $\endgroup$
    – AGML
    Commented Oct 14, 2019 at 20:31
  • $\begingroup$ But this is a well-researched problem, certainly if you only want some examples and you don't insist on constraints such as translational invariance. Just take any 2-body Hamiltonian generating universal gate sets. $\endgroup$ Commented Oct 14, 2019 at 20:40
  • $\begingroup$ Feel free to provide that as an answer, slightly more fleshed out $\endgroup$
    – AGML
    Commented Oct 14, 2019 at 21:09

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Many Hamiltonians will be good. In particular, if you have control over the individual couplings, any interaction, together with a local terms in two directions, will suffice.

As an example, the Ising model with two fields, $$ \sum p_is_z^is_z^{i+1} + \sum p_i' s_z^i + \sum p_i'' s_x^i $$ will do: The local terms allow to implement arbitrary rotations about $X$ and $Z$, and thus any rotation, and the Ising term allows to realize a gate $\mathrm{diag}(1,i,i,1)$ (up to a phase), which is locally equivalent to a CNOT. Together, this yields a universal gate set.

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