# Hamiltonian capable of quantum computation

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?

• Sounds like homework? – Norbert Schuch Oct 14 '19 at 20:27
• It's a research problem. I'm well past my homework days, I'm afraid. – AGML Oct 14 '19 at 20:31
• 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. – Norbert Schuch Oct 14 '19 at 20:40
• Feel free to provide that as an answer, slightly more fleshed out – AGML Oct 14 '19 at 21:09

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.