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What are the known ways to encode or introduce non-linear functions/operators into a quantum circuit?

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I would say that the only way to do anything "nonlinear" on a quantum computer is to make the Hilbert space "big enough" that the linear dynamics on that space are sufficient to approximate something nonlinear in a subspace.

I believe in only one religion and that is the church of the larger Hilbert Space. Although that page specifically refers to the dilation of quantum channels, I think the idea still loosely holds in this context. As such simulations are ultimately realized via some numerical method, I think of the method of carleman linearization as a kind of "dilation theorem" for embedding nonlinear pde's into a (very large) approximate linear system for numerical simulations. With quantum computers, this growth in state space is totally acceptable, but beware there's still plenty of other reasons why such simulations might still be inefficient. Success probabilities, condition numbers, large matrix norms, and the desired observable are important factors in determining the overall efficiency of the simulation. I think a very interesting question is, can every nonlinear theory be put in a Hilbert space "big enough" that they are governed by linear dynamics? Any answer to this question would be very interesting.

Of course Carleman linearization has its own difficulties, and doesn't converge in every case. There are other more exotic methods based on topology, such as the homotopy analysis method. I'm really out of my depth reading that stuff, but I do think it could be very interesting to see if any efficient quantum algorithms can be cooked up out of that formalism. If anyone has expertise here and wants to chat, I would appreciate it.

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  • $\begingroup$ +1 mostly for your Nicenian creed. $\endgroup$ Commented Apr 21 at 0:16

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