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I recently learned of a technique known as "block-encoding" which embeds any $M \times N$ matrix into a unitary matrix, given that the spectral norm is at most $1$. This type of result is pretty astounding--- but with this being able to work, I have had a thought.

Is there a method to embed a nonlinear operator into a quantum circuit? This does not necessarily have to be constrained by the math around quantum circuits, but it can be a physical process perhaps?

I don't see how this is non-reversible, as this type of nonlinear operator can be embedded into a reversible operator, similar to the block encoding technique.

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  • $\begingroup$ I think that you should explain a bit more what you want to do in order to get answers. What do you exactly mean by "embed a nonlinear operator into a quantum circuit". I believe it could mean many different things depending on the context. For your last paragraph: you can encode non-reversible operation in a reversible circuit. The trick is to ignore some outputs. Look at this for instance: quantumcomputing.stackexchange.com/questions/5829/… $\endgroup$ Nov 3, 2022 at 14:13
  • $\begingroup$ But you can't represent arbitrary non-linear maps by matrices. See this post for a discussion along these lines: math.stackexchange.com/questions/450/… $\endgroup$ Nov 3, 2022 at 17:29

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