Assume we are given an $n$-qubit system and complex numbers $a_0, \ldots, a_{m-1}$ with $m = 2^n$. Assume further we start with the initial state $|0 \ldots 0\rangle$ and want to make the transformation $$|0 \ldots 0\rangle \rightarrow \frac{1}{\sqrt{\sum_{i=0}^{m-1} |a_i|^2 }}\sum_{i=0}^{m-1} a_i |i\rangle$$ for arbitrary $a_i$. My question is now: What is the most efficient way (fewest gates resp. smallest circuit depth resp. best robustness on a noisy intermediate-scale quantum computer) currently known to implement that transformation?
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For completely arbitrary coefficients you are out of luck. A simple counting argument says that because:
1) The coefficients are continuous parameters 2) gates implement discrete operations $\to$ There is no finite circuit to prepare the vast majority of states. However, if you're okay with an arbitrarily good approximation to your state, then it can be done with relatively short gate sequences by the Solovay–Kitaev theorem. An algorithm for doing this is given here: https://arxiv.org/pdf/quant-ph/0505030
The most efficient is hard to say since, as you point out, there are many different metrics to choose from and this is an intensive area of active research. So I'd say as long as you're not writing quantum compilers and if you want something that will work for completely arbitrary state preparation, then the Solovay–Kitaev algorithm is a solid choice. If you are writing a quantum compiler you'll need to do a months long, intensive literature review.
Here's a recent paper I read that is cutting edge: https://arxiv.org/pdf/1807.03206
IBM's documentation (IBM Arbitrary Initialization) says it uses the techniques from: https://arxiv.org/pdf/quant-ph/0406176.pdf to accomplish this
I'll also give a nod to dissipative state preparation since that's something I personally work on and it seems to have some promise - but it's still too early for it to be really practical.
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