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I have a classical vector of size $2^{n}$ (normalized) and I want to use this value as amplitude for my $n$ qbits, in the canonical basis $|00000\rangle$, $|00001\rangle$ etc. I want to use real feasible gates ($RX$, $RY$ $RZ$, $H$, Pauli gates, $CNOT$, $CRX$ etc) starting from the $|0000\rangle$ state.

Is there a general algorithm for that?

More Precisely, I'm interested in the length of such circuit and the numbers of gates (in worst case) according to $n$.

Do you have any idea or reference on the subject? I'm a bit lost here...

best regards,

b

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The task of how to perform "Amplitude encoding" - given $\vec{x}$, prepare $|\psi\rangle = \sum_k x_k |k\rangle$ - doesn't have a general answer. It could be accomplished by a depth-one circuit for a "one-hot" vector $x_k = \delta_i^k$. Or it could take an arbitrarily deep circuit to produce a set of amplitudes resembling a discretized gaussian like $x_k \propto e^{-(\bar{x} - k)^2/\mu}$.

In earlier "quantum machine learning" algorithms, its usually proposed that this task will be accomplished by an "oracle" (i.e. black box) or aided by "qRAM", neither of which represent a near-term or mainstream technology. A good reference to check out is Scott Aaronson's critique on these types of devices and algorithms.

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