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We're implementing a QML algorithm in which we have to update the value stored inside a quantum state which is strictly unknown to us, by multiplying it with a real number classically stored with us. One approach is to binary encode the real number too and then multiply both the binary numbers, and there are circuits available for that too. But this takes a lot of extra qubits (one extra register for the new real value to be binary encoded and another extra register for the product). Is there a better method; for example something like the reverse of Amplitude encoding? We obtain the product somehow as the amplitude of a state and binary encode that amplitude back inside the state, without actually ever knowing the product, i.e. complete automation?

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