How to synthesize function $f(x)$ in amplitude encoding

Question

In computational basis encoding, the way to encode $f(x)$ is known - a classical circuit is converted to a quantum circuit which takes $|x\rangle|0\rangle \to |x\rangle|f(x)\rangle $. I wonder how I may be able to encode $f(x)$ into amplitude encoding such that one gets state $|\psi\rangle \propto \sum_{x=0}^{N-1}f(x)|x\rangle$.

For convenience, suppose $f(x)$ is periodic function of period $N$, and suppose classical algorithm for computing $f(x)$ is known.

Should I consider something like LCU - linear combination of unitaries? (section 7.3 of the link)

Answer 1

Any quantum operation (put aside reset and measurement) is a unitary operation and can be represented by a unitary matrix. Any unitary matrix can be built of quantum gates. So if you can encode $f(x)$ into a unitary matrix $U$ then there must be some series of quantum gates that yields $U$.

For example, let $n = 2$ be the number of qubits in the system, and $N = 4$ be the dimension of the Hilbert space that the quantum statevector of the system $|\psi\rangle$ resides in. A simple example is $U_1 = I \otimes Z $:

$U_1 = I \otimes Z = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1 \end{bmatrix}$

Let $f(x) = (-1)^x$. From the matrix of $U_1$ we can deduce that $U_1|x\rangle = (-1)^x|x\rangle = f(x)|x\rangle$, i.e we have encoded $f(x)$ into $U_1$. We can create $U_1$ by applying the $Z$ gate to $q_0$ and do nothing to to $q_1$ so $U_1 = I \otimes Z $ (using little endian).

Another example would be $U_2 = I \otimes S$:

$U_2 = I \otimes S = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 \\ 0 & i \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & i & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & i \end{bmatrix}$

Let $g(x) = i^{x (mod\ 2)}$. From the matrix of $U_2$ wen can deduce that $U_2|x\rangle = i^{x (mod\ 2)}|x\rangle = g(x)|x\rangle$, i.e we have encoded $g(x)$ into $U_2$. We can create $U_2$ by applying the $S$ gate to $q_0$ and do nothing to to $q_1$ so $U_2 = I \otimes S $ (again little endian).

These are just 2 examples and there are more, but I guess (not sure though) that without using extra qubits we are pretty limited (Someone reading this correct me if I am wrong). If we add more qubits then we can do much more.