About index encoding on a quantum state

i have a matrix of data with number of row = 97. i am going to encode the number of row of the matrix of data into a quantum state. How can i do it in Pennylane? 0, 1, 2, 3, …,97. |0> , |1>,… but with log2(N) which N is the number of row.

I’m a little uncertain about what you asking about, but I will try to help the best I can.

Are you trying to encode an Integer between 0 and 97? Or are you trying to encode a vector of data from that row?

Either way, you may be interested in the embedding templates or state preparations.

Do those help solve your problem?

thanks for your answer. a vector of data. Actually, i want to design a quantum circuit for equation (1) of this paper:
https://doi.org/10.1038/s41534-020-0272-6. i know how to encode x , x_bar and y but i don’t know how to encode |m> which are index states.

thanks problem solved.

Glad to hear it @sassan_moradi! Feel free to also share your answer here if you would like — other readers might find it useful, especially if they have the same problem!

A comment about Equation 1 in the paper you mention:

If I understand correctly, the goal is to encode a superposition of data of the form \sum_m | m \rangle |x^m \rangle

If you have an embedding U(x) that encodes inputs x into quantum states |x\rangle, then such a state can be prepared by conditioning U(x) on the states of the qubits in register |m>.

For example, if you want to encode the second data point into the superposition, you have to control U(x) so it is only applied to the branch in the superposition where |m\rangle = |0...01\rangle (since 0..01 is integer notation for 2 and hence tags the second input).

With all the controls it’s a very long circuit and one can only use it for extremely small datasets.

Thanks Maria. yes it is only useful for small data.