In the first paragraph of the 2nd section of this article, it is stated that given a set of linearly independent $n$-qubit state vectors, Alice can perform the Gram-Schmidt procedure to obtain orthogonal quantum states.

But I know the Gram-Shmidt process is not a unitary operation, so is it possible to come up with an algorithm, using ancillary qubits or something?
I.e., if we have $M$ arbitrary, linearly independent $n$-qubit state vectors $\{ |x_1\rangle, |x_2\rangle, |x_3\rangle, \cdots , |x_M\rangle \}$ do we have an algorithm to perform Gram-Schmidt orthogonalization on this set to obtain a set of mutually orthogonal state vectors and can we construct the corresponding circuit?

  • $\begingroup$ You shouldn't link to libgen as it is blocked in many countries. There appears to be an arxiv version of the paper anyway. $\endgroup$
    – Rammus
    Jun 22, 2021 at 15:10
  • $\begingroup$ related: quantumcomputing.stackexchange.com/q/17955/55 and quantumcomputing.stackexchange.com/q/18037/55. Are you asking about a classical or a quantum algorithm here? Could you spell out the difference between this question and quantumcomputing.stackexchange.com/q/17955/55? $\endgroup$
    – glS
    Jun 22, 2021 at 15:48
  • $\begingroup$ @glS Both questions are related but in that, I was interested in the maximum number of steps required only. But here I actually want to understand the mentioned article and interested in an actual quantum circuit to perform such an operation. $\endgroup$
    – IamKnull
    Jun 22, 2021 at 15:54


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