# An algorithm to perform Gram-Schmidt orthogonalization of linearly independent state vectors

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?

• You shouldn't link to libgen as it is blocked in many countries. There appears to be an arxiv version of the paper anyway. Commented Jun 22, 2021 at 15:10
• 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?
– glS
Commented Jun 22, 2021 at 15:48
• @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. Commented Jun 22, 2021 at 15:54