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I want to understand how to orthogonalize a system of qubits.

Suppose I have $n$ sets of quantum states like

$$\{ |1_i\rangle|2_i\rangle|3_i\rangle \cdots|k_i\rangle \mid i=1 \dots n \}$$ where $i=1, \dots, n$ is number of states in the sets with $k$ tensor products.

Now I am interested in orthogonalizing these set of $n$ states. Especially I want to use the Gram-Schmidt orthogonalization procedure here.
But I don't understand how to think from here.


Cross-posted on math.SE

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    $\begingroup$ Crossposted on Mathematics. $\endgroup$
    – Rammus
    Jun 19 at 17:19
  • $\begingroup$ are you simply asking how the Gram-Schmidt procedure works for a set of vectors, or is there something else? I ask because you tagged this with and algorithm, but I don't really see the connection with quantum algorithms here $\endgroup$
    – glS
    Jun 19 at 18:35
  • $\begingroup$ Also crossposted on physics. $\endgroup$ Jun 20 at 6:12
  • $\begingroup$ @gIS: I think the IamKnull is asking how to carry out orthogonalization procedure on a quantum computer. $\endgroup$ Jun 20 at 6:19
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Let us break down the problem into a simpler problem.

Since you have n sets of quantum states, and every set is just a tensor product of the constituent states, each of them can be represented as a vector. For example, take ${|1_{1} \rangle, |2_{1} \rangle, ..., |k_{1} \rangle}$. If you actually evaluate the tensor product ${|1_{1} \rangle \otimes |2_{1} \rangle \otimes ... \otimes |k_{1} \rangle}$ you get a vector which has $m^{k}$ dimensions ($m$ is the dimension of the vector $|j_{1} \rangle ,\ j\ \in\ [1,k]$).

If we take an example, let us say $k = 4$, and each individual state is a 2-dimensional qubit state. Then, we will have $|0_{i} \rangle \otimes |1_{i} \rangle \otimes |3_{i} \rangle \otimes |4_{i} \rangle $ as the state. This particular vector would have $2^{4}$ dimensions and would form one of the vectors which represent one of the state set.

Let us denote the final calculated state of the $i^{th}$ set as $|v_{i} \rangle$. If you may prove that these vectors, $|v_{i} \rangle$ are $n$ linearly independent vectors, you shall be able to apply the Gram Schmidt orthogonalization procedure to make the orthonormal basis.

Hope it helps!

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