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A quaternion can be represented as $a + bi + cj + dk$ where a, b, c, d, are real numbers, and i, j, k, are symbols that can be interpreted as unit-vectors pointing along the three spatial axes. . I am wondering what is the representation of CNOT gate in terms of quaternions. Since it is controlled-X gate, it should be something with $b$. $cj$ and $dk$ should be 0. What about a?

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  • $\begingroup$ Quaternion lies in $\mathbb{R}^4$, which is isomorphic to $\mathbb{C}^2$. They can describe the information of a single 2-level system. To convey information about 2 qubits, you will need 4 Quaternions. Operations on two qubits lie in the operator space of $\mathbb{C}^4$, $\mathcal{B}(\mathbb{C}^4)$, which is $\mathbb{C}^{16}$, which is isomorphic to $\mathbb{R^{32}}$. So technically, I think it should be possible to convey information about operations on 4-level systems with 8 Quaternions, using some arbitrary mathematical model of description. $\endgroup$
    – FDGod
    Commented Sep 28, 2023 at 22:10
  • $\begingroup$ But it seems very inelegant. $\endgroup$
    – FDGod
    Commented Sep 28, 2023 at 22:12

1 Answer 1

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Quaternion lies in $\mathbb{R}^4$, which is isomorphic to $\mathbb{C}^2$. They can describe the information of a single 2-level system. To convey information about 2 qubits, you will need 4 Quaternions. Operations on two qubits lie in the operator space of $\mathbb{C}^4$, $\mathcal{B}(\mathbb{C}^4) = (\mathbb{C}^4)^{\otimes2}$, which is $\mathbb{C}^{16}$, which is isomorphic to $\mathbb{R^{32}}$.

So technically, I think it should be possible to convey information about operations on 4-level systems with 8 Quaternions, using some arbitrary mathematical model of your choice.

For describing 3 qubits ($\mathbb{C}^8 \cong \mathbb{R}^{16}$), you will need 4 quaternions and operations on 3 qubits, i.e., 3 qubit gates ($\mathbb{C}^{64} \cong \mathbb{R}^{128}$) will require 32 quaternions.

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