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I see the quantum gate $U_1^{-1}\circ U_1\circ U_0=U_0$, what is the $\circ$ called and mean?

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  • $\begingroup$ can you share the link from where you get that. $\endgroup$
    – Rayhan
    Commented Oct 11, 2022 at 10:53
  • $\begingroup$ arxiv.org/abs/2201.00752 $\endgroup$
    – karry
    Commented Oct 11, 2022 at 10:54
  • $\begingroup$ Btw, sometimes the $\circ$ symbol is used to represent the Hadamard product between matrices. $\endgroup$ Commented Oct 14, 2022 at 1:56

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The symbol $\circ$ is used in mathematics to denote the composition of maps. In the linked paper (https://arxiv.org/abs/2201.00752) it is used in this sense to denote the composition of quantum channels (e.g. in Eq. (1)). This is common practice.

However, you will often see $\mathcal{U}\mathcal{V}$ instead of $\mathcal{U}\circ\mathcal{V}$. This is a simplified notation resembling matrix multiplication $AB$ for matrices $A$ and $B$. Indeed, matrix multiplication is nothing but the composition of matrices seen as linear maps on a vector space. But it is very uncommon to write $A\circ B$ when dealing with matrices.

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I am one of the authors of this paper and thank you for your attention to our work.

Actually, we often use $\mathcal{U}$ to represent a quantum channel (a linear map from density matrix to density matrix, dimension is $4^N\times 4^N$)), and $U$ as a quantum gate or unitary transformation, whose dimension is $2^N\times 2^N$).

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