# What's the symbol $\circ$ called, and what does it mean?

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

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

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.
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 $$2^N\times 2^N$$).