# Asymmetric Quantum Multicast Network Coding: Asymmetric Optimal Cloning over Quantum Networks
## 2 Materials and Methods
### Quantum Multicast Network Coding
For an outgoing edge \(e\) from the source node \(s\), a unitary operator \(U_{e}\) on \(\mathcal{H}_{s}\otimes\mathcal{H}_{e}\) is defined by means of \(f_{e}:\mathbb{F}_{q}^{\prime}\rightarrow\mathbb{F}_{q}\) as  
\[U_{e}:=\sum_{\vec{x}\in\mathbb{F}_{q}^{\prime},\vec{y}\in\mathbb{F}_{q}}|\vec{ x}\rangle\langle\vec{x}|_{s}\otimes|y+f_{e}(\vec{x})\rangle\langle y|_{e}, \tag{17}\]  
where \(\mathcal{H}_{e}\) is a Hilbert space transmitted through \(e\). Suppose \(In(e)\) is a set of all incoming edges of \(v\), where \(v\) is a tail node of \(e\), and suppose \(\mathcal{H}_{In(e)}:=\bigotimes_{e^{\prime}\in In(e)}\mathcal{H}_{e^{\prime}}\). Then, for an outgoing edge \(e\) from an intermediate node \(v\), a unitary operator \(U_{e}\) on \(\mathcal{H}_{In(e)}\otimes\mathcal{H}_{e}\) is defined by means of \(f_{e}:\mathbb{F}_{q}^{\prime In(e)}\rightarrow\mathbb{F}_{q}\) as  
\[U_{e}:=\sum_{\vec{y}\in\mathbb{F}_{q}^{\prime In(e)},\vec{y}\in\mathbb{F}_{q} }|\vec{y}\rangle\langle\vec{y}|_{In(e)}\otimes|y_{e}+f_{e}(\vec{y})\rangle \langle y_{e}|_{e}. \tag{18}\]Suppose \(\mathcal{V}_{i}\) is a \(q^{\prime}\)-dimensional output Hilbert space on a target node \(t_{i}\). A unitary operator \(V_{i}\) on \(\mathcal{H}_{In(t_{i})}\otimes\mathcal{V}_{i}\) is defined by means of the decoding operation \(g_{i}:\mathbb{F}_{q}^{|In(t_{i})|}\rightarrow\mathbb{F}_{q}^{r}\) as  
\[V_{i}:=\sum_{\vec{y}\in\mathbb{F}_{q}^{|In(t_{i})|},\vec{x}\in\mathbb{F}_{q}^{ r}}|\vec{y}\rangle\langle\vec{y}|_{In(t_{i})}\otimes|\vec{x}+g_{i}(\vec{y}) \rangle\langle\vec{x}|_{i}. \tag{19}\]  
The quantum multicast network coding protocol of [PERSON] et al. is shown in Protocol 1. **Step 1: Initialization**  
At the beginning, the source node \(s\) has an unknown initial state \(|\psi\rangle\) on \(\mathcal{H}_{s}\). Each node \(v\in V\) prepares \(|0\rangle\) on \(\mathcal{H}_{e}\) for an edge \(e\in E\) whose tail node is \(v\). For all \(i\) satisfying \(1\leq i\leq m\), a target node \(t_{i}\) prepares \(|0\rangle\) on \(\mathcal{V}_{i}\). **Step 2: Transmission**  
First, for all edges \(e\in E^{\prime}\) whose tail node is the source node, the source node operates the unitary operator \(U_{e}\) on \(\mathcal{H}_{s}\otimes\mathcal{H}_{e}\) and sends \(\mathcal{H}_{e}\) to the head node of \(e\). Second, all intermediate nodes operate in the order defined by the natural partial ordering on \(E^{\prime}\) as follows: After an intermediate node \(v\) has received Hilbert spaces from all edges whose head node is \(v\), for all edges \(e\in E^{\prime}\) whose tail node is \(v\), node \(v\) operates the unitary operator \(U_{e}\) on \(\mathcal{H}_{In(e)}\otimes\mathcal{H}_{e}\) and sends \(\mathcal{H}_{e}\) to the head node of \(e\). Finally, after all edges have transmitted Hilbert spaces, for all \(i\) satisfying \(1\leq i\leq m\), target node \(t_{i}\) operates the unitary operator \(V_{i}\) on \(\mathcal{H}_{In(t_{i})}\otimes\mathcal{V}_{i}\). **Step 3: Measurement on Fourier basis**  
The source node \(s\) measures the Hilbert space \(\mathcal{H}_{s}\) in the Fourier basis and sends the measurement outcome to all the terminal nodes \(t_{i}\). For all edges \(e\in E^{\prime}\), the head node of \(e\) measures the Hilbert space \(\mathcal{H}_{e}\) in the Fourier basis and sends the measurement outcome to all terminal nodes \(t_{i}\). **Step 4: Recovery**  
All terminal nodes \(t_{i}\) operate \(Z(c_{1})\otimes\cdots\otimes Z(c_{r})\) on \(\mathcal{V}_{i}\). Here, \(\{c_{k}\}_{k=1}^{r}\) is a set of natural numbers that can be determined from the measurement outcomes received in step 3, the classical linear multicast network code \(\{f_{e}\}_{e\in E^{\prime}}\) and the decoding operators \(\{g_{i}\}_{i=1}^{N}\)[44]. In Step 3 of Protocol 1, the Fourier basis of \(\{|\vec{z}\rangle\}_{z\in\mathbb{F}_{q}}\subset\mathcal{H}_{e}\) of the computational basis \(\{|x\rangle\}_{x\in\mathbb{F}_{q}}\subset\mathcal{H}_{e}\) is defined as  
\[|\vec{z}\rangle:=\sum_{x\in\mathbb{F}_{q}}\omega^{\operatorname{Tr}xz}|x\rangle,\]  
where \(\omega:=\exp(-2\pi i/p)\).