\begin{figure}
\centering
\begin{subfigure}{0.45\textwidth}
\centering
\includegraphics{figure_2_1.pdf}
\end{subfigure}
\begin{subfigure}{0.45\textwidth}
\centering
\includegraphics{figure_2_2.pdf}
\end{subfigure}
\caption{The optimal procedure for W and GHZ state. Left: A W state is generated by a combination of direct 2-qubit interaction; Right: A GHZ state is generated by two-qubit interaction between an ancilla qubit and all the other qubits, followed by a measurement on the ancilla qubit}
\label{figure_2}
\end{figure}

symmetric and there is no direct link connecting any two of them since any two qubits can only be indirectly connected by an ancilla, which has been cut off from the GHZ brane by the measurement on it. Similarly it seems the optimal procedure to generate a W state from a product state should achieved by a combination of direct two-qubit interactions as shown in Fig.~\ref{fig:figure_2}.

\subsection{F. A revisit to teleportation}

Teleportation is one of the most famous quantum effect. How the quantum information can be teleported by sending two classical bit is the key issue. Here we will show how teleportation can be described in the QM=GR picture.

According to our QM=GR mechanism, spacetime geometry corresponds to certain quantum operations. Teleportation as a quantum computation procedure, of course it also generate geometries during its implementation.

The standard quantum circuit for quantum teleportation is shown in Fig.~??. Alice, Bob and Tod, each one of them holds a qubit. Starting from a product state $|000\rangle$, Tod generates an unknown quantum state to be teleported. Alice and Bob entangles their qubits. After the interaction between Tod and Alice's qubits, we are at the final stage of the teleportation protocol. Before the joint measurement on Tod and Alice's qubits, the quantum state of the three qubits is generated by the quantum circuit. If we regard this circuit as the geodesic connecting the initial product state to the current quantum state, then the quantum circuit represents a space geometry. According to GM/GR, we can compute the distances between any two points on this geometry based on their mutual information. After the measurement, Tod and Alice's qubits are separated from the network shown by the red dash line in Fig.~??. Now we see in the left geometry, Bob's qubit is connected with Tod's to-be-teleported qubit through a link, on which we have two interactions, $U_{AB}$ and $U_{AT}$. According to QM/GR, this link can be regarded as a kind of wormhole with a length given by $U_{AB}$ and $U_{AT}$. Among the two operations $U_{AB}$ and $U_{AT}$, $U_{AB}$ connects two points with zero distance since after $U_{AB}$, Alice and Bob's qubits are completely correlated. The real problem is $U_{AT}$. We need to reverse this operation to shrink the length of the wormhole so that Tod and Bob stand at the two ends of a wormhole with a length of 0. In fact the classical qubits from measuring Alice and Tod's qubits provide enough information to achieve this. After reversing $U_{AT}$, the length of the wormhole is zero and Bob can access the information of the teleported quantum state if Tod jumps in the wormhole. So from QM=GR, teleportation is achieved by creating a zero length wormhole connecting Tod and Bob.

Another description of teleportation is a fibre bundle or gauge picture, where the total space is the quantum states and the base space is the geometry. We regard this as a fibre bundle structure since different quantum states may correspond to the same geometry. For example, the initial and final product states of the teleportation operation essentially generate the same geometry since they have exactly the same quantum information pattern.

If geometry is determined by the geodesic tensor network, we can see permutation of qubits and local unitary operations on qubits do not change the geometry since they do not change the entanglement pattern of a quantum state. So the fibre or the gauge group is a product of permutation group and all local unitary operations on each qubit. Then obviously the operational procedure of teleportation generates a closed loop in the base space, i.e. the geometry. But in the total space we have an open curve. That's to say, teleportation achieves a kind of gauge transformation.

The initial product state corresponds to a geometry with three separated points and the gauge is fixed by the initial quantum state. After the entanglement of Alice and Bob, the geometry changes to a single point (Tod) and a link connection Alice and Bob. After the scrambling operation between Alice and Tod, all the three parts are connected. The joint measurement on Tod and Bob cuts Tod and Bob out into two separated points. At the same time, the measurement and the four possible measurement results achieve local unitary operations on the three qubit and a permutation of them as well. The post-measurement local operation on Alice inverses the four possible local operations on Bob. So finally the gauge is changed by a pure permutations of the three qubits if the post-measurement also fixes the local operations on Bob and Tod, which is of course achievable.

The standard quantum circuit for quantum teleportation is shown in Fig.~3. Alice, Bob and Tod, each one of them holds a qubit. Starting from a product state $|000\rangle$, Tod generates an unknown quantum state to be teleported. Alice and Bob entangles their qubits. After the interaction between Tod and Alice's qubits, we are at the final stage of the teleportation protocol. Before the joint measurement on Tod and Alice's qubits, the quantum state of the three qubits is generated by the quantum circuit. If we regard this circuit as the geodesic connecting the initial product state to the current quantum state, then the quantum circuit represents a space geometry. According to GM/GR, we can compute the distances between any two points on this geometry based on their mutual information. After the measurement, Tod and Alice's qubits are separated from the network shown by the red dash line in Fig.~??. Now we see in the left geometry, Bob's qubit is connected with Tod's to-be-teleported qubit through a link, on which we have two interactions, $U_{AB}$ and $U_{AT}$. According to QM/GR, this link can be regarded as a kind of wormhole with a length given by $U_{AB}$ and $U_{AT}$. The problem now is to shrink the length of the wormhole so that Bob can dig out the state