How to convert a quantum circuit into a tensor network?

Question

I need to perform tensor network contraction on a maxcut circuit, are there any resources or code available which converts maxcut circuit into a tensor network (preferably the adjacency list or any graph based representation)?

Answer 1

Quantum circuits are effectively already a special type of tensor networks. We just need to interpret them appropriately to see this. Normally, we think of lines as representing qubits and boxes as quantum gates i.e. $2^k\times 2^k$ matrices in the projective unitary group $PU(2^k)$ where $k$ is the number of qubits on which the gate acts. Instead, we can interpret lines as contractions and boxes as tensors with $2k$ indices with each index ranging over $\{0,1\}$. See Example $3$ on page $7$ in this paper for more details.

Note that some gates admit an alternative interpretion as small subnetworks. For example, CNOT may be interpreted as a tensor $$ \text{CNOT}_{i_1j_1i_2j_2}=\begin{cases}\delta^{i_1}_{i_2}\delta^{j_1}_{j_2}\quad&\text{when}\,i_1=0\\ \delta^{i_1}_{i_2}(1-\delta^{j_1}_{j_2})\quad&\text{when}\,i_1=1\\ \end{cases} $$ or it may be interpreted as a contraction of a $\text{COPY}_{ijk}$ tensor with a $\text{XOR}_{ijk}$ tensor. See Example $4$ on page $7$ in the above paper.

This interpretation allows us to compute the unitary (or indeed any isometry obtained by fixing a subset of the input or output qubits including the input or output state) of any quantum circuit. This is the basis of tensor network simulation of quantum circuits.

Answer 2

You may find TensorCircuit package helpful.