CritPt / Challenge_34_main / We will explore a simple example of a quantum tensor network, a tensor network…

# Problem setup:
We will explore a simple example of a quantum tensor network, a tensor network made of unitary tensors that can be executed as a quantum circuit on a quantum computer. In particular, we will perform a numerical computation of a quantum matrix product state (qMPS) with bond dimension χ=2.

Consider a qMPS circuit defined on N+1 qubits labelled 0,…,N, with all qubits starting in the |0⟩ state. In the qMPS circuit, the two-qubit gate
$$
\begin{align*}
U_{jk} &=e^{-i b (X_j X_k + Z_j Z_k)/2}e^{-i a X_k/2},
\end{align*}
$$
which acts on qubits (j,k), is used, where X_(j),Z_(j) are Pauli matrices acting on qubit j. (Note that the above equation describes matrix multiplication, where operations occur in the opposite order as gates in a quantum circuit.) In the circuit that generates the qMPS state, the U_(jk) gate is applied to qubits (0,1),(0,2),(0,3),…,(0,N) in that order. Suppose that 0<a,b<π/2.

# Main problem:
What is the expectation value of the two-point correlation function lim_(N→∞)⟨Z_(N−2)Z_(N)⟩  of the qMPS in the thermodynamic limit as a function of a and b?

# Problem setup:
We will explore a simple example of a quantum tensor network, a tensor network made of unitary tensors that can be executed as a quantum circuit on a quantum computer. In particular, we will perform a numerical computation of a quantum matrix product state (qMPS) with bond dimension $\chi=2$.

Consider a qMPS circuit defined on $N+1$ qubits labelled $0,\ldots,N$, with all qubits starting in the $|0\rangle$ state. In the qMPS circuit, the two-qubit gate
$$
\begin{align*}
U_{jk} &=e^{-i b (X_j X_k + Z_j Z_k)/2}e^{-i a X_k/2},
\end{align*}
$$
which acts on qubits $(j,k)$, is used, where $X_j,Z_j$ are Pauli matrices acting on qubit $j$. (Note that the above equation describes matrix multiplication, where operations occur in the opposite order as gates in a quantum circuit.) In the circuit that generates the qMPS state, the $U_{jk}$ gate is applied to qubits $(0,1),(0,2),(0,3),\ldots,(0,N)$ in that order. Suppose that $0 < a,b < \pi/2.$

# Main problem:
What is the expectation value of the two-point correlation function $\lim_{N\rightarrow \infty} \langle Z_{N-2} Z_{N} \rangle$  of the qMPS in the thermodynamic limit as a function of $a$ and $b$?