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Problem
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problem description
# 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 .
Consider a qMPS circuit defined on qubits labelled , with all qubits starting in the 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 , is used, where are Pauli matrices acting on qubit . (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 gate is applied to qubits in that order. Suppose that
# Main problem:
What is the expectation value of the two-point correlation function of the qMPS in the thermodynamic limit as a function of and ?Plain-text mathematical notation (without MathML)
# 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?Original LaTeX notation
# 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 $b$?code template
Code
import sympy as sp
a, b = sp.symbols('a b')
def answer(a, b):
r"""
Return the expression of the expectation value in SymPy format.
Inputs
----------
a: sympy.Symbol, single-qubit rotation angle in $U_{jk}$, $a$
b: sympy.Symbol, two-qubit entangling angle in $U_{jk}$, $b$
Output
----------
expectation: sympy.Expr, the expectation value of the two-point correlation function of the qMPS in the thermodynamic limit, $\lim_{N\rightarrow \infty} \langle Z_{N-2} Z_{N} \rangle$
"""
# ------------------ FILL IN YOUR RESULT BELOW ------------------
expectation = ... # a SymPy expression of inputs
# ---------------------------------------------------------------
return expectation
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initial import