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If I use the matrix product state method in Qiskit to simulate quantum circuits, I can obtain the exepctation value of an observable by using a finite amount of shots. Is there a way to obtain the exact expectation value wihout using the state vector?

If it is useful, I'm interested in $\langle \psi | Z_1 | \psi \rangle$, i.e. the expectation value of the Pauli Z matrix of the first qubit.

EDIT: I asked this question also in this thread, but it was an EDIT of the original question, so I thought it was better to create another thread.

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By default, the matrix product state method in Qiskit does not compute the state vector when computing the expectation value for a Pauli matrix. For a general matrix, it does compute the density matrix first. For this purpose, should should define your observable as a pauli string, for example (if you are using qiskit runtime):

from qiskit.quantum_info import SparsePauliOp
observable = SparsePauliOp("XZ")
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