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What are the best tensor network-based simulators libraries in Python?

In particular, I'm interested in calculating the expectation value of an observable measured at the end of a quantum circuit.

EDIT: Qiskit has a matrix product state simulation method, but I can't understand if I can compute the exact expectation values (without using the full state vector) or if I am limited to a finite number of shots.

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In Qiskit, if you want to compute the expectation value of a given operator $\hat{O}$ using the matrix_product_state-based simulation, you can use the primitive BackendEstimator as follows:

from qiskit.circuit.random import random_circuit
from qiskit.quantum_info import SparsePauliOp
from qiskit_aer import AerSimulator
from qiskit.primitives import BackendEstimator

num_qubits = 5
psi = random_circuit(num_qubits=num_qubits, depth=10, seed=42)
O = SparsePauliOp('Z' * num_qubits)

mps_simulator = AerSimulator(method='matrix_product_state')
estimator = BackendEstimator(backend=mps_simulator)
job = estimator.run(circuits=[psi], observables=[O])

print(job.result())

However, I don't think it is possible to compute $\langle \psi | \hat{O} | \psi \rangle$ exactly even if you pass shots=None to the method BackendEstimator.run(). The default number of shots is 1024 as you can see by printing the job result.

I also suggest to take a look at tenpy, which is a pure Python library for the simulation of quantum systems with tensor networks.

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