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I have a quantum circuit similar to the ones used in VQE methods. One difference is that the angles of the $R_y$ rotations are limited in the range $[0, 0.2]$ (and chosen randomly). In my case, I have 30 layers of gates, where each layer contains the single-qubit gates $R_y$ and the $CNOT$ gates, so 30 layers of single-qubit gates and 30 layers of 2-qubits gates.

This is an example of this kind of circuit with 6 qubits (but with only 7 layers of gates for visualization): enter image description here

My question is: is there any classical simulation technique that doesn't scale exponentially with the number of qubits that can calculate the expectation values of the $X$ and $Z$ observables for each qubit? It is ok even if there is a small approximation.

I tried the MPS method implemented in Qiskit, and if I ask for a coefficient of determination $R^2\ge 0.99$ for 100 random circuits (that is a sufficient approximation in my case), the bond dimension seems to scale exponentially with the number of qubits, but I can try this only for $N\le 14$, so it is hard to say.

Thank you.

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