I'm studying output expectation values of random quantum circuits. These random circuits are built using random gates from a universal set. I have noticed that the distribution of the output expectation values (of random Pauli observables) of different quanum circuits have mean close to 0 and variance that goes to 0 as the number of qubits increase (I don't know if there is a rigorous explanation for this).
My question is: is there a class of quantum circuits that does not show this behaviour? I'm interested in quantum circuits that have some kind of "parameter" that I can change since I need different instances.
For example, I think that a quantum circuit representing the last step of a VQE (Variatonal Quantum Eigensolver) process should have some expectation value far from 0 (i.e. the ground state energy) since it represents the ground state of an Hamiltonian (but it would be too expensive to run an entire VQE algorithm for each instance). Instead, if I use random angles (without particular constraints) for the same quantum circuit ansatz, I have the same behaviour described at the beginning of the post.