In which known quantum algorithms is sufficient to know $\langle \sigma_1...\sigma_{N} \rangle$ and/or $\langle \sigma_i \rangle$ $\forall i$ in order to solve the problem that the same algorithm would have solved? Where $\sigma=X$,$Y$ or $Z$.
$N$ can be for example the total number of qubits that describe the circuit (see below).
I try to explain the question with some examples:
In Bernstein Vazirani, if I know $\langle Z_i \rangle$ $\forall i$ I get the result since I have only one possible outcome, same with Grover with only 1 searched item.
In VQE (and also in Quantum machine learning algorithms I think), what is important for the algorithm is $\langle \sigma_1...\sigma_{N} \rangle$.
In Shor I think that I can't find the period by knowing the expectation values, same with Grover with 2 or more searched items. I don't know if this is true.
In general $\langle \sigma_1...\sigma_{N} \rangle$ can also not regard all the qubits, but for example all the qubits minus the last one ($\langle \sigma_1...\sigma_{N-1} \rangle$), or also $\langle \sigma_1...\sigma_{N-2} \rangle$, or $\langle \sigma_2...\sigma_{N} \rangle$ and so on. I would like to know the results that i can achieve by knowing some of these expectation values. The impotant thing is that the number of expectation values that I need to know is polynomial with respect to the number of qubits, i.e. is efficient to compute the result knowing these expectation values.