Measuring Pauli strings using generators

Question

I am trying to find the ground state of a Hamiltonian using VQE. I have decomposed the Hamiltonian into a set of Pauli strings. To decrease the number of actual measurements that have to be done, can I just measure the generators of this set of Pauli strings? For example, say if the set is $\{I, Z_1Z_2, Z_2Z_3, Z_1Z_3\}$, can I just measure $Z_1Z_2$, and $Z_2Z_3$, and then find the expectation value of $Z_1Z_3$ from the expectation values of $Z_1Z_2$ and $Z_2Z_3$?

Answer 1

Here I am going to show why $\langle Z_1 Z_3 \rangle$ generally cannot be estimated from $\langle Z_1 Z_2 \rangle$ and $\langle Z_2 Z_3 \rangle$. Let's start with an arbitrary three-qubit state:

\begin{align*} |\psi \rangle = c_{000} &|000\rangle + c_{001} |001\rangle + c_{010} |010\rangle + c_{011} |011\rangle + \\ +c_{100} &|100\rangle + c_{101} |101\rangle + c_{110} |110\rangle + c_{111} |111\rangle \end{align*}

And because of this answer the expectation value for $Z_1 Z_2$, $Z_2 Z_3$ and $Z_1 Z_3$ (I am using this convention for qubit indexes $|q_1 q_2 q_3 \rangle$):

$$ \langle Z_1 Z_2 \rangle = |c_{000}|^2 + |c_{001}|^2 - |c_{010}|^2 - |c_{011}|^2 - |c_{100}|^2 - |c_{101}|^2 + |c_{110}|^2 + |c_{111}|^2 \\ \langle Z_2 Z_3 \rangle = |c_{000}|^2 - |c_{001}|^2 - |c_{010}|^2 + |c_{011}|^2 + |c_{100}|^2 - |c_{101}|^2 - |c_{110}|^2 + |c_{111}|^2 \\ \langle Z_1 Z_3 \rangle = |c_{000}|^2 - |c_{001}|^2 + |c_{010}|^2 - |c_{011}|^2 - |c_{100}|^2 + |c_{101}|^2 - |c_{110}|^2 + |c_{111}|^2 $$

From this expressions one can see that $\langle Z_1 Z_3 \rangle$ generally cannot be estimated from $\langle Z_1 Z_2 \rangle$ and $\langle Z_1 Z_3 \rangle$. For prove let's consider this conterexample when $|c_{000}| = |c_{010}|$, $|c_{001}| = |c_{011}|$:

\begin{equation*} \langle Z_1 Z_2 \rangle = - |c_{100}|^2 - |c_{101}|^2 + |c_{110}|^2 + |c_{111}|^2 \\ \langle Z_2 Z_3 \rangle = |c_{100}|^2 - |c_{101}|^2 - |c_{110}|^2 + |c_{111}|^2 \\ \langle Z_1 Z_3 \rangle = 2|c_{000}|^2 - 2|c_{001}|^2 - |c_{100}|^2 + |c_{101}|^2 - |c_{110}|^2 + |c_{111}|^2 \end{equation*}

So for the same $\langle Z_1 Z_2 \rangle$ and $\langle Z_2 Z_3 \rangle$ there can be many different possible values for $\langle Z_1 Z_3\rangle$ because by varying the values $|c_{000}| = |c_{010}|$ and $|c_{001}| = |c_{011}|$ the expectations $\langle Z_1 Z_2 \rangle$ and $\langle Z_2 Z_3 \rangle$ will not be changed but $\langle Z_1 Z_3 \rangle$ will be changed. So $\langle Z_1 Z_3 \rangle$ cannot be estimated from $\langle Z_1 Z_2 \rangle$ and $\langle Z_2 Z_3 \rangle$ in this conterexample.