I am trying to repeat the results from Peng and Kowalski (Eq 11) which solves the analytical form of an equation consisting of powers of the Hamiltonian (moments):

The $K$-th order PDS formalism is then associated with defining the introduced $K$ real parameters $(a_1,\dots,a_K)$ that minimize the value of $R_K(s,a_1,\dots,a_K)$. In this minimization process the necessary extreme conditions are given by the system of equations $$\frac{\partial R_K(s,a_1,\dots,a_K)}{\partial a_i} = 0, \hspace{4mm} (i=1,\dots,K),$$ which can be alternatively represented by the matrix system of equations for an auxiliary vector $\textbf{X} = (X_1,\dots,X_K)^T$ $$\textbf{MX}=-\textbf{Y}.$$ Here, the matrix elements of $\textbf{M}$ and vector $\textbf{Y}$ are defined as the expectation values of Hamiltonian powers (i.e. moments), $M_{ij} = \langle \phi|H^{2K-i-j}|\phi\rangle,$ $Y_i = \langle \phi|H^{2K-i}|\phi\rangle$ $(i,j=1,\dots,K)$.

Using the Jordan-Wigner transformation, I already get the Hamiltonian and moments in the qubit form but am having trouble finishing the solution. From above, $X=(X_1,\dots, X_K)^T$ ($K=4$ in my case) is the root waiting to be solved.


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