From general Hamiltonian to Ising Hamiltonian

Question

I would like to convert my qubit hamiltonian fom the HeH+ system that I have obtained using Qiskit to an Ising or QUBO model. I have seen multiples examples from QUBO to Qubit Hamiltonian but on the other direction no. Does anyone know how I can do it? Attached my Qubit Hamiltonian.

Answer 1

One way to do it would be to use a transformation, such as this one:

\begin{align} X_i &= \frac{1 - Z_{i,j}Z_{i,k}}{2}\textrm{sgn}(j)\textrm{sgn}(k)\tag{1}\\ Y_i &= \textrm{i}\frac{Z_{i,k}-Z_{i,j}}{2}\textrm{sgn}(j)\textrm{sgn}(k)\tag{2}\\ Z_i &= \frac{Z_{i,j}+Z_{i,k}}{2}\textrm{sgn}(j)\textrm{sgn}(k)\tag{3}\\ I_i &= \frac{1 + Z_{i,j}Z_{i,k}}{2}\textrm{sgn}(j)\textrm{sgn}(k)\tag{4}.\\ \end{align}

Then all of your $X$ and $Y$ operators will be $Z$ operators. In other words, you've transformed a general XYZ model into an Ising model with the help of auxiliary qubits with labels such as $(i,j)$ and $(i,k)$.