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From the VQE paper they claim that a Hamiltonian can be expressed as a polynomial series of pauli operators (equation 1).

While coding up VQE from scratch I made a function which would allow me to specify coefficients up to 2nd order to build the corresponding Hamiltonian (for 1 qubit).

But I noticed that $\sigma_y\sigma_z$ is in fact not hermitian, and so it doesn't give me purely real energy eigenvalues.

So is it not true the other way around? Can I not specify an arbitrary polynomial series of Pauli operators such that the result is a Hamiltonian for a closed system?

**EDIT**

See the accepted answer. I actually misunderstood the equation in the paper, not realising that the higher order terms were actually tensor products and only applicable to more-than-single-qubit systems.

1Alexander, the equation (1) (or slight modification of it) is applicable also for one qubit case: for one qubit $H = \sum_{\alpha} h_{\alpha} \sigma_{\alpha} = h_i I + h_x \sigma_x + h_y \sigma_y + h_z \sigma_z$. – Davit Khachatryan – 2020-05-25T18:28:02.150

And in the second (similarly for the third) sum of the equation (1) we don't have separate $h_{\alpha}$ and $h_{\beta}$, instead, there should be $h_{\alpha \beta}^{ij}$ that is not (necessarily) equal to $h_{\alpha}^i \cdot h_{\beta}^j$. – Davit Khachatryan – 2020-05-25T18:41:14.220

@DavitKhachatryan I'm pretty sure I applied my erroneous thinking in the middle of transcribing it which further reinforced said thinking, lol – Alexander Soare – 2020-05-25T19:22:14.500

Alexander, it is ok :). Sometimes the notations are not clear. BTW here is my Qiskit implementation/tutorial for one qubit VQE that might be interesting: https://github.com/DavitKhach/quantum-algorithms-tutorials/blob/master/variational_quantum_eigensolver.ipynb

– Davit Khachatryan – 2020-05-25T19:45:17.380