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Suppose there is any arbitrary ansatz producing variational energy for any arbitrary Hamiltonian. What is the physical unit corresponding to the energy? Since numerical algorithms only output a numeric value, the unit is missing. I know for quantum chemistry, it is usually taken to be Hartree. However, not all problem Hamiltonians are from chemistry.

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Short answer: The units of your result will be the same as the units of your input problem to VQE (units of your Hamiltonian coefficients).

Long answer: In a VQE calculation, some problem Hamiltonian is mapped to a representation that is suitable for implementation in a quantum circuit. Often, this means mapping all Hamiltonian operators to strings of Pauli operators with appropriate coefficients. Then there are two cases to distinguish.

  1. Given your original problem Hamiltonian corresponds to a physical system, it will have coefficients with units of energy. Take the generic kinetic energy term of a Hamiltonian, $\hbar^2 \nabla^2 / 2m$ has units of energy ($J$). Sometimes energies may be expressed in alternative units (e.g. Hartree, eV, etc.) and it is your choice which unit system you use (see here for an example of a ground state calculation of an electronic structure problem in Hartree, but also of a vibrational problem in $cm^{-1}$). Now, if you do a VQE calculation, the coefficients of your Pauli operators in your qubit-mapped Hamiltonian will have the same units as the coefficients of your original problem Hamiltonian. Your results will then be expressed in the same units.

  2. VQE is essentially an optimization of a cost function. You can also use VQE to optimize an arbitrary cost function that you construct such that it describes an abstract problem (see for instance the Max-Cut problem). In that case, your Hamiltonian coefficients will have arbitrary units and so will the VQE result.

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