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I am having some trouble understanding how the number of simulation qubits are chosen when finding the eigenvalue of a fermionic Hamiltonian.

For the phase-estimation algorithm, is the number of simulation qubits the same as the number of particles you want to simulate, or is it the number of orthogonal single-particle states you want to include?

For example: If I want to find the eigenvalues for a two-particle system and have 10 qubits to spare, do I initialize the 10 qubits to for example $|0011111111\rangle$ or $|1100111111\rangle$ (1s are un-occupied and 0s are occupied states) and let the time evolution operator act on this system? Or do I simply go for a random two-qubit state, for example, $|00\rangle$ and let the time-evolution operator act on this state?

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Typically the second option: you map the fermionic occupation number for each single-particle state to a qubit. (Also, the usual convention is that $0$ denotes unoccupied, $1$ denotes occupied ;) This mapping is usually accomplished via the Jordan-Wigner or Bravyi-Kitaev transformation, or some hybrid of the two: see https://arxiv.org/abs/1208.5986 for a nice overview of both.

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