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If we start with a Hamiltonian $H(t_i)$, and with our qubits prepared in the ground state of this, and then slowly change this to a Hamiltonian $H(t_i)$, the final state of our qubits should be the ground state of the new Hamiltonian. This is due to the adiabatic theorem, and is the basis of quantum annealing.

But what if it's not the ground state we want? Suppose we start with the first excited state of $H(t_i)$. Does the process give us the first excited state of $H(t_f)$? What about for other states?

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In Practice:

Quantum annealing almost always gives excited states in practice. To get the exact ground state at the end, you need the adiabatic passage to be perfect.

The closest thing to a perfect adiabatic passage is probably this recent paper where they get the ground state with 0.975 fidelity, but this is for 3 qubits with a very simple Hamiltonian (see Eq. 5).

However in the D-Wave machine with 2000 qubits there's $2^{2000}-1$ excited states and much higher likelihood that many of them will be near the ground state. Almost every problem D-Wave has worked on recently seeks an approximate solution, not the absolute global minimum.

In Theory:

What if my annealer is perfect and we can stay in the true ground initial state the whole time? Yes it should be possible to prove the adiabatic theorem for any initial state, not just the ground state, but how are you going to initialize the system in some particular excited state?

There is indeed a technique for doing "constrained" annealing described here. The idea is to use a driver Hamiltonian that commutes with the constraints (see the paragraph above Eq. 2)!

More generally you can adiabatically evolve along a particular symmetry sector. For example if the ground state of a molecular Hamiltonian is a singlet state, but you are only interested in a triplet (excited) state, as long as you start with a state that has triplet symmetry and your driver Hamiltonian conserves spin, you can prove a generalization of the "basic" adiabatic theorem that states that you will remain in the triplet state if you evolve slowly enough.

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