Measuring Ising anyons: What is a fusion measurement?

Question

I have several doubts about measuring Ising anyons. Measurement is crucial for quantum computation and even more so for magic state distillation which is necessary to make Ising anyons universal. Several sources, e.g. https://arxiv.org/abs/1501.02813, say that there are basically two ways of measuring Ising anyons, one is a non-topological operation I address in this question, the other is some interferometry experiment which I might tackle in a different question.

The procedure for the measurement is the following: Bring anyons close together so that their degenerate Hilbert space is not degenerate anymore and the energy levels split. I believe I understand this part to some extent so let me sketch what I think I know: In a 4 quasi-particle (e.g. 4 Majorana zero modes) encoding, a qubit can be represented by $$|0\rangle\equiv|(\sigma\times\sigma)_1\times(\sigma \times \sigma)_1\rangle=1 \\ |1\rangle\equiv|(\sigma\times\sigma)_\psi\times(\sigma \times \sigma)_\psi\rangle=1$$ The second state corresponds to two pairs of excited Majorana zero modes whereas the first one does not have these modes. When the quasiparticles $\sigma$ are far apart, then dragging them around each other will cause unitary rotations in the subspace spanned by $|0\rangle $ and $|1\rangle$ which corresponds to the degenerate ground state space of a Hamiltonian that exhibits 4 quasiparticles (I am thinking of a Kitaev chain here). As Majorana zero modes are zero energy excitations when far apart, the space is degenerate. However, bringing close e.g. quasiparticle 1 and 2, suddenly, having the modes present in the system costs a finite amount of energy compared to vacuum, i.e. my logical state $|1\rangle$ sits at a higher energy than $|0\rangle$. Now, of course unitary time evolution by the Hamiltonian of the system will act like $U=e^{-iEt}$ and because of the energy difference $\Delta E$ between $|0\rangle $ and $|1\rangle$, this effectively corresponds to letting a phase gate act on the qubit until the quasi-particles $\sigma$ are pulled apart again. So far so good, only a few minor doubts.

1. When bringing the anonys together, it is claimed that upon measuring the energy of the system, we measure the state of the qubit. Does this mean an energy measurement will collapse the system even if in a superposition of $|0\rangle$ and $|1\rangle$ to either $|0\rangle$ or $|1\rangle$?
2. How can we experimentally measure the energy of the system and how do we know if it corresponds to $|0\rangle$ or $|1\rangle$?
3. When the reference says "detect their charge parity", what does it mean? (My encoding of $|0\rangle$ and $|1\rangle$ is such that both belong to the same fermion parity subspace as this quantity is conserved)

A major doubt however is the following:

Is this related to fusion at all? I read so often that fusion is bringing anyons together and then regarding them as a composite and I even think I came across the term fusion measurement. Until now, I thought fusion is really just viewing two anyons as one composite one but nothing physical.

Answer 1

A bit late, but some comments for the sake of posterity:

• Yes, the claim is that in certain systems, a pair of anyons can be brought close together -- then measuring (for instance) the energy of the pair together should be able to distinguish between their fusion channels.

• I'm not sure about how the energetic measurements would work in practice. I imagine this would largely depend on the specific system used to realise the Ising anyons.

• "I read so often that fusion is bringing anyons together and then regarding them as a composite". That's right. But remember that if two non-abelian anyons are brought together, it's possible that the 'composite' has a charge of form $a+b+\cdots$. That is, the total charge of the composite is in a superposition of charges. Thus, it should be possible to distinguish between said charges with a measurement that projects onto the corresponding subspaces. This is called a fusion measurement, and the measured charge is called the fusion outcome.

• In some systems, it might also be possible to use interferometry to measure fusion outcomes. Ising anyons are actually one of the best candidates for measurement via interferometry: I've written down the details in a blog post.