Topological Superconductors

I've worked in quantum computing (many years ago...) and that statement surprises me as well. Not because it's a caveat, but because, last I checked, there isn't really any such thing as "nonuniversal" quantum computation; there are no well-defined complexity classes in between BQP and BPP (the latter is conjectured to = P). Either you have universal quantum gates, or you don't!

The paper mentions this more specifically a few paragraphs in:

>Majorana Fermions can emerge in a TSC (but not in a conventional metal) because a Majorana Fermion is its own antiparticle. [edit: I don't think this is quite right... Majorana fermions can emerge in a TSC because the spin-statistics theorem allows more freedom in two dimensions]

>[...]

>To achieve quantum computing, a Majorana Fermion normally needs to be bound to a defect (hence called MBS) with zero energy (hence also called MZM), which can contain nontrivial non-Abelian statistics.

>[...]

>Here, an MZM is a type of anyon termed Ising anyon, which contain non-Abelian statistics but by itself is not sufficient to carry out universal quantum computations. (3, 26)

Now Ising anyons are something I actually studied in the context of topological error-correction, and my fuzzy memory says that some of these anyons can perform computations on qubits but are limited to Clifford statistics and so do not provide a useful quantum speedup vs classical computers (in particular, this includes the synthetic anyons on the "Kitaev code" and the "color code" synthetic anyon models). But let's see what the references cited (3, 26) say. First reference (3):

>Unfortunately, such qubit rotations are too restrictive to permit universal quantum computation; two additional processes are needed [12]. The first is a π/8 phase gate that introduces phase factors e±iπ/8 depending on the occupation number corresponding to a given pair of Majoranas. The second is the ability to read out the eigenvalue of the p

Yes, this is exactly the "Clifford group statistics" problem. Incidentally, you don't need specifically the pi/8 gate; a couple of others will do (e.g. Toffoli's gate), but pi/8 is widely believed to be the simplest. When you are "stuck" with Clifford gates, you are able to perform "quantum computations" in the sense that you use qubits to do computing, but your complexity class is BPP.

Now reference (26). This also mentions the Clifford problem (in a more detailed way) but expands:

>Neither of these gates can be applied exactly, which means surrendering some of the protection we have worked so hard to obtain and we need some software error correction. However, it is not necessary for the pi / 8 phase gate or the two-qubit measurement to be extremely accurate in order for error correction to work. The former needs to be accurate to within 14% and the latter to within 38% (Bravyi, 2006). Thus the requisite quantum error correction protocols are not particularly stringent.

So here the hope is that extremely high fidelities (the great promise of topological quantum computing) can be obtained for most of the computations, which reduces the required fidelity needed to obtain stable error-correction algorithms (the "threshold theorems") for the remaining operations that allow universal quantum computation.

Whether this "rescues" the topological superconductor as a practical platform for BQP-class computations is not immediately obvious to me, but I hope this gives an idea of what you get from a TSC and what you need to add to it in order to get a BQP-class quantum computer. For comparison, logic gate error tolerances required for quantum computers lacking any topologically protected subsystem to achieve stable computation are on the order of 0.1% or tighter.

I don’t know for sure, but I’d bet that their construction can only implement “Clifford” gates, which are a subset of the gates needed for arbitrary quantum computation.

This is a common situation. Lots of proposed quantum computing architectures are built to support only Clifford gates, and then they have a separate subsystem dedicated to implanting non-Clifford gates. The keyword to search here is “magic state distillation”.

More info here: https://quantumcomputing.stackexchange.com/questions/13629/w...