For example, given the $R$ & $F$ gates and Toric codes for a given problem, how to convert this code into the conventional circuit model and vice versa. From the literature developed, it seems that they tackle fairly different kinds of problems for these realizations.

How to write, let's say, Grover's algorithm for a topological experiment? What is the correspondence to convert them back and forth?


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This is one of the more subtle points of topological quantum computing (TQC).

TQC is a general idea: make a quantum computer using anyons/topologically ordered systems. There is no a-priori specification of how data should be stored or manipulated. Any given quantum circuit can look very different when performed with different implementations of TQC.

The main reason that you want to have different ways of storing/manipulating data for TQC is that there are lots of different topological orders. Depending on the topological order you're working with, different methods would be appropriate.

By "be more appropriate", I really mean "would allow for universal quantum computation". Once an anyonic theory is specified, you want to store your data/act on your data in the most efficient/protected manner possible that will allow for universal quantum computation. This means that the theory of TQC for very complicated anyons should be simple, and the theory of TQC for very simple anyons should be complicated.

A nice family of examples to get your head around this is the quantum double model. Here, you get an anyonic model based on every finite group $G$. As the group $G$ becomes less-and-less abelian, the anyons become more-and-more powerful. This is quantified below:

  • $G$ non-solvable. Here, you can perform TQC as follows. You store your data in the internal states of pairs of anyons, and compute via braiding/fusion alone. More exactly, for every element $g\in G$ you can create a $g$-subtype quasiparticle $\left| g\right>$, with anti-quasiparticle $\left| g^{-1}\right>$. Your data will be store in qudits isomorphic to $\mathbb{C}^d$, each represented by a pair of sites containing a superposition of the states $\left| g\right>\otimes \left| g^{-1}\right>$ for some $g\in G$. Moving the pair $\left| g\right>\otimes \left|g^{-1}\right>$ through the pair $\left| h \right>\otimes \left| h^{-1}\right>$ will transform the first pair by conjugation, sending it to $\left| hg h^{-1}\right>\otimes \left| (hg h^{-1})^{-1}\right>$. These sorts of qudit actions give universal classical computation by a general mathematical argument about non-solvable groups (at least, this is literally true in the case that $G$ is simple). Measurements can then be performed via fusion, and this is enough for universal quantum computation. All of this is explained in the original article of Mochon.

  • $G$ non-nilpotent. Here, your data is stored in the internal states of anyons corresponding to both conjugacy classes in $G$ and the internal states of anyons corresponding to irreducible representations of $G$, and once again computation is performed via braiding. Now that you are braiding between a wider class of anyons the exact specifications of what braiding you have to perform given a quantum circuit will be different. This is explained in the follow-up article of Mochon.

  • $G=\mathbb{Z}_2$. This is the toric code. Data is stored in the ground states of a $g$-holed system induced with the topological phase corresponding to $\mathbb{Z}_2$, and computation is performed by moving anyons around homotopically non-trivial paths of the surface, as well as transforming the surface (i.e. moving holes around each other). This only implements Clifford gates, however. You also need some non-topologically protected way of implementing the $T$-gate, which ends up being the hardest part of the theory by far. This should be explained in any surface code survey, e.g. here.

  • Fermions. If you want to go all the way down, TQC can be performed with fermions. Information is stored in pairs of sites, an each pair is labeled with a "top" element and "bottom" element. Each site can either be occupied by a fermion or be empty. Every pair of sites will have one site occupied - either the top or the bottom. The key insight is that moving the top site of one pair around the top site of the other pair will add a phase of -1 if both sites are occupied, and do nothing otherwise. That is, this is exactly the controlled phase gate. If you also have a way of performing all one-qubit gates this gives universal TQC. There's no topologically protected way of performing (say,) the Hadamard gate. For this you have to resort to methods like interferometry. This method was proposed by Lloyd.

Of course, there are lots of methods in between these, and many variants on any given method.


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