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The idea of topological quantum computing was introduced by Kitaev in this paper. The basic idea is to build a quantum computer using the properties of exotic types of particles, known as anyons. There are two main properties of anyons that would make them great for this purpose. One is what happens when you use them to create composite particles, a process ...


14

The first thing to do is to think topologically: make sure you understand why a coffee cup is the same thing topologically as a donut. Now, imagine we swap two identical particles, and do it again, so that we are back where we started. Apply this topological thinking to the paths taken by the particles: it is the same as doing nothing. Here I show a ...


9

You are right, it does look like the Wikipedia page needs work, so I will have to update it. But for now I will answer all five questions: 1) What do they mean by "much less restricted than fermions and bosons? The exchange of two fermions or bosons is restricted by: $|\psi_1\psi_2\rangle = \pm|\psi_2\psi_1\rangle$. The "$+$" corresponds to bosons and ...


9

A topological quantum computer could be made by using an exotic phase of matter in which anyons arise as localized effects (such as quasiparticles or defects). In this case, errors typically cost energy, and so the probability is suppressed for small temperatures (though it will never be zero). A topological quantum computer could also be made (or one could ...


7

There were two simultaneous PRLs published by Kitaev & Preskill and Levin & Wen that I think answer your question. These use the area law of entanglement seen by states that can be expressed as ground states of a Hamiltonian with only local interactions. Specifically, suppose you have a 2D system of interacting particles in a pure state. You then ...


6

It depends what you mean by the 'existence' of anyons. One way is to engineer a Hamiltonian which leads to quasiparticles (or other defects) that have anyonic statistics. This will require the Hamiltonian to be implemented, the system to be cooled to sufficiently near the ground state, the anyons to be manipulated, etc. So there's a lot to be done, and I ...


6

Are there other instances of topological QC that do not use anyons? No, that's basically by definition. That said, there are different ways that one could use topological systems in order to achieve quantum computation. In the version you're talking about, you use these anyon pairs to define qubits, and braid them around each other to create quantum gates. ...


5

The Toric code is an error correcting code. The distance of the code (I.e. the number of local operations required to convert one logical state into an orthogonal one) is equal to $N$, where the Toric code is defined on an $N\times N$ grid. One of the places that the performance of the Toric code really wins out is that although it is only distance $N$, the ...


4

There is QTop which is an open-source project that can simulate but also visualize topological quantum codes.


4

When a quantum system, parametrized by a manifold of classical parameters, evolves along a closed path in the parameter space, its state experiences a unitary transformation, which is called a geometric phase. In most applications of quantum computing, this parameter space is usually a set of control parameters used to drive the system. More precisely, the ...


4

From Wikipedia: In physics, topological order is a kind of order in the zero-temperature phase of matter (also known as quantum matter). Macroscopically, topological order is defined and described by robust ground state degeneracy and quantized non-Abelian geometric phases of degenerate ground states (emphasis is mine). Clearly, it is a special ...


4

Microsoft is the only company that is trying to build a topological quantum computer. You mention that topological qubits handle noise far better than other systems, but they are also theoretical. That's the reason Microsoft is applying a topological approach. It's high-risk, high-reward. If Microsoft manages to realize a topological qubit, scaling up a ...


3

The spin-statistics theorem requires a particle's wave function to acquire the same phase when it is rotated by an angle of $2 \pi$ about itself and when exchanged with an identical (indistinguishable) particle. For example, a fermion acquires a phase of $-1$ both in exchange and self-rotation. Thus, when we think of an anyon as a flux-charge entity, we ...


3

Those figures were created manually with sketchup, which is a 3d modelling tool. There was no simulation involved, only careful application of known rules.


3

This is not the first time that there is found 'experimental proof' of (non-Abelian) anyons. Note also that the article does not use the word proof but rather evidence - that's why I also used the quotation marks above. Back in 2012 there supposedly was found the first experimental evidence of Majorana bound states by an observation of the zero bias peak. I ...


2

Another approach to topological quantum computing could be that of topological insulators, and the use of the 1/2 integer quantum Hall effect. These insulators have the potential to be less error-prone. Topological insulators are both insulators, and conductors, at the same time, and being less error-prone, have the potential to provide a robust, quantum ...


1

Microsoft has invested huge resources into engineering topological qubits. Their approach is based on topological Majorana states, which occur at the edges of a topological superconducting chain or at interfaces between such chains. For those who see these words for the first time, a quick mental representation is supplied by a ribbon, which can be twisted a ...


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