At this point you might wonder what all of the previous has to do with topological quantum computing and where it’s all going, so let me sketch an idea which brings the concepts together.

A quantum computer essentially starts with a quantum state of (tangled and non-entangled) elements (atoms, laser, spin…) which via unitary transformation evolves towards a new state. After transformation one can measure the system which then collapses the state in one of the components (basis of the Hilbert space under consideration).
Imagine that we define a qubit as

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and a general 3-qubit states as

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With time flowing upwards you can have transformations $T$ from one state to another

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A quantum algorithm becomes essentially the weaving of a braid. At the end of the weaving process the effect of this can be measured if one uses anyons because the twisting induces a phase shift. At the same time, because a braid is equivalent to a knot one can use the Jones polynomial to identify the quantum algorithm with a class of knots. The class corresponding to all the knots which are the same up to Reidemeister moves. Finally, the physical implementation of anyons and the topological quantum field theory describing it is tightly linked to Chern-Simons theory. The picture above can be seen from different angles but in essence all complentary views of one and the same thing:

  • it’s a physical process where anyons evolve
  • it’s a quantum algorithm
  • it’s a topological quantum field theory
  • it’s a braid (it’s an element of the Artin braid group)
  • it’s a knot
  • it’s an invariant polynomial.

Note that the weaving of the threads across qubits is a beautiful visualization of entanglement. When the braids wildly weaves things across all qubits you get an entangled state one cannot describe as simply a product of 1’s and 0’s.

Hopefully this conceptual sketch brings forward the bridge between the various domains and in the remains of this article we’ll focus on braids and how they are related to knots and the Jones polynomial.

A braid can be described by atomic operations $\sigma_i$ where for example the first three elements are as seen in the picture below

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If you play a bit with these simple switches you will discover that they satisfy the following properties

$$\sigma_i \sigma_{I+1} \sigma_i= \sigma_{I+1} \sigma_i \sigma_{I+1} \\ \sigma_i \sigma_j =\sigma_j \sigma_i \; \text{if} \; |i-j|>1.$$

The $\sigma$’s are closed under multiplication and form a group called the Artin braid group.

Braids are an alternative representation of knots and this simple idea is embodied in the so-called Alexander theorem:

any knot in 3D can be transformed via Reidemeister moves into a closed braid and vice versa.

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Now, since braids and knots are one and the same thing but braids can be generated via the $\sigma$’s defined above, it means there has to be a way to generate the Jones polynomial directly from $\sigma$’s.

If you look at braids and decompose them in the same way as we did with knots you get things that are almost braids:

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and for example

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So, let’s define operators $U$ which like the earlier $\sigma$’s define the basic operattions:

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and if you work out a few simple examples like the one below

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you will discover a pattern which can be summarized in the following rules

$$U_i.U_{i\pm 1}.U_i = U_i \\ U^2_i = \delta.U_i \\ U_i.U_j = U_j.U_i \;\text{if}\; |i-j|>1.$$

These relations define what is known as the Temperley-Lieb algebra.

If you want to make compute the normalized bracket from the Temperly-Lieb algebra you can see the necessary mapping from a simple example. Take the Hopf link and if you compute the normalized bracket you will find

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and noticing that the Hopf link has a braid representation

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with the computation above you can see that the mapping should satisfy

$$(A^2\delta + 2AB)U_1 + B^2 = -A^4-A^{-4}$$

where the $\delta$ comes from the loop in the first term of the expansion.

If one defines

$$\langle 1\rangle = d\\ \langle U_1\rangle = 1 \\ \langle U_1^2\rangle = d$$

and so one you will find that the normalized brackets are identical. So, you can use a braid or a knot and obtain the same polynomial.

The braid representation has the advantage that it fits more a computational model and that one can effectively create simple programs (say, in Java or C#) to represent braids and automatically compute the Jones polynomial from it. Not so with 3D knots.