Topological quantum computer

To live up to its name, a topological quantum computer must provide the unique computation properties promised by a conventional quantum computer design, which uses trapped quantum particles. In 2000, Michael H. Freedman, Alexei Kitaev, Michael J. Larsen, and Zhenghan Wang proved that a topological quantum computer can, in principle, perform any computation that a conventional quantum computer can do, and vice versa.[10][11][12]

They found that a conventional quantum computer device, given an error-free operation of its logic circuits, will give a solution with an absolute level of accuracy, whereas a topological quantum computing device with flawless operation will give the solution with only a finite level of accuracy. However, any level of precision for the answer can be obtained by adding more braid twists (logic circuits) to the topological quantum computer, in a simple linear relationship. In other words, a reasonable increase in elements (braid twists) can achieve a high degree of accuracy in the answer. Actual computation [gates] are done by the edge states of a fractional quantum Hall effect. This makes models of one-dimensional anyons important. In one space dimension, anyons are defined algebraically.

Even though quantum braids are inherently more stable than trapped quantum particles, there is still a need to control for error inducing thermal fluctuations, which produce random stray pairs of anyons which interfere with adjoining braids. Controlling these errors is simply a matter of separating the anyons to a distance where the rate of interfering strays drops to near zero. Simulating the dynamics of a topological quantum computer may be a promising method of implementing fault-tolerant quantum computation even with a standard quantum information processing scheme. Raussendorf, Harrington, and Goyal have studied one model, with promising simulation results.[13]

Fibonacci anyons are defined by three qualities:

1. They have a topological charge of ${\displaystyle \tau }$. In this discussion, we consider another charge called ${\displaystyle 1}$ which is the ‘vacuum’ charge if anyons are annihilated with each-other.
2. Each of these anyons are their own antiparticle. ${\displaystyle \tau =\tau ^{*}}$ and ${\displaystyle 1=1^{*}}$.
3. If brought close to each-other, they will ‘fuse’ together in a nontrivial fashion. Specifically, the ‘fusion’ rules are:
   1. ${\displaystyle 1\otimes 1=1}$
   2. ${\displaystyle 1\otimes \tau =\tau \otimes 1=\tau }$
   3. ${\displaystyle \tau \otimes \tau =1\oplus \tau }$
4. Many of the properties of this system can be explained similarly to that of two spin 1/2 particles. Particularly, we use the same tensor product ${\displaystyle \otimes }$ and direct sum ${\displaystyle \oplus }$ operators.

The last ‘fusion’ rule can be extended this to a system of three anyons:

${\displaystyle \tau \otimes \tau \otimes \tau =\tau \otimes (1\oplus \tau )=\tau \otimes 1\oplus \tau \otimes \tau =\tau \oplus 1\oplus \tau =1\oplus 2\cdot \tau }$

Thus, fusing three anyons will yield a final state of total charge ${\displaystyle \tau }$ in 2 ways, or a charge of ${\displaystyle 1}$ in exactly one way. We use three states to define our basis.[15] However, because we wish to encode these three anyon states as superpositions of 0 and 1, we need to limit the basis to a two-dimensional Hilbert space. Thus, we consider only two states with a total charge of ${\displaystyle \tau }$. This choice is purely phenomenological. In these states, we group the two leftmost anyons into a 'control group', and leave the rightmost as a 'non-computational anyon'. We classify a ${\displaystyle |0\rangle }$ state as one where the control group has total 'fused' charge of ${\displaystyle 1}$, and a state of ${\displaystyle |1\rangle }$ has a control group with a total 'fused' charge of ${\displaystyle \tau }$. For a more complete description, see Nayak.[15]

Following the ideas above, adiabatically braiding these anyons around each-other will result in a unitary transformation. These braid operators are a result of two subclasses of operators:

• The F matrix
• The R matrix

The R matrix can be conceptually thought of as the topological phase that is imparted onto the anyons during the braid. As the anyons wind around each-other, they pick up some phase due to the Aharonov–Bohm effect.

The F matrix is a result of the physical rotations of the anyons. As they braid between each-other, it is important to realize that the bottom two anyons—the control group—will still distinguish the state of the qubit. Thus, braiding the anyons will change which anyons are in the control group, and therefore change the basis. We evaluate the anyons by always fusing the control group (the bottom anyons) together first, so exchanging which anyons these are will rotate the system. Because these anyons are non-abelian, the order of the anyons (which ones are within the control group) will matter, and as such they will transform the system.

The complete braid operator can be derived as:

${\displaystyle B=F^{-1}RF}$

In order to mathematically construct the F and R operators, we can consider permutations of these F and R operators. We know that if we sequentially change the basis that we are operating on, this will eventually lead us back to the same basis. Similarly, we know that if we braid anyons around each-other a certain number of times, this will lead back to the same state. These axioms are called the pentagonal and hexagonal axioms respectively as performing the operation can be visualized with a pentagon/hexagon of state transformations. Although mathematically difficult,[16] these can be approached much more successfully visually.

With these braid operators, we can finally formalize the notion of braids in terms of how they act on our Hilbert space and construct arbitrary universal quantum gates.[17]