uTopological Quantum Computing

AN BRIEF OVERVIEWuZach Forster

©December 15, 2015

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Contents

1 Motivations 3

2 Braid Theory 52.0.1 What is a Braid? . . . . . . . . . . . . . . . . . . . . . . 52.0.2 Equivalence of Braids . . . . . . . . . . . . . . . . . . . 62.0.3 The Braid Group . . . . . . . . . . . . . . . . . . . . . . 9

3 Quantum Computing with Anyons 11

4 Conclusion 13

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1 Motivations

The free lunch is over. This is a sort of slogan which has become quite com-mon in the world of computing during recent years, but what does it mean?What it does not mean is that Moore’s Law is dead. It does mean that, totake advantage of emerging information processing technology, programmersmust learn to be increasingly clever. Hardware designers have been bangingtheir heads against a few physical “walls” for over a decade now. Heat is-sues, power constraints, and a shortage of low hanging fruit with respect toinstruction level processing optimizations have stopped the advance of thesingle-processor machine dead in its tracks [7].

Figure 1: Processing power no longer increases at the same rate as transistorcount [7].

Designers have since turned to parallel computing in order to maintain theexpected annual exponential increases in processing power. However, due tothe sequential nature of many (perhaps most) programming problems, dou-bling the number of cores or processors does not double the processing power.In response, we are seeing the slow death of the personal computer. Manyare investing in massive distributed networks, to which small, lightweightpersonal computers can connect and unload most of their work. This is aninteresting reversal of the move from mainframes and timeshare computingto personal computers in the 1980’s, but it is certainly not an ideal solution.Many believe that computing power must be kept in the hands of individuals.

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With this in mind, one of the most hyped technologies on the horizon seemsto be quantum computing.

Although some “large” strides (many of which are of debatable significance)have been made by researchers of quantum computing techniques, many ma-jor issues continue to persist. One of these is the incredible fragility of aquantum system. Quantum computers make use of the fact that quantumbits may hold both of their two possible states simultaneously, with eachstate’s complex amplitude expressing its probability of being observed. Wecan call these two states 0 and 1, as we do when discussing classical comput-ing bits. A simpler way of visualizing this is to imagine that a qubit may holdonly one state at each instant, but that that state is a superposition of both 0and 1. In other words, we can imagine the states as points along the complexsphere, where the north pole maps to the state 1, the south pole maps to thestate 0, and everything else is a superposition of the two, each weighted byits amplitude [3]. These superpositions are written as α|0 ą `β|1 ą, whereα and β are complex numbers (the amplitudes of 0 and 1).

Figure 2: A spherical representation of a quantum bit.

The issue is that minute fluctuations in temperature, magnetic field, etc.can easily disrupt the states of the qubits [3]. This requires incredibly well-controlled environments to be created, making the personal/portable quan-tum computer quite impractical. However, a topological approach to quan-tum computing yields a possible solution.

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2 Braid Theory

In order to understand how the qubits in a topological quantum computercan be impervious to environmental perturbations, we must first discuss thetopological structure that they represent. We will therefore give a brief intro-duction of a subfield of topology called braid theory, a close cousin of knottheory.

2.0.1 What is a Braid?

One common way to define a braid is by viewing it as a collection of non-intersecting paths in R3, each of which connects a point in tpx,0,1q|x P Zuwith a point in tpx,0,0q|x P Zu [4]. Recall that a path from x to y, wherex,y P X is a continuous function f : r0,1s Ñ X, such that fp0q “ x andfp1q “ y. It may be helpful to point out that, since the paths may not inter-sect, neither may their endpoints intersect. Figure 3 depicts a braid. Noticethat the endpoints are lined up and evenly spaced, since they are fixed in avertical, two-dimensional plane. Conversely, the paths connecting them arefree to travel anywhere in R3, with two restrictions. First, no point along anypath may have a z-coordinate greater than 1, or less than 0 (i.e. the pathsmust stay between the endpoints). Second, the paths must move in the neg-ative z-direction at all times [1]. Figure 4 shows an example of a braid whichbreaks the second rule.

Figure 3: A braid.

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Figure 4: Not a braid.

Instead of imagining braids as a series of paths embedded within R3, it isoften convenient to project them onto the plane. This is typically done in sucha way as to guarantee that there are finitely many points on the projectionwhich correspond to two points on the braid, and none which correspond tomore than two points on the braid. This is called a regular projection, andguarantees that we do not lose information by projecting the braid into alower dimensional space [2]. The method of projection is obvious, as we havealready done it by displaying Figure 3 on a two-dimensional sheet of paper.To project a braid onto the x-z plane, we simply remove the y-coordinatefrom every point in path. In order to avoid losing important topologicalinformation, when two paths intersect, we draw the intersection in such away that it is obvious which path was at a higher y-value (which path crossedin front of the other).

Of course, one might imagine that two paths could run for a while withthe same position in the x-z plane (the y-values must differ), so that onewould be hidden behind the other after projection, thus violating our rules.It turns out that we can always manipulate our paths in such a way that,after projection, they only intersect at a finite number of points, and that atmost two paths intersect at any point. This deformation does not change thebraid topologically, which brings us to our next section.

2.0.2 Equivalence of Braids

Topological spaces are identified, not by any specific representation, nor bytheir appearance when viewed by any specific perspective. Instead, they areidentified by a special topological properties which are preserved by home-omorphisms. Such topological properties are aptly named invariants [2]. Wegive an example with knots before discussing braids, so as to form a connec-tion with a more familiar subject. The connections will continue to emergeas we discuss ambient isotopies and words. Such connections arise becauseevery knot is actually the closure of some braid, formed by connecting theendpoints of the braid as shown in Figure 5.

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Figure 5: A Knot as the Closure of a Braid.

Imagine knotting up a piece of string, and then fusing the ends together.Invariants between knots may be (extremely) loosely defined as loops, coils,ties, etc. which cannot be removed from the knot without cutting it at somepoint, unraveling it, and gluing the severed ends back together. Any featuresof a knot which are not invariants, may be removed from the knot, so that twoknots with the same invariant features, but with any number of non-invariantfeatures, may be uncoiled back into identical states. These knots would thenbe considered equivalent. Figure 6 shows two such equivalent knots.

Figure 6: Equivalent Knots.

Braids, being very closely related to knots, have a very similar equivalencerelation. Here, it helps to imagine that your braid consists of a series ofstrings, tied to two boards. The endpoints of the strings cannot move andyou cannot spin the boards around, but you can otherwise wiggle the stringsaround to your heart’s content without changing the identity of the braid.Figure 7 shows pairs of equivalent braids.

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Figure 7: Equivalent Braids.

A much more formal definition of this equivalence relation is that any twobraids which are ambient isotopic are equivalent. An isotopy is a path in aspace of continuous functions mapping X to Y , connecting two such functionsf : X Ñ Y and g : X Ñ Y in such a way that every point on the path isa homeomorphism from X to Y . An isotopy may also be thought of as acontinuous deformation of the aforementioned space X. This deformationdoes not change the topology on X. An ambient isotopy is an isotopy onthe space of continuous functions from X to itself, acting as a path from theidentity function on X to a function linking two embeddings of another spacein X [2]. What this means is that two braids are equivalent if they can becontinuously deformed to one another. In addition, this definition shows whythe strings of a braid cannot pass through each other during this deformation(if they could, all braids would be equivalent). At the point of intersectionduring such a deformation, our isotopy would not be a homeomorphism.Now we know that the set of all braids can actually be compressed into a setof equivalence classes, but finding ambient isotopies between braids wouldnot be a very practical way to establish equivalence. Before finishing ourdiscussion of braid equivalence, we will need to discuss the braid group.

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2.0.3 The Braid Group

It turns out that braids form a very simple group, which will help us tounderstand braid equivalence. Recall that a group is a set of elements, alongwith an associative binary operation ‹, for which an identity element exists,along with an inverse for every element in the group. In the braid group, thebinary operation is simply concatenation of braids, as shown in Figure 8.

Figure 8: Braid Concatenation.

It is clear that concatenation is associative, and that it results in anotherelement of the braid group. It should also be clear that the identity elementis the trivial braid, represented by a series of vertical lines (i.e. none of thelines are coiled around each other).

Suppose that we are working with the group of all braids with n paths/strings.To see that every braid has an inverse, we should first point out that theentire group is generated from a set of basic braids, and so is denotedă σ1,σ2,...,σn´1 ą. If we assign our paths the numbers 1 to n in order,each σi represents path i crossing over path i` 1. The inverse of σi, denotedσ´1i , represents path i crossing under path i ` 1. It is not difficult to con-vince ourselves that all possible n-braids may be formed from combinationsof these elements and their inverses, and it follows that every such braid hasan inverse composed of the inverses of the generators from which it is made(in reverse sequence) [1]. Figure 9 depicts the generators (and their inverses)of the 3-braid group.

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Figure 9: 3-Braid Group Generators and their Inverses.

Although we have described each σi or σ´1i as a distinct braid, all of which areconcatenated together by the group operation to form more complex braids,it is just as easy to imagine that we have only one braid, and that the σi’sand their inverses are operations that we apply to this braid to modify it.Whichever visualization style you prefer, the point is that we can representany braid with a string of these crossings. Such a string is formally called aword, and we will use them to easily establish equivalence between braids [1].

In order to compare the word representations of braids, we must compile alist of relations, and moves which do not change a braid’s identity. The listis as follows:

1. Adding or removing σiσ´1i or σ´1i σi does not change that word.

2. σi`1σiσi`1 “ σiσi`1σi

3. σiσj “ σjσi when |i´ j| ě 2

The first rule comes from the existence of an element’s inverse in a group.The second relation comes from the third Reidemeister move, an ambientisotopy preserving operation on knots which passes a strand over a crossingof two other strands. When performed on the closure of a braid (which isa knot), it actually is a Type III Reidemeister move [1]. Using the 3-braidgroup as an example, we can imagine starting with the identity and applyingσ2 (reference Figure 9). Then the application of σ1σ2 has the effect of slidingstrand 1 over the crossing of strands 2 and 3. On the other hand, the sequenceσ1σ2σ1 has the effect of simultaneously crossing strands 2 and 3, and slidingthem underneath strand 1.

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If we imagine that the σis are moves applied sequentially to a single braid,the third rule simply states that two moves which do not affect the samepaths may be applied in any order. Be careful not to forget that, in general,the braid group is not abelian. We are now able to determine whether or nottwo braids are topologically equivalent algorithmically, by comparing theirword representations. This allows us to finally begin to discuss the physicalmanifestation of a braid that can be used to simulate a quantum computer.

3 Quantum Computing with Anyons

The first breakthrough in forming a topological approach to quantum com-puting was the discovery of anyons. Anyons are quasiparticles with a frac-tional charge, restricted to movement in two dimensions. Although quantumparticles living in three dimensions are required to be either fermions orbosons, anyons may take on a complex phase which falls into neither cate-gory. This property emerges from the two dimensional world in which anyonslive.

In order to produce anyons, we must create an environment which restrictsmovement to two dimensions. Incredibly, this can be done by pairing galliumarsenide semiconductors, creating what is called a two dimensional electrongas. Essentially, the semiconductors prevent electrons between them frommoving in the third dimension. Such conditions produce what is called thefractional quantum Hall effect, in which quasiparticles appear to be anyons[3].

Recall that numbers of the form eiα are typically taken to represent a rotationby α in two complex dimensions, because the group formed by the membersof the complex unit circle has the map θ Ñ z “ eiθ “ cosθ ` isinθ [8]. Asit turns out, identical anyons can only accumulate phase as they are rotatedaround each other. Each time two identical anyons are swapped in a clockwisedirection, the phase factor added to the system takes the form eiα for some0 ă α ă π. A counterclockwise swapping results in half the phase factor.Notice that eiˆπ “ ´1 by Euler’s formula, and that eiˆ0 “ 1. Traditionally,interchanging bosons can only add a phase factor of 1, and interchangingfermions adds a phase factor of -1 [5].

What is absolutely amazing about all of this is that the phase of a sys-

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tem where anyons are used to represent qubits cannot be affected by smalldeformations in the trajectories of the particles caused by the ambient envi-ronment. In this way, anyons can be considered to be a topological state ofmatter.

Recall that a braid is constructed from a collection of paths. We know thata path connecting two points, a and b, is typically expressed with the pa-rameterized formula fptq “ p1 ´ tq ˆ a ` t ˆ b. If we take t to representtime (fitting, since a braid’s paths must make forward progress at all times,and cannot loop back on themselves), we can imagine that a braid repre-sents the movement of a collection of particles over time. This representationof spatial movement over time is referred to as a particle’s world line. Ananyon’s world line would therefore be imagined as a 2 ` 1 dimensional line[3]. However, since the two dimensional movement of anyons has no effecton the total phase of the system unless two anyons are interchanged, such arepresentation is actually quite simple. We can imagine that the crossing ofbraid path i over path i ` 1 represents a clockwise interchanging of the twoanyons represented by those paths. Likewise, a counterclockwise interchang-ing of anyons can be represented by path i crossing under path j. This givesus the relations σj “ eiθ and σ´1j “ epiθq{2, where θ is the phase factor addedby such an interchanging [6].

Figure 10: The interchanging of anyons represented as a braid.

Due to the inherent topological properties of these anyons, we can be surethat none of the slight deformations which threaten the stability of mostquantum computers will have any effect on this one. Logic gates can be con-structed by sequentially interchanging the positions of a collection of anyons,and measuring the resultant phase of the system. This sequence of inter-changes could be represented by the anyons’ braided world lines, as seen

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in Figure 11. Braids may be distinguished by comparing their word repre-sentations, and these words are simply generator moves applied sequentiallyto the identity braid. Since we have mapped the braid group generators tophase changes, it follows that we can distinguish our braided anyons by thefinal phase, resulting from sequentially applied phase changes. As long as theworld lines of the anyons can be continuously deformed back to their originalstate (if they haven’t been coiled up), the total phase of the system at theend will not be altered.

Figure 11: A CNOT logic gate, built with anyons.

4 Conclusion

In conclusion, it has been shown that two dimensional particles, called anyons,exist as a physical manifestation of the braid group. Within a system, theirinnate topological properties protect the state (or phase) of that system fromalterations caused by the ambient environment. On the other hand, the topo-logical quantum computer falls victim to new problems, such as the appear-ance of stray anyons, generated by thermal fluctuations, which can be caughtup in the braiding of the anyons (this might be avoided by spacing out theanyons). No topological quantum computer has yet been constructed, butthe potential remains. By studying topological braid theory and knot theory,we might be able to gain a better understanding of such a system, and totake another small step into the world of reliable quantum computing.

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References

[1] Colin C. Adams. The Knot Book: An Elementary Introduction to theMathematical Theory of Knots. American Mathematical Society, 2001.

[2] Colin C. Adams and Robert Franzosa. Introduction to Topology: Pureand Applied. Pearson Education, Inc., 2008.

[3] Graham P. Collins. Computing with quantum knots. Scientific American,2006.

[4] Rebecca Hoberg. Knots and braids, 2011.

[5] Sankar D. Sarma, Michael Freedman, and Chetan Nayak. Topologicalquantum computing. American Institute of Physics, 2006.

[6] Michael K. Spillane. An introduction to the theory of topological quan-tum computing.

[7] Herb Sutter. The free lunch is over: A fundamental turn toward concur-rency in software. Dr. Dobb’s Journal, 30(3), 2005.

[8] Wikipedia. Circle group.

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