TTopological Quantum Computing and3-Manifolds
Torsten Asselmeyer-MalugaGerman Aerospace Center (DLR),Rosa-Luxemburg-Str. 2, D-10178 Berlin, Germany;[email protected]: 0000-0003-4813-1010February 10, 2021
Abstract
In this paper, we will present some ideas to use 3D topology for quan-tum computing. Topological quantum computing in the usual sense workswith an encoding of information as knotted quantum states of topologicalphases of matter, thus being locked into topology to prevent decay. Today,the basic structure is a 2D system to realize anyons with braiding oper-ations. From the topological point of view, we have to deal with surfacetopology. However, usual materials are 3D objects. Possible topologiesfor these objects can be more complex than surfaces. From the topo-logical point of view, Thurston’s geometrization theorem gives the maindescription of 3-dimensional manifolds. Here, complements of knots doplay a prominent role and are in principle the main parts to understand3-manifold topology. For that purpose, we will construct a quantum sys-tem on the complements of a knot in the 3-sphere. The whole system de-pends strongly on the topology of this complement, which is determinedby non-contractible, closed curves. Every curve gives a contribution tothe quantum states by a phase (Berry phase). Therefore, the quantumstates can be manipulated by using the knot group (fundamental group ofthe knot complement). The universality of these operations was alreadyshowed by M. Planat et al.
Quantum computing exploits quantum-mechanical phenomena such as super-position and entanglement to perform operations on data, which in many cases,are infeasible to do efficiently on classical computers. The basis of this data isthe qubit, which is the quantum analog of the classical bit. Many of the cur-rent implementations of qubits, such as trapped ions and superconductors, arehighly susceptible to noise and decoherence because they encode information in1 a r X i v : . [ qu a n t - ph ] F e b he particles themselves. Topological quantum computing seeks to implement amore resilient qubit by utilizing non-Abelian forms of matter to store quantuminformation. In such a scheme, information is encoded not in the quasiparticlesthemselves, but in the manner in which they interact and are braided. In topo-logical quantum computing, qubits are initialized as non-abelian anyons, whichexist as their own antiparticles. Then, operations (what we may think of asquantum gates) are performed upon these qubits through braiding the world-lines of the anyons. Because of the non-Abelian nature of these particles, themanner in which they are exchanged matters (similar to non-commutativity).Another important property of these braids to note is that local perturbationsand noise will not impact the state of the system unless these perturbations arelarge enough to create new braids. Finally, a measurement is taken by fusing theparticles. Because anyons are their own antiparticles, the fusion will result inthe annihilation of some of the particles, which can be used as a measurement.We refer to the book in [Wang(2010)] for an introduction of these ideas.However, a limiting factor to use topological quantum computing is the usageof non-abelian anyons. The reason for this is the abelian fundamental group of asurface. Quantum operations are non-commutative operators leading to Heisen-berg’s uncertainty relation, for instance. Non-abelian groups are at the root ofthese operators. Therefore, if we use non-abelian fundamental groups insteadof abelian groups, then (maybe) we do not need non-abelian states to realizequantum computing. In this paper, we discuss the usage of (non-abelian) fun-damental groups of 3-manifolds for topological quantum computing. At first,quantum gates are elements of the fundamental group represented as SU (2) matrices. The set of possible operations depends strongly on the knot, i.e.,the topology. The fundamental group is a topological invariant, thus makingthis representation of quantum gates part of topological quantum computing.In principle, we have two topological ingredients: the knot and the fundamen-tal group of the knot complement. The main problem now is how these twoingredients can be realized in a quantum system. In contrast to topologicalquantum computing with anyons, we cannot directly use 3-manifolds (as sub-manifolds) like surfaces in the fractional Quantum Hall effect. Surfaces (or2-manifolds) embed into a 3-dimensional space like R but 3-manifolds requirea 5-dimensional space like R as an embedding space. Therefore, we cannotdirectly use 3-manifolds. However, as we will argue in the next section, thereis a group-theoretical substitute for a 3-manifolds, the fundamental group of aknot complement also known as knot group. Then, we will discuss the knotgroup of the simplest knot, the trefoil. The knot group is the braid group ofthree strands used for anyons too. The 1-qubit gates are given by the rep-resentation of the knot group into the group SU (2) . Here, one can get all1-qubit operations by this method. For an application of these fundamentalgroup representations to topological quantum computing, we need a realiza-tion of the fundamental group in a quantum systems. Here, we will refer tothe one-to-one connection between the holonomy of a flat SU (2) connection(i.e., vanishing curvature) and the representation of fundamental group into SU (2) . Here, we will use the Berry phase but the corresponding Berry con-2ection admits a non-vanishing curvature. However, we will show that one canrearrange the Berry connection for two-level systems to get a flat SU (2) connec-tion. Then, the holonomy along this connection only depends on the topologyof the knot complement so that the manipulation of states are topologicallyinduced. All SU (2) -representation of the knot group form the so-called charac-ter variety which contains important information about the knot complement(see in [Kirk and Klassen(1990)] for instance). However, the non-triviality ofthe character variety can be interpreted that knot groups give the 1-qubit oper-ations. In Section 6, we will discuss the 2-qubit operations by linking two knots.Here, every link component carries a representation into SU (2) . The relationin the fundamental group induces the interaction term. The interaction termsare known from the Ising model. Therefore, finally we get a complete set of op-erations to realize any quantum circuit: a 1-qubit operation by the knot groupof the trefoil knot and a 2-qubit operation by the complement of the link (Hopflink for instance). For the universality of these operations we refer to the workof M. Planat et al. [Planat et al.(2018)Planat, Aschheim, Amaral, and Irwin ,Planat et al.(2019)Planat, Aschheim, Amaral, and Irwin ], which was the maininspiration of this work. This paper followed the idea to use knots directly forquantum computing. In the focus is the knot complement which is the spaceoutside of a knot. Then, the knot is one system and the space outside is a secondsystem. Currently, the author is working on the concrete realization of this idea.In this paper, I will present the idea in an abstract manner to clarify whetherknot complements are suitable for quantum computing from informational pointof view.The usage of knots in physics but also biology is not new. One of the pio-neers is Louis H. Kauffman, and we refer to his book [Kauffman(1994)] for manyrelations between knot theory and natural science. Furthermore, note his ideasabout topological information [Kauffman(1995)] (see also in [Kauffman(2005),Boi(2005)]). Knots are also important models in quantum gravity, see, for in-stance, in [Baez(2001)], and in particle physics [Bilson-Thompson et al.(2007)Bilson-Thompson, Markopoulou, and Smolin ,Gresnigt(2018), Asselmeyer-Maluga(2019)]. The central concept for the following paper is the concept of a smooth manifold.To present this work as self-contained as possible, we will discuss some resultsin the theory of 2- and 3-dimensional manifolds which is the main motivationfor this paper. At first we will give the formal definition of a manifold:• Let M be a Hausdorff topological space covered by a (countable) family ofopen sets, U , together with homeomorphisms, φ U : U ∈ U → U R , where U R is an open set of R n . This defines M as a topological manifold. Forsmoothness we require that, where defined, φ U · φ − V is smooth in R n , inthe standard multivariable calculus sense. The family A = {U , φ U } is3alled an atlas or a differentiable structure. Obviously, A is not unique.Two atlases are said to be compatible if their union is also an atlas. Fromthis comes the notion of a maximal atlas. Finally, the pair ( M, A ) , with A maximal, defines a smooth manifold of dimension n .• An important extension of this construction yields the notion of smoothmanifold with boundary, M , defined as above, but with the atlas such thatthe range of the coordinate maps, U R , may be open in the half space, R n + ,that is, the subspace of R n for which one of the coordinates is non-positive,say x n ≤ . As a subspace of R n , R n + has a topologically defined boundary,namely, the set of points for which x n = 0 . Use this to define the (smooth)boundary of
M, ∂M, as the inverse image of these coordinate boundarypoints.In the following, we will concentrate on the special theory of 2- and 3-manifolds (i.e., manifolds of dimension 2, surfaces, or 3). The classificationof 2-manifolds has been known since the 19th century. In contrast, the corre-sponding theory for 3-manifolds based on ideas of Thurston around 1980 butwas completed 10 years ago. In both cases—2- and 3-manifolds—the manifoldis decomposed by the operation M N , the connected sum.Let M, N be two n -manifolds with boundaries ∂M, ∂N . The connected sum M N is the procedure of cutting out a disk D n from the interior int ( M ) \ D n and int ( N ) \ D n with the boundaries S n − (cid:116) ∂M and S n − (cid:116) ∂N , respectively,and gluing them together along the common boundary component S n − .For 2-manifolds, the basic elements are the 2-sphere S , the torus T or theKlein bottle R P . Then, one gets for the classification of 2-manifolds:• Every compact, closed, oriented 2-manifold is homeomorphic to either S or the connected sum T T . . . T (cid:124) (cid:123)(cid:122) (cid:125) g of T for a fixed genus g . Every compact, closed, non-oriented 2-manifoldis homeomorphic to the connected sum R P R P . . . R P (cid:124) (cid:123)(cid:122) (cid:125) g of R P for a fixed genus g .• Every compact 2-manifold with boundary can be obtained from one ofthese cases by cutting out the specific number of disks D from one of theconnected sums.A connected 3-manifold N is prime if it cannot be obtained as a connectedsum of two manifolds N N neither of which is the 3-sphere S (or, equiv-alently, neither of which is the homeomorphic to N ). Examples are the 3-torus T and S × S , but also the Poincare sphere. According to the work4n [Milnor(1962)], any compact, oriented 3-manifold is the connected sum ofan unique (up to homeomorphism) collection of prime 3-manifolds (prime de-composition). A subset of prime manifolds are the irreducible 3-manifolds. Aconnected 3-manifold is irreducible if every differentiable submanifold S homeo-morphic to a sphere S bounds a subset D (i.e., ∂D = S ) which is homeomorphicto the closed ball D . The only prime but reducible 3-manifold is S × S .For the geometric properties (to meet Thurston’s geometrization theorem)we need a finer decomposition induced by incompressible tori. A properly em-bedded connected surface S ⊂ N is called 2-sided (The “sides” of S then cor-respond to the components of the complement of S in a tubular neighborhood S × [0 , ⊂ N .) if its normal bundle is trivial, and 1-sided if its normal bundleis nontrivial. A 2-sided connected surface S other than S or D is called in-compressible if for each disk D ⊂ N with D ∩ S = ∂D there is a disk D (cid:48) ⊂ S with ∂D (cid:48) = ∂D . The boundary of a 3-manifold is an incompressible surface.Most importantly, the 3-sphere S , S × S and the 3-manifolds S / Γ with Γ ⊂ SO (4) a finite subgroup do not contain incompressible surfaces. The classof 3-manifolds S / Γ (the spherical 3-manifolds) include cases like the Poincaresphere ( Γ = I ∗ the binary icosaeder group) or lens spaces ( Γ = Z p the cyclicgroup). Let K i be irreducible 3-manifolds containing incompressible surfacesthen we can N split into pieces (along embedded S ) N = K · · · K n n S × S n S / Γ , (1)where n denotes the n -fold connected sum and Γ ⊂ SO (4) is a finite subgroup.The decomposition of N is unique up to the order of the factors. The irreducible3-manifolds K , . . . , K n are able to contain incompressible tori and one cansplit K i along the tori into simpler pieces K = H ∪ T G [Jaco and Shalen(1979)](called the JSJ decomposition). The two classes G and H are the graph manifold G and hyperbolic 3-manifold H .In 1982, W.P. Thurston presented a program intended to classify smooth3-manifolds and solve the Poincare conjecture by investigating the possible ge-ometries on such 3-manifolds. For a survey of this topic see in [Thurston(1997)].The key ingredient of this classification ansatz is the concept of a model geom-etry. Again, in this section, all manifolds are assumed to be smooth.A model geometry ( G, X ) consists of a simply connected manifold X togetherwith a Lie group G of diffeomorphisms acting transitively on X fulfilling certainset of conditions. One of these is that there is a G -invariant Riemannian metric.For example, reducing the dimension, we can consider 2-dimensional modelgeometries of a 2-manifold X . From Riemannian geometry, we know that any G -invariant Riemannian metric on X has constant Gaussian curvature (recall that G must be transitive). A constant scaling of the metric allows us to normalizethe curvature to be , , or − corresponding to the Euclidean ( E ), spherical( S ) and hyperbolic ( H ) space, respectively. Thus, there are precisely threetwo-dimensional model geometries: spherical, Euclidean, and hyperbolic.It is a surprising fact that there are also a finite number of three-dimensionalmodel geometries. It turns out that there are eight geometries: spherical, Eu-clidean, hyperbolic, mixed spherical-Euclidian, mixed hyperbolic-Euclidian, and5hree exceptional cases. A geometric structure on a more general manifold M (not necessarily simply connected) is defined by a model geometry ( G, X ) where X is the universal covering space to M , i.e., M = X/π ( M ) . This is equivalentto a representation π ( M ) → G of the fundamental group into G . Of course ageometric structure on a 3-manifold may not be unique but Thurston exploreddecompositions into pieces each of which admit a unique geometric structure.This decomposition proceeds by splitting M into essentially unique pieces usingembedded 2-spheres and 2-tori in such a way that a model geometry can bedefined on each piece. Thus,• Thurston’s Geometrization conjecture can be stated:The interior of every compact 3-manifold has a canonical decompositioninto pieces (described above), which have one of the eight geometric struc-tures.In short, every 3-manifold can be uniquely decomposed (long 2-spheres)into prime manifolds where some of these prime manifolds can be furthersplit (along 2-tori) into graph G and hyperbolic manifolds H . Then, G, H have a disjoint union of 2-tori as boundary; but how can we constructthese manifolds having a geometric structure? A knot in mathematics isthe embedding of a circle into the 3-sphere S (or in R ), i.e., a closedknotted curve. Let K be a prime knot (a knot not decomposable by asum of two knots). With K × D we denote a thicken knot, i.e. a closedknotted solid torus. The knot complement C ( K ) = S \ (cid:0) K × D (cid:1) is a3-manifold with boundary ∂C ( K ) = T . It was shown that prime knotsare divided into two classes: hyperbolic knots ( C ( K ) admits a hyper-bolic structure) and non-hyperbolic knots ( C ( K ) admits one of the otherseven geometric structures). An embedding of disjoints circles into S is called a link L. Then, C ( L ) is the link complement. Here, the situ-ation is more complicated: C ( L ) can admit a geometric structure or itcan be decomposed into pieces with a geometric structure. C ( L ) , C ( K ) are one of the main models for G or H for suitable knots and links. Ifwe speak about 3-manifolds then we have to consider C ( K ) as one ofthe basic pieces. Furthermore, there is the Gordon–Luecke theorem: iftwo knot complements are homeomorphic, then the knots are equivalent(see in [Gordon and Luecke(1989)] for the statement of the exact theo-rem). Interestingly, knot complements of prime knots are determined byits fundamental group. For the fundamental group, one considers closedcurves which are not contractible. Furthermore, two curves are equiva-lent if one can deform them into each other (homotopy relation). Theconcatenation of curves can be made into a group operation up to defor-mation equivalence (i.e., homotopy). Formally, it is the set of homotopyclasses [ S , X ] of maps S → X (the closed curves) into a space X upto homopy, denoted by π ( X ) . The fundamental group π ( C ( K )) of theknot complement is also known as knot group. Here, we refer to thebooks in [Rolfson(1976), Kauffman(1994), Prasolov and Sossinisky(1997)]for a good introduction into this theory. The main idea of this paper is6he usage of the knot group as substitute for a 3-manifold and try to usethis group for quantum computing. B Any knot can be represented by a projection on the plane with no multiplepoints which are more than double. As an example let us consider the simplestknot, the trefoil knot (knot with three crossings).The plane projection of the trefoil is shown in Figure 1. This projection canbe divided into three arcs, around each arc we have a closed curve as generator of π ( C (3 )) denoted by a, b, c (see also Figure 2 for the definition of the generators a, b ). Now each crossing gives a relation between the corresponding generators: c = a − ba, b = c − ac, a = b − cb , i.e., we get the knot group π ( C (3 )) = (cid:104) a, b, c | c = a − ba, b = c − ac, a = b − cb (cid:105) Then, we substitute the expression c = a − ba into the other relations to geta representation of the knot with two generators and one relation. From relation a = b − cb we will obtain a = b − ( a − ba ) b or bab = aba and the other relation b = c − ac gives nothing new. Finally, we will get the well-known result π ( C (3 )) = (cid:104) a, b | bab = aba (cid:105) However, this group is also well known; it is the braid group B of threestrands. In general, the braid group B n is generated by { , σ , . . . , σ n − } subjectto the relations (see in [Prasolov and Sossinisky(1997)]) σ i σ i +1 σ i = σ i +1 σ i σ i +1 σ i σ j = σ j σ i | i − j | > For B we have two generators σ , σ with one relation σ σ σ = σ σ σ agreeing with π ( C (3 )) . The braid group B has connections to different areas.Notable is the relation to the modular group SL (2 , Z ) (the group of integer × matrices with unit determinant) generated by S = (cid:18) − (cid:19) U = (cid:18) − (cid:19) It is well-known that B maps surjectively onto SL (2 , Z ) via the map σ (cid:55)→ (cid:18) (cid:19) σ (cid:55)→ (cid:18) − (cid:19) However, then σ σ σ = σ σ σ maps to S and σ σ maps to U . It isinteresting to note that ( σ σ ) is in the center Z of B (i.e., this elements com-mutes with all other elements) and B /Z = SL (2 , Z ) / {± } , or B is the centralextension of P SL (2 , Z ) = SL (2 , Z ) / {± } (the (2 , , ∞ ) triangle group) by theintegers Z (see [Asselmeyer-Maluga(2019)] for an application of this relation inphysics). 7igure 1: The simplest knot, trefoil knot .Figure 2: Generators (red circle) a, b of knot group for trefoil .8 Using the Trefoil Knot Complement for Quan-tum Computing
In this section, we will get in touch with quantum computing. The main idea isthe interpretation of the braid group B as operations (gates) on qubits. Fromthe mathematical point of view, we have to consider the representation of B into SU (2) , i.e., a homomorphism φ : B → SU (2) mapping sequences of generators (called words) into matrices as elements of SU (2) . For completeness we will study some representations. Here, we followthe work in [Kauffman and Lomonaco(2008)] to illustrate the general theory.At first, we note that a matrix in SU (2) has the form M = (cid:18) z w − ¯ w ¯ z (cid:19) | z | + | w | = 1 where z and w are complex numbers. Now we choose a well-known basis of SU (2) : = (cid:18) (cid:19) i = (cid:18) i − i (cid:19) j = (cid:18) − (cid:19) k = (cid:18) ii (cid:19) (2)so that M = a + b i + c j + d k with a + b + c + d = 1 (and z = a + bi, w = c + di ). The algebra of , i , j , k are known as quaternions (with relations ij = k , I = j = k = − ). Inthe following, we will switch between the usual basis , i, j, k of the quaternionsand the matrix representation with basis , i , j , k , see (2). Then, the unitquaternions (of length 1) can be identified with the elements of SU (2) . Purequaternions are defined by all expressions b i + c j + d k (i.e., a = 0 ). Now, thehomomorphism φ above is the mapping g = φ ( σ ) , h = φ ( σ ) so that ghg = hgh . Let u, v be pure quaternions of length (unit, pure quater-nions). Now, for g = a + bu and h = c + dv we have to choose g = a + bu, h = a + bv, bu · bv = a − (see in [Kauffman and Lomonaco(2008)] for the proof) for the image of the ho-momorphism φ then the relation of the B is fulfilled, or we have a representationof B into SU (2) .For more practical scenarios this general representation is the following con-struction. Let us choose g = e iθ = a + b i9here a = cos( θ ) and b = sin( θ ) . Let h = a + b (cid:2) ( c − s ) i + 2 cs · k (cid:3) where c + s = 1 and c − s = a − b b . Then we are able to rewrite g, h as matrices G, H . In principle, the matrices
G, H can be obtained from theexpressions above by a switch of the basis, i.e., G = exp( θ i ) H = a + b (cid:2) ( c − s ) i + 2 cs k (cid:3) Here, we choose H = F GF + and G = (cid:18) e iθ e − iθ (cid:19) F = (cid:18) ic isis − ic (cid:19) Among this class of representations, there is the simplest example g = e πi/ , f = iτ + k √ τ , h = f gf − where τ + τ = 1 . Then, g, h satisfy ghg = hgh the relation of B . Thisrepresentation is known as the Fibonacci representation of B to SU (2) . TheFibonacci representation is dense in SU (2) , see [Wang(2010)]. This represen-tation is generated by the 20th root of unity. Other dense representations aregiven by r th roots of unity via recoupling theory, see Sections 1.3 and 1.4 in[Wang(2010)]. In the previous section we discussed, the representations of the knot group into SU (2) to realize the 1-qubit operations. Central point in this paper is therepresentation φ : π ( C ( K )) → SU (2) of the knot group. The fundamental group is a topological invariant and we haveto realize this group in a quantum system. In this section, we will realize these1-qubit gates, the 2-qubit gate will be described in the next section. Here, wewill discuss the direct realization of the knot group, i.e., the fundamental groupof the knot complement. We will not discuss the abstract representation of thegroup π ( C ( K )) by quantum gates which is also possible. For that purpose wehave to define the fundamental group more carefully. Let X be a topologicalspace or a manifold. A map γ : [0 , → X with γ (0) = γ (1) is a closed curve.Two curves γ , γ are homotopic γ (cid:39) γ if there is a one-parameter family ofcontinuous maps which deform γ to γ . The concatenation γ γ of curves(up to homotopy) is the group operation making the set of homotopy classes ofclosed curves to a group, the fundamental group π ( X ) . Now, we will discussthe representation of the fundamental group by the holonomy along a closedcurve, i.e., by an integral of a gauge connection or potential along a closed10urve. As shown by Milnor [Milnor(1958)], there is one-to-one relation betweena homomorphism π ( X ) → G into the Lie group G and the integral ˛ γ A with dA + A ∧ A = 0 of a flat G -connection A , i.e., this integral depends only on the homotopy classof the closed curve γ . In our case, we will interpret the representation φ : π ( C ( K )) → SU (2) up to conjugation as a flat connection of a SU (2) principal bundle over theknot complement C ( K ) . Let P be a SU (2) principal fiber bundle over C ( K ) with connection A locally represented by a 1-form with values in the adjointrepresentation of the Lie algebra su (2) , i.e., A ∈ Λ ( C ( K )) ⊗ ad ( su (2)) . Theconnection is flat if the curvature F = dA + A ∧ A = 0 vanishes. In this case (see Milnor [Milnor(1958)]) the integral ˛ γ A along a closed curve γ : S → C ( K ) depends only on the homotopy class [ γ ] ∈ π ( C ( K )) and the exponential π ( C ( K )) (cid:51) γ → φ ( γ ) = P exp ˛ γ A ∈ SU (2) for varying closed curves ( P path ordering operator) gives a representation π ( C ( K )) → SU (2) . However, which quantum system realized this representa-tion? Let us consider the Hamiltonian H = H + h with a non-adiabatic and adiabatic part. The whole Hamiltonian has to fulfillthe usual Schrödinger equation i (cid:126) ∂∂t | ψ (cid:105) = H | ψ (cid:105) and we have to demand that each eigenstate | k n (cid:105) of the Hamiltonian h withdiscrete spectrum h | k n (cid:105) = E n | k n (cid:105) develops independently in time. Then, we have the decomposition | ψ (cid:105) = (cid:88) n a n | k n (cid:105) a n = exp (cid:18) − i (cid:126) ˆ E n ( τ ) dτ (cid:19) exp (cid:18) ˆ (cid:104) k n | ∂∂τ | k n (cid:105) dτ (cid:19) where the second expression is known as geometric phase or Berry phase. Usu-ally the states are parameterized by some manifold M with coordinates x = x ( t ) and we consider a cyclic evolution x (0) = x ( T ) , i.e., closed curves in M . Then,the Berry phase is given by θ top = ˛ γ (cid:104) k n | d | k n (cid:105) and the expression A Berry = (cid:104) k n | d | k n (cid:105) as Berry connection. This solution iswell known and for completeness we described it here again. At first, the Berryconnection is the connection of U (1) principal bundle. Second, the curvature Ω = dA Berry is non-zero. Therefore, at the first view we cannot use this connec-tion to represent the knot group. We need a connection with values in SU (2) toget a representation for φ via exp( θ top ) . Our idea is now to rearrange the com-ponents of the Berry connection (including the off-diagonal terms) to producea SU (2) connection. For that purpose, we will restrict the system to a 2-levelsystem, | k (cid:105) , | k (cid:105) . Then, we remark that the Lie algebra su (2) is generated bythe three Pauli matrices σ x , σ y , σ z so that every element is given by a linearcombination a · σ x + b · σ y + c · σ z . Now we arrange the possible connectioncomponents ω nm = (cid:104) k n | d | k m (cid:105) into one matrix ω = (cid:18) (cid:104) k | d | k (cid:105) (cid:104) k | d | k (cid:105)(cid:104) k | d | k (cid:105) (cid:104) k | d | k (cid:105) (cid:19) with the decomposition ω = (cid:104) k | d | k (cid:105) + σ z (cid:104) k | d | k (cid:105) − σ z (cid:60) ( (cid:104) k | d | k (cid:105) ) σ x + (cid:61)(cid:104) k | d | k (cid:105) σ y By using the normalization (cid:104) k n | k m (cid:105) = δ nm one gets d ( (cid:104) k n | k m (cid:105) ) = ω ∗ nm + ω nm and dω nm = (cid:88) k ω ∗ nk ∧ ω km = − (cid:88) k ω nk ∧ ω km so that we obtain for the curvature ΩΩ = dω + ω ∧ ω = 0 Obviously ω is a connection of a flat SU (2) bundle and the integral ˛ γ ω γ , as we want. The off-diagonal terms like (cid:104) k | d | k (cid:105) of the connection ω can be calculated with respectto the expectation values of dh . Together with the eigenvalues E , E of h for | k (cid:105) , | k (cid:105) , respectively, we obtain, for instance, (cid:104) k | d | k (cid:105) = (cid:104) k | dh | k (cid:105) E − E Then, the exponential of this integral gives a representation φ of the funda-mental group into SU (2) . Now we go back to the trefoil knot complement C (3 ) with fundamental group π ( C (3 )) = B . Then, the Berry phases along the twogenerators a, b (i.e., two closed curves) of the fundamental group π ( C (3 )) = B generate the 1-Qubit operations. Via the Berry phases, these operations acton the quantum system to influence its state. Keeping this idea in mind, wehave the following scheme: consider a qubit on the trefoil knot and consider thetwo generators a, b of the knot group (see Figure 2).If we do manipulations along these two closed curves we are able to influencethe qubit by using the Berry phase. Here, we refer to the work [Zu et al.(2014)Zu, Wang, He, Zhang, Dai, Wang, and Duan ]for ideas to use the Berry phase for quantum computing. In the previous section, we described the appearance of the braid group B as fundamental group of the trefoil knot complement. Above we described thesituation that the knot complement of the trefoil knot determines the operationsor quantum gates. In this case, the quantum gates are braiding operations (usedfor anyons) for three-strand braids. Unfortunately, the SU (2) − representationsof the braid group B are only 1-qubit gates but one needs at least a 2-qubitgate like CNOT to represent any quantum circuit. It is known that for 2-qubit operations, one needs elements of the braid group B . Is there otherknot complements having braid groups as fundamental groups? Unfortunately,the answer is no. Here, is the line of arguments: every knot complement isdetermined by the fundamental group (aspherical space), then the cohomologyof knot complements is determined by the first two groups ( th and th), allother groups are given by duality. But the braid groups B n for n > have non-trivial cohomology groups in degree 3 or higher which is impossible for knotcomplements.In the previous sections, we describe the knot complement of the simplestknot, the trefoil. However, there are more complicated knots. The complexityof knots is measured by the number of crossings. There is only one knot withthree crossings (trefoil) and with four crossings (figure-8). For the figure-8 knot (see Figure 3), the knot group is given by π ( C (4 )) = (cid:104) a, b | bab − ab = aba − ba (cid:105) admitting a representation φ into SU (2) , see [Kirk and Klassen(1990)]. Here,13igure 3: figure-8 knot .Figure 4: Hopf link L a .we remark that the figure-8 knot is part of a large class, the so-called hyperbolicknots. Hyperbolic knots are characterized by the property that the knot comple-ment admits a hyperbolic geometry. Hyperbolic knot complements have specialproperties, in particular topology and geometry are connected in a special way.We will come back to these ideas in our forthcoming work. As explained above,knot groups admit representations into SU (2) leading to 1-qubit operations.Therefore, we have to change the complexity in another direction by addingmore components, i.e., we have to go from knots to links. The simplest linkis the Hopf link (denoted as L a , see Figure 4), the linking of two unknottedcurves. The knot group is simply π ( C ( L a (cid:104) a, b | aba − b − = [ a, b ] = e (cid:105) = Z ⊕ Z Here, we will discuss a toy model, every component is related to a SU (2) rep-resentation, i.e., the knot group π ( C ( L a is represented as SU (2) ⊗ SU (2) via the Berry connection. Now we associate to each component of the link arepresentation and a generator of SU (2) (i.e., σ x , σ y or σ z ), say σ x to one com-ponent (the generator a ∈ π ( C ( L a ) and σ z to the other component (the14igure 5: Whitehead link W h .generator b ∈ π ( C ( L a ). By using the relation between the group commu-tator and the Lie algebra commutator of the enveloped Lie algebra U ( SU (2)) ,we want to express the relation [ a, b ] = e via the exponential of the commutator σ x ⊗ σ z − σ z ⊗ σ x via the usual relation between the Lie algebra commutatorand this commutator. It induces a representation π ( C ( L a → SU (2) ⊗ SU (2) by using the exponential map exp( su (2) ⊗ su (2)) . The relation in π ( C ( L a can be expressed as Lagrangian multiplier (in the usual way) so that we get theHamiltonian H = σ x ⊗ σ z − σ z ⊗ σ x and we get the qubit operation by the exponential U = exp( it ( σ x ⊗ σ z − σ z ⊗ σ x )) for a suitable time t . Now we see the principle: we associate a term σ x ⊗ σ z to an over-crossing between the two components and a term σ z ⊗ σ x for theunder-crossing. In the last example, we will consider the famous Whiteheadlink (see Figure 5). The knot group is given by π ( C ( W h )) = (cid:104) x, y | [ x, y ][ x, y − ][ x − , y − ][ x − , y ] = e (cid:105) and as described above we will associate the tensor products of the generatorsto the over-crossings or under-crossings between the components. Then, we willget H = 2 σ x ⊗ σ z − σ z ⊗ σ x with the operation U = exp( i t ( σ x ⊗ σ z − σ z ⊗ σ x )) with another choice of the time variable.15 Discussion
In this paper, we presented some ideas to use 3-manifolds for quantum com-puting. A direct usage for surfaces (related to anyons) is not possible, but weexplained above that the best representative is the fundamental group of a man-ifold. The fundamental group is the set of closed curves up to deformation withconcatenation as group operation (also up to deformation). Every 3-manifoldcan be decomposed into simple pieces so that every piece carries a geometricstructure (out of eight classes). In principle, the pieces consist of complementsof knots and links. Then, the fundamental group of the knot complement,known as knot group, is an important invariant of the knot or link. Why notuse this knot group for quantum computing? In [Planat and Zainuddin(2017),Planat and Rukhsan-Ul-Haq(2017), Planat et al.(2018)Planat, Aschheim, Amaral, and Irwin ,Planat et al.(2019)Planat, Aschheim, Amaral, and Irwin ], M. Planat et al. stud-ied the representation of knot groups and the usage for quantum computing.Here, we discussed a direct relation between the knot complement and quan-tum computing via the Berry phase. The knot group determines the operationswhere we fix a suitable SU (2) representation. As a result, we get all 1-qubitoperations for a knot. Then, we discussed the construction of 2-qubit operationsby the linking of two knots. The concrete realization of these ideas by a devicewill be shifted to our forthcoming work. Acknowledgments
I acknowledge useful discussions with Michel Planat. Furthermore, I acknowl-edge the helpful remarks and questions of the referees leading to better read-ability of this paper.
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