Quantum computing, Seifert surfaces and singular fibers

The fundamental group $\pi_1(L)$ of a knot or link $L$ may be used to generate magic states appropriate for performing universal quantum computation and simultaneously for retrieving complete information about the processed quantum states. In this paper, one defines braids whose closure is the $L$ of such a quantum computer model and computes their Seifert surfaces and the corresponding Alexander polynomial. In particular, some $d$-fold coverings of the trefoil knot, with $d=3$, $4$, $6$ or $12$, define appropriate links $L$ and the latter two cases connect to the Dynkin diagrams of $E_6$ and $D_4$, respectively. In this new context, one finds that this correspondence continues with the Kodaira's classification of elliptic singular fibers. The Seifert fibered toroidal manifold $\Sigma'$, at the boundary of the singular fiber $\tilde {E_8}$, allows possible models of quantum computing.


Introduction
To acquire computational advantage over a classical circuit, a quantum circuit needs a non-stabilizer quantum operation for preparing a non-Pauli eigenstate, often called a magic state. The work about qubit magic state distillation [1] was generalized to qudits [2] and multi-qubits (see [3] for a review). Thanks to these methods, universal quantum computation (UQC), the ability to prepare every quantum gate, is possible. A new approach of UQC, based on permutation gates and simultaneously minimal informationally complete positive operator valued measures (MICs), was worked out in [4,5]. It is notable that the structure of the modular group Γ is sufficient for getting most permutation-based magic states [6] useful for UQC and that this can be thought of in terms of the trefoil knot 3-manifold [7].
It is desirable that the UQC approach of [4]- [7] be formulated in terms of braid theory to allow a physical implementation. Braids of the anyon type, that are two-dimensional quasiparticles with world lines creating space-time braids, are nowadays very popular [9,10]. Close to this view of topological quantum computation (TQC) based on anyons, we propose a TQC based on the Seifert surfaces defined over a link L. The links in question will be those able to generate magic states appropriate for performing permutation-based UQC.
In our previous work [7], we investigated the trefoil knot T 1 = 3 1 as a possible source of d-dimensional UQC models through its subgroups of index d (corresponding to d-fold coverings of the T 1 3-manifold) (see [7, Table 1]). More precisely, the link L7n1, corresponding to the congruence subgroup Γ 0 (2) of the modular group Γ, builds a relevant qutrit magic state for UQC and is related to the Hesse configuration. The link L6a3, corresponding to the congruence subgroup Γ 0 (3) of Γ, builds a relevant two-qubit magic state and is related to the figure GQ(2, 2) of two-qubit commutation of Pauli operators. Then the link identified by the software SnapPy as L6n1 (or sometimes L8n3), corresponding to the congruence subgroup Γ(2) of Γ, defines a 6-dit MIC with the figure of Borromean rings as a basic geometry [6,Fig. 4]. As shown below, it turns out that none of the two aforementioned links L6n1 and L8n3 are correctly associated to the subgroup Γ(2) of Γ, but the link 6 3 3 (related to the Dynkin diagram ofD 4 ) is. The possible confusion lies in the fact that all three links share the same link group π 1 (L). Finally, the Dynkin diagram of D 4 (with the icosahedral symmetry of H 3 in the induced permutations) is associated to a 12-dimensional (two-qubit/qutrit) MIC corresponding to the congruence subgroup 10A 1 of Γ [6, Table 1].
As announced in the abstract, we introduce a Seifert surface methodology for converting the UQC models based on the just described links into the appropriate braid representation permitted by Alexander's theorem. These calculations are described in Sec. 2. Then, in Sec. 3, we generalize the building of UQC models to affine Dynkin diagrams of typeD 4 ,Ẽ 6 and E 8 , that are singular fibers of minimal elliptic surfaces. Along the way, topological objects such as the 3-torus, the Poincaré dodecahedral space [11] as well as the first amphicosm [12] are encountered. They are the precursors of 4-manifold topology that is currently under active scrutiny [13]. Its possible role in models of UQC is discussed in the conclusion.
2. Seifert surfaces and braids from d-fold coverings of the trefoil knot manifold (or of hyperbolic 3-manifolds) Alexander's theorem states that every knot or link can be represented as a closed braid [14]. A Seifert surface F of a knot K or a link L is an oriented surface within the 3-sphere S 3 whose boundary ∂F coincides with that knot or link. Given a basis {f k } for the first homology group H 1 (F : Z) of F , one defines a Seifert matrix V whose (i, j)-th entry is the linking number of the component f i and the positive push-off f + j of the component f j along a vector field normal to F . Then a useful invariant of L is the (symmetrized) Alexander polynomial [14], [15,Sec. 2.7] (1) with V T the transpose of V and r the first Betti number F . By definition There exists a remarkable topological property of ∆ L (t) called a skein relation 1 . If L + , L 0 and L − are links in S 3 , with projections differing from each other by a single crossing, as in Fig. 1b, then When L is a knot K, there is a connection of ∆ K (t) with a combinatorial invariant ν of the 3-manifold S 3 K obtained from the 0-surgery along K in S 3 as follows The invariant ν(S 3 K ) is called Milnor (or Reidemeister) torsion [17]. The main interest of ν is its ability to distinguish closed manifolds which are homotopy equivalent while being non homeomorphic. The Seifert surface can be drawn from the braid representation. A good reference is [19] and the related website [20]. The software SeifertView provides a visualization of the Seifert surface. Below, to practically obtain the braid representation and the corresponding Alexander polynomial, we proceed as follows. With the software SnapPy [21], one defines the link from its name [e.g. M=Manifold('3 1 ') for the trefoil knot K = T 1 ] or from its PD representation available after drawing the link in the pink editor [e.g. trefoil=[(6, 4, 1, 3), (4, 2, 5, 1), (2, 6, 3, 5)], L=Link(trefoil) and L.braid word() for obtaining the braid associated to T 1 as [−1, −1, −1] and L.braid matrix() for obtaining the Seifert matrix V ]. Then, with Magma software [22], the Alexander polynomial follows [det: Results are summarized in Table 1.
Before going further, let us recall the homomorphism between the conjugacy classes of subgroups of index d of a group G and the d-fold coverings of Table 1. A few models of universal quantum computation (UQC) [7,8] translated into the language of braids and their Seifert surfaces. The source is a knot (such as the trefoil knot) or a link and the target is a link L associated to a degree d covering of L-manifold that defines an appropriate magic state for UQC and a corresponding MIC (minimal informationally complete) POVM. Cases d = 3, 4, 5, ... correspond to the geometry of the Hesse configuration, to the doily GQ(2, 2) finite geometry, to the Petersen graph . . . The notation for the braids is that of [19]. the notation t ′ means t −1 .
a manifold M whose fundamental group is G = π 1 (M ) [23]. This relationship is not one-to-one (not an isomorphism) in the sense that a π 1 (M ) may characterize distinct manifolds M . A simple example is for the d-coverings of manifold with characteristic ν = 2g − 2 = 0 (with g the genus) whose number is the sum of divisor function σ(d) [24,Sec. 3.4]. The same cardinality structure of finite subgroups of π 1 (L) [denoted η d (L) in our previous paper [7]] is also encountered for the distinct links L6n1, L8n3, 6 3 3 and for the Kirby link introduced later.
2.1. The braids built from the trefoil knot that are associated to the qutrit link L7n1 and the two-qubit link L6a3.
As announced in the introduction, one refers to [7, Table 1] that lists the topological properties of d-fold coverings d = 1 . . . 8 of trefoil knot manifold. as obtained from SnapPy and also identifies the corresponding congruence subgroups of Γ previously investigated in [6]. From now, one denotes A = a −1 , B = b −1 ,. . . and (.,.) means the group theoretical commutator of the entries. The link L7n1 corresponds to the congruence subgroup Γ 0 (2) of Γ, its fundamental group π 1 = a, b|(a, B 2 ) builds a qutrit magic state for UQC of the type (0, 1, ±1) and a MIC with the Hesse geometry [7, Fig. 1a]. Fig. 2a is the drawing of L7n1 and Fig. 2b is that of the Seifert surface for the braid word (ab) 3 b.
The link L6a3 corresponds to the congruence subgroup Γ 0 (3) of Γ, its fundamental group π 1 = a, b|(a, b 3 ) builds a two-qubit magic state for UQC of the type (0, 1, −ω 6 , ω 6 − 1), ω 6 = exp( 2iπ 6 ), as well as a MIC with the geometry of the generalized quadrangle of order two GQ(2, 2) [7, Fig. 1b]. Fig. 2c is the drawing of L6a3 and Fig. 2d is that of the Seifert surface for the braid word ABCDCbaCdEdCBCDCeb. The Alexander polynomials are made explicit in Table 1. 2.2. The braid built from the trefoil knot that is associated to the 6-dit link 6 3 3 and related braids with the same fundamental group. There are eight conjugacy classes of subgroups of index 6 of Γ corresponding to eight 6-fold coverings over the trefoil knot manifold. They are listed and identified in [7, Table 1]. We are first interested in the unique regular covering M of degree 6 with homology Z + Z + Z and three cusps corresponding to the congruence subgroup Γ(2). The cardinality sequence of subgroups for the fundamental group of this particular covering is that of the link 6 It turns out that every degree 6 covering of the trefoil manifold leading to a 6-dit MIC [with magic state of the type (0, 1, ω 6 − 1, 0, −ω 6 , 0)] share the same fundamental group. For M , SnapPy randomly provides several choices such as L6n1 or L8n3 that, of course, share the same cardinality sequence as 6 3 3 . In the Knot Atlas at 'http://katlas.org/wiki/L6n1', one finds the sentence 'L6n1 is 6 3 3 in Rolfsen's table of links'. But that seems to be a wrong statement.  0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0 where 2T is the binary tetrahedral group.
We do not provide the result of performing (−2, 1) surgery on L8n3 which provides a still different result that we could not identify. We conclude that the correct identification of the manifold M should be 6 3 3 although we do not yet have a rigorous proof.
The Seifert surface for the link 6 3 3 is drawn in Fig. 3a and the corresponding Seifert surface associated to braid word (ab) 3 is Fig. 3b. Switching the up/down positions of circles at two points (as shown in Fig. 2c) provides the Kirby link L K drawn in [26, Fig. 3] that we reproduce in Fig. 3c. [Applying the surgeries as L K (4, 1)(1, 1)(2, 1) to red, blue and green circles, one gets the Brieskorn sphere Σ (2,3,5), alias the Poincaré dodecahedral space]. The corresponding Seifert surface associated to the braid word aBabAb is shown in Fig. 3d. It is notable that (−2, 1) surgery on K leads to the Dynkin diagram for A 3 of Weyl group S 4 . Both links 6 3 3 and K are described by the same Alexander polynomial t 2 − t − t ′ + t 2 .
The 6-cover of the trefoil knot manifold corresponding to the congruence subgroup 3C 0 of Γ. Let us conclude this subsection by another observation concerning the 6-cover (that we denote M ′ ) of the trefoil knot manifold identified in [7, Table 1] corresponding to the congruence subgroup 3C 0 . Again, the cardinality sequence of subgroups of π 1 (M ′ ) is that of 6 3 3 , L6n1, L8n3 or L K but M ′ can be distinguished from the manifolds corresponding to these links since one gets under  In [6, Table 1], the first author identified a two-qubit/qutrit MIC corresponding to the congruence subgroup 10A 1 of the modular group Γ. The geometry of this MIC is that of the four-partite graph K (3, 3, 3, 3). More precisely , the subgroup of index 12 corresponding to 10A 1 is generated as G = a, b, c, d|(a, b), (a, c), (a, t) that describes the Dynkin diagram of D 4 . It builds a magic state for UQC of the type (1, 1, 0, 0, 0, 0, −ω 6 , −ω 6 , 0, ω 6 − 1, 0, ω 6 − 1). Under (−2, 1) surgery on all four links one recovers the quaternion group. The permutation representation that organizes the cosets of 10A 1 in Γ is the icosahedral group Z 2 × P SL (2,5), alias the Coxeter group of the Dynkin diagram H 3 . The braid word and the corresponding Alexander polynomial are given in Table 1, the Seifert surface is in Fig. 4.

Braids from d-fold coverings of hyperbolic 3-manifolds.
Models of UQC from MICs are also sometimes associated to links as already recognized in [7,8]. Some of them are listed in Table 1 together as a corresponding braid word and Alexander polynomial. These models happen to be more complicated. We do not describe them in more detail.

Quantum computing from affine Dynkin diagrams
In the previous section, we found that some MICs for UQC (our approach of universal quantum computing with complete quantum information) relate to Coxeter-Dynkin diagrams:Ẽ 6 andD 4 for the 6-dit MIC and D 4 for the two-qubit/qutrit MIC. This is an unexpected observation that we would like to complete by another one: the possibility of defining UQC from the singular fiber II * =Ẽ 8 of Kodairas's classification of minimal elliptic surfaces (see Fig. 5c). This classification is useful in the understanding of 4-manifold topology as shown in [28, p. 320], see also [29] for a different perspective.
Taking π 1 (Ẽ 8 ) as the fundamental group of affine E 8 , the subgroup structure of π 1 (Ẽ 8 ) has the following cardinality list where the bold characters mean that one of the subgroup leads to a MIC, as in [7]. It is worthwhile to observe that the boundary of the manifold associated toẼ 8 is the Seifert fibered toroidal manifold [25], denoted Σ ′ in [7, Table 5]. It may also be obtained by 0-surgery on the trefoil knot T 1 .
For the sequence above the coverings are One observes that the subgroups/coverings are fundamental groups forẼ 8 E 6 ,D 4 or BR 0 , where BR 0 is the manifold obtained by 0-surgery on all circles of Borromean rings. The cardinality sequence of subgroups of BR 0 is which is recognized as A001001 in Sloane's encyclopedia of integer sequences with the title 'Number of sublattices of index d in generic 3-dimensional lattice'. As already given in Sec. 1, the subgroup structure of π 1 (Ẽ 6 ) has the following cardinality list which corresponds to A263825 in Sloane's encyclopedia of integer sequences with the title 'Total number c π 1 (B 1 ) (n) of n-coverings over the first amphicosm B 1 ' [12].
Reidemeister torsion of Σ ′ . As a final note for this section, according to (3), the 'twisted' Reidemeister torsion ν t of the 3-manifold obtained from 0-surgery along a knot K in S 3 is the Alexander polynomial of K. Thus for the trefoil knot T 1 = 3 1 one gets ν t (S 3 T 1 ) = ν t (Σ ′ ) = t − 1 + t ′ . Let BR = L6a4 be the Borromean rings, the manifold BR 0 as above (obtained by 0-surgery on all circles of BR) and BR 1 be the manifold obtained by 0-surgery on two circles of BR. The cardinality sequence η d (BR 0 ) is as in (6) and η d (BR 1 ) is found as in (4). In principle, one can compute the Reidemeister torsion for BR 0 and BR 1 [18,Sec 2.4]. This is left open in this paper.

Conclusion
In the first part of this paper, it has been found that some coverings of the trefoil knot, or of other knots or links useful for informationally complete UQC, are 3-manifolds originating from a link within the 3-sphere. Such links have Seifert surfaces and well defined braids. We pointed out some coverings of the trefoil knot (of index 6 and 12, respectively) connecting to the symmetry of Dynkin diagrams ( affine E 6 and D 4 ).
In the second part, it has been found that the singular fiberẼ 8 may be used to generate UQC and that its coverings are the singular fibersẼ 6 and D 4 (alias the first amphicosm), as well as the manifold BR 0 obtained by 0-surgery on all circles of Borromean rings. Generalizing the analysis to coverings ofD 4 and BR 0 , one finds that the 3-torus enters the game [11]. The full understanding of these facts needs the theory of 4-manifolds [26,28].
There exists surface braids wrapped around singularities within 4-manifolds that support quasiparticles closely related to anyon states [30]. More work is necessary to connect our observations to this latter work. Another connection seems to be the theory of quasicrystals that uses E 8 symmetry, e.g. [31,32].