Loading ...
Klee Irwin
Science
QGR Research papers
12
0
Try Now
Log In
Pricing
Citation: Planat, M.; Chester, D.; Amaral, M.M.; Irwin, K. Fricke Topological Qubits. Quantum Rep. 2022, 4, 523–532. https://doi.org/ 10.3390/quantum4040037 Academic Editor: Antonio Manzalini Received: 7 October 2022 Accepted: 9 November 2022 Published: 14 November 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). quantum reports Article Fricke Topological Qubits Michel Planat 1,*,† , David Chester 2,† , Marcelo M. Amaral 2,† and Klee Irwin 2,† 1 Institut FEMTO-ST CNRS UMR 6174, Université de Bourgogne/Franche-Comté, 15 B Avenue des Montboucons, F-25044 Besançon, France 2 Quantum Gravity Research, Los Angeles, CA 90290, USA * Correspondence: michel.planat@femto-st.fr † These authors contributed equally to this work. Abstract: We recently proposed that topological quantum computing might be based on SL(2,C) representations of the fundamental group π1(S3 \ K) for the complement of a link K in the three- sphere. The restriction to links whose associated SL(2,C) character variety V contains a Fricke surface κd = xyz− x2 − y2 − z2 + d is desirable due to the connection of Fricke spaces to elementary topology. Taking K as the Hopf link L2a1, one of the three arithmetic two-bridge links (the Whitehead link 521, the Berge link 6 2 2 or the double-eight link 6 2 3) or the link 7 2 3, the V for those links contains the reducible component κ4, the so-called Cayley cubic. In addition, the V for the latter two links contains the irreducible component κ3, or κ2, respectively. Taking ρ to be a representation with character κd (d < 4), with |x|, |y|, |z| ≤ 2, then ρ(π1) fixes a unique point in the hyperbolic space H3 and is a conjugate to a SU(2) representation (a qubit). Even though details on the physical implementation remain open, more generally, we show that topological quantum computing may be developed from the point of view of three-bridge links, the topology of the four-punctured sphere and Painlevé VI equation. The 0-surgery on the three circles of the Borromean rings L6a4 is taken as an example. Keywords: topological quantum computing; SL(2,C) character variety; knot theory 1. Introduction Building a quantum computer is still challenging. However, progress has been made using natural and artificial atoms [1], superconducting technology [2] and other physical techniques [3,4]. One of the greatest challenges involved with constructing quantum computers is controlling or removing quantum decoherence. One possible solution is to create a topological quantum computer. The paper describes progress towards an understanding and possibly an implemen- tation of quantum computation based on algebraic surfaces. In the orthodox acceptation, a topological quantum computer deploys two-dimensional quasiparticles called anyons that are braids in three dimensions. The braids lead to logic gates used for computation. The topological nature of the braids makes the quantum computation less sensitive to the decoherence errors than in a standard quantum computer [5,6]. One theoretical proposal of universal quantum computation is based on Fibonacci anyons that are non-Abelian anyons with fusion rules. In particular, a fractional quantum Hall device would, in principle, realize a topological qubit. Owing to the lack of evidence that such quantum Hall-based anyons have been obtained, other theoretical proposals are worthwhile to develop. A recent paper of our group proposed a correspondence between the fusion Hilbert space of Fibonacci anyons and the tiling two-dimensional space of the one-dimensional Fibonacci chain [7]. In this paper, following our recent proposal [8] (see also [9]), we propose a non- anyonic theory of a topological quantum computer based on surfaces in a three-dimensional topological space. Such surfaces are part of the SL(2,C) character variety underlying the symmetries of a properly chosen manifold. In our earlier work, we were interested in basing topological quantum computing on three- or four-manifolds defined from the Quantum Rep. 2022, 4, 523–532. https://doi.org/10.3390/quantum4040037 https://www.mdpi.com/journal/quantumrep Quantum Rep. 2022, 4 524 complement of a knot or link. In [10,11], our goal was to define informationally complete quantum measurements from three-manifolds and, in [12], from four-manifolds, seeing the embedding four-dimensional ‘exotic’ space R4 of the manifold as a the physical Euclidean space–time. In the later paper, exotism means that one can define homeomorphic but non- diffeomorphic four-dimensional manifolds to interpret a type of ‘many-world’ quantum measurements. Our concepts in [8] and in the present paper are different in the sense that the SL(2,C) character variety is the three-dimensional locus of the supposed qubit prior to its measure- ment. The Lorentz group SL(2,C) reads the symmetries of the selected topology like that of the punctured torus, the quadruply punctured sphere or the topology obtained from the complement of a knot or a link. Our work in [8] focused on the complement of the Hopf link—the linking of two unknotted curves—where the character variety consists of the Cay- ley cubic κ4(x, y, z). Here, we took the broader context of Fricke surfaces, whose compact bounded component consists of the SU(2) representations [13]. Such representations are our proposed model of the topological qubits. In Section 2, we recall the definition of the SL(2,C)-character variety for a manifold M whose fundamental group is π1(M) and the method used to build it in an explicit way. In Section 3, we focus on the character variety κ2(x, y, z) for the fundamental group F2 (the free group of rank 2) of the once-punctured torus S1,1 and on the character variety κ4(x, y, z) attached to the fundamental group of the Hopf link L2a1. The former case is found to be related to the two-bridge link L7a4. The role of the extended mapping class group Mod±(S1,1) on a character variety of type κd(x, y, z), d ∈ C, is emphasized. We also introduce the concept of a topological qubit associated to the bounded SU(2) component of the surface κ2(x, y, z). In Section 4, we focus on the character variety Va,b,c,d(x, y, z) for the fundamental group (the free group of index 3) F3 of the quadruply punctured sphere S4,2. In particular, we recall the conditions that separate the compact (and bounded) SU(2) component and the non-compact SL(2,R) component. The investigation of the coverings of the four-manifold Ẽ8 (the Kodaira singular fiber I I∗) allows for an application of the theory. In the same section, we describe the Riemann–Hilbert correspondence for the case of S4,2, as well as the so-called Painlevé-Okomoto correspondence. The Painlevé VI equation plays a special role. In Section 5, we apply some perspectives of the present research toward topological quantum computing related to cosmology. 2. The SL(2,C)-Character Variety of a Manifold M Let π1 be the fundamental group of a topological surface S. We describe the represen- tations of π1 in the Lorentz group SL(2,C), the group of (2× 2) matrices with complex entries and determinant 1. Such a group contains representations as degrees of freedom for all quantum fields and is the gauge group for the Einstein–Cartan theory, which contains the Einstein–Hilbert action and Einstein’s field equations [14]. Topological formulations of gravity have been introduced [15–17] and the relationship of entanglement with spacetime has been articulated [18,19], which motivates the exploration of SL(2,C) character varieties for quantum computation. Representations of π1 in SL(2,C) are homomorphisms ρ : π1 → SL(2,C) with char- acter κρ(g) = tr(ρ(g)), g ∈ π1. The set of characters allows us to define an algebraic set by taking the quotient of the set of representations ρ by the group SL2(C), which acts by conjugation on representations [13,20]. Below, we need the distinction between the two real forms SU(2) and SL(2,R) of the group SL(2,C). The SU(2) representations are those that fix a point in the three- dimensional hyperbolic space H3 and SL(2,R) representations are those that preserve a two-dimensional hyperbolic space H2 in H3, as well as an orientation of H2. Both real forms of SL(2,C) play an important role in our attempt to define and stabilize the potential Fricke topological qubits. Quantum Rep. 2022, 4 525 A Sage Program for Computing the SL(2,C)-Character Variety of π1(S) The SL(2,C) character variety of a manifold M defined in SnapPy may be calculated in Sage using a program [21] written by the last author of [20] as follows: from snappy import Manifold M = Manifold(‘M’) G = M.fundamental_group() I = G.character_variety_vars_and_polys(as_ideal=True) I.groebner_basis() In some cases, a Groebner basis is not obtained from Sage. One may also use Magma [22] to obtain the Groebner basis or a small basis with the shortest length of the ideal I. 3. The Cubic Surface κd(x, y, z) and Two-Bridge Links Following [23], in this section, we describe the special case of representations for the punctured torus S1,1 and the relevance of the extended mapping class group Mod±(S1,1) in its action on surfaces of type κd(x, y, z), d ∈ C. Then, we find that surfaces κ2(x, y, z) and κ3(x, y, z) are contained in the character variety for the fundamental group of links L7a4 and L6a1, respectively. The SU(2) representations and the concept of a Fricke topological qubit is outlined. 3.1. The SL(2,C)-Character Variety for a Once Punctured Torus Let us take the example of the punctured torus T1,1 whose fundamental group π1 is the free group F2 = 〈a, b|∅〉 on two generators a and b. The boundary component of T1,1 is a single loop around the puncture expressed by the commutator [a, b] = abAB with A = a−1 and B = b−1. We introduce the traces x = tr(ρ(a)), y = tr(ρ(b)), z = tr(ρ(ab)). The trace of the commutator is the surface [13,23] tr([a, b]) = κ2(x, y, z) = x2 + y2 + z2 − xyz− 2. Another noticeable surface is obtained from the character variety attached to the fundamental group of the Hopf link L2a1 that links two unknotted curves. For the Hopf link, the fundamental group is π1(S3 \ L2a1) = 〈a, b|[a, b]〉 = Z2, and the corresponding character variety is the Cayley cubic [8] κ4(x, y, z) = x2 + y2 + z2 − xyz− 4. Both surfaces κ4 and κ2 are shown in Figure 1. Surfaces κ2 and κ4 have been obtained from two different mathematical concepts, from topological and algebraic concepts in dimension two, respectively. To relate them, one makes use of the Dehn–Nielsen–Baer theorem applied to the once-punctured torus [24]. According to this theorem, for a surface of genus g ≥ 1, we have Mod±(Sg) ∼= Out(π1(Sg)), where the mapping class group Mod(S) denotes the group of isotopy classes of orientation- preserving diffeomorphisms of S (that restrict to the identity on the boundary ∂S if ∂S 6= ∅), the extended mapping class group Mod±(S) denotes the group of isotopy classes of all homeomorphisms of S (including the orientation-reversing ones) and Out(π1) denotes the outer automorphism group of π1(S). This leads to the (topological) action of Mod± on the punctured torus as follows: Quantum Rep. 2022, 4 526 Mod±(S1,1) = Out(F2) = GL(2,Z). (1) Figure 1. Top left: the Cayley cubic κ4(x, y, z), Top right: the surface κ2(x, y, z), Down, the surface κd(x, y, z) with d = − 116 . The automorphism group Aut(F2) acts by composition on the representations ρ and induces an action of the extended mapping class group Mod± on the character variety by polynomial diffeomorphisms of the surface κd defined by [25] κd(x, y, z) = xyz− x2 − y2 − z2 + d. (2) 3.2. The Surface κ2, the Link L7a4 = 723 and Fricke Topological Qubits The surface κ2 corresponds to representations ρ : π1(κ2)→ SL(2,C) of the group ([25] Section 4.2) π1(κ2) = 〈 a, b|[a, b]4 〉 . (3) Since the surface κ4(x, y, z) is the character variety of the Hopf link, we would also like to obtain a link whose character variety contains the surface κ2(x, y, z). Making use of the Thistlethwaite link table [26], we find that the only two-bridge link that has this property is the link L7a4 = 723; see Figure 2. Taking 0-surgery on both cusps of L7a4, Snappy calculates the fundamental group as π1(S3 \ L7a4(0, 1)(0, 1)) = 〈a, b|aBAbabAbabABaBAB, abAbaabAbabABBBBBBAb〉. The corresponding Groebner base for the character variety is κL7a4(x, y, z) = xyz(z2 − 2)κ4(x, y, z)κ2(x, y, z), whose factorization contains both surfaces κ4(x, y, z) and κ2(x, y, z). Quantum Rep. 2022, 4 527 Figure 2. The 0-surgery on both pieces of links L7a4 in (a) and L6a1 in (b). The Groebner base for the corresponding character varieties contains the surfaces κ2(x, y, z) and κ4(x, y, z) for the former case, and κ3(x, y, z) for the latter case. Topological Qubits from κ2(x, y, z) Qubits are the elements of group SU(2). There is an interesting connection of the group π1(κ2) in Equation (3) to SU(2) representations. According to [25] [Theorem 1.1], there exists a representation ρ : π1(κ2) → SL(2,C) such that the closure of the orbit of its conjugacy class κ(ρ) under the action of the extended mapping class group Out(F2) in Equation (1) contains the whole set of SU(2) representa- tions of π1(κ2). The subset of the real surface κ2(x, y, z) consisting of SU(2) representations is the unique bounded connected component of κ2(x, y, z) homeomorphic to a sphere; see Figure 1 (Right). The bounded component is invariant under the mapping class Ψ = ( 2 1 1 1 ) and there are two fixed points of the polynomial transformation fΨ(x, y, z) = (z, yz− x, z(yz− x)− y) made of points (x, x/(x− 1), x) with irrational values x ∼ 0.52 and x ∼ −1.1 ([25], p. 19). 3.3. The Surface κ3(x; y, z) and the Link 623 = L6a1 We would like to obtain a link whose character variety contains the surface κ3(x, y, z). Making use of the Thistlethwaite link table [26], we find that the only two-bridge link that has this property is the link 623 = L6a1; see Figure 2. Taking 0-surgery on both cusps of L6a1, Snappy calculates the fundamental group as π1(S3 \ L6a1(0, 1)(0, 1)) = 〈a, b|abbaBAbaabAB, aBabABabaBAb〉. The corresponding Groebner base for the character variety is κL6a1(x, y, z) = xκ3(x, y, z)(x2 + y2 − xyz)(−xy2z+ y3 + x2y+ xz− 2y) ∗ f9(x, y, z) whose factorization contains surface κ3(x, y, z) and a ninth-order trivariate polynomial f9(x, y, z) not made explicit here. 4. The Fricke Cubic Surface and Three-Bridge Links Our main object in this section is the four-punctured sphere S4,2, for which, the fundamental group is the free group F3 of rank three whose character variety generalizes the Fricke cubic surface (2) to the hypersurface Va,b,c,d(C) in C7. It is shown how this hypersurface is realized in the variety of a covering of index 6 of the four-manifold Ẽ8, the 0-surgery on all circles of the Borromean rings BR0. The Okamoto–Painlevé correspondence is re-examined in terms of Dynkin diagrams of the appropriate four-manifolds. Quantum Rep. 2022, 4 528 4.1. The SL(2,C)-Character Variety for the Quadruply Punctured Sphere S4,2 We follow the work of references [13,25,27]. The fundamental group for S4,2 can be expressed in terms of the boundary components A, B, C and D as π1(S4,2) = 〈A, B, C, D|ABCD〉 ∼= F3. A representation π1 → SL(2,C) is a quadruple α = ρ(A), β = ρ(B) γ = ρ(C), δ = ρ(D) ∈ SL(2,C) where αβγδ = I. Let us associate the seven traces a = tr(ρ(α)), b = tr(ρ(β)), c = tr(ρ(γ)), d = tr(ρ(δ)) x = tr(ρ(αβ)), y = tr(ρ(βγ)), z = tr(ρ(γα)), where a, b, c and d are boundary traces and x, y and z are traces of elements AB, BC and CA representing simple loops on S4,2. The character variety for S4,2 satisfies the equation ([13] Section 5.2, [25] Section 2.1, [27] Section 3B, [28] Equation (1.9) or [29] Equation (39)) Va,b,c,d(C) = Va,b,c,d(x, y, z) = x2 + y2 + z2 + xyz− θ1x− θ2y− θ3z− θ4 = 0 (4) with θ1 = ab+ cd, θ2 = ad+ bc, θ3 = ac+ bd and θ4 = 4− a2 − b2 − c2 − d2 − abcd. 4.2. A Compact Component of SL(2,R) As shown in the previous section, for the real surface κ2(x, y, z), the compact compo- nent is made of SU(2) representations. For the real surface Va,b,c,d(R), there exists a compact component if and only if [27], Proposition 1.4 ∆(a, b, c, d) = (2(a2 + b2 + c2 + d2)− abcd− 16)2 − (4− a2)(4− b2)(4− c2)(4− d2) > 0 and 16− abcd− 2(a2 + b2 + c2 + d2) > 0 (5) When Equation (5) is satisfied and (a, b, c, d) ∈ (−2, 2), then Va,b,c,d(R) contains a compact component made of SL(2,R) representations. Otherwise, each element of the component is the character of an SU(2) representation; see also ([23], Theorem 9.6). The former case occurs in the following example, a, b, c = 32 and d = − 3 2 , for which, the surface is Va,b,c,d = x2 + y2 + z2 + xyz− 116 ([27], p. 102). A sketch of this surface is given in Figure 1. 4.3. The SL(2,C) Character Variety for the Manifold Ẽ8 and for Its Covering Manifolds In a recent paper ([8], Section 3.2), we noticed connections between the coverings of the manifold Ẽ8 and the matter of topological quantum computing. The affine Coxeter– Dynkin diagram Ẽ8 corresponds to the fiber I I∗ in Kodaira’s classification of minimal elliptic surfaces ([30], p. 320). Alternatively, one can see Ẽ8 as the 0-surgery on the trefoil knot 31. The boundary of the manifold associated to Ẽ8 is the Seifert fibered toroidal manifold [31,32]. The coverings of the fundamental group π1(S4 \ 31(0, 1)) are fundamental groups of the manifolds in the following sequence: [Ẽ8, Ẽ6, {D̃4, Ẽ8}, {Ẽ6, Ẽ8}, Ẽ8, {BR0, D̃4, Ẽ6}, {Ẽ8}, {Ẽ6}, {D̃4, Ẽ8}, Ẽ6, · · · ] The subgroups/coverings are fundamental groups for Ẽ8 Ẽ6, D̃4 or BR0, where BR0 is the manifold obtained by 0-surgery on all circles of Borromean rings. Quantum Rep. 2022, 4 529 A Groebner base for the SL(2,C) character variety of π1(Ẽ8) is z(x− z)(y− z2 + 2)(y+ z2 − 1), where the latter two factors are quadrics. A Groebner base for the SL(2,C) character variety of π1(Ẽ6) is κ4(x, y, z)(x− y)(xy− z+ 1)(x2 + xy+ y2 − 3) f1(x, y, z) f2(x, y, z), where κ4(x, y, z) is the SL(2,C) character variety for the fundamental group of the Hopf link complement, f1(x, y, z) = xy3 − y2z− x2 − 2xy + z + 2 and f2(x, y, z) = y4 − x2z + xy− 4y2 + z+ 2. A plot of the latter surfaces is in ([8], Figure 4). In the three-dimensional projective space, the two surfaces are birationally equivalent to a conic bundle and to the projective plane P2, respectively. Both show a Kodaira dimension zero characteristic of K3 surfaces. A Groebner base for the SL(2,C) character variety of π1(D̃4) contains the five- dimensional hypersurface f (x, y, z, w, k) = κ4(x, y, z)− wxk− 2k2, which is close to (but different from) the Fricke form V0,0,w,k(C) = κ4(x, y, z)−wxk+w2 + k2. Finally, for π1(BR0), a Groebner base obtained from Magma contains 28 polynomials. However, a simpler small basis with 10 polynomials, like the size of I, is available. The ideal ring for π1(BR0) is I = {36 fBR0(x, y, z, u, v, w, k), xκ4(x, y, z), yκ4(x, y, z), xκ4(x, u, v), yκ4(y, u, v), −xyk+ xv+ yw+ zk− 2u, xu2 − uzw+ yz− uv+ wk− 2x, .} (6) where the seventh variable polynomial reads fBR0(x, y, z, u, v, w, k) = −xyz+ x 2 + y2 + z2 + xyuk− θ1x− θ2y− θ03z+ θ04 and θ1 = uv+ wk, θ2 = uw+ vk, θ03 = uk− vw, θ04 = u2 + v2 + w2 + k2 − 4. Taking the new variable z′ = −z+ k, the polynomial fBR0 transforms into the Fricke form (4). Vu,v,w,k(x, y, z) = xyz+ x2 + y2 + z2 − θ1x− θ2x− θ3z+ θ4, (7) with θ3 = uk+ vw and θ4 = θ04 + uvwk. The missing term in (6) is a fifth-order polynomial. 4.4. Painlevé VI and the Riemann-Hilbert Correspondence Equation (7) corresponds to a four-punctured sphere with four singular points and a monodromy group π1 isomorphic to the free group on three-generators. The existence of a certain class of linear differential equations with such singular points and a monodromy group is known as Hilbert’s twenty first problem, the original setting of Riemann–Hilbert correspondence. For the present case of the four-punctured sphere, the searched differential (dynamical) equation is the sixth Painlevé equation (or Painlevé VI) [23] qtt = 12 ( 1 q + 1 q−1 + 1 q−t )q 2 t − ( 1t + 1 t−1 + 1 q−t )qt + q(q−1)(q−t) 2t2(t−1)2 {α 2 4 − α21 t q2 + α22 t−1 (q−1)2 + (1− α 2 3) t(t−1) (q−t)2 } (8) with complex parameters α1, α2, α3, α4. The Painlevé property is the absence of movable singular points. The essential singularities of all solutions q(t) of Equation (8) only appear when t ∈ {0, 1,∞}. Quantum Rep. 2022, 4 530 Analyzing the nonlinear monodromy of Painlevé VI leads to the relation between parameters a, b, c and d of the family of cubic surfaces Va,b,c,d(x, y, z) given in (7) and parameters αi, i = 1 . . . 4 of Painlevé VI equation ([29], Section 4.2): (a, b, c, d) = [2 cos(πα1), 2 cos(πα2), 2 cos(πα3),−2 cos(πα4)]. (9) The relation between the two classes of parameters has been found to be controlled by the so-called Okamoto–Painlevé pairs. The Painlevé equation corresponding to Ẽ8 is Painlevé I, the Painlevé equation corresponding to Ẽ7 is Painlevé II, the Painlevé equation corresponding to Ẽ6 is Painlevé IV and the Painlevé equation corresponding to D̃4 is Painlevé VI ([33] Table 1, [23] Section 9.1.2). These mathematical results fit our approach developed in the previous subsection. Incidentally, the Painlevé equation corresponding to the manifold D̃5 is Painlevé V. We find that the Groebner base for the SL(2,C) character variety of π1(D̃5) contains the surface κd(x, y, z) defined in Equation (2) (apart from trivial quadratic factors). Finally, Painlevé III corresponds to one of the three types D̃6, D̃7 or D̃8. We find that, for Ẽ7 and D̃7, the character variety is trivial (up to quadratic factors), for D̃6, it is of type κd(x, y, z) and, for D̃8, it is close (but different from the form V0,0,c,d(C), as for D̃4 investigated in Section 4.3. The Okamoto–Painlevé correspondence and the type of main factor in the related Groebner base is summarized in Table 1. Table 1. The manifold type according to the Dynkin diagram (row 1 ), the corresponding Painlevé equation (row 2) and the main factor in the Groebner base for the corresponding SL(2,C) variety. The symbol T means that the variety is trivial (up to quadratic factors). manifold Ẽ8 Ẽ7 D̃8 D̃7 D̃6 Ẽ6 D̃5 D̃4 Painlevé type PI PI I PD̃8 I I I PD̃7 I I I PD̃6 I I I PIV PV PVI char var T T ≈Vo,o,c,d T κd κ4 κd ≈Vo,o,c,d 5. Discussion and Conclusions In this paper, using the SL(2,C) character variety of the punctured torus S1,1 and of the quadruply punctured sphere S4,2, we focused on the interest in defining topolog- ical qubits from the cubic surface κd(x, y, z) in (2) or Va,b,c,d(x, y, z) in (4) in the compact bounded domain of real variables x, y and z. We explored the connection of such real surfaces to the character variety of some two- and three-bridge links. We pointed out their relationship to Painlevé VI transcendents through Okamoto Equation (9). While possible experimental directions remain open for further investigation, recent advances in the field are noteworthy [34–37]. Let us now add that there exists a link between Painlevé transcendents and Ein- stein’s equations of cosmology when the metric is chosen to be self-dual. The six Painlevé equations are ‘essentially’ equivalent to SL(2,C) self-dual Yang–Mills equations with ap- propriate three-dimensional Abelian groups of conformal symmetries [38]. The symmetry groups are taken to be groups of conformal transformations of the complex Minkowski space–time with the metric ds2 = dτdτ̄ − dξdξ̄. For Painlevé VI, the Higgs fields Pi = Φ(Xi), i = 0, 1, t are sl(2,C) valued functions of the time variable t = ξξ̄ ττ̄ . The self-dual equations S′ = 0, tP′0 + [P0, Pt], (t− 1)P′1 + [P1, Pt] = 0, with S = −(P0 + P1 + Pt) are equivalent to Painlevé VI with parameters calculated from the constant determinants of the Pi and S ([38], p. 573). As a result, the Fricke sur- faces that we investigated in this paper correspond to relevant solutions of self-dual Einstein’s equations. Quantum Rep. 2022, 4 531 Author Contributions: Conceptualization, M.P.; methodology, M.P. and M.M.A.; software, M.P.; validation, D.C., M.M.A. and K.I.; formal analysis, M.P. and D.C.; investigation, M.M.A., D.C. and M.M.A.; data curation, M.P.; writing—original draft preparation, M.P.; writing—review and editing, M.P. and M.M.A.; visualization, M.M.A.; supervision, M.P.; project administration, K.I.; funding acquisition, K.I. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Informed Consent Statement: Not applicable. Data Availability Statement: Data are available from the authors after a reasonable demand. Conflicts of Interest: The authors declare no conflict of interest. References 1. Buluta, I.; Ashhab, S.; Nori, F. Natural and artificial atoms for quantum computation. Rep. Prog. Phys. 2011, 74, 104401. [CrossRef] 2. Obada, A.S.F.; Hessian, H.A.; Mohamed, A.B.A.; Homid, A.H. A proposal for the realization of universal quantum gates via superconducting qubits inside a cavity. Ann. Phys. 2013, 334, 47–57. [CrossRef] 3. Top 10 Quantum Computing Experiments of 2019. Available online: https://medium.com/swlh/top-quantum-computing- experiments-of-2019-1157db177611 (accessed on 1 November 2022). 4. Timeline of Quantum Computing and Communication. Available online: https://en.wikipedia.org/wiki/Timeline_of_quantum_ computing_and_communication (accessed on 1 November 2022). 5. Topological Quantum Computer. Available online: https://en.wikipedia.org/wiki/Topological_quantum_computer (accessed on 1 January 2021). 6. Pachos, J.K. Introduction to Topological Quantum Computation; Cambridge University Press: Cambridge, UK, 2012. 7. Amaral, M.; Chester, D.; Fang, F.; Irwin, K. Exploiting anyonic behavior of quasicrystals for topological quantum computing. Symmetry 2022, 14, 1780. [CrossRef] 8. Planat, M.; Aschheim, R.; Amaral, M.M.; Fang, F.; Chester, D.; Irwin, K. Character varieties and algebraic surfaces for the topology of quantum computing. Symmetry 2022, 14, 915. [CrossRef] 9. Asselmeyer-Maluga, T. Topological quantum computing and 3-manifolds. Quantum Rep. 2021, 3, 153–165. [CrossRef] 10. Planat, M.; Aschheim, R.; Amaral, M.M.; Irwin, K. Universal quantum computing and three-manifolds. Symmetry 2018, 10, 773. [CrossRef] 11. Planat, M.; Aschheim, R.; Amaral, M.M.; Irwin, K. Group geometrical axioms for magic states of quantum computing. Mathematics 2019, 7, 948. [CrossRef] 12. Planat, M.; Aschheim, R.; Amaral, M.M.; Irwin, K. Quantum computation and measurements from an exotic space-time R4. Symmetry 2020, 12, 736. [CrossRef] 13. Goldman, W.M. Trace coordinates on Fricke spaces of some simple hyperbolic surfaces. In Handbook of Teichmüller Theory; European Mathematical Society: Zürich, Switzerland, 2009; Volume 13, pp. 611–684. 14. Hehl, F.W.; von der Heyde, P.; Kerlick, G.D.; Nester, J.M. General relativity with spin and torsion: Foundations and prospects. Rev. Mod. Phys. 1976, 48, 393–416. [CrossRef] 15. Yang, C.N. Integral Formalism for Gauge Fields. Phys. Rev. Lett. 1974, 33, 445; Erratum in Phys. Rev. Lett. 1975, 35, 1748. [CrossRef] 16. MacDowell, S.W.; Mansouri, F. Unified geometric theory of gravity and supergravity. Phys. Rev. Lett. 1977, 38, 739–742; Erratum in Phys. Rev. Lett. 1977, 38, 1376. [CrossRef] 17. Trautman, A. The geometry of gauge fields. Czechoslov. J. Phys. B 1979, B29, 107–116. [CrossRef] 18. Ryu, S.; Takayanagi, T. Holographic derivation of entanglement entropy from AdS/CFT. Phys. Rev. Lett. 2006, 96, 181602. [CrossRef] [PubMed] 19. Van Raamsdonk, M. Building up spacetime with quantum entanglement. Gen. Relativ. Gravit. 2010, 42, 2323. [CrossRef] 20. Ashley, C.; Burelle J.P.; Lawton, S. Rank 1 character varieties of finitely presented groups. Geom. Dedicata 2018, 192, 1–19. [CrossRef] 21. Python Code to Compute Character Varieties. Available online: http://math.gmu.edu/~slawton3/Main.sagews (accessed on 1 May 2021). 22. Bosma, W.; Cannon, J.J.; Fieker, C.; Steel, A. (Eds.) Handbook of Magma Functions; Edition 2.23; University of Sydney: Sydney, Australia 2017; 5914p. 23. Cantat, S.; Loray, F. Holomorphic dynamics, Painlevé VI equation and character varieties. arXiv 2007, arXiv:0711.1579. 24. Farb, B.; Margalit, D. A Primer on Mapping Class Groups; Princeton University Press: Princeton, NJ, USA, 2012. 25. Cantat, S. Bers and Hénon, Painlevé and Schrödinger. Duke Math. J. 2009, 149, 411–460. [CrossRef] 26. The Thistlethwaite Link Table. Available online: http://katlas.org/wiki/The_Thistlethwaite_Link_Table (accessed on 1 September 2021). 27. Benedetto, R.L.; Goldman, W.M. The topology of the relative character varieties of a quadruply-punctured sphere. Exp. Math. 1999, 8, 85–103. [CrossRef] Quantum Rep. 2022, 4 532 28. Iwasaki, K. An area-preserving action of the modular group on cubic surfaces and the Painlevé VI. Commun. Math. Phys. 2003, 242, 185–219. [CrossRef] 29. Inaba, M.; Iwasaki, K.; Saito, M.H. Dynamics of the sixth Painlevé equation. arXiv 2005, arXiv:math.AG/0501007. 30. Scorpian, A. The Wild World of 4-Manifolds; American Mathematical Society: Providence, RI, USA, 2005. 31. Planat, M.; Aschheim, R.; Amaral, M.M.; Irwin, K. Quantum computing, Seifert surfaces and singular fibers. Quantum Rep. 2019, 1, 12–22. [CrossRef] 32. Wu, Y.-Q. Seifert fibered surgery on Montesinos knots. arXiv 2012, arXiv:1207.0154. 33. Saito, M.H.; Terajima, H. Nodal curves and Riccati solutions of Painlevé equations. J. Math. Kyoto Univ. 2004, 44, 529–568. [CrossRef] 34. Deng, D.-L.; Wang, S.-T.; Sun, K.; Duan, L.-M. Probe Knots and Hopf Insulators with Ultracold Atoms. Chin. Phys. Lett. 2018, 35, 013701. [CrossRef] 35. Lubatsch, A.; Frank, R. Behavior of Floquet Topological Quantum States in Optically Driven Semiconductors. Symmetry 2019, 11, 1246. [CrossRef] 36. Smalyukh, I.I. Review: Knots and other new topological effects in liquid crystals and colloids. Rep. Prog. Phys. 2020, 83, 106601. [CrossRef] 37. Stalhammar, M. Knots and Transport in Topological Matter. Ph.D. Thesis, Stockholm University, Stockholm, Switzerland, 2022. 38. Mason, L.J.; Woodhouse, N.M.J. Self-duality and the Painlevé transcendents. Nonlinearity 1993, 6, 569–581. [CrossRef]