QUANTUM COMPUTATION AND MEASUREMENTS FROM AN EXOTIC SPACE-TIME R4 MICHEL PLANAT†, RAYMOND ASCHHEIM‡, MARCELO M. AMARAL‡ AND KLEE IRWIN‡ Abstract. The authors previously found a model of universal quan- tum computation by making use of the coset structure of subgroups of a free group G with relations. A valid subgroup H of index d in G leads to a ‘magic’ state |ψi in d-dimensional Hilbert space that encodes a minimal informationally complete quantum measurement (or MIC), possibly carrying a finite ‘contextual’ geometry. In the present work, we choose G as the fundamental group π1(V ) of an exotic 4-manifold 4 V , more precisely a ‘small exotic’ (space-time) R (that is homeomor- 4 phic and isometric, but not diffeomorphic to the Euclidean R ). Our selected example, due to to S. Akbulut and R. E. Gompf, has two re- markable properties: (a) it shows the occurence of standard contextual geometries such as the Fano plane (at index 7), Mermin’s pentagram (at index 10), the two-qubit commutation picture GQ(2, 2) (at index 15) as well as the combinatorial Grassmannian Gr(2, 8) (at index 28) , (b) it allows the interpretation of MICs measurements as arising from 4 such exotic (space-time) R ’s. Our new picture relating a topological quantum computing and exotic space-time is also intended to become an approach of ‘quantum gravity’. PACS: 03.65.Fd, 03.67.Lx, 03.65.Aa, 03.65.Ta, 02.20.-a, 02.10.Kn, 02.40.Pc, 02.40.-k, 03.65.Wj MSC codes: 81P68, 57R65, 57M05, 32Q55, 57M25, 14H30 Keywords: Topological quantum computing, 4-manifolds, Akbulut cork, exotic R4, fundamental group, finite geometry, Cayley-Dickson algebras 1. Introduction Concepts in quantum computing and the mathematics of 3-manifolds could be combined in our previous papers. In the present work, 4-manifolds are our playground and the four dimensions are interpreted as a space-time. This offers a preliminary connection to the field called ‘quantum gravity’ with unconventional methods. In four dimensions, the topological and the arXiv:2001.09091v1 [math.GT] 22 Jan 2020 smooth structure of a manifold are apart. There exist infinitely many 4- manifolds that are homeomorphic but non diffeomorphic to each other [1]- [4]. They can be seen as distinct copies of space-time not identifiable to the ordinary Euclidean space-time. A cornerstone of our approach is an ‘exotic’ 4-manifold called an Akbulut cork W that is contractible, compact and smooth, but not diffeomorphic to the 4-ball [4]. In our approach, we do not need the full toolkit of 4- manifolds since we are focusing on W and its neighboors only. All what we need is to understand the handlebody decomposition of a 4-manifold, the fundamental group π1(∂W ) of the 3-dimensional boundary ∂W of W , and related fundamental groups. Following the methodology of our previous work, the subgroup structure of such π1’s corresponds to the Hilbert spaces 1 2MICHEL PLANAT , RAYMOND ASCHHEIM , MARCELO M. AMARAL AND KLEE IRWIN † ‡ ‡ ‡ of interest. One gets a many-manifold view of space-time -reminiscent of the many-worlds- where the many-manifolds can be considered as many- quantum generalized measurements, the latter being POVM’s (positive op- erator valued measures). In quantum information theory, the two-qubit configuration and its prop- erties: quantum entanglement and quantum contextuality have been dis- cussed at length as prototypes of peculiarities or resources in the quantum world. We are fortunate that the finite geometrical structures sustaining two-qubit commutation, the projective plane or Fano plane P G(2, 2) over the two-element field F2, the generalized quadrangle of order two GQ(2, 2), its embedding 3-dimensional projective space P G(3, 2), and other structures relevant to quantum contextuality, are ingredients in our quantum comput- ing model based on an Akbulut cork. Even more appealing, is the recovery of the Grassmannian configuration Gr(2, 8) on 28 vertices and 56 lines that connects to Cayley-Dickson algebras up to level 8. Our description will be as follows. In Sec. 2, one sums up the peculiarities of small exotic 4-manifolds of type R4. This includes the handlebody structure of an arbitrary 4-manifold in Sec. 2.1, the concept of an Akbulut cork in Sec. 2.2 and examples of small exotic 4-manifolds that are homeomorphic but not diffeomorphic to the Euclidean space R4 in Sec. 2.3. In Sec. 3, one first gives an account of the model of quantum compu- tation developed by the authors in their previous papers. The connection to a manifold M is through the fundamental group π1(M). The subgroup structure of π1(M) is explored when M is the boundary ∂W of an Akbulut cork (in Sec. 3.1), the manifold W¯ in Akbulut h-cobordism (in Sec. 3.2) and the middle level Q between diffeomorphic connnected sums involving the exotic R4’s (in Sec. 3.3). Sec. 4 is a discussion of the obtained results and their potential interest for connecting quantum information and the structure of space-time. Calculations in this paper are performed on the softwares Magma [5] and SnapPy [6]. 2. Excerpts on the theory of 4-manifolds and exotic R4’s 2.1. Handlebody of a 4-manifold. Let us introduce some excerpts of the theory of 4-manifolds needed for our paper [1, 2, 3]. It concerns the decomposition of a 4-manifold into one- and two-dimensional handles as shown in Fig. 1 [1, Fig. 1.1 and Fig. 1.2]. Let Bn and Sn be the n- dimensional ball and the n-dimensional sphere, respectively. An observer is placed at the boundary ∂B4 = S3 of the 0-handle B4 and watch the attaching regions of the 1- and 2-handles. The attaching region of 1-handle is a pair of balls B3 (the yellow balls), and the attaching region of 2-handles is a framed knot (the red knotted circle) or a knot going over the 1-handle (shown in blue). Notice that the 2-handles are attached after the 1-handles. For closed 4-manifolds, there is no need of visualizing a 3-handle since it can be directly attached to the 0-handle. The 1-handle can also be figured out as a dotted circle S1 × B3 obtained by squeezing together the two three- dimensional balls B3 so that they become flat and close together [2, p. 169] QUANTUM COMPUTATION AND MEASUREMENTS FROM AN EXOTIC SPACE-TIME R34 Figure 1. (a) Handlebody of a 4-manifold with the struc- ture of 1- and 2-handles over the 0-handle B4 , (b) the struc- ture of a 1-handle as a dotted circle S1 × B3. as shown in Fig. 1b. For the attaching region of a 2- and a 3-handle one needs to enrich our knowledge by introducing the concept of an Akbulut cork to be described later on. The surgering of a 2-handle to a 1-handle is illustrated in Fig. 2a (see also [2, Fig. 5.33]). The 0-framed 2-handle (left) and the ‘dotted’ 1-handle (right) are diffeomorphic at their boundary ∂. The boundary of a 2- and a 3-handle is intimately related to the Akbulut cork shown in Fig 2b as described at the subsection 2.3. 2.2. Akbulut cork. A Mazur manifold is a contractible, compact, smooth 4-manifold (with boundary) not diffeomorphic to the standard 4-ball B4 [1]. Its boundary is a homology 3-sphere. If one restricts to Mazur manifolds that have a handle decomposition into a single 0-handle, a single 1-handle and a single 2-handle then the manifold has to be of the form of the dotted circle S1 × B3 (as in Fig. 2a) (right) union a 2-handle. Recall that, given p, q, r (with p ≤ q ≤ r), the Brieskorn 3-manifold Σ(p, q, r) is the intersection in the complex 3-space C3 of the 5-dimensional 5 p q r sphere S with the surface of equation z1 + z2 + z3 = 0. The smallest known Mazur manifold is the Akbulut cork W [4, 7] pictured in Fig 2b and its boundary is the Brieskorn homology sphere Σ(2, 5, 7). According to [8], there exists an involution f : ∂W → ∂W that surgers the dotted 1-handle S1 × B3 to the 0-framed 2-handle S2 × B2 and back, in the interior of W . Akbulut cork is shown in Fig. 2b. The Akbulut cork has a simple definition in terms of the framings ±1 of (−3, 3, −3) pretzel knot also called K = 946 [8, Fig. 3]. It has been shown that ∂W = Σ(2, 5, 7) = K(1, 1) and W = K(−1, 1). 4MICHEL PLANAT , RAYMOND ASCHHEIM , MARCELO M. AMARAL AND KLEE IRWIN † ‡ ‡ ‡ Figure 2. (a) A 0-framed 2-handle S2 × B2 (left) and a dotted 1-handle S1 × B3 (right) are diffeomorphic at their boundary ∂ = S2 × S1 , (b) Two equivalent pictures of the Akbulut cork W . 2.3. Exotic manifold R4. An exotic R4 is a differentiable manifold that is homeomorphic but not diffeomorphic to the Euclidean space R4. An exotic R4 is called small if it can be smoothly embedded as an open subset of the standard R4 and is called large otherwise. Here we are concerned with an example of a small exotic R4. Let us quote Theorem 1 of [4]. There is a smooth contractible 4-manifold V with ∂V = ∂W , such that V is homeomorphic but not diffeomorphic to W relative to the boundary. Sketch of proof [4]: Let α be a loop in ∂W as in Fig.