S S symmetry Article Quantum Computation and Measurements from an Exotic Space-Time R4 Michel Planat 1,* , Raymond Aschheim 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; [email protected] (R.A.); [email protected] (M.M.A.); [email protected] (K.I.) * Correspondence: [email protected] Received: 17 March 2020; Accepted: 16 April 2020; Published: 5 May 2020 Abstract: The authors previously found a model of universal quantum 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 jyi 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 p1(V) of an exotic 4-manifold V, more precisely a ‘small exotic’ (space-time) R4 (that is homeomorphic and isometric, but not diffeomorphic to the Euclidean R4). Our selected example, due to S. Akbulut and R. E. Gompf, has two remarkable properties: (a) it shows the occurrence 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), and the combinatorial Grassmannian Gr(2, 8) (at index 28); and (b) it allows the interpretation of MICs measurements as arising from such exotic (space-time) R4s. Our new picture relating a topological quantum computing and exotic space-time is also intended to become an approach of ‘quantum gravity’. Keywords: topological quantum computing; 4-manifolds; akbulut cork; exotic R4; fundamental group; finite geometry; Cayley–Dickson algebras 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: 81P68; 57R65; 57M05; 32Q55; 57M25; 14H30 1. Introduction Concepts in quantum computing and the mathematics of 3-manifolds could be combined, as shown in our previous papers [1–4]. In the present work, we consider that quantum computing is performed dynamically in four dimensions interpreted as the Euclidean space-time R4. This offers a preliminary connection to the field called ‘quantum gravity’ with unconventional methods. In four dimensions, the topological and the smooth structure of a manifold are apart. There exist infinitely many 4-manifolds that are homeomorphic but non-diffeomorphic to each other [5–8]. 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 [8]. In our approach, we do not need the full toolkit of 4-manifolds since we are focusing on W and its neighboors only. All we need is to understand the handlebody decomposition of a 4-manifold, the fundamental group p1(¶W) of the three-dimensional boundary ¶W of W, and related fundamental groups. Following the methodology of Symmetry 2020, 12, 736; doi:10.3390/sym12050736 www.mdpi.com/journal/symmetry Symmetry 2020, 12, 736 2 of 13 our previous work, the subgroup structure of such p1s corresponds to the Hilbert spaces of interest. Our view is close to the many-worlds interpretation of quantum mechanics where all possible outcomes of quantum measurements are realized in some ‘world’ and are objectively real [9]. One arrives at a many-manifolds view of quantum computing—reminiscent of the many-worlds—where the many-manifolds are in an exotic class and can be seen as many-quantum generalized measurements, the latter being POVMs (positive operator valued measures). In quantum information theory, the two-qubit configuration and its properties, namely quantum entanglement and quantum contextuality, have been discussed at length as prototypes of peculiarities or resources in the quantum world. Our model of quantum computing is based on the concept of a magic state—a state that has to be added to the eigenstates of the d-dimensional Pauli group—to allow universal quantum computation. This was started by Bravyi and Kitaev in 2005 [10] for qubits (d = 2). A subset of magic states consists of states associated to minimal informationally complete measurements, which we called MIC states [1] (see AppendixA for a definition). We require that magic states should be MIC states as well. To obtain the candidate MIC states, one uses the fact that a permutation may be realized as a permutation matrix/gate and that mutually commuting matrices share eigenstates. They are either of the stabilizer type (as elements of the Pauli group) or of the magic type. One keeps magic states that are MIC states in order to preserve complete information during the computation and measurements. The detailed process of quantum computation is not revealed at this stage and will depend on the physical support for the states. A further step in our quantum computing model is to introduce a three-dimensional manifold M3 3 whose fundamental group G = p1(M ) would be the source of MIC states [2,3]. Recall that G is a free group with relations and that a d-dimensional MIC state may be obtained from the permutation group that organizes the cosets of an appropriate subgroup of index d of G. It was considered by us quite remarkable that two group geometrical axioms very often govern the MIC states of interest [4]: (i) the normal (or conjugate) closure fg−1hgjg 2 Gandh 2 Hg of the appropriate subgroup H of G equals G itself; and (ii) there is no geometry (a triple of cosets does not produce equal pairwise stabilizer subgroups). See [4] (Section 1.1) for our method of building a finite geometry from coset classes and AppendixB for the special case of the Fano plane. However, these rules had to be modified by allowing either the simultaneous falsification of (i) and (ii) or by tolerating a few exceptions. The latter item means a case of geometric contextuality, the parallel to quantum contextuality [11]. We are fortunate that the finite geometrical structures sustaining two-qubit commutation, the projective plane or Fano plane PG(2, 2) over the two-element field F2, the generalized quadrangle of order two GQ(2, 2), its embedding three-dimensional projective space PG(3, 2), and other structures relevant to quantum contextuality, are ingredients in our quantum computing 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 is organized as follows. Section2 summarizes the peculiarities of small exotic 4-manifolds of type R4. This includes the handlebody structure of an arbitrary 4-manifold in Section 2.1, the concept of an Akbulut cork in Section 2.2, and examples of small exotic 4-manifolds that are homeomorphic but not diffeomorphic to the Euclidean space R4 in Section 2.3. In Section3, the connection of quantum computing to a manifold M is through the fundamental group p1(M). The subgroup structure of p1(M) is explored when M is the boundary ¶W of an Akbulut cork (in Section 3.1), the manifold W¯ in Akbulut h-cobordism (in Section 3.2), and the middle level Q between diffeomorphic connected sums involving the exotic R4s (in Section 3.3). Section4 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 were performed using Magma [12] and SnapPy [13]. Symmetry 2020, 12, 736 3 of 13 2. Excerpts on the Theory of 4-Manifolds and Exotic R4s 2.1. Handlebody of a 4-Manifold Let us introduce some excerpts of the theory of 4-manifolds needed for our paper [5–7]. It concerns the decomposition of a 4-manifold into one- and two-dimensional handles, as shown in Figure1, see [5] (Figure 1.1 and Figure 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 watches 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 for 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 [6] (p. 169), as shown in Figure1b. 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, as described below. The surgering of a 2-handle to a 1-handle is illustrated in Figure2a (see also [ 6] (Figure 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 Figure2b, as described in Section 2.3.