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arXiv:2001.09091 · Symmetry

Quantum Computation and Measurements from an Exotic Space-Time R4

Michel R. P. Planat, Raymond Aschheim, Marcelo M. Amaral, Klee Irwin

Published May 5, 2020• 10 citations
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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 ψ 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 π 1 ( V ) of an exotic 4-manifold V, more precisely a ‘small exotic’ (space-time) R 4 (that is homeomorphic and isometric, but not diffeomorphic to the Euclidean R 4 ). 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 G Q ( 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) R 4 s. Our new picture relating a topological quantum computing and exotic space-time is also intended to become an approach of ‘quantum gravity’.

DOI
10.3390/sym12050736

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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 ψ 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 π 1 ( V ) of an exotic 4-manifold V, more precisely a ‘small exotic’ (space-time) R 4 (that is homeomorphic and isometric, but not diffeomorphic to the Euclidean R 4 ). 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 G Q ( 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) R 4 s. Our new picture relating a topological quantum computing and exotic space-time is also intended to become an approach of ‘quantum gravity’.
Let H be a nontrivial subgroup of index d of a free group G and N be the normal closure of H in G . The coset organization in a subgroup H of G provides a group P of permutation gates whose common eigenstates are either stabilizer states of the Pauli group or magic states for universal quantum computing. A subset of magic states consists of states associated to minimal informationally complete measurements, called MIC states. It is shown that, in most cases, the existence of a MIC state entails the two conditions (i) N = G and (ii) no geometry (a triple of cosets cannot produce equal pairwise stabilizer subgroups) or that these conditions are both not satisfied. Our claim is verified by defining the low dimensional MIC states from subgroups of the fundamental group G = π 1 ( M ) of some manifolds encountered in our recent papers, e.g., the 3-manifolds attached to the trefoil knot and the figure-eight knot, and the 4-manifolds defined by 0-surgery of them. Exceptions to the aforementioned rule are classified in terms of geometric contextuality (which occurs when cosets on a line of the geometry do not all mutually commute).
Let H be a nontrivial subgroup of index d of a free group G and N be the normal closure of H in G. The coset organization in a subgroup H of G provides a group P of permutation gates whose common eigenstates are either stabilizer states of the Pauli group or magic states for universal quantum computing. A subset of magic states consists of states associated to minimal informationally complete measurements, called MIC states. It is shown that, in most cases, the existence of a MIC state entails the two conditions (i) N = G and (ii) no geometry (a triple of cosets cannot produce equal pairwise stabilizer subgroups) or that these conditions are both not satisfied. Our claim is verified by defining the low dimensional MIC states from subgroups of the fundamental group G = π 1 ( M ) of some manifolds encountered in our recent papers, e.g., the 3-manifolds attached to the trefoil knot and the figure-eight knot, and the 4-manifolds defined by 0-surgery of them. Exceptions to the aforementioned rule are classified in terms of geometric contextuality (which occurs when cosets on a line of the geometry do not all mutually commute).
It has been shown that the concept of a magic state (in universal quantum computing: uqc) and that of a minimal informationally completepositive operator valued measure: MIC-POVMs (in quantum measurements) are in good agreement when such a magic state is selected in the set of nonstabilizer eigenstates of permutation gates with the Pauli group acting on it [1].Further work observed that most found low-dimensional MICs may be built from subgroups of the modular group PS L(2, Z) [2] and that this can be understood from the picture of the trefoil knot and related 3-manifolds [3]. Here one concentrates on Bianchi groups PS L(2,Od) (with Od the integer ring over the imaginary quadratic field) whose torsion-free subgroups define the appropriate knots and links leading to MICs and the related uqc. One finds a chain of Bianchi congruence n-cusped links playing a significant role [4].
It has been shown that the concept of a magic state (in universal quantum computing: uqc) and that of a minimal informationally complete positive operator valued measure: MIC-POVMs (in quantum measurements) are in good agreement when such a magic state is selected in the set of non-stabilizer eigenstates of permutation gates with the Pauli group acting on it [1]. Further work observed that most found low-dimensional MICs may be built from subgroups of the modular group PS L(2, Z) [2] and that this can be understood from the picture of the trefoil knot and related 3-manifolds [3]. Here one concentrates on Bianchi groups PS L(2, O10) (with O10 the integer ring over the imaginary quadratic field) whose torsion-free subgroups define the appropriate knots and links leading to MICs and the related uqc. One finds a chain of Bianchi congruence n-cusped links playing a significant role [4].
A single qubit may be represented on the Bloch sphere or similarly on the 333-sphere S3S^3S3. Our goal is to dress this correspondence by converting the language of universal quantum computing (uqc) to that of 333-manifolds. A magic state and the Pauli group acting on it define a model of uqc as a POVM that one recognizes to be a 333-manifold M3M^3M3. E. g., the ddd-dimensional POVMs defined from subgroups of finite index of the modular group PSL(2,Z)PSL(2,\mathbb{Z})PSL(2,Z) correspond to ddd-fold M3M^3M3- coverings over the trefoil knot. In this paper, one also investigates quantum information on a few \lq universal' knots and links such as the figure-of-eight knot, the Whitehead link and Borromean rings, making use of the catalog of platonic manifolds available on SnapPy. Further connections between POVMs based uqc and M3M^3M3's obtained from Dehn fillings are explored.
Minimal Informationally Complete quantum measurements, or MICs, illuminate the structure of quantum theory and how it departs from the classical. Central to this capacity is their role as tomographically complete measurements with the fewest possible number of outcomes for a given finite dimension. Despite their advantages, little is known about them. We establish general properties of MICs, explore constructions of several classes of them, and make some developments to the theory of MIC Gram matrices. These Gram matrices turn out to be a rich subject of inquiry, relating linear algebra, number theory and probability. Among our results are some equivalent conditions for unbiased MICs, a characterization of rank-1 MICs through the Hadamard product, several ways in which immediate properties of MICs capture the abandonment of classical phase space intuitions, and a numerical study of MIC Gram matrix spectra. We also present, to our knowledge, the first example of an unbiased rank-1 MIC which is not group covariant. This work provides further context to the discovery that the symmetric informationally complete quantum measurements (SICs) are in many ways optimal among MICs. In a deep sense, the ideal measurements of quantum physics are not orthogonal bases.