scholarly journals Group Geometrical Axioms for Magic States of Quantum Computing

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 948 ◽  
Author(s):  
Michel Planat ◽  
Raymond Aschheim ◽  
Marcelo M. Amaral ◽  
Klee Irwin
Keyword(s):  
Quantum Computing ◽  
Fundamental Group ◽  
Free Group ◽  
Normal Closure ◽  
Trefoil Knot ◽  
Universal Quantum ◽  
Nontrivial Subgroup ◽  
Low Dimensional ◽  
Figure Eight Knot ◽  
Stabilizer States

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).

scholarly journals Quantum computing thanks to Bianchi groups

2019 ◽  
Vol 198 ◽  
pp. 00012 ◽  
Author(s):  
Michel Planat
Keyword(s):  
Quantum Computing ◽  
Trefoil Knot ◽  
Integer Ring ◽  
Universal Quantum ◽  
A Chain ◽  
Low Dimensional ◽  
Free Subgroups ◽  
Knots And Links ◽  
Good Agreement ◽  
Positive Operator Valued Measure

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].

scholarly journals Quantum Computing, Seifert Surfaces, and Singular Fibers

2019 ◽  
Vol 1 (1) ◽  
pp. 12-22 ◽  
Author(s):  
Michel Planat ◽  
Raymond Aschheim ◽  
Marcelo M. Amaral ◽  
Klee Irwin
Keyword(s):  
Quantum Computing ◽  
Quantum Computer ◽  
Complete Information ◽  
Singular Fiber ◽  
Dynkin Diagrams ◽  
Singular Fibers ◽  
Seifert Surfaces ◽  
Trefoil Knot ◽  
Universal Quantum

The fundamental group π 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 braid-induced 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 Kodaira’s classification of elliptic singular fibers. The Seifert fibered toroidal manifold Σ ′ , at the boundary of the singular fiber E 8 ˜ , allows possible models of quantum computing.

scholarly journals Universal Quantum Computing and Three-Manifolds

2018 ◽  
Author(s):  
Michel Planat ◽  
Raymond Aschheim ◽  
Marcelo Amaral ◽  
Klee Irwin
Keyword(s):  
Quantum Computing ◽  
Modular Group ◽  
Bloch Sphere ◽  
Borromean Rings ◽  
Single Qubit ◽  
Whitehead Link ◽  
Trefoil Knot ◽  
Universal Quantum ◽  
Dehn Fillings ◽  
Knots And Links

A single qubit may be represented on the Bloch sphere or similarly on the $3$-sphere $S^3$. Our goal is to dress this correspondence by converting the language of universal quantum computing (uqc) to that of $3$-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 $3$-manifold $M^3$. E. g., the $d$-dimensional POVMs defined from subgroups of finite index of the modular group $PSL(2,\mathbb{Z})$ correspond to $d$-fold $M^3$- 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 $M^3$'s obtained from Dehn fillings are explored.

scholarly journals Low dimensional homotopy for monoids II: groups

1999 ◽  
Vol 41 (1) ◽  
pp. 1-11 ◽  
Author(s):  
STEPHEN J. PRIDE
Keyword(s):  
Free Group ◽  
Equivalence Classes ◽  
Normal Closure ◽  
Group Presentation ◽  
One To One ◽  
Low Dimensional ◽  
Reduced Words

Consider a group presentation: $$\hat{[Pscr ]}\tfrm{=<\tfbf{x};}\tfbf{r}\tfrm{>}$$. Here x is a set and r is a set of non-empty, cyclically reduced words on the alphabet x ∪ x−1 (where x−1 is a set in one-to-one correspondence x[harr ]x−1 with x). We assume throughout that $\hat{[Pscr ]}$ is finite. Let $\hat{F}$ be the free group on x (thus $\hat{F}$ consists of free equivalence classes [W] of word on x∪x−1), and let N be the normal closure of {[R] : R∈r} in $\hat{F}$. Then the group G=G($\hat{[Pscr ]}$) defined by $\hat{[Pscr ]}$ is $\hat{F}\tfrm{/}N$. We will write W1 =GW2 if [W1]N=[W2]N.

scholarly journals Universal Quantum Computing and Three-Manifolds

2018 ◽  
Author(s):  
Michel Planat ◽  
Raymond Aschheim ◽  
Marcelo Amaral ◽  
Klee Irwin
Keyword(s):  
Quantum Computing ◽  
Bloch Sphere ◽  
Borromean Rings ◽  
Single Qubit ◽  
Whitehead Link ◽  
Trefoil Knot ◽  
Universal Quantum ◽  
Dehn Fillings ◽  
Knots And Links ◽  
Positive Operator Valued Measure

A single qubit may be represented on the Bloch sphere or similarly on the 3-sphere S3. Our goal is to dress this correspondence by converting the language of universal quantum computing (UQC) to that of 3-manifolds. A magic state and the Pauli group acting on it define a model of UQC as a positive operator-valued measure (POVM) that one recognizes to be a 3-manifold M3. More precisely, the d-dimensional POVMs defined from subgroups of finite index of the modular group PSL(2, Z) correspond to d-fold M3- coverings over the trefoil knot. In this paper, one also investigates quantum information on a few &lsquo;universal&rsquo; 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 the software SnapPy. Further connections between POVMs based UQC and M3&rsquo;s obtained from Dehn fillings are explored.

scholarly journals Quantum Computation and Measurements from an Exotic Space-Time R4

Symmetry ◽  
2020 ◽  
Vol 12 (5) ◽  
pp. 736
Author(s):  
Michel Planat ◽  
Raymond Aschheim ◽  
Marcelo M. Amaral ◽  
Klee Irwin
Keyword(s):  
Hilbert Space ◽  
Quantum Computing ◽  
Fundamental Group ◽  
Quantum Computation ◽  
Quantum Measurement ◽  
Space Time ◽  
Fano Plane ◽  
Topological Quantum ◽  
Universal Quantum ◽  
Complete Quantum

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’.

scholarly journals Universal Quantum Computing and Three-Manifolds

Symmetry ◽  
2018 ◽  
Vol 10 (12) ◽  
pp. 773 ◽  
Author(s):  
Michel Planat ◽  
Raymond Aschheim ◽  
Marcelo Amaral ◽  
Klee Irwin
Keyword(s):  
Quantum Computing ◽  
Bloch Sphere ◽  
Borromean Rings ◽  
Single Qubit ◽  
Whitehead Link ◽  
Trefoil Knot ◽  
Universal Quantum ◽  
Dehn Fillings ◽  
Knots And Links ◽  
Positive Operator Valued Measure

A single qubit may be represented on the Bloch sphere or similarly on the 3-sphere S 3 . Our goal is to dress this correspondence by converting the language of universal quantum computing (UQC) to that of 3-manifolds. A magic state and the Pauli group acting on it define a model of UQC as a positive operator-valued measure (POVM) that one recognizes to be a 3-manifold M 3 . More precisely, the d-dimensional POVMs defined from subgroups of finite index of the modular group P S L ( 2 , Z ) correspond to d-fold M 3 - coverings over the trefoil knot. In this paper, we also investigate quantum information on a few ‘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 the software SnapPy. Further connections between POVMs based UQC and M 3 ’s obtained from Dehn fillings are explored.

scholarly journals The normal closure of a power of a half-twist has infinite index in the mapping class group of a punctured sphere

2019 ◽  
Vol 11 (02) ◽  
pp. 273-292
Author(s):  
Charalampos Stylianakis
Keyword(s):  
Free Group ◽  
Braid Group ◽  
Mapping Class Group ◽  
Abelian Subgroup ◽  
Infinite Order ◽  
Class Group ◽  
Normal Closure ◽  
Mapping Class ◽  
Infinite Index ◽  
Half Twist

In this paper we show that the normal closure of the [Formula: see text]th power of a half-twist has infinite index in the mapping class group of a punctured sphere if [Formula: see text] is at least five. Furthermore, in some cases we prove that the quotient of the mapping class group of the punctured sphere by the normal closure of a power of a half-twist contains a free abelian subgroup. As a corollary we prove that the quotient of the hyperelliptic mapping class group of a surface of genus at least two by the normal closure of the [Formula: see text]th power of a Dehn twist has infinite order, and for some integers [Formula: see text] the quotient contains a free group. As a second corollary we recover a result of Coxeter: the normal closure of the [Formula: see text]th power of a half-twist in the braid group of at least four strands has infinite index. Our method is to reformulate the Jones representation of the mapping class group of a punctured sphere, using the action of Hecke algebras on [Formula: see text]-graphs, as introduced by Kazhdan–Lusztig.

scholarly journals Additive-error fine-grained quantum supremacy

Quantum ◽  
2020 ◽  
Vol 4 ◽  
pp. 329
Author(s):  
Tomoyuki Morimae ◽  
Suguru Tamaki
Keyword(s):  
Quantum Computing ◽  
Complexity Theory ◽  
Polynomial Time ◽  
Boolean Circuits ◽  
Exponential Time ◽  
Fine Grained ◽  
Additive Error ◽  
Universal Quantum ◽  
The One ◽  
Computing Models

It is known that several sub-universal quantum computing models, such as the IQP model, the Boson sampling model, the one-clean qubit model, and the random circuit model, cannot be classically simulated in polynomial time under certain conjectures in classical complexity theory. Recently, these results have been improved to ``fine-grained" versions where even exponential-time classical simulations are excluded assuming certain classical fine-grained complexity conjectures. All these fine-grained results are, however, about the hardness of strong simulations or multiplicative-error sampling. It was open whether any fine-grained quantum supremacy result can be shown for a more realistic setup, namely, additive-error sampling. In this paper, we show the additive-error fine-grained quantum supremacy (under certain complexity assumptions). As examples, we consider the IQP model, a mixture of the IQP model and log-depth Boolean circuits, and Clifford+T circuits. Similar results should hold for other sub-universal models.

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