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Quantum computing with Bianchi groups
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Abstract
It has been shown that non-stabilizer eigenstates of permutation gates are appropriate for allowing d-dimensional universal quantum computing (uqc) based on minimal informationally complete POVMs. The relevant quantum gates may be built from subgroups of finite index of the modular group [M. Planat, Entropy 20, 16 (2018)] or more generally from subgroups of fundamental groups of 3-manifolds [M. Planat, R. Aschheim, M.~M. Amaral and K. Irwin, arXiv 1802.04196(quant-ph)]. In this paper, previous work is encompassed by the use of torsion-free subgroups of Bianchi groups for deriving the quantum gate generators of uqc. A special role is played by a chain of Bianchi congruence n-cusped links starting with Thurston's link.
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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.
It has been shown that classes of (minimal asymmetric) informationally complete POVMs in dimension d can be built using the multiparticle Pauli group acting on appropriate fiducial states [M. Planat and Z. Gedik, R. Soc. open sci. 4, 170387 (2017)]. The latter states may also be derived starting from the Poincar\'e upper half-plane model H. For doing this, one translates the congruence (or non-congruence) subgroups of index d of the modular group into groups of permutation gates whose some of the eigenstates are the seeked fiducials. The structure of some IC-POVMs is found to be intimately related to the Kochen-Specker theorem.
Eigenstates of permutation gates are either stabilizer states (for gates in the Pauli group) or magic states, thus allowing universal quantum computation [M. Planat and Rukhsan-Ul-Haq, Preprint 1701.06443]. We show in this paper that a subset of such magic states, when acting on the generalized Pauli group, define (asymmetric) informationally complete POVMs. Such IC-POVMs, investigated in dimensions 2 to 12, exhibit simple finite geometries in their projector products and, for dimensions 4 and 8 and 9, relate to two-qubit, three-qubit and two-qutrit contextuality.
Recently, several intriguing conjectures have been proposed connecting symmetric informationally complete quantum measurements (SIC POVMs, or SICs) and algebraic number theory. These conjectures relate the SICs and their minimal defining algebraic number field. Testing or sharpening these conjectures requires that the SICs are expressed exactly, rather than as numerical approximations. While many exact solutions of SICs have been constructed previously using Gr\"obner bases, this method has probably been taken as far as is possible with current computer technology. Here we describe a method for converting high-precision numerical solutions into exact ones using an integer relation algorithm in conjunction with the Galois symmetries of a SIC. Using this method we have calculated 69 new exact solutions, including 9 new dimensions where previously only numerical solutions were known, which more than triples the number of known exact solutions. In some cases the solutions require number fields with degrees as high as 12,288. We use these solutions to confirm that they obey the number-theoretic conjectures and we address two questions suggested by the previous work.
The role of permutation gates for universal quantum computing is investigated. The \lq magic' of computation is clarified in the permutation gates, their eigenstates, the Wootters discrete Wigner function and state-dependent contextuality (following many contributions on this subject). A first classification of main types of resulting magic states in low dimensions is performed.
A new method for calculating the number of conjugacy classes of subgroups in any finitely generated group is described. According to the general theory of covering spaces, any n fold covering is uniquely determined by a subgroup of index, moreover, two coverings are equivalent if the corresponding subgroups are conjugate. The new method can also be used to count nonisomorphic maps on a Riemann surface and it makes possible to solve it for a surface of any genus. All manifolds are connected and have finitely generated fundamental groups and no constraints are imposed on their dimensions. Manifolds may be closed or open or may not have boundary.
Let O be the ring of integers in an imaginary quadratic numberfield. The group PSL2(O) acts discontinuously on hyperbolic 3-space H . If Γ≤ PSL2(O) is a torsionfree subgroupof finite index then the manifold Γ\H can be compactified to a manifold Mr sothat the inclusion Γ\H≤ MΓ is a homotopy equivalence. MΓ is a compact with boundary. The boundary being a union of finitely many 2-tori. This paper contains a computer-aided study of subgroups of low index in PSL2 for various O. The explicit description of these subgroups leads to a study of the homeomorphism types of the MΓ.
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There have been great strides made over the past 20 years in the understanding of three-dimensional topology, by translating topology into geometry. Even though a lot remains to be done, we already have an excellent working understanding of 3-manifolds. Our spatial imagination, aided by computers, is a critical tool, for the human mind is surprisingly well equipped with a bit of training and suggestion, to `see' the kinds of geometry that are needed for 3-manifold topology.
This paper is not about the theory but instead about the phenomenology of 3-manifolds, addressing the question `What are 3-manifolds like?' rather than `What facts can currently be proven about 3-manifolds?'
The best currently available experimental tool for exploring 3-manifolds is Jeff Weeks' program SnapPea. Experiments with SnapPea suggest that there may be an overall structure for the totality of 3-manifolds whose backbone is made of lattices contained in .
We consider in this paper the minimally twisted chain link with 5 components
in the 3-sphere, and we analyze the Dehn surgeries on it, namely the Dehn
fillings on its exterior M5. The 3-manifold M5 is a nicely symmetric hyperbolic
one, filling which one gets a wealth of hyperbolic 3-manifolds having 4 or
fewer (including 0) cusps. In view of Thurston's hyperbolic Dehn filling
theorem it is then natural to face the problem of classifying all the
exceptional fillings on M5, namely those yielding non-hyperbolic 3-manifolds.
Here we completely solve this problem, also showing that, thanks to the
symmetries of M5 and of some hyperbolic manifolds resulting from fillings of
M5, the set of exceptional fillings on M5 is described by a very small amount
of information.
Congruence link complements -a 3-dimensional Rademacher conjecture
- M D Baker
- A W Reid
M. D. Baker and A. W. Reid, Congruence link complements -a 3-dimensional
Rademacher conjecture, Proc. of the 66-th Birthday Conference for Joachim Schwermer (2016).
On the quantumness of a Hibert space
- Chris A Fuchs
Chris A. Fuchs, On the quantumness of a Hibert space, Quant. Inf. Comp. 4 467-478
(2004).