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 GEOMETRIC PHASES AND TOPOLOGICAL QUANTUM COMPUTATION

2003 ◽  
Vol 01 (01) ◽  
pp. 1-23 ◽  
Author(s):  
VLATKO VEDRAL
Keyword(s):  
Quantum Computation ◽  
Large Scale ◽  
Quantum Computer ◽  
Fault Tolerant ◽  
Geometric Phases ◽  
Quantum Phases ◽  
Topological Quantum ◽  
Universal Quantum ◽  
The Subject ◽  
Topological Quantum Computation

In the first part of this review we introduce the basics theory behind geometric phases and emphasize their importance in quantum theory. The subject is presented in a general way so as to illustrate its wide applicability, but we also introduce a number of examples that will help the reader understand the basic issues involved. In the second part we show how to perform a universal quantum computation using only geometric effects appearing in quantum phases. It is then finally discussed how this geometric way of performing quantum gates can lead to a stable, large scale, intrinsically fault-tolerant quantum computer.

THE QUANTUM WORLD AND QUANTUM COMPUTING: FUNDAMENTAL ISSUES

2011 ◽  
Vol 20 (01) ◽  
pp. 179-202
Author(s):  
WILLIAM C. PARKE ◽  
ALI ESKANDARIAN
Keyword(s):  
Quantum Computing ◽  
Quantum Theory ◽  
Quantum Information ◽  
Information Transfer ◽  
Space Time ◽  
Topological Quantum ◽  
Non Locality

Some of the fundamental issues in quantum theory related to quantum computing are reviewed and discussed. Particularly emphasized is the need to be diligent in what quantum theory predicts, and what it does not. The non-intuitive features of quantum theory, that are often associated with aspects of non-locality and that also arise in quantum computing and quantum information transfer, are described. Some discussion of topological quantum computing using space-time strings is presented, as well as general notions about quantum computing.

MEASUREMENT, IRREVERSIBILITY AND DECOHERENCE IN QUANTUM MECHANICS

1994 ◽  
Vol 09 (22) ◽  
pp. 3913-3924
Author(s):  
BELAL E. BAAQUIE
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Quantum System ◽  
Quantum Measurement ◽  
State Vector ◽  
Space Time ◽  
Measuring Apparatus ◽  
Rigged Hilbert Space ◽  
Friedrichs Model ◽  
Time Irreversibility

We review Prigogine's model of quantum measurement. The measuring apparatus is considered to be an unstable quantum system with its state vector belonging to a rigged Hilbert space. Time irreversibility arises due to the dissipative nature of the measuring apparatus (an unstable quantum system) which induces decoherence in the system being measured. Friedrichs' model is used to concretely illustrate these ideas.

scholarly journals Obscure Qubits and Membership Amplitudes

2021 ◽  
Author(s):  
Steven Duplij ◽  
Raimund Vogl
Keyword(s):  
Hilbert Space ◽  
Quantum Computing ◽  
Density Matrix ◽  
Tensor Product ◽  
Quantum Computation ◽  
Membership Function ◽  
Quantum Probability ◽  
Quantum States ◽  
Born Rule ◽  
Quantum Amplitude

We propose a concept of quantum computing which incorporates an additional kind of uncertainty, i.e. vagueness (fuzziness), in a natural way by introducing new entities, obscure qudits (e.g. obscure qubits), which are characterized simultaneously by a quantum probability and by a membership function. To achieve this, a membership amplitude for quantum states is introduced alongside the quantum amplitude. The Born rule is used for the quantum probability only, while the membership function can be computed from the membership amplitudes according to a chosen model. Two different versions of this approach are given here: the “product” obscure qubit, where the resulting amplitude is a product of the quantum amplitude and the membership amplitude, and the “Kronecker” obscure qubit, where quantum and vagueness computations are to be performed independently (i.e. quantum computation alongside truth evaluation). The latter is called a double obscure-quantum computation. In this case, the measurement becomes mixed in the quantum and obscure amplitudes, while the density matrix is not idempotent. The obscure-quantum gates act not in the tensor product of spaces, but in the direct product of quantum Hilbert space and so called membership space which are of different natures and properties. The concept of double (obscure-quantum) entanglement is introduced, and vector and scalar concurrences are proposed, with some examples being given.

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).

Both Toffoli and Controlled-NOT need little help to universal quantum computing

2003 ◽  
Vol 3 (1) ◽  
pp. 84-92
Author(s):  
Y-Y Shi
Keyword(s):  
Quantum Computing ◽  
Quantum Computation ◽  
Quantum Circuit ◽  
Single Qubit ◽  
Universal Quantum ◽  
Qubit Gate ◽  
Computational Basis ◽  
Universal Gates

What additional gates are needed for a set of classical universal gates to do universal quantum computation? We prove that any single-qubit real gate suffices, except those that preserve the computational basis. The Gottesman-Knill Theorem implies that any quantum circuit involving only the Controlled-NOT and Hadamard gates can be efficiently simulated by a classical circuit. In contrast, we prove that Controlled-NOT plus any single-qubit real gate that does not preserve the computational basis and is not Hadamard (or its like) are universal for quantum computing. Previously only a generic gate, namely a rotation by an angle incommensurate with \pi, is known to be sufficient in both problems, if only one single-qubit gate is added.

scholarly journals Efficient universal quantum computation with auxiliary Hilbert space

2013 ◽  
Vol 88 (3) ◽  
Author(s):  
Wen-Dong Li ◽  
Yong-Jian Gu ◽  
Kai Liu ◽  
Yuan-Harng Lee ◽  
Yao-Zhong Zhang
Keyword(s):  
Hilbert Space ◽  
Quantum Computation ◽  
Universal Quantum

scholarly journals Systematic distillation of composite Fibonacci anyons using one mobile quasiparticle

2012 ◽  
Vol 12 (9&10) ◽  
pp. 876-892
Author(s):  
Ben W. Reichardt
Keyword(s):  
Quantum Computation ◽  
Quantum Computer ◽  
Fault Tolerant ◽  
Success Probability ◽  
Quantum Algorithm ◽  
Systematic Construction ◽  
Topological Quantum ◽  
Universal Quantum ◽  
Measurement Devices ◽  
Topological Quantum Computation

A topological quantum computer should allow intrinsically fault-tolerant quantum computation, but there remains uncertainty about how such a computer can be implemented. It is known that topological quantum computation can be implemented with limited quasiparticle braiding capabilities, in fact using only a single mobile quasiparticle, if the system can be properly initialized by measurements. It is also known that measurements alone suffice without any braiding, provided that the measurement devices can be dynamically created and modified. We study a model in which both measurement and braiding capabilities are limited. Given the ability to pull nontrivial Fibonacci anyon pairs from the vacuum with a certain success probability, we show how to simulate universal quantum computation by braiding one quasiparticle and with only one measurement, to read out the result. The difficulty lies in initializing the system. We give a systematic construction of a family of braid sequences that initialize to arbitrary accuracy nontrivial composite anyons. Instead of using the Solovay-Kitaev theorem, the sequences are based on a quantum algorithm for convergent search.

Sign in / Sign up

Export Citation Format

Share Document