Quantum Computation of Universal Link Invariants

2006 ◽  
Vol 13 (04) ◽  
pp. 373-382 ◽  
Author(s):  
Silvano Garnerone ◽  
Annalisa Marzuoli ◽  
Mario Rasetti
Keyword(s):  
Tensor Algebra ◽  
Network Simulator ◽  
Spin Network ◽  
Quantum Automata ◽  
Link Invariants ◽  
Root Of Unity ◽  
Braid Index ◽  
Finite State ◽  
Jones Polynomials ◽  
The One

In the framework of the spin-network simulator based on the SU q(2) tensor algebra, we implement families of finite state quantum automata capable of accepting the language generated by the braid group, and whose transition amplitudes are coloured Jones polynomials. The automaton calculation of the polynomial of a link L on n strands at any fixed root of unity q is bounded from above by a linear function of the number of crossings of the link, on the one hand, and polynomially bounded in terms of the braid index n, on the other.

scholarly journals Quantum automata, braid group and link polynomials

2007 ◽  
Vol 7 (5&6) ◽  
pp. 479-503
Author(s):  
S. Garnerone ◽  
A. Marzuoli ◽  
M. Rasetti
Keyword(s):  
Braid Group ◽  
Theoretical Background ◽  
Simons Theory ◽  
Tensor Algebra ◽  
Spin Network ◽  
Quantum Automata ◽  
Low Dimensional Topology ◽  
Root Of Unity ◽  
Jones Polynomials ◽  
Low Dimensional

The spin--network quantum simulator model, which essentially encodes the (quantum deformed) $SU(2)$ Racah--Wigner tensor algebra, is particularly suitable to address problems arising in low dimensional topology and group theory. In this combinatorial framework we implement families of finite--states and discrete--time quantum automata capable of accepting the language generated by the braid group, and whose transition amplitudes are colored Jones polynomials. The automaton calculation of the polynomial of (the plat closure of) a link $L$ on $2N$ strands at any fixed root of unity is shown to be bounded from above by a linear function of the number of crossings of the link, on the one hand, and polynomially bounded in terms of the braid index $2N$, on the other. The growth rate of the time complexity function in terms of the integer $k$ appearing in the root of unity $q$ can be estimated to be (polynomially) bounded by resorting to the field theoretical background given by the Chern--Simons theory.

scholarly journals Estimating Jones and HOMFLY polynomials with one clean qubit

2009 ◽  
Vol 9 (3&4) ◽  
pp. 264-289
Author(s):  
S.P. Jordan ◽  
P. Wocjan
Keyword(s):  
Quantum Computing ◽  
Quantum Computer ◽  
Jones Polynomial ◽  
Single Variable ◽  
Initial State ◽  
Link Invariants ◽  
Root Of Unity ◽  
Complete Problem ◽  
The One ◽  
Homfly Polynomials

The Jones and HOMFLY polynomials are link invariants with close connections to quantum computing. It was recently shown that finding a certain approximation to the Jones polynomial of the trace closure of a braid at the fifth root of unity is a complete problem for the one clean qubit complexity class\cite{Shor_Jordan}. This is the class of problems solvable in polynomial time on a quantum computer acting on an initial state in which one qubit is pure and the rest are maximally mixed. Here we generalize this result by showing that one clean qubit computers can efficiently approximate the Jones and single-variable HOMFLY polynomials of the trace closure of a braid at \emph{any} root of unity.

scholarly journals Estimating Jones polynomials is a complete problem for one clean qubit

2008 ◽  
Vol 8 (8&9) ◽  
pp. 681-714
Author(s):  
P.W. Shor ◽  
S.P. Jordan
Keyword(s):  
Mixed State ◽  
Jones Polynomial ◽  
Quantum Circuit ◽  
Quantum Computers ◽  
Standard Quantum ◽  
Root Of Unity ◽  
Complete Problem ◽  
Intractable Problems ◽  
Jones Polynomials ◽  
The One

It is known that evaluating a certain approximation to the Jones polynomial for the plat closure of a braid is a BQP-complete problem. That is, this problem exactly captures the power of the quantum circuit model. The one clean qubit model is a model of quantum computation in which all but one qubit starts in the maximally mixed state. One clean qubit computers are believed to be strictly weaker than standard quantum computers, but still capable of solving some classically intractable problems. Here we show that evaluating a certain approximation to the Jones polynomial at a fifth root of unity for the trace closure of a braid is a complete problem for the one clean qubit complexity class. That is, a one clean qubit computer can approximate these Jones polynomials in time polynomial in both the number of strands and number of crossings, and the problem of simulating a one clean qubit computer is reducible to approximating the Jones polynomial of the trace closure of a braid.

scholarly journals AN EFFICIENT QUANTUM ALGORITHM FOR COLORED JONES POLYNOMIALS

2008 ◽  
Vol 06 (supp01) ◽  
pp. 773-778 ◽  
Author(s):  
MARIO RASETTI ◽  
SILVANO GARNERONE ◽  
ANNALISA MARZUOLI
Keyword(s):  
Jones Polynomial ◽  
Quantum Algorithm ◽  
Spin Network ◽  
Colored Jones Polynomial ◽  
Oriented Link ◽  
Root Of Unity ◽  
Jones Polynomials ◽  
Colored Jones Polynomials

We construct a quantum algorithm to approximate efficiently the colored Jones polynomial of the plat presentation of any oriented link L at a fixed root of unity q. The construction exploits the q-deformed spin network as computational background. The complexity of such algorithm is bounded above linearly by the number of crossings of the link, and polynomially by the number of link strands.

scholarly journals Analytical Model of Induction Machines with Multiple Cage Faults Using the Winding Tensor Approach

Sensors ◽  
2021 ◽  
Vol 21 (15) ◽  
pp. 5076
Author(s):  
Javier Martinez-Roman ◽  
Ruben Puche-Panadero ◽  
Angel Sapena-Bano ◽  
Carla Terron-Santiago ◽  
Jordi Burriel-Valencia ◽  
...  
Keyword(s):  
Computational Cost ◽  
Induction Machines ◽  
Diagnostic Techniques ◽  
Analytical Models ◽  
Tensor Algebra ◽  
Diagnostic Systems ◽  
Technical Literature ◽  
Twin Model ◽  
Ring Segment ◽  
The One

Induction machines (IMs) are one of the main sources of mechanical power in many industrial processes, especially squirrel cage IMs (SCIMs), due to their robustness and reliability. Their sudden stoppage due to undetected faults may cause costly production breakdowns. One of the most frequent types of faults are cage faults (bar and end ring segment breakages), especially in motors that directly drive high-inertia loads (such as fans), in motors with frequent starts and stops, and in case of poorly manufactured cage windings. A continuous monitoring of IMs is needed to reduce this risk, integrated in plant-wide condition based maintenance (CBM) systems. Diverse diagnostic techniques have been proposed in the technical literature, either data-based, detecting fault-characteristic perturbations in the data collected from the IM, and model-based, observing the differences between the data collected from the actual IM and from its digital twin model. In both cases, fast and accurate IM models are needed to develop and optimize the fault diagnosis techniques. On the one hand, the finite elements approach can provide highly accurate models, but its computational cost and processing requirements are very high to be used in on-line fault diagnostic systems. On the other hand, analytical models can be much faster, but they can be very complex in case of highly asymmetrical machines, such as IMs with multiple cage faults. In this work, a new method is proposed for the analytical modelling of IMs with asymmetrical cage windings using a tensor based approach, which greatly reduces this complexity by applying routine tensor algebra to obtain the parameters of the faulty IM model from the healthy one. This winding tensor approach is explained theoretically and validated with the diagnosis of a commercial IM with multiple cage faults.

LIMITED AUTOMATA AND REGULAR LANGUAGES

2014 ◽  
Vol 25 (07) ◽  
pp. 897-916 ◽  
Author(s):  
GIOVANNI PIGHIZZINI ◽  
ANDREA PISONI
Keyword(s):  
Finite Automata ◽  
Upper And Lower Bounds ◽  
Finite State Automata ◽  
Turing Machines ◽  
Double Exponential ◽  
Finite State ◽  
A Cell ◽  
Fixed Constant ◽  
The One ◽  
Context Free

Limited automata are one-tape Turing machines that are allowed to rewrite the content of any tape cell only in the first d visits, for a fixed constant d. In the case d = 1, namely, when a rewriting is possible only during the first visit to a cell, these models have the same power of finite state automata. We prove state upper and lower bounds for the conversion of 1-limited automata into finite state automata. In particular, we prove a double exponential state gap between nondeterministic 1-limited automata and one-way deterministic finite automata. The gap reduces to a single exponential in the case of deterministic 1-limited automata. This also implies an exponential state gap between nondeterministic and deterministic 1-limited automata. Another consequence is that 1-limited automata can have less states than equivalent two-way nondeterministic finite automata. We show that this is true even if we restrict to the case of the one-letter input alphabet. For each d ≥ 2, d-limited automata are known to characterize the class of context-free languages. Using the Chomsky-Schützenberger representation for contextfree languages, we present a new conversion from context-free languages into 2-limited automata.

scholarly journals q-DEFORMED SPIN NETWORKS, KNOT POLYNOMIALS AND ANYONIC TOPOLOGICAL QUANTUM COMPUTATION

2007 ◽  
Vol 16 (03) ◽  
pp. 267-332 ◽  
Author(s):  
LOUIS H. KAUFFMAN ◽  
SAMUEL J. LOMONACO
Keyword(s):  
Quantum Algorithms ◽  
Jones Polynomial ◽  
Braid Groups ◽  
Spin Networks ◽  
Spin Network ◽  
Topological Quantum ◽  
Jones Polynomials ◽  
Topological Quantum Computation ◽  
Bracket Polynomial ◽  
Colored Jones Polynomials

We review the q-deformed spin network approach to Topological Quantum Field Theory and apply these methods to produce unitary representations of the braid groups that are dense in the unitary groups. Our methods are rooted in the bracket state sum model for the Jones polynomial. We give our results for a large class of representations based on values for the bracket polynomial that are roots of unity. We make a separate and self-contained study of the quantum universal Fibonacci model in this framework. We apply our results to give quantum algorithms for the computation of the colored Jones polynomials for knots and links, and the Witten–Reshetikhin–Turaev invariant of three manifolds.

Glueing operation for R-matrices, quantum groups and link-invariants of Hecke type

1996 ◽  
Vol 119 (1) ◽  
pp. 139-166 ◽  
Author(s):  
Shahn Majid ◽  
Martin Markl
Keyword(s):  
Quantum Groups ◽  
Vector Spaces ◽  
Matrix Group ◽  
Link Invariant ◽  
One Dimensional ◽  
Quantum Matrices ◽  
Link Invariants ◽  
Quantum Vector ◽  
The One ◽  
Quantum Matrix

AbstractWe introduce an associative glueing operation ⊕q on the space of solutions of the Quantum Yang–Baxter Equations of Hecke type. The corresponding glueing operations for the associated quantum groups and quantum vector spaces are also found. The former involves 2×2 quantum matrices whose entries are themselves square or rectangular quantum matrices. The corresponding glueing operation for link-invariants is introduced and involves a state-sum model with Boltzmann weights determined by the link invariants to be glued. The standard su(n) solution, its associated quantum matrix group, quantum space and link-invariant arise at once by repeated glueing of the one-dimensional case.

Transmission Control Protocol and Active Queue Management together against congestion: cross-comparison through simulations

SIMULATION ◽  
2018 ◽  
Vol 95 (10) ◽  
pp. 979-993
Author(s):  
Carlo Augusto Grazia ◽  
Natale Patriciello ◽  
Martin Klapez ◽  
Maurizio Casoni
Keyword(s):  
Congestion Control ◽  
Transmission Control Protocol ◽  
Active Queue Management ◽  
Queue Management ◽  
Network Simulator ◽  
Transmission Control ◽  
Control Protocol ◽  
Trip Time ◽  
Multiple Round ◽  
The One

Most Internet traffic is carried by the Transmission Control Protocol (TCP) nowadays, even in the case of real-time services. Detecting and mitigating the congestion is one of the primary tasks of this protocol, in fact, different TCP versions are defined by their congestion control algorithms. Furthermore, Active Queue Management (AQM) algorithms share the same goal of congestion mitigation with TCP; in particular, the most efficient congestion control occurs when AQM and TCP work together. This paper presents a brief survey and a cross-comparison of the latest and most important TCP and AQM variants, then provides an evaluation of a different kind of performance on the ns-3 network simulator over various types of environments (multiple Round Trip Time, long delay, different congestion levels, etc.). In any shared bottleneck, the choice of the TCP-AQM couple to adopt is crucial. We will show that the results are not univocal and the “one size fits all” solution does not exist. Moreover, the proper couple depends on the performance that we want to boost and on the environment that we have to deal with.

scholarly journals Spin networks, quantum automata and link invariants

2006 ◽  
Vol 33 ◽  
pp. 95-106 ◽  
Author(s):  
Silvano Garnerone ◽  
Annalisa Marzuoli ◽  
Mario Rasetti
Keyword(s):  
Spin Networks ◽  
Quantum Automata ◽  
Link Invariants
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