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 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 OPTIMISTIC LIMITS OF THE COLORED JONES POLYNOMIALS AND THE COMPLEX VOLUMES OF HYPERBOLIC LINKS

2016 ◽  
Vol 100 (3) ◽  
pp. 303-337 ◽  
Author(s):  
JINSEOK CHO
Keyword(s):  
Classical Limit ◽  
Mathematical Formulation ◽  
Jones Polynomial ◽  
Physical Method ◽  
Saddle Point Method ◽  
Colored Jones Polynomial ◽  
Root Of Unity ◽  
Jones Polynomials ◽  
Colored Jones Polynomials ◽  
Actual Limit

The optimistic limit is a mathematical formulation of the classical limit, which is a physical method to estimate the actual limit by using the saddle-point method of a certain potential function. The original optimistic limit of the Kashaev invariant was formulated by Yokota, and a modified formulation was suggested by the author and others. This modified version is easier to handle and more combinatorial than the original one. On the other hand, it is known that the Kashaev invariant coincides with the evaluation of the colored Jones polynomial at a certain root of unity. This optimistic limit of the colored Jones polynomial was also formulated by the author and others, but it is very complicated and needs many unnatural assumptions. In this article, we suggest a modified optimistic limit of the colored Jones polynomial, following the idea of the modified optimistic limit of the Kashaev invariant, and show that it determines the complex volume of a hyperbolic link. Furthermore, we show that this optimistic limit coincides with the optimistic limit of the Kashaev invariant modulo $4{\it\pi}^{2}$. This new version is easier to handle and more combinatorial than the old version, and has many advantages over the modified optimistic limit of the Kashaev invariant. Because of these advantages, several applications have already appeared and more are in preparation.

scholarly journals P Systems with Evolutional Communication and Division Rules

Axioms ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 327
Author(s):  
David Orellana-Martín ◽  
Luis Valencia-Cabrera ◽  
Mario J. Pérez-Jiménez
Keyword(s):  
Computational Complexity ◽  
Complexity Theory ◽  
Membrane Computing ◽  
P Systems ◽  
Membrane Systems ◽  
Sat Problem ◽  
Complete Problem ◽  
Intractable Problems ◽  
Np Complete ◽  
The One

A widely studied field in the framework of membrane computing is computational complexity theory. While some types of P systems are only capable of efficiently solving problems from the class P, adding one or more syntactic or semantic ingredients to these membrane systems can give them the ability to efficiently solve presumably intractable problems. These ingredients are called to form a frontier of efficiency, in the sense that passing from the first type of P systems to the second type leads to passing from non-efficiency to the presumed efficiency. In this work, a solution to the SAT problem, a well-known NP-complete problem, is obtained by means of a family of recognizer P systems with evolutional symport/antiport rules of length at most (2,1) and division rules where the environment plays a passive role; that is, P systems from CDEC^(2,1). This result is comparable to the one obtained in the tissue-like counterpart, and gives a glance of a parallelism and the non-evolutionary membrane systems with symport/antiport rules.

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.

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 A divide-and-conquer algorithm for quantum state preparation

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Israel F. Araujo ◽  
Daniel K. Park ◽  
Francesco Petruccione ◽  
Adenilton J. da Silva
Keyword(s):  
Computational Cost ◽  
Quantum Circuit ◽  
Divide And Conquer ◽  
Computational Time ◽  
Quantum Computers ◽  
Dimensional Vector ◽  
Quantum Devices ◽  
Divide And Conquer Algorithm ◽  
Quantum State Preparation ◽  
Quantum Device

AbstractAdvantages in several fields of research and industry are expected with the rise of quantum computers. However, the computational cost to load classical data in quantum computers can impose restrictions on possible quantum speedups. Known algorithms to create arbitrary quantum states require quantum circuits with depth O(N) to load an N-dimensional vector. Here, we show that it is possible to load an N-dimensional vector with exponential time advantage using a quantum circuit with polylogarithmic depth and entangled information in ancillary qubits. Results show that we can efficiently load data in quantum devices using a divide-and-conquer strategy to exchange computational time for space. We demonstrate a proof of concept on a real quantum device and present two applications for quantum machine learning. We expect that this new loading strategy allows the quantum speedup of tasks that require to load a significant volume of information to quantum devices.

scholarly journals Practical Quantum Computing

2021 ◽  
Vol 2 (1) ◽  
pp. 1-35
Author(s):  
Adrien Suau ◽  
Gabriel Staffelbach ◽  
Henri Calandra
Keyword(s):  
Wave Equation ◽  
Quantum Computer ◽  
Quantum Algorithms ◽  
Quantum Algorithm ◽  
Quantum Circuit ◽  
Equation Solving ◽  
Small Quantum ◽  
Quantum Wave ◽  
Variational Algorithms ◽  
The One

In the last few years, several quantum algorithms that try to address the problem of partial differential equation solving have been devised: on the one hand, “direct” quantum algorithms that aim at encoding the solution of the PDE by executing one large quantum circuit; on the other hand, variational algorithms that approximate the solution of the PDE by executing several small quantum circuits and making profit of classical optimisers. In this work, we propose an experimental study of the costs (in terms of gate number and execution time on a idealised hardware created from realistic gate data) associated with one of the “direct” quantum algorithm: the wave equation solver devised in [32]. We show that our implementation of the quantum wave equation solver agrees with the theoretical big-O complexity of the algorithm. We also explain in great detail the implementation steps and discuss some possibilities of improvements. Finally, our implementation proves experimentally that some PDE can be solved on a quantum computer, even if the direct quantum algorithm chosen will require error-corrected quantum chips, which are not believed to be available in the short-term.

scholarly journals Deep neural networks for quantum circuit mapping

2021 ◽  
Author(s):  
Giovanni Acampora ◽  
Roberto Schiattarella
Keyword(s):  
Neural Networks ◽  
Deep Neural Networks ◽  
Quantum Algorithms ◽  
Quantum Algorithm ◽  
Quantum Circuit ◽  
Machine Learning Techniques ◽  
Quantum Computers ◽  
Physical Constraints ◽  
Proper Mapping ◽  
Correct Execution

AbstractQuantum computers have become reality thanks to the effort of some majors in developing innovative technologies that enable the usage of quantum effects in computation, so as to pave the way towards the design of efficient quantum algorithms to use in different applications domains, from finance and chemistry to artificial and computational intelligence. However, there are still some technological limitations that do not allow a correct design of quantum algorithms, compromising the achievement of the so-called quantum advantage. Specifically, a major limitation in the design of a quantum algorithm is related to its proper mapping to a specific quantum processor so that the underlying physical constraints are satisfied. This hard problem, known as circuit mapping, is a critical task to face in quantum world, and it needs to be efficiently addressed to allow quantum computers to work correctly and productively. In order to bridge above gap, this paper introduces a very first circuit mapping approach based on deep neural networks, which opens a completely new scenario in which the correct execution of quantum algorithms is supported by classical machine learning techniques. As shown in experimental section, the proposed approach speeds up current state-of-the-art mapping algorithms when used on 5-qubits IBM Q processors, maintaining suitable mapping accuracy.

Jones polynomial invariants for knots and satellites

1991 ◽  
Vol 109 (1) ◽  
pp. 83-103 ◽  
Author(s):  
H. R. Morton ◽  
P. Strickland
Keyword(s):  
Quantum Group ◽  
Linear Combination ◽  
Simple Formula ◽  
Jones Polynomial ◽  
Polynomial Invariants ◽  
Satellite Link ◽  
Representation Ring ◽  
Parallel Links ◽  
Jones Polynomials ◽  
Special Case

AbstractResults of Kirillov and Reshetikhin on constructing invariants of framed links from the quantum group SU(2)q are adapted to give a simple formula relating the invariants for a satellite link to those of the companion and pattern links used in its construction. The special case of parallel links is treated first. It is shown as a consequence that any SU(2)q-invariant of a link L is a linear combination of Jones polynomials of parallels of L, where the combination is determined explicitly from the representation ring of SU(2). As a simple illustration Yamada's relation between the Jones polynomial of the 2-parallel of L and an evaluation of Kauffman's polynomial for sublinks of L is deduced.

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