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.

Efficient reversible quantum design of sig-magnitude to two's complement converters

2020 ◽  
Vol 20 (9&10) ◽  
pp. 747-765
Author(s):  
F. Orts ◽  
G. Ortega ◽  
E.M. E.M. Garzon
Keyword(s):  
Quantum Computing ◽  
Scientific Community ◽  
Quantum Computer ◽  
Fault Tolerant ◽  
State Of The Art ◽  
The Other ◽  
Quantum Computers ◽  
Binary Number ◽  
Quantum Cost ◽  
The One

Despite the great interest that the scientific community has in quantum computing, the scarcity and high cost of resources prevent to advance in this field. Specifically, qubits are very expensive to build, causing the few available quantum computers are tremendously limited in their number of qubits and delaying their progress. This work presents new reversible circuits that optimize the necessary resources for the conversion of a sign binary number into two's complement of N digits. The benefits of our work are two: on the one hand, the proposed two's complement converters are fault tolerant circuits and also are more efficient in terms of resources (essentially, quantum cost, number of qubits, and T-count) than the described in the literature. On the other hand, valuable information about available converters and, what is more, quantum adders, is summarized in tables for interested researchers. The converters have been measured using robust metrics and have been compared with the state-of-the-art circuits. The code to build them in a real quantum computer is given.

ON THE QUANTUM COMPLEXITY OF EVALUATING THE TUTTE POLYNOMIAL

2010 ◽  
Vol 19 (06) ◽  
pp. 727-737
Author(s):  
HAMED AHMADI ◽  
PAWEL WOCJAN
Keyword(s):  
Braid Group ◽  
Quantum Computer ◽  
Tutte Polynomial ◽  
Roots Of Unity ◽  
Homfly Polynomial ◽  
Root Of Unity ◽  
Quantum Complexity ◽  
Homfly Polynomials ◽  
Alternating Links ◽  
Alternating Link

We show that the problem of approximately evaluating the Tutte polynomial of triangular graphs at the points (q, 1/q) of the Tutte plane is BQP-complete for (most) roots of unity q. We also consider circular graphs and show that the problem of approximately evaluating the Tutte polynomial of these graphs at the point (e2πi/5, e-2πi/5) is DQC1-complete and at points [Formula: see text] for some integer k is in BQP. To show that these problems can be solved by a quantum computer, we rely on the relation of the Tutte polynomial of a planar G graph with the Jones and HOMFLY polynomial of the alternating link D(G) given by the medial graph of G. In the case of our graphs the corresponding links are equal to the plat and trace closures of braids. It is known how to evaluate the Jones and HOMFLY polynomial for closures of braids. To establish the hardness results, we use the property that the images of the generators of the braid group under the irreducible Jones–Wenzl representations of the Hecke algebra have finite order. We show that for each braid b we can efficiently construct a braid [Formula: see text] such that the evaluation of the Jones and HOMFLY polynomials of their closures at a fixed root of unity leads to the same value and that the closures of [Formula: see text] are alternating links.

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.

Can We Build a Large-Scale Quantum Computer Using Semiconductor Materials?

MRS Bulletin ◽  
2005 ◽  
Vol 30 (2) ◽  
pp. 105-110 ◽  
Author(s):  
B. E. Kane
Keyword(s):  
Quantum Computing ◽  
San Francisco ◽  
Large Scale ◽  
Quantum Computer ◽  
Research Society ◽  
University Of Maryland ◽  
The Road ◽  
Materials Research ◽  
The One ◽  
Conventional Computer

AbstractThe following article is based on the Symposium X presentation given by Bruce E. Kane (University of Maryland) at the 2004 Materials Research Society Spring Meeting in San Francisco. Quantum computing has the potential to revolutionize our ability to solve certain classes of difficult problems. A quantum computer is able to manipulate individual two-level quantum states (“qubits”) in the same way that a conventional computer processes binary ones and zeroes. Here, Kane discusses some of the most promising proposals for quantum computing, in which the qubit is associated with single-electron spins in semiconductors. While current research is focused on devices at the one- and two-qubit level, there is hope that cross-fertilization with advancing conventional computer technology will enable the eventual development of a large-scale (thousands of qubits) semiconductor quantum computer.The author focuses on materials issues that will need to be surmounted if large-scale quantum computing is to be realizable. He argues in particular that inherent fluctuations in doped semiconductors will severely limit scaling and that scalable quantum computing in semiconductors may only be possible at the end of the road of Moore's law scaling, when devices are engineered and fabricated at the atomic level.

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.

The Realization of New Computing Model and System Research Related to Quantum Computer

2014 ◽  
Vol 1078 ◽  
pp. 413-416
Author(s):  
Hai Yan Liu
Keyword(s):  
Quantum Computing ◽  
High Performance ◽  
Quantum Physics ◽  
Quantum Computer ◽  
Quantum Computers ◽  
Information Coding ◽  
Computing Model ◽  
Mechanics Model ◽  
Electronic Spin ◽  
Classical Computer

The ultimate goal of quantum calculation is to build high performance practical quantum computers. With quantum mechanics model of computer information coding and computational principle, it is proved in theory to be able to simulate the classical computer is currently completely, and with more classical computer, quantum computation is one of the most popular fields in physics research in recent ten years, has formed a set of quantum physics, mathematics. This paper to electronic spin doped fullerene quantum aided calculation scheme, we through the comprehensive use of logic based network and based on the overall control of the two kinds of quantum computing model, solve the addressing problem of nuclear spin, avoids the technical difficulties of pre-existing. We expect the final realization of the quantum computer will depend on the integrated use of in a variety of quantum computing model and physical realization system, and our primary work shows this feature..

scholarly journals What is Quantum Computing and How it Works

2020 ◽  
pp. 1-5
Author(s):  
Bahman Zohuri ◽  
◽  
Farhang Mossavar Rahmani ◽  
Keyword(s):  
Artificial Intelligence ◽  
Quantum Computing ◽  
Quantum Computer ◽  
Time Frame ◽  
Basic Unit ◽  
Next Generation ◽  
Intel Corporation ◽  
Encrypt Data ◽  
Near Term ◽  
Incremental Step

Companies such as Intel as a pioneer in chip design for computing are pushing the edge of computing from its present Classical Computing generation to the next generation of Quantum Computing. Along the side of Intel corporation, companies such as IBM, Microsoft, and Google are also playing in this domain. The race is on to build the world’s first meaningful quantum computer—one that can deliver the technology’s long-promised ability to help scientists do things like develop miraculous new materials, encrypt data with near-perfect security and accurately predict how Earth’s climate will change. Such a machine is likely more than a decade away, but IBM, Microsoft, Google, Intel, and other tech heavyweights breathlessly tout each tiny, incremental step along the way. Most of these milestones involve packing more quantum bits, or qubits—the basic unit of information in a quantum computer—onto a processor chip ever. But the path to quantum computing involves far more than wrangling subatomic particles. Such computing capabilities are opening a new area into dealing with the massive sheer volume of structured and unstructured data in the form of Big Data, is an excellent augmentation to Artificial Intelligence (AI) and would allow it to thrive to its next generation of Super Artificial Intelligence (SAI) in the near-term time frame.

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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