arXiv subjects

Samuel J. Lomonaco Jr

Publications and source records attributed to Samuel J. Lomonaco Jr.

16 recordsLinked to original sources

arXiv2017

Quantum Fourier Sampling is Guaranteed to Fail to Compute Automorphism Groups of Easy Graphs

The quantum hidden subgroup approach is an actively studied approach to solve combinatorial problems in quantum complexity theory. With the success of the Shor's algorithm, it was hoped that similar approach may be useful to solve the other combinatorial problems. One such problem is the graph isomorphism problem which has survived decades of efforts using the hidden subgroup approach. This paper provides a systematic approach to create arbitrarily large classes of classically efficiently solvable graph automorphism problems or easy graph automorphism problems for which hidden subgroup approach is guaranteed to always fail irrespective of the size of the graphs no matter how many copies of coset states are used. As the graph isomorphism problem is believed to be at least as hard as the graph automorphism problem, the result of this paper entails that the hidden subgroup approach is also guaranteed to always fail for the arbitrarily large classes of graph isomorphism problems. Combining these two results, it is argued that the hidden subgroup approach is essentially a dead end and alternative quantum algorithmic approach needs to be investigated for the graph isomorphism and automorphism problems.

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arXiv2014

Quantum Diagrams and Quantum Networks

This paper is an introduction to diagrammatic methods for representing quantum processes and quantum computing. We review basic notions for quantum information and quantum computing. We discuss topological diagrams and some issues about using category theory in representing quantum computing and teleportation. We analyze very carefully the diagrammatic meaning of the usual representation of the Mach-Zehnder interferometer, and we show how it can be generalized to associate to each composition of unitary transformations a "laboratory quantum diagram" such that particles moving though the many alternate paths in this diagram will mimic the quantum process represented by the composition of unitary transformations. This is a finite dimensional way to think about the Feynman Path Integral. We call our representation result the Path Theorem. Then we go back to the basics of networks and matrices and show how elements of quantum measurement can be represented with network diagrams.

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arXiv2010

Quantum Algorithms for the Jones Polynomial

This paper gives a generalization of the AJL algorithm and unitary braid group representation for quantum computation of the Jones polynomial to continuous ranges of values on the unit circle of the Jones parameter. We show that our 3-strand algorithm for the Jones polynomial is a special case of this generalization of the AJL algorithm. The present paper uses diagrammatic techniques to prove these results. The techniques of this paper have been used and will be used in the future in work with R. Marx, A. Fahmy, L. H. Kauffman, S. J. Lomonaco Jr.,A. Sporl, N. Pomplun, T. Schulte Herbruggen, J. M. Meyers, and S. J. Glaser on NMR quantum computation of the Jones polynomial.

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arXiv2008

Quantum Knots and Mosaics

In this paper, we give a precise and workable definition of a quantum knot system, the states of which are called quantum knots. This definition can be viewed as a blueprint for the construction of an actual physical quantum system. Moreover, this definition of a quantum knot system is intended to represent the "quantum embodiment" of a closed knotted physical piece of rope. A quantum knot, as a state of this system, represents the state of such a knotted closed piece of rope, i.e., the particular spatial configuration of the knot tied in the rope. Associated with a quantum knot system is a group of unitary transformations, called the ambient group, which represents all possible ways of moving the rope around (without cutting the rope, and without letting the rope pass through itself.) Of course, unlike a classical closed piece of rope, a quantum knot can exhibit non-classical behavior, such as quantum superposition and quantum entanglement. This raises some interesting and puzzling questions about the relation between topological and quantum entanglement. The knot type of a quantum knot is simply the orbit of the quantum knot under the action of the ambient group. We investigate quantum observables which are invariants of quantum knot type. We also study the Hamiltonians associated with the generators of the ambient group, and briefly look at the quantum tunneling of overcrossings into undercrossings. A basic building block in this paper is a mosaic system which is a formal (rewriting) system of symbol strings. We conjecture that this formal system fully captures in an axiomatic way all of the properties of tame knot theory.

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arXiv2007

Spin Networks and Anyonic Topological Computing II

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. The simplest case of these models is the Fibonacci model, itself universal for quantum computation. We here formulate these braid group representations in a shape suitable for computation and algebraic work.

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arXiv2007

$q$ - Deformed Spin Networks, Knot Polynomials and Anyonic Topological Quantum Computation

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.

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arXiv2006

Spin Networks and Anyonic Topological Computing

We review the q-deformed spin network approact to Topological Quantum Field Theory and apply these methods to produce unitary representations of the braid groups that are dense in the unitary groups. These methods produce a concise proof that quantum computation can be performed within a single representation of the Artin Braid Group.

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arXiv2004

Braiding Operators are Universal Quantum Gates

This paper explores of the role of unitary braiding operators in quantum computing. We show that a single specific solution R (the Bell basis change matrix) of the Yang-Baxter Equation is a universal gate for quantum computing, in the presence of local unitary transformations. We show that this same R generates a new non-trivial invariant of braids, knots, and links. Other solutions of the Yang-Baxter Equation are also shown to be universal for quantum computation. The paper discusses these results in the context of comparing quantum and topological points of view. In particular, we discuss quantum computation of link invariants, the relationship between quantum entanglement and topological entanglement, teleportation, and the structure of braiding in a topological quantum field theory.

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arXiv2004

Quantum Knots

This paper proposes the definition of a quantum knot as a linear superposition of classical knots in three dimensional space. The definition is constructed and examples are discussed. Then the paper details extensions and also limitations of the Aravind Hypothesis for comparing quantum measurement with classical topological measurement. We propose a separate, network model for quantum evolution and measurement, where the background space is replaced by an evolving network. In this model there is an analog of the Aravind Hypothesis that promises to directly illuminate relationships between physics, topology and quantum knots.

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arXiv2004

Distributed quantum computing: A distributed Shor algorithm

We present a distributed implementation of Shor's quantum factoring algorithm on a distributed quantum network model. This model provides a means for small capacity quantum computers to work together in such a way as to simulate a large capacity quantum computer. In this paper, entanglement is used as a resource for implementing non-local operations between two or more quantum computers. These non-local operations are used to implement a distributed factoring circuit with polynomially many gates. This distributed version of Shor's algorithm requires an additional overhead of O((log N)^2) communication complexity, where N denotes the integer to be factored.

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arXiv2004

Generalized GHZ States and Distributed Quantum Computing

A key problem in quantum computing is finding a viable technological path toward the creation of a scalable quantum computer. One possible approach toward solving part of this problem is distributed computing, which provides an effective way of utilizing a network of limited capacity quantum computers. In this paper, we present two primitive operations, cat-entangler and cat-disentangler, which in turn can be used to implement non-local operations, e.g. non-local CNOT and quantum teleportation. We also show how to establish an entangled pair, and use entangled pairs to efficiently create a generalized GHZ state. Furthermore, we present procedures which allow us to reuse channel qubits in a sequence of non-local operations. These non-local operations work on the principle that a cat-like state, created by cat-entangler, can be used to distribute a control qubit among multiple computers. Using this principle, we show how to efficiently implement non-local control operations in many situation, including a parallel implementation of a certain kind of unitary transformation. Finally, as an example, we present a distributed version of the quantum Fourier transform.

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arXiv2003

Entanglement Criteria - Quantum and Topological

This paper gives a criterion for detecting the entanglement of a quantum state, and uses it to study the relationship between topological and quantum entanglement. It is fundamental to view topological entanglements such as braids as entanglement operators and to associate to them unitary operators that are capable of creating quantum entanglement. The entanglement criterion is used to explore this connection. The paper discusses non-locality in the light of this criterion.

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arXiv2003

Countinuous Quantum Hidden Subgroup Algorithms

In this paper we show how to construct two continuous variable and one continuous functional quantum hidden subgroup (QHS) algorithms. These are respectively quantum algorithms on the additive group of reals R, the additive group R/Z of the reals R mod 1, i.e., the circle, and the additive group Paths of L^2 paths x:[0,1]-->R^n in real n-space R^n. Also included is a curious discrete QHS algorithm which is dual to Shor's algorithm.

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arXiv2001

A Talk on Quantum Cryptography, or How Alice Outwits Eve

Alice and Bob wish to communicate without the archvillainess Eve eavesdropping on their conversation. Alice, decides to take two college courses, one in cryptography, the other in quantum mechanics. During the courses, she discovers she can use what she has just learned to devise a cryptographic communication system that automatically detects whether or not Eve is up to her villainous eavesdropping. Some of the topics discussed are Heisenberg's Uncertainty Principle, the Vernam cipher, the BB84 and B92 cryptographic protocols. The talk ends with a discussion of some of Eve's possible eavesdropping strategies, opaque eavesdropping, translucent eavesdropping, and translucent eavesdropping with entanglement.

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arXiv2001

An Entangled Tale of Quantum Entanglement

This paper gives an overview from the perspective of Lie group theory of some of the recent advances in the rapidly expanding research area of quantum entanglement. This paper is a written version of the last of eight one hour lectures given in the American Mathematical Society (AMS) Short Course on Quantum Computation held in conjunction with the Annual Meeting of the AMS in Washington, DC, USA in January 2000. More information about the AMS Short Course can be found at http://www.csee.umbc.edu/~lomonaco/ams/Announce.html

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arXiv2000

Grover's quantum search algorithm

This paper is a written version of a one hour lecture given on Lov Grover's database search algorithm. Table Of Contents 1 ... Problem Definition 2 ... The quantum mechanical perspective 3 ... Properties of the inversion I_|psi> 4 ... The method in Lov's "madness" 5 ... Summary of Grover's algorithm 6 ... An example of Grover's algorithm References

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