arXiv subjects

Samuel J. Lomonaco

Publications and source records attributed to Samuel J. Lomonaco.

11 recordsLinked to original sources

arXiv2022

How viable is quantum annealing for solving linear algebra problems?

With the increasing popularity of quantum computing and in particular quantum annealing, there has been growing research to evaluate the meta-heuristic for various problems in linear algebra: from linear least squares to matrix and tensor factorization. At the core of this effort is to evaluate quantum annealing for solving linear least squares and linear systems of equations. In this work, we focus on the viability of using quantum annealing for solving these problems. We use simulations based on the adiabatic principle to provide new insights for previously observed phenomena with the D-wave machines, such as quantum annealing being robust against ill-conditioned systems of equations and scaling quite well against the number of rows in a system. We then propose a hybrid approach which uses a quantum annealer to provide a initial guess of the solution $x_0$, which would then be iteratively improved with classical fixed point iteration methods.

quant-ph↗

arXiv2021

Quantum Approximate Optimization for Hard Problems in Linear Algebra

The Quantum Approximate Optimization Algorithm (QAOA) by Farhi et al. is a quantum computational framework for solving quantum or classical optimization tasks. Here, we explore using QAOA for Binary Linear Least Squares (BLLS); a problem that can serve as a building block of several other hard problems in linear algebra, such as the Non-negative Binary Matrix Factorization (NBMF) and other variants of the Non-negative Matrix Factorization (NMF) problem. Most of the previous efforts in quantum computing for solving these problems were done using the quantum annealing paradigm. For the scope of this work, our experiments were done on noiseless quantum simulators, a simulator including a device-realistic noise-model, and two IBM Q 5-qubit machines. We highlight the possibilities of using QAOA and QAOA-like variational algorithms for solving such problems, where trial solutions can be obtained directly as samples, rather than being amplitude-encoded in the quantum wavefunction. Our numerics show that Simulated Annealing can outperform QAOA for BLLS at a QAOA depth of $p\leq3$ for the probability of sampling the ground state. Finally, we point out some of the challenges involved in current-day experimental implementations of this technique on cloud-based quantum computers.

quant-ph↗

arXiv2018

Analyzing the Quantum Annealing Approach for Solving Linear Least Squares Problems

With the advent of quantum computers, researchers are exploring if quantum mechanics can be leveraged to solve important problems in ways that may provide advantages not possible with conventional or classical methods. A previous work by O'Malley and Vesselinov in 2016 briefly explored using a quantum annealing machine for solving linear least squares problems for real numbers. They suggested that it is best suited for binary and sparse versions of the problem. In our work, we propose a more compact way to represent variables using two's and one's complement on a quantum annealer. We then do an in-depth theoretical analysis of this approach, showing the conditions for which this method may be able to outperform the traditional classical methods for solving general linear least squares problems. Finally, based on our analysis and observations, we discuss potentially promising areas of further research where quantum annealing can be especially beneficial.

quant-ph↗

arXiv2016

Braiding Majorana Fermions

In this paper we study a Clifford algebra generalization of the quaternions and its relationship with braid group representations related to Majorana fermions. The Fibonacci model for topological quantum computing is based on the fusion rules for a Majorana fermion. Majorana fermions can be seen not only in the structure of collectivies of electrons, as in the quantum Hall effect, but also in the structure of single electrons both by experiments with electrons in nanowires and also by the decomposition of the operator algebra for a fermion into a Clifford algebra generated by two Majorana operators. The purpose of this paper is to discuss these braiding representations, important for relationships among physics, quantum information and topology. A new result in this paper is the Clifford Braiding Theorem. This theorem shows that the Majorana operators give rise to a particularly robust representation of the braid group that is then further represented to find the phases of the fermions under their exchanges in a plane space. The more robust representation in our braiding theorem will be the subject of further work.

cond-mat.str-el↗

arXiv2013

Symbolic Arithmetic and Integer Factorization

In this paper, we create a systematic and automatic procedure for transforming the integer factorization problem into the problem of solving a system of Boolean equations. Surprisingly, the resulting system of Boolean equations takes on a "life of its own" and becomes a new type of integer, which we call a generic integer. We then proceed to use the newly found algebraic structure of the ring of generic integers to create two new integer factoring algorithms, called respectively the Boolean factoring (BF) algorithm, and the multiplicative Boolean factoring (MBF) algorithm. Although these two algorithms are not competitive with current classical integer factoring algorithms, it is hoped that they will become stepping stones to creating much faster and more competitive algorithms, and perhaps be precursors of a new quantum algorithm for integer factoring.

math.NT↗

arXiv2011

Quantizing Knots and Beyond

This paper formulates a generalization of our work on quantum knots to explain how to make quantum versions of algebraic, combinatorial and topological structures. We include a description of previous work on the construction of Hilbert spaces from the states of the bracket polynomial with applications to algorithms for the Jones polynomial and relations with Khovanov homology. The purpose of this paper is to place such constructions in a general context of the quantization of mathematical structures.

quant-ph↗

arXiv2011

Quantizing Braids and Other Mathematical Objects: The General Quantization Procedure

Extending the methods from our previous work on quantum knots and quantum graphs, we describe a general procedure for quantizing a large class of mathematical structures which includes, for example, knots, graphs, groups, algebraic varieties, categories, topological spaces, geometric spaces, and more. This procedure is different from that normally found in quantum topology. We then demonstrate the power of this method by using it to quantize braids. This general method produces a blueprint of a quantum system which is physically implementable in the same sense that Shor's quantum factoring algorithm is physically implementable. Mathematical invariants become objects that are physically observable.

quant-ph↗

arXiv2009

Quantum Knots and Lattices, or a Blueprint for Quantum Systems that Do Rope Tricks

Using the cubic honeycomb (cubic tessellation) of Euclidean 3-space, we define a quantum system whose states, called quantum knots, represent a closed knotted piece of rope, i.e., represent the particular spatial configuration of a knot tied in a rope in 3-space. This quantum system, called a quantum knot system, is physically implementable in the same sense as Shor's quantum factoring algorithm is implementable. To define a quantum knot system, we replace the standard three Reidemeister knot moves with an equivalent set of three moves, called respectively wiggle, wag, and tug, so named because they mimic how a dog might wag its tail. We argue that these moves are in fact more "physics friendly" because, unlike the Reidemeister moves, they respect the differential geometry of 3-space, and moreover they can be transformed into infinitesimal moves. These three moves wiggle, wag, and tug generate a unitary group, called the lattice ambient group, which acts on the state space of the quantum system. The lattice ambient group represents all possible ways of moving a rope around in 3-space without cutting the rope, and without letting the rope pass through itself. We then investigate those quantum observables of the quantum knot system which are knot invariants. We also study Hamiltonians associated with the generators of the lattice ambient group. We conclude with a list of open questions.

quant-ph↗

arXiv2002

Comparing Quantum Entanglement and Topological Entanglement

This paper discusses relationships between topological entanglement and quantum entanglement. Specifically, we propose that for this comparison 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.

quant-ph↗

arXiv2000

A Rosetta Stone for Quantum Mechanics with an Introduction to Quantum Computation

The purpose of these lecture notes is to provide readers, who have some mathematical background but little or no exposure to quantum mechanics and quantum computation, with enough material to begin reading the research literature in quantum computation and quantum information theory. This paper is a written version of the first 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, and will appear in the AMS PSAPM volume entitled "Quantum Computation." Part 1 of the paper is an introduction the to the concept of the qubit. Part 2 gives an introduction to quantum mechanics covering such topics as Dirac notation, quantum measurement, Heisenberg uncertainty, Schrodinger's equation, density operators, partial trace, multipartite quantum systems, the Heisenberg versus the Schrodinger picture, quantum entanglement, EPR paradox, quantum entropy. Part 3 gives a brief introduction to quantum computation, covering such topics as elementary quantum computing devices, wiring diagrams, the no-cloning theorem, quantum teleportation, Shor's algorithm, Grover's algorithm. Many examples are given. A table of contents as well as an index are provided for readers who wish to "pick and choose." Since this paper is intended for a diverse audience, it is written in an informal style at varying levels of difficulty and sophistication from the very elementary to the more advanced.

quant-ph↗

arXiv1998

A Quick Glance at Quantum Cryptography

The recent application of the principles of quantum mechanics to cryptography has led to a remarkable new dimension in secret communication. As a result of these new developments, it is now possible to construct cryptographic communication systems which detect unauthorized eavesdropping should it occur, and which give a guarantee of no eavesdropping should it not occur. CONTENTS P3. Cryptographic systems before quantum cryptography P7. Preamble to quantum cryptography P10. The BB84 quantum cryptographic protocol without noise P16. The BB84 quantum cryptographic protocol with noise P19..The B92 quantum cryptographic protocol P21. EPR quantum cryptographic protocols P25. Other protocols P25. Eavesdropping stategies and counter measures P26. Conclusion P29. Appendix A. The no cloning theorem P30. Appendix B. Proof that an undetectable eavesdropper can obtain no information from the B92 protocol P31. Appendix C. Part of a Rosetta stone for quantum mechanics P44. References

quant-ph↗