At first sight, quantum computing is completely different from classical computing. Nevertheless, a link is provided by reversible computation.
Whereas an arbitrary quantum circuit, acting on w qubits, is described by an n x n unitary matrix with n=2w, a reversible classical circuit, acting on w bits, is described by a 2w x 2w permutation matrix. The permutation matrices are studied in group theory of finite groups (in particular the symmetric group Sn); the unitary matrices are discussed in group theory of continuous groups (a.k.a. Lie groups, in particular the unitary group U(n).
Both the synthesis of a reversible logic circuit and the synthesis of a quantum logic circuit take advantage of the decomposition of a matrix: the former of a permutation matrix, the latter of a unitary matrix. In both cases the decomposition is into three matrices. In both cases the decomposition is not unique.
Author Biography:
Alexis De Vos is an electrical engineer, physicist, and doctor in applied sciences and graduated from the Universiteit Gent (Belgium). He is currently a part-time professor in the Department of Electronics of the Universiteit Gent. His research is concerned with material science (polymers, semiconductors, metals, liquid crystals), microelectronics (thin films, chips, neural networks, reversible circuits), and energy sciences (thermodynamics, solar energy, endoreversible engines). He is author of the books Thermodynamics of Solar Energy Conversion (Wiley-VCH, 2008) and Reversible Computing (Wiley-VCH, 2010). He designed and produced several prototype integrated circuits for reversible computers such as adders, multipliers, and linear transformers. He currently investigates quantum computing. Stijn De Baerdemacker is a physicist and doctor in sciences and graduated from the Universiteit Gent (Belgium). He has been a visiting scientist at the University of Toronto (ON, Canada), University of Notre Dame (IN, USA) and Universiteit Amsterdam (The Netherlands). He is currently a post-doctoral researcher in the Department of Physics and Astronomy of the Universiteit Gent. His research is concerned with the development of accurate quantum many-body methods in quantum physics, quantum chemistry and quantum computing. For this, he uses and develops techniques from Lie algebra theory and notions from (quantum) integrability. In his free time, he is a painter and explores the boundaries between science and art. Yvan Van Rentergem is an electrical engineer and doctor in applied sciences and graduated from the Universiteit Gent (Belgium). He obtained his Ph.D. in 2008 in the subject of reversible computing. During his research, he developed algorithms for the synthesis of reversible circuits. These methods were applied for real-life prototype chips. His research led to ten articles presented at international conferences or published in international journals. After earning his Ph.D., he went to work at ArcelorMittal Gent as operations research specialist, developing models to optimize the logistical flow of the steel shop. These models are successfully applied at ArcelorMittal Gent and several other sites of ArcelorMittal. He currently is slabyard support manager at ArcelorMittal Gent. Mitchell A. (Mitch) Thornton received the BS degree in electrical engineering from Oklahoma State University in Stillwater, OK, in 1985, the MS degree in electrical engineering from the University of Texas-Arlington in Arlington, Texas, in 1990, the MS degree in computer science from Southern Methodist University in 1993, and the PhD degree in computer engineering from Southern Methodist University in 1995. He was a senior electronic systems engineer at E-Systems, Inc. in Greenville, TX from 1986 through 1991. He was employed as a design engineer at Cyrix Corporation from 1992 through 1993. He has served as a full-time faculty member in the University of Arkansas from 1995 to 1999, and Mississippi State University from 1999 to 2002. Currently he is the Cecil H. Green Chair of Engineering and Professor at Southern Methodist University in the Computer Science and Engineering Department, and by courtesy, in the Electrical Engineering Department. He also serves as the Technical Director of the Darwin Deason Institute for Cyber Security.
Dr. Thornton has research and teaching interests in the general area of digital circuits and systems design with specific emphasis in EDA/CAD methods including asynchronous circuit and computer arithmetic circuit synthesis, formal verification/validation and simulation of digital systems, multiple-valued logic, and spectral techniques. Dr. Thornton also works in the area of emerging technology in clouding reversible and quantum logic and computing.
Dr. Thornton is an active industrial consultant and is a licensed professional engineer in the States of Arkansas, Mississippi, and Texas. He has served numerous roles in professional societies such as IEEE, NSPE, and NCEES as well as an organizing committee member for symposia and conferences.
