The adiabatic quantum computation (AQC) is based on the adiabatic theorem to approximate solutions of the Schrodinger equation. The design of an AQC algorithm involves the construction of a Hamiltonian that describes the behavior of the quantum system. This Hamiltonian is expressed as a linear interpolation of an initial Hamiltonian whose ground state is easy to compute, and a final Hamiltonian whose ground state corresponds to the solution of a given combinatorial optimization problem. The adiabatic theorem asserts that if the time evolution of a quantum system described by a Hamiltonian is large enough, then the system remains close to its ground state. An AQC algorithm uses the adiabatic theorem to approximate the ground state of the final Hamiltonian that corresponds to the solution of the given optimization problem. In this book, we investigate the computational simulation of AQC algorithms applied to the MAX-SAT problem. A symbolic analysis of the AQC solution is given in order to understand the involved computational complexity of AQC algorithms. This approach can be extended to other combinatorial optimization problems and can be used for the classical simulation of an AQC algorithm where a Hamiltonian problem is constructed. This construction requires the computation of a sparse matrix of dimension 2

^{n} * 2

^{n}, by means of tensor products, where n is the dimension of the quantum system. Also, a general scheme to design AQC algorithms is proposed, based on a natural correspondence between optimization Boolean variables and quantum bits. Combinatorial graph problems are in correspondence with pseudo-Boolean maps that are reduced in polynomial time to quadratic maps. Finally, the relation among NP-hard problems is investigated, as well as its logical representability, and is applied to the design of AQC algorithms. It is shown that every monadic second-order logic (MSOL) expression has associated pseudo-Boolean maps that can be obtained by expanding the given expression, and also can be reduced to quadratic forms.

### Table of Contents

Preface

Acknowledgments

Introduction

Approximability of NP-hard Problems

Adiabatic Quantum Computing

Efficient Hamiltonian Construction

AQC for Pseudo-Boolean Optimization

A General Strategy to Solve NP-Hard Problems

Conclusions

Bibliography

Authors' Biographies

### About the Author(s)

**William Cruz-Santos**, CU UAEM Valle de Chalco, Universidad Aut

William Cruz-Santos is a full-time professor of Mathematics and Computer Science at the Computer Engineering at the Universidad Autonoma del Estado de Mexico. Dr. Cruz-Santos' research interests include design of adiabatic quantum algorithms for solving NP-hard problems and simulation of quantum systems, as well as computational complexity analysis and algorithm design of classical algorithms. Dr. Cruz-Santos is particularly interested in the development of adiabatic quantum algorithms applied to computer vision problems from a combinatorial optimization point of view. Dr. Cruz-Santos holds a B.Sc. in Computer Science from the Universidad Juarez Autonoma de Tabasco, as well as M.Sc. and Ph.D. degrees in Computer Science, both degrees from the Centro de Investigacion y de Estudios Avanzados del IPN (Cinvestav-IPN).

**Guillermo Morales-Luna**, Computer Science Department, Cinvestav-IPN, Mexico City, Mexico

Guillermo Morales-Luna received the BSc degree in Mathematics from the Mexican National Polytechnic Institute in 1977, the MSc degree in Mathematics from Mexican Cinvestav-IPN, in 1978, and the PhD degree from the Mathematics Institute of the Polish Academy of Sciences in 1984. Since 1985 he is a researcher at Cinvestav-IPN. His research interests include cryptography, complexity theory, and mathematical logic. He is a Mexican national and he also holds Polish citizenship.

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