QUANTUM INFORMATION AND COMPUTATIONFOR CHEMISTRY

ADVANCES IN CHEMICAL PHYSICS

VOLUME 154

EDITORIAL BOARD

KURT BINDER, Condensed Matter Theory Group, Institut Für Physik, Johannes Gutenberg-Universität, Mainz, Germany

WILLIAM T. COFFEY, Department of Electronic and Electrical Engineering, Printing House,Trinity College, Dublin, Ireland

KARL F. FREED, Department of Chemistry, James Franck Institute, University of Chicago,Chicago, Illinois, USA

DAAN FRENKEL, Department of Chemistry, Trinity College, University of Cambridge,Cambridge, UK

PIERRE GASPARD, Center for Nonlinear Phenomena and Complex Systems, UniversitéLibre de Bruxelles, Brussels, Belgium

MARTIN GRUEBELE, Departments of Physics and Chemistry, Center for Biophysics andComputational Biology, University of Illinois at Urbana-Champaign, Urbana, Illinois,USA

GERHARD HUMMER, Theoretical Biophysics Section, NIDDK-National Institutes of Health,Bethesda, Maryland, USA

RONNIE KOSLOFF, Department of Physical Chemistry, Institute of Chemistry and FritzHaber Center for Molecular Dynamics, The Hebrew University of Jerusalem, Israel

KA YEE LEE, Department of Chemistry, James Franck Institute, University of Chicago,Chicago, Illinois, USA

TODD J. MARTINEZ, Department of Chemistry, Photon Science, Stanford University,Stanford, California, USA

SHAUL MUKAMEL, Department of Chemistry, School of Physical Sciences, University ofCalifornia, Irvine, California, USA

JOSE N. ONUCHIC, Department of Physics, Center for Theoretical Biological Physics, RiceUniversity, Houston, Texas, USA

STEPHEN QUAKE, Department of Bioengineering, Stanford University, Palo Alto,California, USA

MARK RATNER, Department of Chemistry, Northwestern University, Evanston, Illinois,USA

DAVID REICHMAN, Department of Chemistry, Columbia University, New York City,New York, USA

GEORGE SCHATZ, Department of Chemistry, Northwestern University, Evanston, Illinois,USA

STEVEN J. SIBENER, Department of Chemistry, James Franck Institute, University ofChicago, Chicago, Illinois, USA

ANDREI TOKMAKOFF, Department of Chemistry, James Franck Institute, University ofChicago, Chicago, Illinois, USA

DONALD G. TRUHLAR, Department of Chemistry, University of Minnesota, Minneapolis,Minnesota, USA

JOHN C. TULLY, Department of Chemistry, Yale University, New Haven, Connecticut,USA

QUANTUM INFORMATION AND COMPUTATIONFOR CHEMISTRY

ADVANCES IN CHEMICAL PHYSICS

VOLUME 154

Edited by

SABRE KAISPurdue University

QEERI, QatarSanta Fe Institute

Series Editors

STUART A. RICEDepartment of Chemistry

andThe James Franck InstituteThe University of Chicago

Chicago, Illinois

AARON R. DINNERDepartment of Chemistry

andThe James Franck InstituteThe University of Chicago

Chicago, Illinois

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CONTRIBUTORS TO VOLUME 154

ALAN ASPURU-GUZIK, Department of Chemistry and Chemical Biology, HarvardUniversity, 12 Oxford Street, Cambridge, MA 02138, USA

JONATHAN BAUGH, Institute for Quantum Computing, University of Waterloo,Waterloo, Ontario, N2L 3G1, Canada; Departments of Physics and Astron-omy, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada;Department of Chemistry, University of Waterloo, Waterloo, Ontario, N2L3G1, Canada

JACOB BIAMONTE, ISI Foundation, Via Alassio 11/c, 10126, Torino, Italy; Centrefor Quantum Technologies, National University of Singapore, Block S15,3 Science Drive 2, Singapore 117543, Singapore

SERGIO BOIXO, Department of Chemistry and Chemical Biology, Harvard Univer-sity, 12 Oxford Street, Cambridge, MA 02138, USA; Google, 340 Main St,Venice, CA 90291, USA

KENNETH R. BROWN, School of Chemistry and Biochemistry, School of Compu-tational Science and Engineering, School of Physics, Georgia Institute ofTechnology, Ford Environmental Science and Technology Building, 311Ferst Dr, Atlanta, GA 30332-0400, USA

GARNET KIN-LIC CHAN, Department of Chemistry and Chemical Biology, CornellUniversity, Ithaca, NY 14850, USA

ROBIN COTE, Department of Physics, U-3046, University of Connecticut, 2152Hillside Road, Storrs, CT 06269-3046, USA

BEN CRIGER, Institute for Quantum Computing, University of Waterloo, Water-loo, Ontario, N2L 3G1, Canada; Departments of Physics and Astronomy,University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada

BORIVOJE DAKIC, Faculty of Physics, University of Vienna, Boltzmanngasse 5,A-1090 Vienna, Austria

R. DE VIVIE-RIEDLE, Department Chemie, Ludwig-Maximilians-Universitat,Butenandt-Str. 11, 81377 München, Germany

FRANK GAITAN, Laboratory for Physical Sciences, 8050 Greenmead Drive, CollegePark, MD 20740-4004, USA

C. GOLLUB, Department Chemie, Ludwig-Maximilians-Universitat, Butenandt-Str. 11, 81377 München, Germany

v

vi CONTRIBUTORS TO VOLUME 154

SABRE KAIS, Department of Chemistry and Physics, Purdue University, 560 OvalDrive, West Lafayette, IN 47907, USA; Qatar Environment & EnergyResearch Institute (QEERI), Doha, Qatar; Santa Fe Institute, Santa Fe, NM87501, USA

GRAHAM KELLS, Dahlem Center for Complex Quantum Systems, FachbereichPhysik, Freie Universitat Berlin, Arminallee 14, D-14195 Berlin, Germany

JESSE M. KINDER, Department of Chemistry and Chemical Biology, Cornell Uni-versity, Ithaca, NY 14850, USA

M. KOWALEWSKI, Department Chemie, Ludwig-Maximilians-Universitat,Butenandt-Str. 11, 81377 München, Germany

DANIEL A. LIDAR, Departments of Electrical Engineering, Chemistry, and Physics,and Center for Quantum Information Science & Technology, University ofSouthern California, Los Angeles, CA 90089, USA

PETER J. LOVE, Department of Physics, Haverford College, 370 Lancaster Avenue,Haverford, PA 19041, USA

XIAO-SONG MA, Faculty of Physics, University of Vienna, Boltzmanngasse 5, A-1090 Vienna, Austria; Institute for Quantum Optics and Quantum Information(IQOQI), Austrian Academy of Sciences, Boltzmanngasse 3, A-1090 Vienna,Austria

DAVID A. MAZZIOTTI, Department of Chemistry and the James Franck Institute,The University of Chicago, Chicago, IL 60637, USA

J. TRUE MERRILL, School of Chemistry and Biochemistry, School of ComputationalScience and Engineering, School of Physics, Georgia Institute of Technology,Ford Environmental Science and Technology Building, 311 Ferst Dr, Atlanta,GA 30332-0400, USA

SEBASTIAN MEZNARIC, Clarendon Laboratory, University of Oxford, Parks Road,Oxford OX1 3PU, UK

FRANCO NORI, CEMS, RIKEN, Saitama 351-0198, Japan; Physics Department,University of Michigan, Ann Arbor, MI 48109-1040, USA

A. PAPAGEORGIOU, Department of Computer Science, Columbia University, NewYork, NY 10027, USA

DANIEL PARK, Institute for Quantum Computing, University of Waterloo, Water-loo, Ontario, N2L 3G1, Canada; Departments of Physics and Astronomy,University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada

JIRÍ PITTNER, J. Heyrovsky Institute of Physical Chemistry, Academy of Sciencesof the Czech Republic, v.v.i., Dolejškova 3, 18223 Prague 8, Czech Republic

CLAIRE C. RALPH, Department of Chemistry and Chemical Biology, Cornell Uni-versity, Ithaca, NY 14850, USA

CONTRIBUTORS TO VOLUME 154 vii

GEHAD SADIEK, Department of Physics, King Saud University, Riyadh, SaudiArabia; Department of Physics, Ain Shams University, Cairo 11566, Egypt

NOLAN SKOCHDOPOLE, Department of Chemistry and the James Franck Institute,The University of Chicago, Chicago, IL 60637, USA

DAVID G. TEMPEL, Department of Chemistry and Chemical Biology, HarvardUniversity, 12 Oxford Street, Cambridge, MA 02138, USA; Department ofPhysics, Harvard University, 17 Oxford Street, Cambridge, MA 02138, USA

J. F. TRAUB, Department of Computer Science, Columbia University, New York,NY 10027, USA

U. TROPPMANN, DepartmentChemie,Ludwig-Maximilians-Universitat, Butenandt-Str. 11, 81377 München, Germany

JIRÍ VALA, Department of Mathematical Physics, National University of IrelandMaynooth, Maynooth, Co. Kildare, Ireland; School of Theoretical Physics,Dublin Institute for Advanced Studies, 10 Burlington Road, Dublin 4, Ireland

LIBOR VEIS, J. Heyrovsky Institute of Physical Chemistry, Academy of Sciencesof the Czech Republic, v.v.i., Dolejškova 3, 18223 Prague 8, Czech Republic;Department of Physical and Macromolecular Chemistry, Faculty of Science,Charles University in Prague, Hlavova 8, 12840 Prague 2, Czech Republic

P. VON DEN HOFF, Department Chemie, Ludwig-Maximilians-Universitat,Butenandt-Str. 11, 81377 München, Germany

PHILIP WALTHER, Faculty of Physics, University of Vienna, Boltzmanngasse 5,A-1090 Vienna, Austria; Institute for Quantum Optics and Quantum Infor-mation (IQOQI), Austrian Academy of Sciences, Boltzmanngasse 3, A-1090Vienna, Austria

PAUL WATTS, Department of Mathematical Physics, National University of IrelandMaynooth, Maynooth, Co. Kildare, Ireland; School of Theoretical Physics,Dublin Institute for Advanced Studies, 10 Burlington Road, Dublin 4, Ireland

JAMES D. WHITFIELD, Department of Chemistry and Chemical Biology, HarvardUniversity, 12 Oxford Street, Cambridge, MA 02138, USA; NEC Laborato-ries America, 4 Independence Way, Princeton, NJ 08540, USA; Departmentof Physics, Columbia University, 538 West 120th Street, New York, NY10027, USA

QING XU, Department of Chemistry, Purdue University, 560 Oval Drive, WestLafayette, IN 47907, USA

MAN-HONG YUNG, Center for Quantum Information, Institute for Interdisci-plinary Information Sciences, Tsinghua University, Beijing 100084, P. R.China; Department of Chemistry and Chemical Biology, Harvard University,12 Oxford Street, Cambridge, MA 02138, USA

FOREWORD

Quantum mechanics and information theory were key new areas for scientificprogress in the 20th century. Toward the end of the 1900s, when both of thesefields were generally regarded as mature, a small community of researchers inphysics, computer science, mathematics, and chemistry began to explore the fer-tile ground at the intersection of these two areas. Formulation of a quantum versionof information theory and analysis of computational and communication proto-cols with quantum, rather than classical states, led to a powerful new paradigm forcomputation. Dramatic theoretical results were achieved in the mid-1990s with thepresentation of both a quantum algorithm for solving factorization problems withan exponential speedup relative to the best-known classical algorithm and a proto-col for performing fault-tolerant quantum computation (i.e., computation allowingfor errors but in which these errors are guaranteed not to propagate). These twokey results, both due to Peter Shor, motivated the launch of experimental efforts torealize physical implementations of quantum computations. Initially restricted toa small set of candidate physical systems, experimental studies have since multi-plied to encompass an increasingly wide range of physical qubit systems, rangingfrom photons to solid-state quantum circuits. As the study of physical candidatesfor viable qubit arrays for computation has grown and as attention turns to morecomplex quantum systems that promise scalability, many quantum theorists activein the field are exploring the use of information theoretic concepts to bring newinsights and understanding to the study of quantum systems. Theorists and exper-imentalists alike are also increasingly heavily focused on quantum simulation, theart of making one Hamiltonian emulate another, according to Feynman’s originalproposal (1982) for a quantum computer.

Today, the field of quantum information science covers an increasingly broadrange of physical systems in addition to theoretical topics in mathematics, com-puter science, and information theory. As a result, the community of scientistsworking in this field is quite diverse and interdisciplinary. The unifying featureof the community is interest in a fully quantum description of both informationprocessing and the physical systems that might enable it to be realized in a con-trollable and programmable fashion. Chemical physics, rooted in quantum physicsand the intellectual home of molecular quantum mechanics, is centrally located inthis interdisciplinary community. The chemical physicist’s interest and expertisein analysis and control of molecular systems bring key tools and perspectives tothe twin challenges of experimentally realizing quantum computation and of usingquantum information theory for analysis of chemical problems in the full quantum

ix

x FOREWORD

regime. I use the latter term instead of the simpler and oft-used term quantumchemistry, which has unfortunately become associated almost exclusively with thespecific subdiscipline of electronic structure of atoms and molecules. For quan-tum information science however, the field quantum chemistry covers much more,namely, the set of molecular phenomena that require a full quantum descriptionfor energetics, structure, and dynamics, whether electronic or nuclear, local orcollective in nature.

This volume brings together chapters from the quantum information sciencecommunity that describe topics in quantum computation and quantum informa-tion related to or overlapping with key topics in chemical physics. The motivationfor this volume may be summarized by two questions. First, what can chem-istry contribute to quantum information? Second, what can quantum informationcontribute to the study of chemical systems? The contributions in this volumeaddress both perspectives, while surveying theoretical and experimental quantuminformation-related research within chemical physics.

In Chapter 1, Sabre Kais introduces the fields of quantum information, quantumcomputation, and quantum simulation, outlining their relevance and impact forchemistry. In Chapter 2, Peter Love presents ideas from electronic structure theorythat might facilitate quantum computation and quantum simulation. Alan Aspuru-Guzik and Joseph Traub describe quantum algorithms that are relevant for efficientsolution of a range of problems in both physics and chemistry, in Chapters 3 and 6,respectively. Libor Veis and Jirí Pittner discuss the solution for both nonrelativisticand relativistic electronic energies via quantum computations in Chapter 4. Severalcontributions specifically address the relation between quantum computation andelectronic structure calculations in terms of the two motivating questions statedearlier. Two contributions examine the use and impact of electronic structurecalculations for quantum computation. In Chapter 5, Frank Gaitan and Franco Norireview an approach based on use of density functional theory to generate efficientcalculation of energy gaps for adiabatic quantum computation. In Chapter 7, JesseKinder, Claire Ralph, and Garnet Chan discuss the use of matrix product statesfor understanding the dynamics of complex electronic states. The understandingoffered by quantum information theoretic concepts for realizing correlations incomplex quantum systems is also the focus of Gehad Sadiek and Sabre Kais’sdiscussion of entanglement for spin systems in Chapter 15.

Other authors address the realization of high-fidelity quantum operations, akey requirement of implementation of all quantum information processing. InChapter 10, True Merrill and Ken Brown discuss the use of compensating pulsesequences from NMR for correcting unknown errors in the quantum bits (qubits).In Chapter 11, Daniel Lidar compares the benefits of passive versus active errorcorrection, with analysis of decoherence-free subspaces, noiseless subsystems, anddynamical decoupling pulse sequences. Fault tolerance, the ability to effectivelycompute despite a bounded rate of error, is also a powerful driver for development

FOREWORD xi

of the topological materials that Jirí Vala describes in Chapter 16. The exotic“topological” phases of these materials are characterized by massive ground-statedegeneracy, an energy gap for all local excitations and anyonic excitations—properties that mathematicians have shown guarantee a remarkable natural faulttolerance for quantum computation.

Several experimentalists have contributed chapters describing specific physicalimplementations of qubits and outlining experimental schemes for the realiza-tion of quantum computation, quantum simulation, or quantum communication.Ben Criger, Daniel Park, and Jonathan Baugh describe spin-based quantum infor-mation processing in Chapter 8. Xiao-Song Ma, Borivoje Drakic, and PhilipWalther describe the use of photonic systems for quantum simulation of chemi-cal phenomena in Chapter 9. Regina de Vivie-Riedle describes how informationmay be transferred through molecular chains by taking advantage of vibrationalenergy transfer (Chapter 13), and Robin Côté summarizes the application of therapidly growing field of ultracold molecules to quantum information processing(Chapter 14). Finally, David Mazziotti examines the roles of electron correla-tion, entanglement, and redundancy in the energy flow within the pigment–proteinstructure of a light-harvesting complex in Chapter 12.

The contributions to this volume represent just a partial overview of the synergythat has developed over the past 10 years between quantum information sciencesand chemical physics. Many more areas of overlap have not been addressed indetail here, notably coherent quantum control of atoms and molecules. However,we hope this selection of chapters will provide a useful introduction and perspectiveto current directions in quantum information and its relation to the diverse setof quantum phenomena and theoretical methods that are central to the chemicalphysics community.

University of California, Berkeley BIRGITTA WHALEY

PREFACE TO THE SERIES

Advances in science often involve initial development of individual specializedfields of study within traditional disciplines, followed by broadening and overlap,or even merging, of those specialized fields, leading to a blurring of the linesbetween traditional disciplines. The pace of that blurring has accelerated in thepast few decades, and much of the important and exciting research carried outtoday seeks to synthesize elements from different fields of knowledge. Examplesof such research areas include biophysics and studies of nanostructured materials.As the study of the forces that govern the structure and dynamics of molecularsystems, chemical physics encompasses these and many other emerging researchdirections. Unfortunately, the flood of scientific literature has been accompaniedby losses in the shared vocabulary and approaches of the traditional disciplines.Scientific journals are exerting pressure to be ever more concise in the descriptionsof studies, to the point that much valuable experience, if recorded at all, is hiddenin supplements and dissipated with time. These trends in science and publishingmake this series, Advances in Chemical Physics, a much needed resource.

Advances in Chemical Physics is devoted to helping the reader obtain generalinformation about a wide variety of topics in chemical physics, a field that weinterpret broadly. Our intent is to have experts present comprehensive analyses ofsubjects of interest and to encourage the expression of individual points of view.We hope this approach to the presentation of an overview of a subject will bothstimulate new research and serve as a personalized learning text for beginners ina field.

STUART A. RICE

AARON R. DINNER

xiii

CONTENTS

CONTRIBUTORS TO VOLUME 154 v

FOREWORD ix

PREFACE TO THE SERIES xiii

1INTRODUCTION TO QUANTUM INFORMATION AND COMPUTATION FOR CHEMISTRY

By Sabre Kais

39BACK TO THE FUTURE: A ROADMAP FOR QUANTUM SIMULATION FROM VINTAGE

QUANTUM CHEMISTRY

By Peter J. Love

67INTRODUCTION TO QUANTUM ALGORITHMS FOR PHYSICS AND CHEMISTRY

By Man-Hong Yung, James D. Whitfield, Sergio Boixo,David G. Tempel, and Alan Aspuru-Guzik

107QUANTUM COMPUTING APPROACH TO NONRELATIVISTIC AND RELATIVISTIC

MOLECULAR ENERGY CALCULATIONS

By Libor Veis and Jirí Pittner

137DENSITY FUNCTIONAL THEORY AND QUANTUM COMPUTATION

By Frank Gaitan and Franco Nori

151QUANTUM ALGORITHMS FOR CONTINUOUS PROBLEMS AND THEIR APPLICATIONS

By A. Papageorgiou and J. F. Traub

179ANALYTIC TIME EVOLUTION, RANDOM PHASE APPROXIMATION, AND GREEN

FUNCTIONS FOR MATRIX PRODUCT STATES

By Jesse M. Kinder, Claire C. Ralph, and Garnet Kin-Lic Chan

193FEW-QUBIT MAGNETIC RESONANCE QUANTUM INFORMATION PROCESSORS:

SIMULATING CHEMISTRY AND PHYSICS

By Ben Criger, Daniel Park, and Jonathan Baugh

xv

xvi CONTENTS

229PHOTONIC TOOLBOX FOR QUANTUM SIMULATION

By Xiao-Song Ma, Borivoje Dakic, and Philip Walther

241PROGRESS IN COMPENSATING PULSE SEQUENCES FOR QUANTUM COMPUTATION

By J. True Merrill and Kenneth R. Brown

295REVIEW OF DECOHERENCE-FREE SUBSPACES, NOISELESS SUBSYSTEMS,

AND DYNAMICAL DECOUPLING

By Daniel A. Lidar

355FUNCTIONAL SUBSYSTEMS AND STRONG CORRELATION IN PHOTOSYNTHETIC

LIGHT HARVESTING

By David A. Mazziotti and Nolan Skochdopole

371VIBRATIONAL ENERGY TRANSFER THROUGH MOLECULAR CHAINS:

AN APPROACH TOWARD SCALABLE INFORMATION PROCESSING

By C. Gollub, P. von den Hoff, M. Kowalewski, U. Troppmann,and R. de Vivie-Riedle

403ULTRACOLD MOLECULES: THEIR FORMATION AND APPLICATION TO

QUANTUM COMPUTING

By Robin Cote

449DYNAMICS OF ENTANGLEMENT IN ONE- AND TWO-DIMENSIONAL SPIN SYSTEMS

By Gehad Sadiek, Qing Xu, and Sabre Kais

509FROM TOPOLOGICAL QUANTUM FIELD THEORY TO TOPOLOGICAL MATERIALS

By Paul Watts, Graham Kells, and Jirí Vala

567TENSOR NETWORKS FOR ENTANGLEMENT EVOLUTION

By Sebastian Meznaric and Jacob Biamonte

AUTHOR INDEX 581

SUBJECT INDEX 615

INTRODUCTION TO QUANTUM INFORMATIONAND COMPUTATION FOR CHEMISTRY

SABRE KAIS

Department of Chemistry and Physics, Purdue University, 560 Oval Drive,West Lafayette, IN 47907, USA; Qatar Environment & Energy Research Institute

(QEERI), Doha, Qatar; Santa Fe Institute, Santa Fe, NM 87501, USA

I. IntroductionA. Qubits and GatesB. Circuits and AlgorithmsC. Teleportation

II. Quantum SimulationA. IntroductionB. Phase Estimation Algorithm

1. General Formulation2. Implementation of Unitary Transformation U

3. Group Leaders Optimization Algorithm4. Numerical Example5. Simulation of the Water Molecule

III. Algorithm for Solving Linear Systems A�x = �bA. General FormulationB. Numerical Example

IV. Adiabatic Quantum ComputingA. Hamiltonians of n-Particle SystemsB. The Model of Adiabatic ComputationC. Hamiltonian Gadgets

V. Topological Quantum ComputingA. AnyonsB. Non-Abelian Braid GroupsC. Topological Phase of MatterD. Quantum Computation Using Anyons

VI. EntanglementVII. Decoherence

VIII. Major Challenges and OpportunitiesReferences

Advances in Chemical Physics, Volume 154: Quantum Information and Computation for Chemistry,First Edition. Edited by Sabre Kais.© 2014 John Wiley & Sons, Inc. Published 2014 by John Wiley & Sons, Inc.

1

2 SABRE KAIS

I. INTRODUCTION

The development and use of quantum computers for chemical applications hasthe potential for revolutionary impact on the way computing is done in the fu-ture [1–7]. Major challenge opportunities are abundant (see next fifteen chapters).One key example is developing and implementing quantum algorithms for solvingchemical problems thought to be intractable for classical computers. Other chal-lenges include the role of quantum entanglement, coherence, and superpositionin photosynthesis and complex chemical reactions. Theoretical chemists have en-countered and analyzed these quantum effects from the view of physical chemistryfor decades. Therefore, combining results and insights from the quantum infor-mation community with those of the chemical physics community might lead to afresh understanding of important chemical processes. In particular, we will discussthe role of entanglement in photosynthesis, in dissociation of molecules, and in themechanism with which birds determine magnetic north. This chapter is intendedto survey some of the most important recent results in quantum computation andquantum information, with potential applications in quantum chemistry. To startwith, we give a comprehensive overview of the basics of quantum computing(the gate model), followed by introducing quantum simulation, where the phaseestimation algorithm (PEA) plays a key role. Then we demonstrate how PEA com-bined with Hamiltonian simulation and multiplicative inversion can enable us tosolve some types of linear systems of equations described by A�x = �b. Then oursubject turns from gate model quantum computing (GMQC) to adiabatic quantumcomputing (AQC) and topological quantum computing, which have gained in-creasing attention in the recent years due to their rapid progress in both theoreticaland experimental areas. Finally, applications of the concepts of quantum infor-mation theory are usually related to the powerful and counter intuitive quantummechanical effects of superposition, interference, and entanglement.

Throughout history, man has learned to build tools to aid computation. Fromabacuses to digital microprocessors, these tools epitomize the fact that laws ofphysics support computation. Therefore, a natural question arises: “Which physi-cal laws can we use for computation?” For a long period of time, questions suchas this were not considered relevant because computation devices were built ex-clusively based on classical physics. It was not until the 1970s and 1980s whenFeynmann [8], Deutsch [9], Benioff [10], and Bennett [11] proposed the idea ofusing quantum mechanics to perform calculation that the possibility of building aquantum computing device started to gain some attention.

What they conjectured then is what we call today a quantum computer. Aquantum computer is a device that takes direct advantage of quantum mechani-cal phenomena such as superposition and entanglement to perform calculations[12]. Because they compute in ways that classical computers cannot, for certainproblems quantum algorithms provide exponential speedups over their classical

INTRODUCTION TO QUANTUM INFORMATION 3

counterparts. As an example, in solving problems related to factoring large num-bers [13] and simulation of quantum systems [14–28], quantum algorithms areable to find the answer exponentially faster than classical algorithms. Recently,it has also been proposed that a quantum computer can be useful for solving lin-ear systems of equations with exponential speedup over the best-known classicalalgorithms [29]. In the problem of factoring large numbers, the quantum exponen-tial speedup is rooted in the fact that a quantum computer can perform discreteFourier transform exponentially faster than classical computers [12]. Hence, anyalgorithm that involves Fourier transform as a subroutine can potentially be spedup exponentially on a quantum computer. For example, efficient quantum algo-rithms for performing discrete sine and cosine transforms using quantum Fouriertransform have been proposed [30]. To illustrate the tremendous power of theexponential speedup with concrete numbers, consider the following example: theproblem of factoring a 60-digit number takes a classical computer 3 × 1011 years(about 20 times the age of universe) to solve, while a quantum computer can beexpected to factor a 60-digit number within 10−8 seconds. The same order ofspeedup applies for problems of quantum simulation.

In chemistry, the entire field has been striving to solve a number of “Holy Grail”problems since their birth. For example, manipulating matter on the atomic andmolecular scale, economic solar splitting of water, the chemistry of consciousness,and catalysis on demand are all such problems. However, beneath all these prob-lems is one common problem, which can be dubbed as the “Mother of All HolyGrails: exact solution of the Schrodinger equation. Paul Dirac pointed out thatwith the Schrodinger equation, “the underlying physical laws necessary for themathematical theory of a large part of physics and the whole of chemistry are thuscompletely known and the difficulty is only that the exact application of these lawsleads to equations much too complicated to be soluble” [31]. The problem of solv-ing the Schrodinger equation is fundamentally hard [32,33] because as the numberof particles in the system increases, the dimension of the corresponding Hilbertspace increases exponentially, which entails exponential amount of computationalresource.

Faced with the fundamental difficulty of solving the Schrodinger equations ex-actly, modern quantum chemistry is largely an endeavor aimed at finding approx-imate methods. Ab initio methods [34] (Hartree–Fock, Moller–Plesset, coupledcluster, Green’s function, configuration interaction, etc.), semi-empirical methods(extended Huckel, CNDO, INDO, AM1, PM3, etc.), density functional methods[35] (LDA, GGA, hybrid models, etc.), density matrix methods [36], algebraicmethods [37] (Lie groups, Lie algebras, etc.), quantum Monte Carlo methods[38] (variational, diffusion, Green’s function forms, etc.), and dimensional scalingmethods [39] are all products of such effort over the past decades. However, allthe methods devised so far have to face the challenge of unreachable computa-tional requirements as they are extended to higher accuracy to larger systems. For

4 SABRE KAIS

example, in the case of full CI calculation, for N orbitals and m electrons thereare Cm

N ≈ Nm

m! ways to allocate electrons among orbitals. Doing full configurationinteraction (FCI) calculations for methanol (CH3OH) using 6-31G (18 electronsand 50 basis functions) requires about 1017 configurations. This task is impossibleon any current computer. One of the largest FCI calculations reported so far hasabout 109 configurations (1.3 billion configurations for Cr2 molecules [40]).

However, due to exponential speedup promised by quantum computers, suchsimulation can be accomplished within only polynomial amount of time, which isreasonable for most applications. As we will show later, using the phase estimationalgorithm, one is able to calculate eigenvalues of a given Hamiltonian H in timethat is polynomial in O(log N), where N is the size of the Hamiltonian. So inthis sense, quantum computation and quantum information will have enormousimpact on quantum chemistry by enabling quantum chemists and physicists tosolve problems beyond the processing power of classical computers.

The importance of developing quantum computers derives not only from thediscipline of quantum physics and chemistry alone, but also from a wider contextof computer science and the semiconductor electronics industry. Since 1946, theprocessing power of microprocessors has doubled every year simply due to theminiaturization of basic electronic components on a chip. The number of transistorson a single integrated circuit chip doubled every 18 months, which is a fact knownas Moore’s law. This exponential growth in the processing power of classical com-puters has spurred revolutions in every area of science and engineering. However,the trend cannot last forever. In fact, it is projected that by the year 2020 the size ofa transistor would be on the order of a single atom. At that scale, classical lawsof physics no longer hold and the behavior of the circuit components obeys lawsof quantum mechanics, which implies that a new paradigm is needed to exploitthe effects of quantum mechanics to perform computation, or in a more generalsense, information processing tasks. Hence, the mission of quantum computing isto study how information can be processed with quantum mechanical devices aswell as what kinds of tasks beyond the capabilities of classical computers can beperformed efficiently on these devices.

Accompanying the tremendous promises of quantum computers are the exper-imental difficulties of realizing one that truly meets its above-mentioned theoret-ical potential. Despite the ongoing debate on whether building a useful quantumcomputer is possible, no fundamental physical principles are found to prevent aquantum computer from being built. Engineering issues, however, remain. Theimprovement and realization of quantum computers are largely interdisciplinaryefforts. The disciplines that contribute to quantum computing, or more generallyquantum information processing, include quantum physics, mathematics, com-puter science, solid-state device physics, mesoscopic physics, quantum devices,device technology, quantum optics, optical communication, and nuclear magneticresonance (NMR), to name just a few.

INTRODUCTION TO QUANTUM INFORMATION 5

A. Qubits and Gates

In general, we can think of information as something that can be encoded in thestate of a physical system. If the physical system obeys classical laws of physics,such as a classical computer, the information stored there is of “classical” nature.To quantify information, the concept of bit has been introduced and defined as thebasic unit of information. A bit of information stored in a classical computer is avalue 0 or 1 kept in a certain location of the memory unit. The computer is able tomeasure the bit and retrieve the information without changing the state of the bit.If the bit is at the same state every time it is measured, it will yield the same results.A bit can also be copied and one can prepare another bit with the same state. Astring of bits represents one single number.

All these properties of bits seem trivial but in the realm of quantum informationprocessing, this is no longer true (Table I). The basic unit of quantum information isa qubit. Physically, a qubit can be represented by the state of a two-level quantumsystem of various forms, be it an ion with two accessible energy levels or a photonwith two states of polarization. Despite the diverse physical forms that a qubit cantake, for the most part the concept of “qubit” is treated as an abstract mathematicalobject. This abstraction gives us the freedom to construct a general theory ofquantum computation and quantum information that does not depend on a specificsystem for its realization [12].

Unlike classical bits, a qubit can be not only in state |0〉 or |1〉, but also asuperposition of both: α|0〉 + β|1〉. If a qubit is in a state of quantum superposition,a measurement will collapse the state to either one of its component states |0〉 or|1〉, which is a widely observed phenomenon in quantum physics. Suppose werepetitively do the following: prepare a qubit in the same state α|0〉 + β|1〉 andthen measure it with respect to the basis state {|0〉, |1〉}. The measurement outcomeswould most probably be different—we will get |0〉 in some measurements and |1〉in the others—even the state of the qubit that is measured is identical each time.Furthermore, unlike classical bits that can be copied, a qubit cannot be copied due tothe no-cloning theorem, which derives from a qubit’s quantum mechanical nature

TABLE IComparison Between Classical Bits and Qubits

Classical Bit Qubit

State 0 or 1 |0〉, |1〉, or superpositionMeasurement does not change Measurement changes the system

the state of the bitDeterministic result Obtain different results with the same systemCan make a copy of bit (eavesdrop) Cannot clone the qubit (security)One number for a string bit Store several numbers simultaneously due to superposition

6 SABRE KAIS

(see Ref. [12], p. 24 for details). Such no-cloning property of a qubit has been usedfor constructing security communication devices, because a qubit of information isimpossible to eavesdrop. In terms of information storage, since a qubit or an arrayof qubits could be in states of quantum superposition such as α00|00〉 + α01|01〉 +α10|10〉 + α11|11〉, a string of qubits is able to store several numbers α00, α01, . . .

simultaneously, while a classical string of bits can only represent a single number.In this sense, n qubits encode not n bits of classical information, but 2n numbers.In spite of the fact that none of the 2n numbers are efficiently accessible becausea measurement will destroy the state of superposition, this exponentially largeinformation processing space combined with the peculiar mathematical structureof quantum mechanics still implies the formidable potential in the performance ofsome of computational tasks exponentially faster than classical computers.

Now that we have introduced the basic processing units of quantumcomputers—the qubits, the next question is: How do we make them compute?From quantum mechanics we learned that the evolution of any quantum systemmust be unitary. That is, suppose a quantum computation starts with an initialstate |�initial〉, then the final state of the computation |�final〉 must be the result ofa unitary transformation U, which gives |�final〉 = U|�initial〉. In classical com-puting, the basic components of a circuit that transforms a string {0, 1}n to anotherstring {0, 1}m are called gates. Analogously, in quantum computing, a unitarytransformation U that transforms a system from |�initial〉 to |�final〉 can also be de-composed into sequential applications of basic unitary operations called quantum

gates (Table II). Experimentally, the implementation of a quantum gate largelydepends on the device and technique used for representing a qubit. For example,if a qubit is physically represented by the state of a trapped ion, then the quantumgate is executed by an incident laser pulse that perturbs the trapped atom(s) andalters its state; if the qubit states are encoded in the polarization states of photons,then a quantum gate consists of optical components that interact with photons andalter their polarization states as they travel through the components.

If we use vectors to describe the state of a qubit, that is, using |0〉 to represent(1, 0)T and |1〉 to represent (0, 1)T, a single-qubit quantum gate can be represented

TABLE IIComparison Between Classical and Quantum Gates

Classical Logic Gates Quantum Gates

Each gate corresponds to Each quantum gate corresponds to a transformationa mapping {0, 1}m → {0, 1}n |�〉 → |�′〉 or a rotation on the surface

of Bloch sphere ([12], p. 15)Nonunitary UnitaryIrreversible Reversible

INTRODUCTION TO QUANTUM INFORMATION 7

using a 2 × 2 matrix. For example, a quantum NOT gate can be represented bythe Pauli X matrix

UNOT = X =(

0 1

1 0

)(1)

To see how this works, note that X|0〉 = |1〉 and X|1〉 = |0〉. Therefore, thesheer effect of applying X to a qubit is to flip its state from |0〉 to |1〉. This isjust one example of single-qubit gates. Other commonly used gates include theHadamard gate H , Z rotation gate, phase gate S, and π

8 gate T :

H = 1√2

(1 1

1 −1

), S =

(1 0

0 i

), T =

(1 0

0 eiπ/4

)(2)

If a quantum gate involves two qubits, then it is represented by a 4 × 4 matrix.The state of a two-qubit system is generally in form of α00|00〉 + α01|01〉 +α10|10〉 + α11|11〉, which can be written as a vector (α00, α01, α10, α11)T. In matrixform, the CNOT gate is defined as

UCNOT =

⎛⎜⎜⎜⎝

1 0 0 0

0 1 0 0

0 0 0 1

0 0 1 0

⎞⎟⎟⎟⎠ (3)

It is easy to verify that applying CNOT gate to a state |�0〉 = α00|00〉 +α01|01〉 + α10|10〉 + α11|11〉 results in a state

UCNOT|�0〉 = α00|00〉 + α01|01〉 + α10|11〉 + α11|10〉 (4)

Hence, the effect of a CNOT gate is equivalent to a conditional X gate: If thefirst qubit is in |0〉, then the second qubit remains intact; on the other hand, if thefirst qubit is in |1〉, then the second qubit is flipped. Generally, the first qubit iscalled the control and the second is the target.

In classical computing, an arbitrary mapping {0, 1}n → {0, 1}m can be executedby a sequence of basic gates such as AND, OR, NOT, and so on. Similarly inquantum computing, an arbitrary unitary transformation U can also be decomposedas a product of basic quantum gates. A complete set of such basic quantum gates isa universal gate set. For example, Hadamard, phase, CNOT, and π/8 gates forma universal gate set ([12], p. 194).

Now that we have introduced the concepts of qubits and quantum gates andcompared them with their classical counterpart, we can see that they are the verybuilding blocks of a quantum computer. However, it turns out that having qubitsand executable universal gates is not enough for building a truly useful quantum

8 SABRE KAIS

computer that delivers its theoretical promises. So what does it really take to buildsuch a quantum computer? A formal answer to this question is the following sevencriteria proposed by DiVincenzo [41]:

• A scalable physical system with well-characterized qubits.• The ability to initialize the state of the qubits to a simple fiducial state.• Long (relative) decoherence times, much longer than the gate operation time.• A universal set of quantum gates.• A qubit-specific measurement capability.• The ability to inter convert stationary and flying qubits.• The ability to faithfully transmit flying qubits between specified locations.

For a detailed review of state-of-the-art experimental implementation based onthe preceding criteria, refer to Ref. [42]. The take-home message is that it is clearthat we can gain some advantage by storing, transmitting, and processing informa-tion encoded in systems that exhibit unique quantum properties, and a number ofphysical systems are currently being developed for quantum computation. How-ever, it remains unclear which technology, if any, will ultimately prove successfulin building a scalable quantum computer.

B. Circuits and Algorithms

Just as in classical computing, logic gates are cascaded to form a circuit. A quantumcircuit is a sequence of quantum gates. When an algorithm needs to be implementedwith a quantum computer, it must first be translated to a quantum circuit in orderto be executed on the quantum hardware (qubits). Figure 1 is an example of aquantum circuit. Each horizontal line represents a qubit and every box on the lineis a quantum gate applied on that qubit. If the box is connected with a verticalline that joins it to the line(s) with solid circles, then the box is a controlled gateoperation and the qubit(s) that it is joined to are the control qubit(s). Just like aCNOT gate, only when the control qubit (s) is (are all) in |1〉 state will the controlledoperation be applied onto the target qubit.

x3〉 • • H y1〉||

|

|

|

|

x2〉 • H Rπ/2 y2〉

x1〉 H Rπ/2 Rπ/4 y3〉

Figure 1. Quantum circuit for quantum Fourier transform on the quantum state |x1, x2, x3〉.Starting from the left, the first gate is a Hadamard gate H that acts on the qubit in the state |x1〉, andthe second gate is a |x2〉-controlled phase rotation Rθ(a|0〉 + b|1〉) → (a|0〉 + beiθ |1〉) on qubit |x1〉,where θ = π/2. The rest of the circuit can be interpreted in the same fashion.

INTRODUCTION TO QUANTUM INFORMATION 9

C. Teleportation

Quantum teleportation exploits some of the most basic and unique features of quan-tum mechanics, which is quantum entanglement, essentially implies an intriguingproperty that two quantum correlated systems cannot be considered independenteven if they are far apart. The dream of teleportation is to be able to travel bysimply reappearing at some distant location. Teleportation of a quantum state en-compasses the complete transfer of information from one particle to another. Thecomplete specification of a quantum state of a system generally requires an infiniteamount of information, even for simple two-level systems (qubits). Moreover, theprinciples of quantum mechanics dictate that any measurement on a system imme-diately alters its state, while yielding at most one bit of information. The transferof a state from one system to another (by performing measurements on the firstand operations on the second) might therefore appear impossible. However, it wasshown that the property of entanglement in quantum mechanics, in combinationwith classical communication, can be used to teleport quantum states. Althoughteleportation of large objects still remains a fantasy, quantum teleportation hasbecome a laboratory reality for photons, electrons, and atoms [43–52].

More precisely, quantum teleportation is a quantum protocol by which the in-formation on a qubit A is transmitted exactly (in principle) to another qubit B. Thisprotocol requires a conventional communication channel capable of transmittingtwo classical bits, and an entangled pair (B, C) of qubits, with C at the locationof origin with A and B at the destination. The protocol has three steps: measureA and C jointly to yield two classical bits; transmit the two bits to the other endof the channel; and use the two bits to select one of the four ways of recovering B

[53,54].Efficient long-distance quantum teleportation is crucial for quantum commu-

nication and quantum networking schemes. Ursin and coworkers [55] have per-formed a high-fidelity teleportation of photons over a distance of 600 m across theRiver Danube in Vienna, with the optimal efficiency that can be achieved using lin-ear optics. Another exciting experiment in quantum communication has also beendone with one photon that is measured locally at the Canary Island of La Palma,whereas the other is sent over an optical free-space link to Tenerife, where the Op-tical Ground Station of the European Space Agency acts as the receiver [55,56].This exceeds previous free-space experiments by more than an order of magni-tude in distance, and is an essential step toward future satellite-based quantumcommunication.

Recently, we have proposed a scheme for implementing quantum teleportationin a three-electron systems [52]. For more electrons, using Hubbard Hamiltonian,in the limit of the Coulomb repulsion parameter for electrons on the same siteU → +∞, there is no double occupation in the magnetic field; the system isreduced to the Heisenberg model. The neighboring spins will favor the anti parallel

10 SABRE KAIS

configuration for the ground state. If the spin at one end is flipped, then the spins onthe whole chain will be flipped accordingly due to the spin–spin correlation. Suchthat the spins at the two ends of the chain are entangled, a spin entanglement canbe used for quantum teleportation, and the information can be transferred throughthe chain. This might be an exciting new direction for teleportation in molecularchains [57].

II. QUANTUM SIMULATION

A. Introduction

As already mentioned, simulating quantum systems by exact solution of theSchrodinger equation is a fundamentally hard task that the quantum chemistry com-munity has been trying to tackle for decades with only approximate approaches.The key challenges of quantum simulation include the following (see next fivechapters) [28]:

1. Isolate qubits in physical systems. For example, in a photonic quantum com-puter simulating a hydrogen molecule, the logical states |0〉 and |1〉 corre-spond to horizontal |H〉 and vertical |V 〉 polarization states [58].

2. Represent the Hamiltonian H . This is to write H as a sum of Hermitianoperators, each to be converted into unitary gates under the exponentialmap.

3. Prepare the states |ψ〉. By direct mapping, each qubit represents the fermionicoccupation state of a particular orbital. Fock space of the system is mappedonto the Hilbert space of qubits.

4. Extract the energy E.

5. Read out the qubit states.

A technique to accomplish challenge 2 in a robust fashion is presented in Sec-tion II.B.2. Challenge 4 is accomplished using the phase estimation quantum al-gorithm (see details in Section II.B). Here, we can mention some examples ofalgorithms and their corresponding quantum circuits that have been implementedexperimentally: (a) the IBM experiment, which factors the number 15 with nu-clear magnetic resonance (NMR) (for details see Ref. [59]); (b) using quantumcomputers for quantum chemistry [58].

B. Phase Estimation Algorithm

The phase estimation algorithm (PEA) takes advantage of quantum Fourier trans-form ([12], see chapter by Gaitan and Nori) to estimate the phase ϕ in the eigenvaluee2πiϕ of a unitary transformation U. For a detailed description of the algorithm,refer to Ref. [60]. The function that the algorithm serves can be summarized as the

INTRODUCTION TO QUANTUM INFORMATION 11

following: Let |u〉 be an eigenstate of the operator U with eigenvalue e2πiϕ. The al-gorithm starts with a two-register system (a register is simply a group of qubits) inthe state |0〉⊗t|u〉. Suppose the transformation U2k

can be efficiently performed forinteger k, then this algorithm can efficiently obtain the state |ϕ〉|u〉, where |ϕ〉 ac-curately approximates ϕ to t − log(2 + 1

2ε)� bits with probability of at least 1 − ε.

1. General Formulation

The generic quantum circuit for implementing PEA is shown in Fig. 2. Section 5.2of Ref. [12] presents a detailed account of how the circuit functions mathematicallyto yield the state |ϕ〉, which encodes the phase ϕ. Here we focus on its capabilityof finding the eigenvalues of a Hermitian matrix, which is of great importance inquantum chemistry where one often would like to find the energy spectrum of aHamiltonian.

Suppose we let U = eiAt0/2tfor some Hermitian matrix A, then eiAt0 |uj〉 =

eiλjt|uj〉, where λj and |uj〉 are the j-th eigenvalue and eigenvector of matrixA. Furthermore, we replace the initial state |u〉 of register b (Fig. 2) with anarbitrary vector |b〉 that has a decomposition in the basis of the eigenvectors of A:|b〉 = ∑n

j βj|uj〉. Then the major steps of the algorithm can be summarized as thefollowing.

1. Transform the t-qubit register C (Fig. 2) from |0〉⊗t to 1√2t

∑2t−1τ=0 |τ〉 state

by applying Hadamard transform on each qubit in register C.

2. Apply the U2kgates to the register b, where each U2k

gate is controlledby the (k − 1)th qubit of the register C from bottom. This series of con-trolled operations transforms the state of the two-register system from

1√2t

∑2t−1τ=0 |τ〉 ⊗ |b〉 to 1√

2t

∑2t−1τ=0 |τ〉 ∑n

j=1 eiλjτt/2tβj|uj〉.

3. Apply inverse Fourier transform FT† to the register C. Because every basisstate |τ〉 will be transformed to 1√

2t

∑2t−1k=0 e−2πiτk/2t |k〉 by FT†, the final

|0〉 H . . . •

†

.........

Reg. C FT|0〉 H • . . . |ϕ〉

|0〉 H • . . .

|0〉 H • . . .

Reg. b |u〉 / U20U21

U22 . . . U2t−1 |u〉

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎭

Figure 2. Schematic of the quantum circuit for phase estimation. The quantum wire with a “/”symbol represents a register of qubits as a whole. FT† represents inverse Fourier transform, whosecircuit is fairly standard ([12], see chapter by Gaitan and Nori).

12 SABRE KAIS

state of the PEA is proportional to∑2t−1

k=0∑n

j=1 ei(λjt0−2πk)τ/2tβj|k〉|uj〉.

Due to a well-known property of the exponential sum, in which sums ofthe form

∑N−1k=0 exp(2πik r

N) vanish unless r = 0 mod N, the values of k

are concentrated on those whose value is equal or close to t02π

λj . If we lett0 = 2π, the final state of system is

∑j βj|λj〉|uj〉 up to a normalization

constant.

In particular, if we prepare the initial state of register b to be one of matrixA’s eigenvector |ui〉, according to the procedure listed above, the final state ofthe system will become |λi〉|ui〉 up to a constant. Hence, for any |ui〉 that we canprepare, we can find the eigenvalue λj of A corresponding to |ui〉 using a quantumcomputer. Most importantly, it has been shown that [17] quantum computers areable to solve the eigenvalue problem significantly more efficiently than classicalcomputers.

2. Implementation of Unitary Transformation U

Phase estimation algorithm is often referred to as a black box algorithm becauseit assumes that the unitary transformation U and its arbitrary powers can beimplemented with basic quantum gates. However, in many cases U has a structurethat renders finding the exact decomposition U = U1U2...Um either impossible orvery difficult. Therefore, we need a robust method for finding approximate circuitdecompositions of unitary operators U with minimum cost and minimum fidelityerror.

Inspired by the optimization nature of the circuit decomposition problem,Daskin and Kais [61,62] have developed an algorithm based on group leaderoptimization technique for finding a circuit decomposition U = U1U2...Um withminimum gate cost and fidelity error for a particular U. Hence, there are two fac-tors that need to be optimized within the optimization: the error and the cost ofthe circuit. The costs of a one-qubit gate and a control gate (two-qubit gate) aredefined as 1 and 2, respectively. Based on these two definitions, the costs of otherquantum gates can be deduced. In general, the minimization of the error to an ac-ceptable level is more important than the cost in order to get more reliable resultsin the optimization process. The circuit decompositions for U = eiAt presented inFig. 3b for the particular instance of A in Eq. (6) are found by the algorithm suchthat the error ||U ′ − U|| and the cost of U ′ are both minimized.

3. Group Leaders Optimization Algorithm

The group leaders optimization algorithm (GLOA) described in more detail inRefs [61,62] is a simple and effective global optimization algorithm that models theinfluence of leaders in social groups as an optimization tool. The algorithm startswith dividing the randomly generated solution population into several disjunctgroups and assigning for each group a leader (the best candidate solution inside