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Information and Computation: Classical and Quantum Aspects

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Information and Computation: Classical and Quantum Aspects Information and Computation: Classical and Quantum Aspects A. Galindo† and M.A. Mart´ın-Delgado‡ Departamento de F´ısica T´eorica I. Facultad de Ciencias F´ısicas. Universidad Complutense. 28040 Madrid. Spain. Quantum theory has found a new field of applications in the realm of information and computation during the recent years. This paper reviews how quantum physics allows information coding in classically unexpected and subtle nonlocal ways, as well as information processing with an efficiency largely surpassing that of the present and foreseeable classical computers. Some outstanding aspects of classical and quantum information theory will be addressed here. Quantum teleportation, dense coding, and quantum cryptography are discussed as a few samples of the impact of quanta in the transmission of information. Quantum logic gates and quantum algorithms are also discussed as instances of the improvement in information processing by a quantum computer. We provide finally some examples of current experimental realizations for quantum computers and future prospects. PACS numbers: 03.67.-a, 03.67.Lx CONTENTS A.The Ion-Trap QC 57 B.NMR Liquids: Quantum Ensemble Computation 61 I.Introduction 1 C.Solid-State Quantum Computers 66 II.Classical Information 2 XII.Conclusions 71 A.The Theorems of Shannon 2 Acknowledgments 71 B.Classical Error Correction 4 List of Symbols and Acronyms 71 III.Quantum Information 6 Appendix: Computational Complexity 72 A.Entanglement and Information 8 A.Classical Complexity Classes 72 B.Quantum Coding and Schumacher’s Theorem 10 B.Quantum Complexity Classes 74 C.Capacities of a Quantum Channel 10 References 74 D.Quantum Error Correction 11 E.Entanglement Distillation 13 IV.Quantum Teleportation 14 I. INTRODUCTION V.Dense Coding 16 VI.Cryptography 17 A.Classical Cryptography 17 The twentieth century we have just left behind opened B.Quantum Cryptography 20 with the discovery of quanta by Planck (1900) and fol- C.Practical Implementation of QKD 22 lowed with the formulation of the quantum theory during VII.Quantum Computation 23 the first decades. As the century went by, we have wit- VIII.Classical Computers 23 A.The Turing Machine 23 nessed a continuous and growing increase in the number B.The von Neumann Machine 28 of applications of quantum mechanics, which began with C.Classical Parallelism 28 atomic physics and then the number kept growing (nu- D.Classical Logic Gates and Circuits 30 clearandparticlephysics,optics,condensedmatter,...) IX.Principles of Quantum Computation 32 and became countless. As the century was closing we A.The Quantum Turing Machine 33 B.Quantum Logic Gates 36 have come across an unexpected new field of applications C.Quantum Circuits 39 that have given quantum physics a refreshing twist, keep- X.Quantum Algorithms 44 ing the pace even with the newest trends of discoveries, A.Deutsch-Jozsa Algorithm 45 such as the field of new technologies of information and B.Simon Algorithm 46 computation. In a sense and having in mind the times we C.Grover Algorithm 47 D.Shor Algorithm 51 live, those of the information era and the new technolo- E.On the Classification of Algorithms 55 gies, it seems inevitable that physics gets affected by the XI.Experimental Proposals of Quantum Computers 56 presence of computers all over around, which are more and more powerful and have revolutionized many areas of science. What is more surprising is the fact that quan- tum physics may influence the field of information and †Electronic address: [email protected] computation in a new and profound way, getting at the ‡Electronic address: [email protected] very root of their foundations. For instance, fundamental aspects of quantum mechanics such as those entering the scribing the notion of a quantum Turing machine and its EPR (Einstein, Podolsky and Rosen, 1935) states have practical implementation with quantum circuits. We de- found unexpected applications in information transmis- scribe the notion of elementary quantum gates for univer- sion and cryptography. sal computation and how this extends the classical coun- But, why has this happened? It all begun by realizing terpart. We also provide a discussion of the basic quan- that information has physical nature (Landauer, 1991; tum algorithms and finally we give a general overview 1996; 1961). It is printed on a physical support (the of some of the possible physical realizations of quantum rocky wall of a cave, a clay tablet, a parchment, a sheet computers. of paper, a magneto-optic disk, etc.), it cannot be trans- Both in the information and computation parts we mitted faster than light in vacuum, and it abides by the make special emphasis in presenting first an introduc- natural laws. The statement that information is physical tion to the classical aspects of these disciplines in order does not simply mean that a computer is a physical ob- to better clarify what quantum theory adds to them in ject, but in addition that information itself is a physical the new formulations of these theories. Actually, this is entity. In turn, this implies that the laws of informa- also what we do in physics. tion are restricted or governed by the laws of physics. In particular, those of quantum physics. In fact these ones, through their linearity, entanglement of states, nonlocal- II. CLASSICAL INFORMATION ity and indetermination principle make possible new and powerful transmission tools and information treatments, Information is discretized: it comes in irreducible pack- as well as a really prodigious efficiency of computation. ages. The elementary unit of classical information is the A typical computation is implemented through an al- bit (or cbit, for classic bit), a classical system with only gorithm in a computer. This algorithm is now regarded twostates0and1(FalseandTrue,NoandYes,...). Any as a set of physical operations and the registers of the text can be coded into a string of bits: for instance, it quantum computer are considered to be states of a quan- is enough to assign to each symbol its ASCII code num- tum system. Moreover, the familiar operation of initial- ber in binary form, appended with a parity check bit. izing the data for a program to run is replaced by the Example: quanta can be coded as preparation of an initial quantum state, and the usual tasks of writing programs and running them correspond, 11100010 11101011 11000011 11011101 11101000 11000011 in the new formulation, to finding appropriate Hamilto- Each bit can be stored physically; in classical comput- nians for their time evolution operators to lead to the ers, each bit is registered as a charge state of a capacitor desired output. This output is retrieved by a quantum (0 = discharged, 1 = charged). They are distinguishable measurement of the register, and this fact has deep im- macroscopic states, and robust enough or stable. They plications on the way quantum information must be han- are not spoiled when they are read in (if carefully done) dled. and they can be cloned or replicated without any prob- We shall see that information and computation blend lem. well with quantum mechanics. Their combination brings Information is not only stored; it is usually transmit- unexpected results on the way information can be trans- ted (communication), and sometimes processed (compu- mitted and processed, extending the capabilities known tation). so far in the field of classical information to unsuspected limits, sometimes entering the realm of science-fiction, sometimes surpassing it. A. The Theorems of Shannon The advance has been remarkable mainly in the field of cryptography, where it has provided systems absolutely The classical theory of information is due to Shannon secure for the quantum distribution of keys. Quantum (1948,1949), who in two seminal works definitively laid computation is also one of the hot research fields in cur- down its principles in 1948. With his celebrated noise- rent physics; the same applies to the challenge posed less coding theorem he showed how much compressible a by the experimental realization of a computer complex message can be, or equivalently, how much redundancy enough to implement the new algorithms that exploit it has. Likewise with his coding theorem in a noisy chan- the fantastic possibilities of the massive parallelism char- nel he also found what is the minimum redundancy that acterizing those quantum computers, and that would must be present into a message in order to be compre- amount to a dramatic improvement for solving hard or hensible when reaching the receiver, despite of the noise. classically untractable problems. Let A := a1, ..., a A be a finite alphabet, endowed We first review the essentials of quantum information { | |} with a probability distribution pA : ai pA(ai), with theory and then discuss several of their consequences and p (a ) = 1. Sometimes we shall7→ be write this applications, some of them specifically quantum such as 1 i A A i ≤ ≤| | A quantum teleportation, dense coding; some of them with Pas A := ai,pA(ai) i|=1| . Let us consider messages or { } n a classical echo such as quantum cryptography. Next character strings x1x2...xn A , originating from a ∈ we review the fundamentals of quantum computation de- memoryless source, i.e., a symbol a appears in a given place with probability pA(a), independently of the sym- Let X be the alphabet of the transmitter station (of bols entering the remaining sites in the chain.1 The first a memoryless source), and Y be the one of the receiver Shannon’s theorem asserts that, if n 1, the informa- station. Let (pY X (yj xi)) be the stochastic matrix for tion supplied by a generic message ofn characters (and that channel, with| entries| given by the probabilities that thus (n log2 A )-bits long) essentially coincides with that the input symbol xi X appears as yi Y on out- transmitted| by| another shorter message, of bit length put.
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