To Vera Rae
3 Contents
1 Preliminaries 14 1.1 Elements of Linear Algebra ...... 17 1.2 Hilbert Spaces and Dirac Notations ...... 23 1.3 Hermitian and Unitary Operators; Projectors...... 27 1.4 Postulates of Quantum Mechanics ...... 34 1.5 Quantum State Postulate ...... 36 1.6 Dynamics Postulate ...... 42 1.7 Measurement Postulate ...... 47 1.8 Linear Algebra and Systems Dynamics ...... 50 1.9 Symmetry and Dynamic Evolution ...... 52 1.10 Uncertainty Principle; Minimum Uncertainty States ...... 54 1.11 Pure and Mixed Quantum States ...... 55 1.12 Entanglement; Bell States ...... 57 1.13 Quantum Information ...... 59 1.14 Physical Realization of Quantum Information Processing Systems ...... 65 1.15 Universal Computers; The Circuit Model of Computation ...... 68 1.16 Quantum Gates, Circuits, and Quantum Computers ...... 74 1.17 Universality of Quantum Gates; Solovay-Kitaev Theorem ...... 79 1.18 Quantum Computational Models and Quantum Algorithms ...... 82 1.19 Deutsch, Deutsch-Jozsa, Bernstein-Vazirani, and Simon Oracles ...... 89 1.20 Quantum Phase Estimation ...... 96 1.21 Walsh-Hadamard and Quantum Fourier Transforms ...... 102 1.22 Quantum Parallelism and Reversible Computing ...... 107 1.23 Grover Search Algorithm ...... 111 1.24 Amplitude Amplification and Fixed-Point Quantum Search ...... 123 1.25 Error Models and Quantum Algorithms ...... 130 1.26 History Notes ...... 132 1.27 Summary ...... 136 1.28 Exercises and Problems ...... 137
2 Measurements and Quantum Information 142 2.1 Measurements and Physical Reality ...... 144 2.2 Copenhagen Interpretation of Quantum Mechanics ...... 147 2.3 Mixed States and the Density Operator ...... 149 2.4 Purification of Mixed States ...... 156 2.5 Born Rule ...... 158 2.6 Measurement Operators ...... 159 2.7 Projective Measurements ...... 161 2.8 Positive Operator Valued Measures (POVM) ...... 164 2.9 Neumark Theorem ...... 167 2.10 Gleason Theorem ...... 168 2.11 Mixed Ensembles and their Time Evolution ...... 172 2.12 Bipartite Systems; Schmidt Decomposition ...... 174
4 2.13 Measurements of Bipartite Systems ...... 176 2.14 Operator-Sum (Kraus) Representation ...... 182 2.15 Entanglement; Monogamy of Entanglement ...... 185 2.16 Einstein-Podolski-Rosen (EPR) Thought Experiment ...... 189 2.17 Hidden Variables ...... 193 2.18 Bell and CHSH Inequalities ...... 199 2.19 Violation of Bell Inequality ...... 202 2.20 Entanglement and Hidden Variables ...... 206 2.21 Quantum and Classical Correlations ...... 208 2.22 Measurements and Quantum Circuits ...... 210 2.23 Measurements and Ancilla Qubits ...... 214 2.24 Measurements and Distinguishability of Quantum States ...... 217 2.25 Measurements and an Axiomatic Quantum Theory ...... 221 2.26 History Notes ...... 223 2.27 Summary and Further Readings ...... 225 2.28 Exercises and Problems ...... 228
3 Classical and Quantum Information Theory 230 3.1 The Physical Support of Information ...... 233 3.2 Entropy; Thermodynamic Entropy ...... 236 3.3 Shannon Entropy ...... 241 3.4 Shannon Source Coding ...... 251 3.5 Mutual Information; Relative Entropy ...... 255 3.6 Fano Inequality; Data Processing Inequality ...... 259 3.7 Classical Information Transmission through Discrete Channels ...... 261 3.8 Trace Distance and Fidelity ...... 267 3.9 von Neumann Entropy ...... 269 3.10 Joint, Conditional, and Relative von Neumann Entropy ...... 274 3.11 Trace Distance and Fidelity of Mixed Quantum States ...... 275 3.12 Accessible Information in a Quantum Measurement; Holevo Bound ...... 282 3.13 No Broadcasting Theorem for General Mixed States ...... 292 3.14 Schumacher Compression ...... 295 3.15 Quantum Channels ...... 297 3.16 Quantum Erasure ...... 301 3.17 Classical Information Capacity of Noiseless Quantum Channels ...... 306 3.18 Entropy Exchange, Entanglement Fidelity, and Coherent Information...... 312 3.19 Quantum Fano and Data Processing Inequalities ...... 318 3.20 Reversible Extraction of Classical Information from Quantum Information . . 322 3.21 Noisy Quantum Channels ...... 324 3.22 Holevo-Schumacher-Westmoreland Noisy Quantum Channel Encoding Theorem 329 3.23 Capacity of Noisy Quantum Channels ...... 334 3.24 Entanglement-Assisted Capacity of Quantum Channels ...... 338 3.25 Additivity and Quantum Channel Capacity ...... 342 3.26 Applications of Information Theory ...... 345 3.27 History Notes ...... 347
5 3.28 Summary and Further Readings ...... 348 3.29 Exercises and Problems ...... 351
4 Classical Error Correcting Codes 355 4.1 Informal Introduction to Error Detection and Error Correction ...... 357 4.2 Block Codes. Decoding Policies ...... 359 4.3 Error Correcting and Detecting Capabilities of a Block Code ...... 363 4.4 Algebraic Structures and Coding Theory ...... 366 4.5 Linear Codes ...... 375 4.6 Syndrome and Standard Array Decoding of Linear Codes ...... 383 4.7 Hamming, Singleton, Gilbert-Varshamov, and Plotkin Bounds ...... 387 4.8 Hamming Codes ...... 392 4.9 Proper Ordering, and the Fast Walsh-Hadamard Transform ...... 394 4.10 Reed-Muller Codes ...... 400 4.11 Cyclic Codes ...... 405 4.12 Encoding and Decoding Cyclic Codes ...... 410 4.13 The Minimum Distance of a Cyclic Code; BCH Bound ...... 421 4.14 Burst Errors. Interleaving ...... 424 4.15 Reed-Solomon Codes ...... 427 4.16 Convolutional Codes ...... 438 4.17 Product Codes ...... 445 4.18 Serially Concatenated Codes and Decoding Complexity ...... 446 4.19 Parallel Concatenated Codes - Turbo Codes ...... 449 4.20 History Notes ...... 453 4.21 Summary and Further Readings ...... 454 4.22 Exercises and Problems ...... 457
5 Quantum Error Correcting Codes 461 5.1 Quantum Error Correction ...... 463 5.2 A Necessary Condition for the Existence of a Quantum Code ...... 468 5.3 Quantum Hamming Bound ...... 469 5.4 Scale-up and Slow-down ...... 470 5.5 A Repetitive Quantum Code for a Single Bit-flip Error ...... 471 5.6 A Repetitive Quantum Code for a Single Phase-flip Error ...... 478 5.7 The Nine Qubit Error Correcting Code of Shor ...... 483 5.8 The Seven Qubit Error Correcting Code of Steane ...... 485 5.9 An Inequality for Representations in Different Bases ...... 490 5.10 Calderbank-Shor-Steane (CSS) Codes ...... 494 5.11 The Pauli Group ...... 500 5.12 Stabilizer Codes ...... 503 5.13 Stabilizers for Perfect Quantum Codes ...... 512 5.14 Quantum Restoration Circuits ...... 515 5.15 Quantum Codes over GF (pk) ...... 518 5.16 Quantum Reed-Solomon Codes ...... 521 5.17 Concatenated Quantum Codes ...... 527
6 5.18 Quantum Convolutional and Quantum Tail-Biting Codes ...... 528 5.19 Correction of Time-Correlated Quantum Errors ...... 538 5.20 Quantum Error Correcting Codes as Subsystems ...... 541 5.21 Bacon-Shor Code ...... 544 5.22 Operator Quantum Error Correction ...... 549 5.23 Stabilizers for Operator Quantum Error Correction ...... 553 5.24 Correction of Systematic Errors Based on Fixed-Point Quantum Search ....555 5.25 Reliable Quantum Gates and Quantum Error Correction ...... 557 5.26 History Notes ...... 560 5.27 Summary and Further Readings ...... 560 5.28 Exercises and Problems ...... 562
6 Physical Realization of Quantum Information Processing Systems 565 6.1 Requirements for Physical Implementations of Quantum Computers ...... 567 6.2 Cold Ion Traps ...... 573 6.3 First Experimental Demonstration of a Quantum Logic Gate ...... 583 6.4 Trapped Ions in Thermal Motion ...... 588 6.5 Entanglement of Qubits in Ion Traps ...... 590 6.6 Nuclear Magnetic Resonance - Ensemble Quantum Computing ...... 596 6.7 Liquid-State NMR Quantum Computer ...... 598 6.8 NMR Implementation of Single-Qubit Gates ...... 605 6.9 NMR Implementation of Two-Qubit Gates ...... 606 6.10 The First Generation NMR Computer ...... 612 6.11 Quantum Dots ...... 614 6.12 Fabrication of Quantum Dots ...... 621 6.13 Quantum Dot Electron Spins and Cavity QED ...... 624 6.14 Quantum Hall Effect ...... 628 6.15 Fractional Quantum Hall Effect ...... 631 6.16 Alternative Physical Realizations of Topological Quantum Computers .....641 6.17 Photonic Qubits ...... 643 6.18 Summary and Further Readings ...... 649
7 Appendix. Observable Algebras and Channels 652
8 Glossary 688
7 “I want to know God’s thoughts... the rest are details. ” Albert Einstein.
Preface
A new discipline, Quantum Information Science, has emerged in the last two decades of the twentieth century at the intersection of Physics, Mathematics, and Computer Science. Quan- tum Information Processing (QIP) is an application of Quantum Information Science which covers the transformation, storage, and transmission of quantum information; it represents a revolutionary approach to information processing We have witnessed the development of microprocessors, high-speed optical communication, high-density storage technologies, followed by the widespread use of sensors, and more recently multi- and many-core processors and spintronics technology. We are now able to collect humongous amounts of information, process the information at high speeds, transmit the information through high-bandwidth and low-latency channels, store it on digital media, and share it using numerous applications built around the World Wide Web. Thus, the full cycle at the heart of information revolution was closed, Figure 1 [285], and this revolution became a reality that profoundly affects our daily life. Now, at the beginning of the twenty first century, information processing is facing new challenges: heat dissipation, leakage, and other physical phenomena limit our ability to build
8 SENSORS DIGITAL CAMERAS (2000s)
WORLD WIDE WEB (1990s) MICROPROCESSORS (1980s) GOOGLE, YouTube (2000s) MULTI-CORE MICROPROCESSORS (2000s)
COLLECT
MILESTONES IN INFORMATION PROCESSING
BOOLEAN ALGEBRA (1854) DISSEMINATE DIGITAL COMPUTERS (1940s) PROCESS INFORMATION THEORY (1948)
Quantum Computing Quantum Information Theory
COMMUNICATE STORE
FIBER OPTICS (1990s) WIRELESS (2000s) OPTICAL STORAGE HIGH DENSITY SOLID-STATE (1990s) SPINTRONICS (2000s)
Figure 1: Our ability to collect, process, store, communicate, and disseminate information has increased considerably during the last two decades of the twentieth century. 1980s was the decade of microprocessors; advances in solid state technologies allowed the increase of the number of transistors on a chip by three order of magnitude and a substantial reduction of the cost of a microprocessor. In 1990s we have seen major breakthroughs in optical storage, high density solid-state storage technologies, fiber optics communication, and the widespread acceptance of the Word Wide Web. The first decade of the twenty first century is the decade of sensors, rapid information dissemination, and multi-core microprocessors. increasingly faster and, implicitly, increasingly smaller solid-state devices; it is very difficult to ensure the security of our communication; we are overwhelmed by the volume of information we are bombarded with, and it is increasingly more difficult to extract useful information from the vast ocean of information surrounding us. Information, either classical or quantum, is physical; this is the mantra repeated through- out the book. Therefore, we must understand the physical processes that affect the state
9 of the systems used to carry information. The physical processes for the storage, transfor- mation, and transport of classical information are governed by the laws of classical Physics which limit our ability to process information increasingly faster using present day solid-state technologies. The speed of charge carriers in semiconductors is finite; to increase the speed of the device we have to pack the logic gates as tightly as possible. The heat dissipated by a device increases with the clock rate to the power of 2 or 3, depending upon the solid-state technology. Heat removal is a hard problem for densely packed devices; the heat produced by a solid-state device is proportional to the number of gates thus, to the volume of the device. If we pack the gates into a sphere, the heat dissipated is proportional to the volume of the sphere and can be removed through the surface of the sphere; while the amount of the heat increases as the cube of the radius, our ability to remove it only increases as the square of the radius of the sphere. We are thus limited in our ability to increase the speed and density of classical circuits. These facts provide a serious motivation to search for alternative physical realization of computing and communication systems. Scientists are now exploring revolutionary means to overcoming the limitations of computing and communication systems based upon the laws of classical Physics. Quantum and biological information processing provide a glimpse of hope in overcoming some of the limitations we mentioned and could revolutionize computing and communication in the third millennium. DNA computing together with quantum computing and quantum communication are the most promising avenues explored nowadays. While a significant progress has been made in understanding the properties of quantum information, fundamental questions regarding biological information are still waiting for answers. For example,how to explain the semantic aspect of biological information; how is information from a damaged region of the brain recovered? Quantum information is information stored as a property of a quantum system e.g., the polarization of a photon, or the spin of an electron. Quantum information can be transmitted, stored, and processed following the laws of Quantum Mechanics. Several physical embodi- ments of quantum information are possible; for example, quantum communication involves a source that supplies quantum systems in a given state, a noisy channel that “transports” the quantum system, and the recipient that receives and decodes the quantum information. The source could be a laser producing monochromatic photons, the channel could be an op- tical fiber and the recipient a photocell; the source could also be an ion trap controlled by laser pulses, the channel a series of trapped ions, and the receiver a photo detector reading out the state of the ions via laser-induced fluorescence [276]. The diversity of the processes and technologies to process quantum information gives us hope that practical applications of quantum information will emerge sooner rather than later. The physical processes for photonic, ion-traps, quantum dots, NMR, and other quantum systems are very different and could distract us from the goal of discovering the common properties of quantum information independent of its physical support. To study the proper- ties of quantum information we use an abstract model which captures the critical aspects of quantum behavior; this model, Quantum Mechanics, describes the properties of physical sys- tems as entities in a finite-dimensional Hilbert space. Therefore, quantum information theory requires a basic understanding of Quantum Mechanics and familiarity with the mathematical apparatus used by Quantum Mechanics and information theory. Quantum information has special properties: the state of a quantum system cannot be
10 measured or copied without disturbing it; the quantum state of two systems can be entangled, the two-system ensemble has a definite state, though neither individual system has a well defined state of its own; we cannot reliably distinguish non-orthogonal states of a quantum system. Charles Bennett noted that “Speaking metaphorically, quantum information is like the information in a dream: attempting to describe your dream to someone else changes your memory of it, so you begin to forget the dream and remember only what you said about it.” [49]. The properties of quantum information are remarkable and could be exploited for infor- mation processing: in quantum computing systems an exponential increase in parallelism requires only a linear increase in the amount of space needed thus, in principle, a quan- tum computer will be able to solve problems that cannot be solved with today’s computers; reversible quantum computers avoid logically irreversible operations and can, in principle, dis- sipate arbitrarily little energy for each logic operation. Quantum information theory allows us to design algorithms for quantum key distribution and for quantum teleportation. Eaves- dropping on a quantum communication channel can be detected with very high probability. Decoherence, the randomization of the internal state of a quantum computer due to interac- tions with the environment, is a major problem in quantum information processing; quantum computers rely on undisturbed evolution of quantum coherence. Quantum error correction allows reliable communication over noisy quantum channels, provided that the channels are not too noisy. We should caution the reader that the complexity of the circuits involved in quantum error correction is far beyond today’s technological possibilities; a fault-tolerant implementation of Shor’s quantum factoring algorithm would most likely require thousands of physical qubits, at least two orders of magnitude more qubits than the systems reported in the literature have been able to harness. It may be possible though to resort to techniques which exploit the specific properties of individual physical realizations of quantum devices to manage the complexity of the quantum circuits for fault-tolerant systems. Fault-tolerant quantum computing still requires many more years of research. Quantum information processing involves several areas including: quantum algorithms, quantum complexity theory, quantum information theory, quantum error correcting codes, quantum cryptography, and quantum reliability. This book covers basic concepts in quantum computing, quantum information theory, and quantum error correcting codes. Classical information theory is a mathematical model for the transfer, storage and process- ing of information based on the laws of classical Physics. In the late 1940s Claude Shannon proved that it is possible to reliably transmit information over noisy classical communication channels; this discovery triggered the search for classical error correcting codes and the first codes were discovered by Richard Hamming in the early 1950s. Error correction is a critical component of modern technologies for reliable transfer, storage and processing of classical information. Quantum information theory (QIT) combines classical information theory with Quantum Mechanics to model information-related processes in quantum systems. The foun- dations of quantum information theory were established in the late 1980s by Charles Bennett and others and the interest in quantum information increased dramatically in mid 1990s after Peter Shor and Andrew Steane showed that quantum error correction is feasible and, together with Robert Calderbank, demonstrated that good quantum error correcting codes exist. New discoveries add to the excitement of quantum information science: topological quan- tum computing proposed by Kitaev in 1997 and further developed by Friedman, Kitaev,
11 Larsen, and Wang has the potential to revolutionize fault-tolerance; in 2005 Grover discov- ered the fixed-point quantum search. In 2008 Smith and Yard showed that communication is possible over zero capacity quantum channels and in 2009 Hastings provided an answer to one of the most important open question in quantum information theory showing that the minimum entropy output of a quantum communication channel is not additive. In 1999 Knill, Laflamme, and Viola reformulated quantum error correction and proposed to view quantum error correcting codes as subsystems where the information resides in noiseless subspaces rather than considering a quantum code a subspace of a larger Hilbert space; in 2004 Kribs, Laflamme, and Poulin proposed a unified approach to quantum error correction and extended the concept of noiseless subsystems and their work led to the introduction of operator quan- tum error subsystems. These theoretical developments are mirrored by advances in quantum communication e.g., applications of Quantum Cryptography are close to commercialization. The book organization is summarized in Figure 2; we first discuss classical concepts and then, gradually, we move to the corresponding concepts for quantum information. We adopted this philosophy for several reasons. First, the classical concepts are easier to grasp. Natural sciences develop increasingly more accurate and, at the same time, more complex models of physical reality; the level of abstraction makes it harder to develop the intuition behind the formalism and it is more difficult to master the mathematical apparatus the models are based on. The second reason why we discuss first the classical concepts is because the targeted audience for this book are not physicists familiar with Quantum Mechanics, but the larger population of scientists, engineers, students, or ordinary people puzzled by the “strange” properties of quantum information. Some of them are familiar with the classical information theory concepts and with classical error correcting codes; for them the significant leap is to transpose their intuition, and knowledge to a different frame of reference. We follow the same philosophy in the presentation of quantum algorithms; we analyze first quantum oracles, the easier to understand algorithms for “toy” problems proposed by Deutsch, Jozsa, Bernstein and Vazirani, and Simon followed by an in depth analysis of phase estimation and of Grover search algorithm. The chapter covering information theory starts with the thermodynamic and Shannon entropy and classical channels and then we introduce von Neumann entropy and quantum channels. We discuss first linear codes and gradually move to more sophisticated cyclic, convolutional, and other families of classical codes; sim- ilarly, we analyze first the Shor, Steane, and CSS quantum error correcting codes before introducing stabilizer and subsystem codes. We hope that the numerous examples will facil- itate the understanding of the more abstract concepts introduced throughout the book and will make the book accessible to a larger audience. Whenever possible we use the traditional notations in the literature or in the original papers which introduced the basic concepts. This required a careful selection of characters and fonts; for example, an 2n-dimensional Hilbert space is denoted as H2n , Shannon entropy is H, the parity check matrix of a code is H the Hadamard transform is H and the transfer matrix of a Hadamard gate is H. The authors are indebted to several colleagues who have read the manuscript and have made many constructive suggestions. Among them special thanks are due to Professors Dan Burghelea from the Mathematics Department at Ohio State University, Eduardo Mucciolo from the Physics Department, and Pawel Wocjan from the Computer Science Department at University of Central Florida. Of course, the authors are responsible for the errors that, in spite of our efforts, may still be found in the text.
12 Chapter 1 Chapter 2 Preliminaries Measurements
Mathematical Foundations von Neumann, POVM, Measurements
Quantum Mechanics Mixed States & Bipartite Concepts Systems
Quantum Gates, Circuits, Entanglement, EPR, Quantum Computers Bell & CSHS Inequalities
Quantum Algorithms Measurements of Quantum Circuits
Chapter 3 Chapter 4 Chapter 5 Information Theory Classical ECC Quantum ECC Shannon Entropy and Shor, Steane, Block Codes Coding CSS Codes
von Neumann Entropy Linear Codes. Bounds Stabilizer Codes
Noiseless Quantum RS, Concatenated & Cyclic Codes Channels Convolutional Codes
Noisy Quantum Convolutional, Product & Subsystem Codes Channels Concatenated Codes
Chapter 6 Physical Realization
Ion Traps Anyons Requirements Nuclear Magnetic Resonance Photons Quantum Dots
Figure 2: Book organization at a glance.
The artwork was created by George Dima, a gifted artist, concertmaster of the Bucharest Philharmonic and accomplished creator of computer-generated graphics (see http://picasaweb.google.com/degefe2008). We express our thanks to Patricia Osborne, Gavin Becker, and the editorial staff from Elsevier for their constructive suggestions.
13 “No amount of experiments can ever prove me right; a single experiment can prove me wrong.” Albert Einstein.
1 Preliminaries
What is information? Carl Friederich von Weizs¨acker’s answer, information is what is understood, implies that information has a sender and a receiver who have a common under- standing of the representation and the means to convey information using some properties of the physical systems [447]. He adds, “Information has no absolute meaning; it exists relatively between two semantic levels” [448]. Once asked the question what is time, Richard Feynman answered: “time is what happens when nothing else happens.” Unfortunately, history did not record Feynman’s answer to the question “what is information” and thus we do not have a crisp, witty, and insightful answer to a question central to the 21st century science. Indeed, the questions what is information and what is its relationship with the physical world become more important as we try to better understand physical phenomena at quantum scale and the behavior of biological systems. It is easy to understand why there is no simple answer to the question we posed at the very beginning of this section; like matter and energy, information is a primitive concept thus,
14 it is rather difficult to rigorously define it. Informally, we can state that information abstracts properties of and allows us to distinguish among objects/entities/phenomena/thoughts; infor- mation is a common denominator for the very diverse contents of our material and spiritual world. There is a common expression of information as strings of bits, regardless of the ob- jects/entities/processes/thoughts it describes. Moreover, these bits are independent of their physical embodiment. Information can be expressed using pebbles on the beach, mechanical relays, electronic circuits, and even atomic and subatomic particles. Classical information is information encoded to some property of a physical system obeying the laws of classical Physics. Classical information is transformed using logic operations. Classical gates implement logic operations and allow for processing of classical information with classical computing devices. Quantum information is information encoded to some property of quantum particles and obeys the laws of Quantum Mechanics. Quantum information is transformed using quantum gates, the building blocks for quantum circuits, which, in turn, can be assembled to build quantum computing and communication devices. The societal impact of information increases if the physical embodiments of bits and gates become smaller and we need less energy to process, store, and transmit information. This justifies our interest in quantum information. This book. This book covers topics in quantum computing, quantum information the- ory, and quantum error correction, three important areas of quantum information processing. Quantum information theory and quantum error correction build on the scope, concepts, methodology, and techniques developed in the context of their close relatives, classical infor- mation theory and classical error correcting codes. It seems natural to follow the historical evolution of the concepts, and in this book, we first introduce the classical version of the concepts and techniques which are often simpler and easier to grasp, and then discuss in de- tail the significant leaps forward necessary to apply the concepts and techniques to quantum information Information theory is a mathematical model for transmission and manipulation of classical information. Quantum information theory studies fundamental problems related to transmis- sion of quantum information over classical and quantum communication channels such as: the entropy of quantum systems, the capacity of classical and quantum channels, the effect of the noise, fidelity, and optimal information encoding. Quantum information theory promises to lead to a deeper understanding of fundamental properties of nature and, at the same time, support new and exciting applications. Error correcting codes allow us to detect and then correct errors during transmission of classical information over classical channels and to build fault-tolerant computing and communication systems which obey the laws of classical Physics. Quantum error correcting codes exploit the fundamental properties of quantum information investigated by quantum information theory and play an important role in the fault-tolerance of quantum computing and communication systems. Quantum error correcting codes are critical for the practical use of quantum computing and communication systems. The first chapter of the book provides basic concepts from Mathematics, Quantum Me- chanics, and Computer Science necessary for understanding the properties of quantum infor- mation. Then we discuss the building blocks of a quantum computer, the quantum circuits and quantum gates and survey some of the properties of quantum algorithms. Figure 3 pro- vides a structured view of the topics covered in this chapter: (1) the mathematical apparatus
15 Mathematical Foundations Quantum Mechanics Quantum Information Processing
Quantum Oracles - Phase Estimation - W-HT and QFT - Quantum Parallelism & Reversibility - Grover Search - Amplitude Amplification
Figure 3: Chapter 1 at a glance. used by Quantum Mechanics; (2) the fundamental ideas of Quantum Mechanics; (3) the circuits and algorithms for quantum computing devices.
16 1.1 Elements of Linear Algebra
Familiarity with complex numbers, algebraic structures such as groups, Abelian groups, and fields [58] and linear algebra [170] is required to understand the mathematical formalism of Quantum Mechanics. A review of algebraic structures used in coding theory is given in Section 4.4; in this section we review concepts such as vector spaces, inner product, norm, distance, orthogonality, basis, orthonormal basis, dimension of a vector space, linear transformation and matrices, eigenvectors and eigenvalues, and trace. A vector space is an algebraic structure consisting of:
1. An Abelian group (V,+) whose elements {v¯i} are called “vectors” and whose binary operation “+” is called addition;
2. A field F of numbers whose elements are called “scalars”; we restrict F to be either R (the field of real numbers) or C (the field of complex numbers).
3. An operation called “multiplication with scalars” and denoted by “·”, which associates to any scalar c ∈ F and vectorv ¯i ∈ V a new vectorv ¯j = c · v¯i ∈ V . F acts linearly on V :ifa, b ∈ F andu, ¯ v¯ ∈ V then a · (¯u +¯v)=a · u¯ + a · v¯ and (a + b) · u¯ = a · u¯ + b · u¯.
m×n Assume that F ≡ C; it is easy to show that C , the set of all matrices A =[aij] with entries aij ∈ C, 1 ≤ i ≤ n, 1 ≤ j ≤ m is a vector space where addition of two matrices A =[aij] and B =[bij] is defined as A + B =[aij + bij], the inverse with respect to addition of A =[aij]is−A =[−aij] and the identity element is E = [0]. A set B of vectors is called a basis in V if: (i) every vectorv ¯ ∈ V can be expressed as a linear combination of vectors from B; (ii) the vectors in B are linearly independent. The dimension of a vector space is the cardinality of B. We consider only finite-dimensional vector spaces and in this case the cardinality is the number of elements of B.Ann-dimensional vector space will be denoted as Vn.
An inner product in the vector space Vn over the field F is a mapping g : Vn × Vn → F with several properties; ∀v¯i, v¯j, v¯k ∈ Vn and c ∈ F:
1. Obeys the addition rule in Vn:
g(¯vi +¯vj, v¯k)=g(¯vi, v¯k)+g(¯vj, v¯k) and g(¯vi, v¯j +¯vk)=g(¯vi, v¯j)+g(¯vi, v¯k).
2. Obeys the multiplication with a scalar rule in Vn:
∗ g(c · v¯i, v¯j)=c × g(¯vi, v¯j) and g(¯vi,c· v¯j)=c × g(¯vi, v¯j) with c∗ the complex conjugate of c when F ≡ C.
3. Satisfies the following relations
∗ g(¯vi, v¯j)=g(¯vj, v¯i)ifF = R and g(¯vi, v¯j)=g (¯vj, v¯i)ifF ≡ C.
17 4. The inner product is non-degenerate, g(¯vi, v¯i) ≥ 0 and g(¯vi, v¯i) = 0 if and only ifv ¯i =0;
To simplify the notation the inner product g(¯vi, v¯j) will be written as v¯i, v¯j and we shall use this notation from now on.
If an inner product in V is provided, then the norm || v¯ || of the vectorv ¯ ∈ Vn is the square root of the inner product of the vector with itself: || v¯ ||= v,¯ v¯ .
The distance d(¯vi, v¯j) of two vectorsv ¯i, v¯j ∈ Vn is
d(¯vi, v¯j)=|| v¯i − v¯j ||= (¯vi − v¯j), (¯vi − v¯j) .
Two vectors,v ¯i, v¯j ∈ Vn are orthogonal if
v¯i, v¯j =0. ¯ ¯ ¯ The vectors of {b1, b2,...bn}∈Vn form an orthonormal basis if the inner product of any ¯ ¯ two of them is zero, bibj =0, ∀(i, j) ∈{1,n},i = j, and the norm of a vector is equal to ¯ ¯ ¯ unity, || bi ||= bi, bi =1, ∀i ∈{1,n}. A linear operator A between two vector spaces V and W over the field F is any mapping from V to W , A : V → W , linear in its inputs A civ¯i = ciA(¯vi). i i
The identity operator I mapsv ¯ ∈ Vn to itself, I(¯v)=¯v. A linear transformation is a linear operator with V = W .
The dual of a vector space V , denoted as V ∗, is the set of all scalar-valued linear maps ϕ : V → F. If we define the addition and scalar multiplications in V ∗ as
(ϕ + φ)(¯v)=ϕ(¯v)+φ(¯v) and (cϕ)(¯v)=cϕ(¯v) withv ¯ ∈ V,c ∈ F and ϕ, φ ∈ V ∗, then the dual is also a vector space over the field F.IfV is ∗ ¯ ¯ ¯ ¯ an n-dimensional vector space so is V and if the vectors {b1, b2,...,bj,...,bn} form a basis ∗ ¯1 ¯2 ¯i ¯n for Vn then V is also an n-dimensional vector space and the vectors {b , b ,...,b ,...,b } defined by the property: 1ifi = j ¯bi(¯bj)=δ = ij 0ifi = j form a basis for V ∗. An inner product in V provides an isomorphism, i.e., an invertible linear operator, A : V → V ∗, A(v)(w)= v, w . ¯ ¯ ¯ If we choose a basis B = {b1, b2,...,bn} then the linear transformation A is represented by an n × n matrix, A =[aij], 1 ≤ i, j ≤ n. Let the vectorsv ¯ andw ¯ be
18 ¯ ¯ v¯ = vibi andw ¯ = wibi i i n andw ¯ = Av¯. Then wi = j=1 aijvj which can be written as ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ w1 a11 a12 ... a1 v1 ⎜ ⎟ ⎜ n ⎟ ⎜ ⎟ ⎜ w2 ⎟ ⎜ a21 a22 ... a2n ⎟ ⎜ v2 ⎟ ⎜ . ⎟ = ⎜ . . . . ⎟ ⎜ . ⎟ . ⎝ . ⎠ ⎝ . . . . ⎠ ⎝ . ⎠ wn an1 an2 ... ann vn with the right side of the equality representing the product of the n×n matrix whose elements are aij’s with the n × 1 matrix whose elements are the vi’s. Addition and multiplication of matrices satisfy standard algebraic laws:
A + B = B + AA+(B + C)=(A + B)+CA(B + C)=AB + AC A(BC)=(AB)C (A + B)C = AC + BC AI = IA = A with I the identity matrix; In is an n × n matrix with main diagonal elements equal to one and off-diagonal elements equal to zero. Non-zero matrices do not always have inverses and the product of two matrices is in general noncommutative, AB = BA. The determinant of the n × n matrix A, det(A), is a number calculated from the elements of matrix A; it vanishes if and only if the matrix represents a linear transformation which is not one-to-one. The determinant can be written as a polynomial a1 a2 a3 ··· b1 b2 b3 ··· = ··· a b c ··· c1 c2 c3 ··· ijk i j k . . . . ijk··· . . . .. where (ijk....) a permutation of indices {1, 2, 3,...} and 1 for even permutations, ··· = ijk −1 for odd permutations.
The determinants of two n × n matrices A and B have the following property
det(AB) = det(A) det(B).
An eigenvector,¯v, of a linear transformation A is a non-zero vector such that
Av¯ = λv.¯ The scalar λ is called an eigenvalue corresponding to the eigenvectorv ¯ of A. The previous expression can be also written as
Av¯ = λIv.¯ Thus:
19 (A − λI)¯v =0. ¯ ¯ This equation can be transformed to a matrix equation by choosing a basis {bi}, for Vn. With respect to this basisv ¯ can be expressed as ¯ v¯ = cibi. i Then, ¯ (A − λI) cibi =0 i where the coefficients must satisfy the equation (A − λI)i,jci = 0 for any fixed j. i A nontrivial solution exists only if the determinant
det(A − λ I)=0. The scalar λ for which that happens is an eigenvalue of A. This condition becomes: (a11 − λ) a12 a13 ··· a1n a21 (a22 − λ) a23 ··· a2n a31 a32 (a33 − λ) ··· a3 n =0. . . . .. . . . . an1 an2 an3 ··· (ann − λ) This is called the characteristic equation. The characteristic equation above is a polynomial of degree n in λ, where n is the dimension of the vector space. If F is either R or C then the polynomial has n possibly complex numbers as roots and by the “fundamental theorem of algebra” can be expressed as a product of linear factors:
(λ1 − λ)(λ2 − λ)(λ3 − λ) ···(λn − λ)=0.
If F = C then the n roots, λ1,λ2,λ3,...,λn are the eigenvalues of the operator and are independent of the basis chosen to represent the operator as a matrix. If F = R then the real roots are eigenvalues. If the characteristic equation has less than n distinct roots, there are multiple roots; the root λ of multiplicity larger than one is said to be multiple. The multiplicity m of a root λi is the number of times the factor (λi − λ) appears in the product above. For a multiple eigenvalue it is possible to have more than one eigenvector, all linearly independent and the corresponding linear space is of dimension ≤ m. If there are multiple eigenvalues of A and if for each multiple eigenvalue the multiplicity ¯ ¯ ¯ ¯ m is equal to the dimension d then it is possible to find a basis {b1, b2,...,bj,...,bn} for A such that each basis vector is an eigenvector of A:
20 ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ Ab1 = λ1b1, Ab2 = λ2b2, Ab3 = λ3b3, ... ,Abn = λnbn. ¯ ¯ ¯ ¯ Then A with respect to the basis {b1, b2,...,bj,...,bn} can be expressed by the diagonal matrix ⎛ ⎞ λ1 00··· 0 ⎜ ⎟ ⎜ 0 λ2 0 ··· 0 ⎟ ⎜ ⎟ ⎜ 00λ3 ··· 0 ⎟ . ⎜ . . . . ⎟ ⎝ . . . .. ⎠ 000··· λn
Example. Let F = C and consider the matrix: −12i A = −2i 2 representing a linear transformation A : C → C2. λ is an eigenvalue of A if the determinant of the matrix: −1 − λ 2i A − λI = −2i 2 − λ is zero. The resulting characteristic polynomial
λ2 − λ − 6=0 has the roots λ1 = −2 and λ2 = 3. Hence, the eigenvectors of A satisfy one or the other of the following systems of equations: −x1 − 2iy1 = −2x1 −x2 − 2iy2 =3x2 or, 2ix1 +2y1 = −2y1 2ix2 +2y2 =3y2 where x1,x2 and y1,y2 represent the components corresponding to the basis vectors used to represent matrix A. The equations can be rewritten as x1 − 2iy1 =0 2x2 + iy2 =0 or, ix1 +2y1 =0 2ix2 − y2 =0 − 1 Solving these systems of equations we obtain the eigenvectors (1, 2 i) and (1, 2i). These eigenvectors are not unique, (λ, 2iλ) are also eigenvectors. These eigenvectors can be used as a new basis and the transformed matrix A has a diagonal form: −20 03− λ with respect to this basis. Now we review several important properties of square matrices with elements from C useful for the proof of the proposition presented at the end of this section.
21 The trace of a square matrix A =[aij] , 1 ≤ i, j ≤ n, aij ∈ C, is the sum of the elements on the main diagonal of A, tr(A)=a11 + a22 + ...+ ann. From this definition it is easy to prove several properties of the trace. (i) The trace is a linear map. Given a scalar c ∈ R | C then tr(A + B) = tr(A) + tr(B) and d tr(cA)=c tr(A). As a consequence, if A(t) is a matrix-valued function and dt [A(t)] denotes the matrix whose entries are the derivatives of the entries of A(t) then d d [tr(A(t))] = tr [A(t)] . dt dt
T (ii) The trace is invariant to transposition, tr(A) = tr(A ). Indeed, the diagonal elements aii of a square matrix A =[aij] are invariant to transposition. (iii) The trace is not affected by the order of the matrices in a product of two matrices
tr(AB) = tr(BA). ≤ ≤ If A =[aij] and B =[ bij] , 1 i, j n, the diagonal elements of the products AB and BA n n × are i=1 akibik and i=1 bkiaik; thus, tr(AB) = tr(BA). Consequently if U is a square n n matrix and it is invertible then
tr(U −1AU) = tr(A). Given three n × n matrices A, B, and C we have tr(ABC) = tr(CAB) = tr(BCA). Another consequence, if λ1,λ2,...λn are the eigenvalues of a square matrix A then