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1v4721150/hp-tnauq:viXra Quantum Computation:
A Computer Science Perspective 1
Anders K.H. Bengtsson 2
February 22, 2005
1Work supported by theSwedish KK-foundation underthePromoteIT program.
2e-mail: anders.bengtsson@hb.se
Abstract
Thetheoryofquantumcomputationispresentedinaselfcontainedwayfroma
computerscienceperspective. Thebasicsofclassicalcomputationandquantum
mechanicsis reviewed. The circuitmodel ofquantumcomputationis presented
indetail. Throughoutthereisanemphasisonthephysicalaswellastheabstract
aspects of computation and the interplay between them.
ThisreportispresentedasaMaster’sthesisatthedepartmentofComputer
Science and Engineering at G¨oteborg University, G¨oteborg, Sweden.
Thetextispartofalargerworkthatisplannedtoincludechaptersonquan-
tum algorithms, the quantum Turing machine model and abstract approaches
to quantum computation.
Contents
1 Introduction 7
1.1 The theory of computation . . . . . . . . . . . . . . . . . . . . . 8
1.2 The input/output model of physics and computation . . . . . . . 9
1.3 Classical physics and the computer . . . . . . . . . . . . . . . . . 10
1.4 Quantum computation . . . . . . . . . . . . . . . . . . . . . . . . 11
2 Classical computation 13
2.1 Some definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.1.1 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.1.2 Computation . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.1.3 Program. . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.1.4 Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.1.5 Alphabets, Strings and Numbers . . . . . . . . . . . . . . 16
2.1.6 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.1.7 Decision procedures and Computation procedures . . . . 19
2.2 A note on the connection to everyday computing . . . . . . . . . 20
2.3 The classical Turing machine model of computation . . . . . . . 20
2.3.1 Informal description of Turing machines . . . . . . . . . . 21
2.3.2 Formal definition of a Turing Machine Model . . . . . . . 22
2.3.3 Syntax and semantics . . . . . . . . . . . . . . . . . . . . 30
2.3.4 Decision procedures and Computation procedures revisited 30
2.3.5 The Church-Turing Thesis . . . . . . . . . . . . . . . . . . 31
2.3.6 Computability . . . . . . . . . . . . . . . . . . . . . . . . 32
2.3.7 Universal Turing machines. . . . . . . . . . . . . . . . . . 34
2.3.8 The halting problem is undecidable . . . . . . . . . . . . . 35
2.4 The classical circuit model of computation . . . . . . . . . . . . . 37
2.4.1 The circuit model and non-computable functions . . . . . 40
2.4.2 Reversible gates . . . . . . . . . . . . . . . . . . . . . . . 42
2.4.3 Reversible circuits and un-computation . . . . . . . . . . 45
2.4.4 Reversible computation and physics . . . . . . . . . . . . 46
2.5 Comparison to real computers . . . . . . . . . . . . . . . . . . . . 47
2.6 Non-deterministic Turing Machines . . . . . . . . . . . . . . . . . 48
2.6.1 A note on classical parallelism . . . . . . . . . . . . . . . 49
2.7 Probabilistic Turing machines . . . . . . . . . . . . . . . . . . . . 49
1
2.8 Some Complexity Theory . . . . . . . . . . . . . . . . . . . . . . 51
2.8.1 Measures of complexity . . . . . . . . . . . . . . . . . . . 52
2.8.2 Complexity classes . . . . . . . . . . . . . . . . . . . . . . 54
3 Algebra of quantum bits 57
3.1 Classical and quantum physical systems . . . . . . . . . . . . . . 57
3.2 Two-state quantum systems and the quantum bit . . . . . . . . . 58
3.3 Multiple qubit states . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.4 Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4 Introduction to quantum mechanics 67
4.1 Quantum mechanics in one space dimension . . . . . . . . . . . . 69
4.1.1 Separation of space and time . . . . . . . . . . . . . . . . 71
4.1.2 Particle in a potential well. . . . . . . . . . . . . . . . . . 72
4.2 Linear harmonic oscillator . . . . . . . . . . . . . . . . . . . . . . 75
4.2.1 Quantization of the oscillator . . . . . . . . . . . . . . . . 76
4.2.2 Operators for momentum and position . . . . . . . . . . . 77
4.2.3 Commutators . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.2.4 A note on classical dynamics . . . . . . . . . . . . . . . . 78
4.2.5 Quantization . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.2.6 Dirac notation, a case of abstraction . . . . . . . . . . . . 81
4.2.7 Summary of the classical harmonic oscillator . . . . . . . 82
4.2.8 Creation and annihilation operators . . . . . . . . . . . . 82
4.3 Angular momentum and spin . . . . . . . . . . . . . . . . . . . . 89
4.3.1 Spin 1/2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5 General quantum theory 97
5.1 State spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.1.1 Vector spaces . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.1.2 Hilbert spaces. . . . . . . . . . . . . . . . . . . . . . . . . 100
5.1.3 Dirac notation . . . . . . . . . . . . . . . . . . . . . . . . 101
5.1.4 Tensor products . . . . . . . . . . . . . . . . . . . . . . . 102
5.2 Operators and dynamical variables . . . . . . . . . . . . . . . . . 103
5.2.1 Linear operators . . . . . . . . . . . . . . . . . . . . . . . 103
5.2.2 Outer products . . . . . . . . . . . . . . . . . . . . . . . . 105
5.2.3 Projectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.2.4 Adjoints . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.2.5 Composition of operators . . . . . . . . . . . . . . . . . . 109
5.3 Transformations and symmetries . . . . . . . . . . . . . . . . . . 109
5.4 Eigenvectors and eigenvalues . . . . . . . . . . . . . . . . . . . . 111
5.4.1 Spectral decomposition . . . . . . . . . . . . . . . . . . . 112
5.5 Quantum dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.5.1 Schr¨odinger picture. . . . . . . . . . . . . . . . . . . . . . 116
5.5.2 Heisenberg picture . . . . . . . . . . . . . . . . . . . . . . 116
5.6 Quantum measurement . . . . . . . . . . . . . . . . . . . . . . . 118