Introduction to Quantum Information Processing — Draft Technical University of Clausthal SS 2001 – SS 2005 W. L¨ucke 3 Preface Quantum information processing is one of the most fascinating and active fields of contemporary physics. Its central topic is the coherent control of quantum states in order to perform tasks — like quantum teleportation, absolutely secure data trans- mission and efficient factorization of large integers — that do not seem possible by means of classical systems alone. The vast possibilities of physical implementations are currently being extensively studied and evaluated. Various proof-of-principle experiments have already been performed. However, in the present note only some possiblities can be indicated. Main emphasis will be on quantum optical methods, indispensable for transmission of quantum information. For more complete information on achivements and latest proposals concerning quantum information processing the Los Alamos preprint server http://xxx.lanl.gov/archive/quant-ph is highly recommended. Recommended Literature: (Alber et al., 2001; Bowmeester et al., 2000; Ekert et al., 2000; Nielsen and Chuang, 2001; Preskill, 01; Shannon, 1949; Bertlmann and Zeilinger, 2002; Audretsch, 2002; Bruß, 2003) 4 Contents I Idealized Quantum Gates and Algorithms 9 1 Basics of Quantum Computation 11 1.1 Classical Logic Circuits .......................... 11 1.2 Quantum Computational Networks ................... 17 1.2.1 Quantum Gates .......................... 17 1.2.2 Quantum Teleportation ..................... 24 1.2.3 Universality ............................ 27 2 QuantumAlgorithms 35 2.1 Quantum Data Base Search ....................... 35 2.1.1 Grover’s Algorithm ....................... 35 2.1.2 Network for Grover’s Algorithm ................ 37 2.1.3 Details and Generalization .................... 38 2.2 Factoring Large Integers ......................... 39 2.2.1 Basics ............................... 39 2.2.2 The Quantum Fourier Transform ............... 43 2.2.3 Quantum Order Finding ..................... 46 3 Physical Realizations of Quantum Gates 53 3.1 Quantum Optical Implementation .................... 53 3.1.1 Photons .............................. 54 3.1.2 Photonic n-Qubit Systems .................... 56 3.1.3 Nonlinear Optics Quantum Gates ................ 60 3.1.4 Linear Optics Quantum Gates .................. 62 3.2 Measurement-Based Quantum Computation .............. 70 3.3 Cold Trapped Ions ............................ 73 3.3.1 General Considerations ...................... 74 3.3.2 Linear Paul Trap ......................... 75 3.3.3 Implementing Quantum Gates by Laser Pulses ......... 81 3.3.4 Laser Cooling ........................... 90 5 6 CONTENTS II Fault Tolerant Quantum Information Processing 91 4 General Aspects of Quantum Information 93 4.1 Introduction ................................ 93 4.2 Quantum Channels ............................ 95 4.2.1 Open Quantum Systems and Quantum Operations ...... 95 4.2.2 Quantum Noise and Error Correction ..............103 4.3 Error Correcting Codes ..........................105 4.3.1 General Apects ..........................105 4.3.2 Classical Codes ..........................110 4.3.3 Quantum Codes ..........................112 4.3.4 Reliable Quantum Computation .................118 4.4 Entanglement Assisted Channels ....................122 4.4.1 Quantum Dense Coding .....................122 4.4.2 Quantum Teleportation .....................123 4.4.3 Entanglement Swapping .....................125 4.4.4 Quantum Cryptography .....................126 5 Quantifying Quantum Information 129 5.1 Shannon Theory for Pedestrians .....................129 5.2 Adaption to Quantum Communication .................133 5.2.1 Von Neumann Entropy ......................133 5.2.2 Accessible Information ......................137 5.2.3 Distance Measures for Quantum States .............139 5.2.4 Schumacher Encoding ......................145 5.2.5 A la Nielsen/Chuang .......................147 5.2.6 Entropy ..............................147 6 Handling Entanglement 149 6.1 Detecting Entanglement .........................149 6.1.1 Entanglement Witnesses .....................149 6.1.2 Examples .............................152 6.1.3 Other Criteria ...........................155 6.2 Local Operations and Classical Communication ............158 6.2.1 General Aspects ..........................158 6.2.2 Entanglement Dilution ......................164 6.2.3 Entanglement Distillation ....................164 6.3 Quantification of Entanglement .....................164 A 167 A.1 Turing’s Halting Problem ........................167 A.2 Some Remarks on Quantum Teleportation ...............168 A.3 Quantum Phase Estimation and Order Finding ............169 A.4 Finite-Dimensional Quantum Kinematics ................173 A.4.1 General Description ........................173 CONTENTS 7 A.4.2 Qubits ...............................177 A.4.3 Bipartite Systems .........................178 Bibliography 185 Index 202 8 CONTENTS Part I Idealized Quantum Gates and Algorithms 9 Chapter 1 Basics of Quantum Computation 1.1 Classical Logic Circuits The smallest entity of classical information theory (Shannon, 1949) is the bit (binary digit), i.e. the decision on a classical binary alternative. Usually bits are identified with the numbers 0 (for wrong) or 1 (for true) and typically correspond to the position of some simple switch. Every definite statement may be encoded into a 1 sufficiently long but finite sequence (b1, . , bn) of bits. In this sense the essence of a calculations may be described as the transformations of a finite sequence of input bits (encoding the task) into a finite sequence of output bits (encoding the result). This suggests the following model for actual calculators: 1. An input register (array of switches) will be put into a state corresponding to the n -tuple (b , . , b ) 0, 1 n1 encoding the task. 1 1 n1 ∈ { } 2. A computational circuit, the elementary components of which are called gates,2 transforms (b1, . , bn1 ) into an n2-tuple (b1′ , . , bn′ 2 ) of bits encoding the result to be stored into an output register. From the mathematical point of view it is only important which element of n1,n2 , denoting the set of all mappings from 0, 1 n1 into 0, 1 n2 , is implementedF by the circuit. Therefore, computational circuits{ } implement{ ing} the same mapping are called equivalent. Every element of n1,n2 can be implemented by some assembly of gates listed in Table 1.1: F DRAFT, June 26, 2009 1An important consequence of this fact is the halting problem (see Appendix A.1). 2For simple hardware implementations see (P¨utz, 1971, pp. 244–252). 11 12 CHAPTER 1. BASICS OF QUANTUM COMPUTATION Name Symbol Class Action ID b b F1,1 7→ FANOUT r b (b, b) F1,2 7→ HH NOT ¨¨ b b 1 b F1,1 7→ − AND & (b , b ) b b F2,1 1 2 7→ 1 2 1 OR ≥ (b , b ) b + b b b F2,1 1 2 7→ 1 2 − 1 2 Table 1.1: Elementary gates Lemma 1.1.1 For arbitrary positive integer n1, n2 all elements of n1,n2 can be represented as compositions of tensor products of functions from TabularF 1.1. Proof: See below. Thus every classical logic circuit corresponds to a graph consisting of symbols from Tabular 1.1. For instance, the graph ... & ....... ............. ............ ....... u 1 ≥ & corresponds to the mapping SWITCH def= OR (AND AND) (IDNOT ID ID) (ID FANOUT ID) , ◦ ⊗ ◦ ⊗ ⊗ ⊗ ◦ ⊗ ⊗ acting as b0 if s = 0 , (b0, s, b1) 7−→ b1 if s = 1 . ½ Another example is the graph u & .. ....... .............. ............ & ....... u 1 . ≥ u & corresponding to XOR def= OR (AND AND) (ID FANOUT ID) ◦ ⊗ ◦ ⊗ ⊗ ID (NOT AND) ID (FANOUT FANOUT) ◦ ⊗ ◦ ⊗ ◦ ⊗ ³ ´ 1.1. CLASSICAL LOGIC CIRCUITS 13 Name Symbol Class Action s CNOT 2,2 (b1, b2) (b1, b1 b2) h F 7→ ⊕ h TCNOT 2,2 (b1, b2) (b1 b2, b2) s F 7→ ⊕ SWAP \ (b , b ) (b , b ) \ F2,2 1 2 7→ 2 1 s (0, b , b ) (0, b , b ) 3 1 2 7→ 1 2 CSWAP \ 3,3 \ F (1, b , b ) (1, b , b ) 1 2 7→ 2 1 s 4 CCNOT s (b , b , b ) (b , b , b b b ) F3,3 1 2 3 7→ 1 2 1 2 ⊕ 3 h Table 1.2: Some reversible gates and acting as def NOT(b ) if b = 1 (b , b ) b b = 2 1 1 2 7−→ 1 ⊕ 2 b if b = 0 ½ 2 1 = b + b 2b b 1 2 − 1 2 = b1 + b2 mod 2 . Of course, also for the gates listed in Table 1.2 there are equivalent networks, e.g.:5 CNOT = (ID XOR) (FANOUT ID) , (1.1) ⊗ ◦ ⊗ DRAFT, June 26, 2009 3The CSWAP gate is also called Fredkin gate. 4The CCNOT gate is also called Toffoli gate. 5See (Tucci, 2004) for more equivalences of classical and/or quantum networks. 14 CHAPTER 1. BASICS OF QUANTUM COMPUTATION h \ s \ \ \ , (1.2) s ≡ h \ s h s \ . (1.3) ≡ h s h s s \ h s h (1.4) \ ≡ s h s Now we are prepared for the Proof of Lemma 1.1.1: Thanks to FANOUT and SWAP it is sufficient to proof the lemma for decision functions, i.e. for n2 = 1 . Obviously, then, the statement of the lemma holds for n = 1 , since the four elements of are ID, 1 F1,1 TRUE def= OR (ID NOT) FANOUT , ◦ ⊗ ◦ and their compositions with NOT (applied last). Now, assume that the statement of the lemma has already been proved for n = n and consider an arbitrary f . 1 ∈Fn+1,1 Then both f0 and f1 , where def fs(b1,...,bn) = f(b1,...,bn,s) , can be represented as compositions of tensor products of functions from Tabular 1.1. There is a composition of FANOUTs and SWAPs acting as (b ,...,b ,s) (b ,...,b ,s,b ,...,b ). 1 n 7−→ 1 n 1 n Composing this with SWITCH (f ID f ) ◦ 0 ⊗ ⊗ 1 (to be applied last) gives f . This proves the statement of the lemma for n1 = n + 1 . According to Lemma 1.1.1 we may perform arbitrarily complex computations by composing simple hardware components of very small variety. Of course, given f n1,n2 , there are infinitely many representations of f as composition of tensor products∈ F of elementary components. Therefore, the interesting problem arises how to simplify a given gate (logic circuit) without changing its action.6 From the technological point of view it is also of interest that NAND def= NOT AND ◦ DRAFT, June 26, 2009 6 See (Lindner et al., 1999, Sect. 8.2.3) for n1 6 , n2 = 1 and (Lee et al., 1999; Shende et al., 2003) for quantum gates.