I I Mladen Paviˇci´C: Companion to Quantum Computation and Communication — 2013/3/5 — Page 331

i i Mladen Paviˇci´c: Companion to Quantum Computation and Communication — 2013/3/5 — page 331 — le-tex i i

331

Index

a – nuclear 258 A-gate 218 – quantum number 258 abacus 10 – total 258 adiabatic passage 255 annihilation Adleman, Leonard 22 – operator 195 AlGaAs 223 anti-diagonally polarized photon 53 algorithm anti-Hermitian – Bernstein–Vazirani 286 –operator 43 – Deutsch–Jozsa 284 anti-Stokes – exponential speedup 286 – photon 265 – Deutsch’s 282 Antikythera mechanism 10 –eigenvalue argument – exponential speedup 293 –bit 27 – exponential time 293 astrolabe 10 – Euclid’s 288 atom –feasible 20 – interference 269 – ﬁeld sieve 22, 287 atom–cavity – general number ﬁeld sieve 155 – coupling 252, 256, 273 – GNFS 155, 287 – constant 256 – Grover’s 19, 293 atomic –intractable 20 – dipole matrix element 198 – Shor’s 19, 157, 214, 287, 288 – ensemble 262 – all optical 192 auxiliary qubit 181 – exponential speedup 288 axis – exponential time 288 –fast 53 – NMR 214, 287, 290 –slow 53 – Simon’s 287 – exponential speedup 287 b – subexponential 22 B92 165 Alice 81 B92 protocol 170, 171 – classical 141, 152 balanced function 282 – quantum 160 bandwidth 16 alkali-metal atoms 257 Bardeen all-optical 176 – John 236 analog computer 12 basis ancilla 103, 138, 146, 181 – computational 34 angular BB84 protocol 167, 170 – momentum BBN Technologies 157 – electron 215 BBO crystal 55

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i i i i Mladen Paviˇci´c: Companion to Quantum Computation and Communication — 2013/3/5 — page 332 — le-tex i i

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beam splitter c –BS 14 Calderbank–Shor–Steane code 144 – polarization-nonpreserving 71 carrier frequency 203 – polarization-preserving 71 carry 151 – polarizing 57, 260 carry bit 152 Bell cavity 37, 256 –inequality 77 – dark state 257 – state 64–66 – optical 252 – preparation 81 – spherical mirror 253 Bennett–Brassard protocol 167 – QED 176 Bernstein–Vazirani algorithm 286 CC-Uˆ gate biexciton 225 – quantum 138 bit 7 CCNOT –argument 27 –gate 28 – control 27 CCNOT gate –garbage 27 – quantum 145 – parity 141 cesium 257 – quantum 34 charge –result 27 – measurements 220 –sink 27 check matrix 143 –source 27 chip –target 27 –optical –train – integrated 192 – Kane computer 221 Chuang bit-ﬂip 141 – Isaac 188 – correction 148 Church thesis 2 – quantum 144 Church’s general recursiveness 3 Bletchley Park 1 Cirac–Zoller phase gate 192, 230, 248, 306 blue sideband frequency 203 circuit Bob 81 – classical – classical 152 – reversible 138 – quantum 160 – entanglement 140 Bohm – quantum 145–147, 149, 206, 284, 289 – David 303 – diagram 289 Bohr circularly polarized photon 52 – magneton 233 classical –Niels 97 – Alice 141 Boltzmann cloning –entropy 26 –aqubit 94 Bomba 1 – no, theorem 109 Bombes 1 cluster state 300 Boolean – continuous variable 306 – algebra 3, 4, 6 CNOT – lattice model of classical logic 8 –operator 49 –logic 6 CNOT gate 28, 49, 140, 180 –operations 6 – destructive 183 bra 37 – f- 282 bra-ket notation 14 – full 186 broadcasting – interaction-free 266 – no, theorem 109 – pseudo 266 BS – quantum 138 – beam splitter 14 – ion trap 204 – semitransparent mirror 14 – Kane 221, 222

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i i i i Mladen Paviˇci´c: Companion to Quantum Computation and Communication — 2013/3/5 — page 333 — le-tex i i

Index 333

– NMR 214 – Doppler 194 – silicon-based spin 221, 222 – Sisyphus 202 code 142 Cooper codeword 142 – Leon Neil 236 coding – pair 237 –dense 85 – box 240 –superdense 85 Cooper-pair – controlled 89 – number operator 242 coherence 54 coprime 288 collection efficiency 178, 251 correction communication – bit-flip 148 – quantum coset 172 – deterministic 161 counterfactual commutator 40 – measurement 110 complement 6 – reasoning 110 complementary subspace 44 counting rods complexities 17 – Chinese 10 – computational 19 coupler complexity – fiber 170 –class 18 coupling constant 214 – computational 19 creation –oforderO(f(n)) 21 – operator 195 – subexponential 22, 156, 287 critical – superpolynomial 22, 287 – Kochen–Specker set 120 computation cryptography – one-way 298 – interaction-free 280 –physical 17 – quantum 157, 167 –time 19 – continuous variables 177 computational – deterministic 161 –basis 34 – entangled pairs 177 condensation – free space transmission 177 – superconducting 236 – interaction-free 280 conjunction 6 – phase-coding 170 contextuality – roadmap 177 – quantum 110 – single-photon sources 177, 178 continued fraction expansion 292 – weak laser pulses 177 continuous variable crystal – cluster state 306 – BBO 55 continuous variable computation 302 – birefringent 53 continuous wave laser 16 –KDP 55 control –uniaxial 53 –bit 27 CSS code 144 – qubit 135, 138 CW laser 16 control mode 162 CZ 192 controlled-controlled-NOT gate CZ gate 192, 230, 248, 306 – quantum 145 controlled-controlled-Û gate d – quantum 138 dark controlled-NOT gate – counts 178 – quantum 138 – state 255–257 controlled superdense coding 89 – cavity 257 cooling – mixing angle 255 –laser – teleportation 257

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i i i i Mladen Paviˇci´c: Companion to Quantum Computation and Communication — 2013/3/5 — page 334 — le-tex i i

334 Index

DARPA quantum network 158 edge 20 Decidable theories 3 eigenbra 44 decoherence 175 eigenket 44 decomposition of unity 45 eigenvalue 44, 293 decoy – algorithm – pulse 178 – exponential speedup 293 delay – exponential time 293 – gate 152 eigenvector 44, 293 demand Einstein – photons on 178 –Albert 77 dense coding 85 – “element of physical reality” 77 density operator 47 Einstein–Podolsky–Rosen 303 destructive CNOT gate 183 – Gedankenexperiment (EPR) 78 deterministic Ekert protocol 165 – quantum electric dipole moment 256 – communication 161 electric ﬁeld vector 51 – cryptography 161 electron Deutsch–Jozsa algorithm 284 –angular – exponential speedup 286 – momentum 215 Deutsch’s algorithm 282 –magnetic diagonally polarized photon 52 – moment 215 diode laser 177 –single dipole – transistor 220 – approximation 197, 199 – spin 215 –matrixelement electron commutation relations 199 – atomic 198 emission, spontaneous 54 –moment encoder – electric 256 – gate 186 direct product 40 – quantum 183 discrete Fourier transform 281 – quantum 182 disjunction 6 encoding distributivity 9 – quantum 182 DiVincenzo entangled – Criteria 176 –photons 50 – D. 231 – state 65, 140, 147 DLCZ protocol 262 – on demand 204 donor entanglement – phosphorus 218 – circuit 140 Doppler cooling 194 – swapping 262, 263 Doppler laser cooling 194 entropy 26 dot –Boltzmann 26 – quantum 179 EPR (Einstein–Podolsky–Rosen) 78, 303 down-conversion error – parametric 55 – correction –type-I 55 – classical 141 –type-II 56 – Hadamard gate 145 Duan–Lukin–Cirac–Zoller protocol 262 – quantum 144 dyad 40 – weight 143 Euclid’s algorithm 288 e Euler tour 20 Earnshaw’s theorem 193 Eve 81 eavesdropping – quantum 167 – quantum 167 evolution operator 130

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i i i i Mladen Paviˇci´c: Companion to Quantum Computation and Communication — 2013/3/5 — page 335 — le-tex i i

Index 335

exciton 223, 225 free space transmission – state 226 – cryptography exponential – quantum 177 – matrix 134 Frege – speedup – Gottlob 4 – Deutsch–Jozsa algorithm 286 frequency – eigenvalue algorithm 293 –angular 51 – Shor’s algorithm 288 –ofawave 51 – Simon’s algorithm 287 full –time – adder 151 – eigenvalue algorithm 293 – CNOT gate 186 – Shor’s algorithm 288 EXPTIME 19 g extraordinary ray 166 GaAs 223 GaInP 224 f garbage f-CNOT gate 282 –bit 27 factoring a number 287 gas factoring number 17 – van der Waals 296 false (0, ?)6 fan-out gate 6 –gate 30 – A 218 fault-tolerant computation 149 –AND 7 feasible – CCNOT 28 – algorithm 20 – classical 6 –problem 20 – CNOT 28, 49, 140 Fermi – destructive 183 – operator 200 – full 186 Fert, Albert 225 – interaction-free 266 Feynman – ion trap 204 – Richard 28 – Kane 221, 222 fiber 191 – NMR 214 – coupler 170 – quantum 138 fictitious magnetic – silicon-based spin 221, 222 –dipole – CZ 192, 230, 248, 306 – ion trap 200 – delay 152 –field – encoder 186 – ion trap 200 – quantum 183 field –fan-out 30 – sieve algorithm 287 –Fredkin 27 Field Sieve algorithm 22 – H 132 flip – Hadamard 81, 132, 248, 283, 285, 292 – parity 304 – ion trap 203 – spatial 304 – NMR 213 Fock – J 218, 219 – state –logic 7 – single-photon 178 –operations 7 Fourier transform 10, 292 –symbols 7 – discrete 281 – NMR 213 – quantum 281, 290 – NOT 7, 28, 132 FP7 158 p– ion trap 203 Franson, James D. 180 – NOT 132 Fredkin –OR 7 –gate 27 – parity check 180

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– phase 132, 230, 248 – distance 142 – phase shift 132 – rule 142 – quantum 6, 222 – scheme 142 – rotation 132 harmonic oscillator 195 – S 132, 221 – Hamiltonian 195 – Toffoli 28, 138 Heisenberg microscope 270 –universal Hermitian – quantum 136 –matrix 43 – X 132 –operator 43 –XNOR 7 hexagon – Y 132 – lattice model of classical logic 8 – Z 132 Hilbert gcd 288 –David 4 general number field sieve algorithm –program 4 155, 287 Holevo limit 165 generator home qubit 161 – rotation gate 133 Hong–Ou–Mandel dip 90, 304 giant magnetoresistance 225 HWP (half-wave plate) 80 Ginzburg hyperfine interaction 215, 218, 258 – Vitaly 237 hyperfine structure 36 GNFS algorithm 155, 287 hypergraph 20, 115 Goldberg–Vaidman protocol 165 – MMP 116 Gottesman Hänsch, T.W. 194 – Daniel 188 i graph 20 identity operator 40, 130 – state 306 idler 55 greatest common divisor 288 idQuantique 157 Grover’s algorithm 19, 293 inconsistent mathematics 4 Grünberger, Peter 225 information Gödel, Kurt 3, 4 –garbage 27 Gödel’s reckonability 3 Integrated – chip h – optical 192 H gate 132 Intel 27 Hadamard gate 81, 132, 248, 283, 285, 292 interaction – error correction 145 – picture 202 – ion trap 203 – strong 296 – NMR 213 – weak 296 half interaction-free – adder 151 – CNOT gate 266 – wave plate (HWP) 80, 260 – cryptography 280 halting problem 18 – resonance detection 269 Hamilton operator 130 interferometer Hamiltonian 130 – Mach–Zehnder 17, 170 – harmonic oscillator 195 – Michelson 69 – ion trap 199 Internet 152, 155 – Jaynes–Cummings 252 interval analysis 117 – Kane computer 219 intractable – local 293 – algorithm 20 – NMR 213, 215 –problem 20 Hamming inverted population 252 – code 142, 144 ion – codeword 145 – trap 192

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i i i i Mladen Paviˇci´c: Companion to Quantum Computation and Communication — 2013/3/5 — page 337 — le-tex i i

Index 337

ion trap – 60-75 class 122 – computer 203 – parity proof 120 – scalable 206 – set 113 – fictitious magnetic dipole 200 – critical 120 – fictitious magnetic field 200 –sets – Hamiltonian 199 – billions of 118 – laser beam 196 – computer generation of 120 Ising model 231, 233 – stripping technique 99 isomorph-free generation of graphs and – theorem 112 hypergraphs 20 – vector 113 Kronecker product 40 j Königsberg bridges 20 J-coupling 214 J-gate 218, 219 l Jaynes–Cummings Laflamme – Hamiltonian 252 – Raymond 180 – model 199 Lamb–Dicke Jones – limit 202 –vector 52 – parameter 201 Josephson Landau –charge – Lev 237 – qubit 246 Landauer – current 238 – principle 26 – critical 239 Larmor frequency – effect 238 – resonance 218 – island 240 laser 251 – junction 238 –beam – atom interaction 197 k – ion trap 196 Kane – beam-electron–phonon interaction 199 – CNOT gate 221 – beam-ion interaction 199 – computer 218, 221 – continuous wave 16 – bit train 221 – cooling 202 – Hamiltonian 219 – diode 177 – radio frequency magnetic field 218 – phase-coding 170 – spin subspace 221 – Doppler cooling 194 –magnetic – one-atom 251 – field, external 217 –pulse KDP crystal 55 – quantum cryptography 177 ket 14, 34 –pumping 99 key lattice 212 – distribution – distributive 6 – classical 170 –model – quantum 170 – of classical logic 8 – RSA 155 law – sifted 168 – Moore’s 18 Kleene’s λ-definability 3 Liar paradox 4 Knill linear – Emanuel 180 – ion trap 193 Knill–Laflamme–Milburn approach 180 linewidth 16 Kochen–Specker local Hamiltonian 293 – 24-24 class 120 logic 6 – 60-105 class 126 – classical

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i i i i Mladen Paviˇci´c: Companion to Quantum Computation and Communication — 2013/3/5 — page 338 — le-tex i i

338 Index

– Boolean lattice model of 8 – hypergraph 115 –gate 6 – hypergraphs 116 – hexagon lattice model of 8 model –operationsin 6 –o6 8 modulo 143, 146, 282, 288 m momentum Mach–Zehnder – space 295 – interferometer 17, 170, 179 Moore – modified 304 –Gordon 12 MagiQ Technologies 157 –law 18 magnetic Moore’s law 12, 18 – fictitious dipole Mortensen, C. 4 – ion trap 200 multigraph 20 – fictitious field – ion trap 200 n –field Network – Kane, external 217 – QKD 160 – radio frequency, Kane 218 – Tokyo 160 – radio frequency, MNR 212 neutral atom 176 –moment NMR 176, 208, 209, 213, 287 – electron 215 – computer 213, 214 – nuclear 215 – radio frequency magnetic field 212 magneto-optical trap 269 – Hadamard gate 213 magnetoresistance – Hamiltonian 213, 215 – giant 225 p – NOT gate 213 Malus law 52 Mathematica 40 – Shor’s algorithm 214, 287, 290 mathematics no broadcasting theorem 109 – inconsistent 4 no cloning Mathieu’s functions 243 – theorem 109 matrix no-cloning – check 143 – principle 31, 182 – exponential 130, 134 no-cloning principle 94 – generation 143 no-cloning theorem 94 –Hermitian 43 NOT –self-adjoint 43 –gate 28 MatrixExp 213, 234, 248 – ion trap 203 Maxwell demon 26 pNOT gate 132 mean value 46 NOT gate 132 measurable values 46 nuclear memory –magnetic – quantum 191, 262 – moment 215 message mode 161 – magnetic resonance 176, 208, 209 Meyer, R.K. 4 nuclear-electron interaction 36 Michelson interferometer 69 number microchannel plate detector 269 –fieldsieve Milburn – general algorithm 155 – Gerard J. 180 – operator 195 mixing angle – state – dark state 255 – notation 200 mixture 46 number field sieve MMP – general – diagrams 115 – algorithm 155

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Index 339

o phonon 194, 196, 237 o6 8 – anticommutation relations 196 –model 9 – state 196 O’Brien phosphorus donor 218 – Jeremy L. 191 photon observable 44 –entangled 50 one-atom laser 251 – extraordinary 56 one-time pad 152, 154 – gun 178 one-way computation 122, 298 – on demand 107, 178, 252 operational amplifier (op-amp) 10 –ordinary 56 operator physical computation 17 – anti-Hermitian 43 pickup coil 212 –CNOT 49 ping–pong 161 –density 47 plane wave 52 – evolution 130 plug-and-play quantum cryptography 171 – Fermi 200 Poincaré sphere 305 – Hamilton 130 Poisson distribution 178 –Hermitian 43 polarization – identity 40, 130 – circular 53, 258 – unitary 43, 293 – horizontal 34 optic axis 53 –left 52 optical – linear 53, 99, 166, 167, 258 – cavity 251, 252 – right 52 – spherical mirror 253 – vertical 34, 52 order 288 polarization-nonpreserving beam splitter 71 ordinary ray 166 polarization-preserving beam splitter 71 polarizing beam splitter 57, 260 p population parametric down-conversion 55 – inverted 252 parity 141, 181 Post’s canonical and normal systems 3 – bit 141 precession 212 – check 181 – frequency 213 – even 181 Presburger arithmetic 3 – flip 304 prime – odd 181 – relatively 154, 288 – proof principle – Kochen–Specker 120 –Landauer 26 parity-spatial space 305 privacy amplification 173 Paul trap 192 problem Pauli –feasible 20 –spin –intractable 20 – matrices 41, 233 –satiability(SAT) 20 PBS (polarizing beam splitter) 57 product Peano arithmetic 3 –direct 40 Penning trap 192 –Kronecker 40 phase –tensor 40 –coding projector 45 – quantum cryptography 170 propagation vector 51 – gate 132, 230, 248 protocol – shift 144 – B92 170, 171 – correction 148 – BB84 167, 170 – gate 132 – Bennett–Brassard 167 – shifter 80 – six-state 170

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i i i i Mladen Paviˇci´c: Companion to Quantum Computation and Communication — 2013/3/5 — page 340 — le-tex i i

340 Index

pseudo CNOT gate 266 – memory 191, 262 public –network –keycryptography – DARPA 158 – classical 153 – nondemolition 306 – key protocol –number – classical 153 – angular momentum 258 – RSA 153 – magnetic 258 pump beam 55, 273 –phase pure state 45 – gate 230 – register 289 q – repeater 149, 261 QED quantum bits 8 – cavity 176 quarter wave plate 53, 260 QIPC 157 qubit 8, 34 QKD 178 –clone 94 QKD Network 160 – control 135, 138 – Tokyo 160 – flying 175 QND 306 – home 161 quantum – stationary 175 –bit 34 – target 135, 138 – bit-flip 144 – travel 161 – CCNOT gate 145 – work 138 – circuit 145–147, 149, 206, 284, 289 – diagram 289 r – coin 282 Rabi – communication – flopping frequency 198 – deterministic 161 – frequency 199, 201 – computation radio frequency 209, 212, 218 – roadmap 176 –magneticfield – computer – Kane computer 218 – cilicon-based 218 – NMR computer 212 – ion trap 203 Raman – Kane 218 – adiabatic passage – NMR 213 – stimulated 255 – contextuality 110 – emission 263 – controlled-controlled-NOT gate 145 – passage – cryptography 157, 167 – stimulated 273 – companies 157 – scheme 196 – continuous variables 177 ray – deterministic 161 – extraordinary 166 – entangled pairs 177 – ordinary 166 – free space transmission 177 red sideband frequency 203 – interaction-free 280 reflectance 58 – phase-coding 170 register – roadmap 177 – quantum 289 – single-photon sources 177, 178 relatively prime numbers 154, 288 – weak laser pulses 177 repeaters – dot 179, 223 – quantum 149 – spin qubit 232 resonance – spintronics 232 – interaction-free detection 269 – encoding 182 resonant pulses 212 – Fourier transform 281, 290 result – gate 222 –bit 27

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Index 341

reversible Shor’s algorithm 19, 157, 192, 214, 287, 288 – circuit 138 – all optical 192 RF 209, 212, 218 – exponential speedup 288 – coil 212 – exponential time 288 – electric field 193 – NMR 214, 287, 290 – field 218 Si semiconductor 218 – generator 212 Si substrate 218 – pulses 212 sideband frequency 203 – signals 212 sifted key 168 Rice, H.G. 4 signal 55 Rice’s theorem 4 silica 191 Rivest, Ron 22 silicon 191 roadmap silicon-based nuclear spins 215 – quantum computation 176 Simon’s algorithm 287 – quantum cryptography 177 – exponential speedup 287 rotating wave approximation 198, 202 single electron transistor (SET) 24, 220 rotation single-photon – gate 132 – detector 177 – generator 133 – Fock state 178 RSA 22 singlet 61 – key 155 sink – public key protocol 153 –bit 27 rubidium 258 Sisyphus cooling 202 rubidium atom (Rb) 36 six-state protocol 170 Russel slide rule 11 –Bertrand 4 SmartQuantum 157 solid state 176 s source S gate 132 –bit 27 S-gate 221 spatial flip 304 SAT 20 spectrum satiability problem (SAT) 20 – of an observable 45 scalable spherical mirror – ion trap computer 206 – optical cavity 253 Schawlow, A. L. 194 spin Schrieffer –down 34 – John Robert 236 – electron 215 Schrödinger –subspace – equation 33, 130, 197, 253, 294 – Kane computer 221 – picture 202 –up 34 SECOQC 158 spin-electronics 225 security spin–orbit interaction 218, 258 – unconditional 169 spintronics 225 selection frequency 268 SQUID 241 self-adjoint matrix 43 state semantic – dark 255–257 – valuation 9 – cavity 257 semiconductor 223 – teleportation 257 – Si 218 – entangled 65, 147 semitransparent mirror – on demand 204 – beam splitter (BS) 14 –Fock SET 24, 220 – single-photon 178 Shamir, Adi 22 – graph 306

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342 Index

– number 200 tensor product 40 –pure 45 time – unknown 94, 251 – complexity 19 – vacuum 200 – evolution operator 130 statistical mixture 46 Toffoli stimulated Raman adiabatic passage –gate 28 255, 273 Toffoli gate 138 STIRAP 37, 255, 256, 259, 273 Tokyo QKD Network 160 Stokes trace 47 – laser 255 transistor – beam 254, 273 – single electron 220 – ﬁeld 255 transmittance 58 – photon 263 transmon 243 strong trapped ion 176, 192 – interaction 296 – computer 192 subexponential travel qubit 161 – algorithm 22 traveling salesman problem (TSP) 21 – complexity 22, 156, 287 trial divisions 17 subspace trichloroethylene 213 – complementary 44 triplet 64 –spin Trotter formula 222 – Kane computer 221 true (1, T)6 substrate truth tables 6 – Si 218 truth values 6 super-Turing TSP 21 – computation 4 Turing – machines 5 –Alan 1 superatoms 223 – machine 1, 3 superconducting 176 – super-, machines 5 – condensation 236 – super-Turing computation 4 – devices 235 – 2012 Year of 5 – quantum type-I down-conversion 55 – interference device 241 type-II down-conversion 56 superdense coding 85 – controlled 89 u superpolynomial complexity 287 unconditional security of quantum superposition 7 cryptography 158, 169, 171 SWAP undecidable theories 3 – operator 235 unitary operator 43, 293 universal p – square root of 235 SWAP 235 – quantum gate 136 sweet spot 244 unknown state 94, 251 syndrome 143 v t vacuum target – state 200 –bit 27 valence band holes 223 – qubit 135, 138 valuation 9 Tarski, Alfred 3 –semantic 9 Teich van der Waals gas 296 – Malvin 303 vector teleportation 99, 260, 261 –Jones 52 – deterministic 204 Vernam ciphers 152 – trick 180, 188 vertex 20

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i i i i Mladen Paviˇci´c: Companion to Quantum Computation and Communication — 2013/3/5 — page 343 — le-tex i i

Index 343

w winner (would be quantum computer) 176 water work qubit 138 – analog computer 11 – computation 11 x –integrator 10 X gate 132 wave XNOR 6 – approximation, rotating 198 XOR 141, 143, 153, 173, 282 – plane 52 – plate, quarter 53 y –vector 51 Y gate 132 waveguide 191 weak z – interaction 296 Z gate 132 Weierstrass lenses 179 Zeeman weight 143 – effect welcher-weg 270 – nuclear 210 which way 270 –magnetic Whitehead – Hamiltonian part 219 –AlfredNorth 4 – splitting 196, 254 Wigner function 303 – sublevels 36, 258

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