ei ole lubatud
ei ole lubatud
Digitaalõiguste kaitse (DRM)
Kirjastus on väljastanud selle e-raamatu krüpteeritud kujul, mis tähendab, et selle lugemiseks peate installeerima spetsiaalse tarkvara. Samuti peate looma endale Adobe ID Rohkem infot siin. E-raamatut saab lugeda 1 kasutaja ning alla laadida kuni 6'de seadmesse (kõik autoriseeritud sama Adobe ID-ga).
Vajalik tarkvara
Mobiilsetes seadmetes (telefon või tahvelarvuti) lugemiseks peate installeerima selle tasuta rakenduse: PocketBook Reader (iOS / Android)
PC või Mac seadmes lugemiseks peate installima Adobe Digital Editionsi (Seeon tasuta rakendus spetsiaalselt e-raamatute lugemiseks. Seda ei tohi segamini ajada Adober Reader'iga, mis tõenäoliselt on juba teie arvutisse installeeritud )
Seda e-raamatut ei saa lugeda Amazon Kindle's.
Preface 7
1.1 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
I Quantum machine learning and Tensorflow* 11
2 Introduction 13
2.1 Fusion between QM and NN . . . . . . . . . . . . . . . . . . . . . 13
2.2 The quantum advantage in boson sampling and NN . . . . . . . 13
2.3 The background of a quantum engineer . . . . . . . . . . . . . . 13
2.4 Impact on the foundation of quantum mechanics . . . . . . . . . 15
3 Quantum hardware 17
4 Review on quantum machine learning and related 19
4.1 Neural networks in physics beyond quantum mechanics . . . . . . 20
4.2 Further readings . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
5 Coding fundamentals 21
5.1 Matrix manipulation in Python . . . . . . . . . . . . . . . . . . . 21
5.2 What is Tensorflow . . . . . . . . . . . . . . . . . . . . . . . . . 21
5.3 Tensor and variables in Tensorflow . . . . . . . . . . . . . . . . 21
5.4 Objects in Tensorflow . . . . . . . . . . . . . . . . . . . . . . . . 21
5.5 Models in Tensorflow . . . . . . . . . . . . . . . . . . . . . . . . . 21
5.5.1 Automatic Graph building . . . . . . . . . . . . . . . . . . 21
5.5.2 Automatic differentiation . . . . . . . . . . . . . . . . . . 216 Neural networks model 23
6.1 Examples by tensorflow . . . . . . . . . . . . . . . . . . . . . . . 23
7 Reservoir computing 25
7.1 Examples by tensorflow . . . . . . . . . . . . . . . . . . . . . . . 25
II Neural networks and phase space 27
8 Phase-space representation 29
8.1 The characteristic function with real variables . . . . . . . . . . . 30
8.2 Gaussian states . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
8.3 Vacuum state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
8.4 Coherent state . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
9 Linear transformations 35
9.1 The U and M matrices* . . . . . . . . . . . . . . . . . . . . . . . 36
9.2 Generating a symplectic matrix for a random medium . . . . . . 39
10 Gaussian density matrix as a neural network layer 41
10.1 The vacuum layer . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
11 Pullback 45
11.1 Pullback of Gaussian states . . . . . . . . . . . . . . . . . . . . . 46
11.2 Coding the linear layer . . . . . . . . . . . . . . . . . . . . . . . . 46
11.3 Pullback cascading . . . . . . . . . . . . . . . . . . . . . . . . . . 48
11.4 The Glauber displacement layer . . . . . . . . . . . . . . . . . . . 51
11.5 A linear layer for a complex medium . . . . . . . . . . . . . . . . 52
12 Quantum reservoir computing examples 55
12.1 Observables as derivatives of ? . . . . . . . . . . . . . . . . . . . 55
12.2 A coherent state in a complex medium . . . . . . . . . . . . . . . 56
12.3 Training a complex medium for an arbitrary coherent state . . . 57
12.3.1 Training to fit a target characteristic function . . . . . . . 59
12.3.2 Training by first derivatives . . . . . . . . . . . . . . . . . 61
12.3.3 Training by second derivatives . . . . . . . . . . . . . . . 63
12.3.4 The CovarianceLayer . . . . . . . . . . . . . . . . . . . . 63
12.4 Proof of Eq. (12.3) . . . . . . . . . . . . . . . . . . . . . . . . . . 65
12.5 Two trainable media and a reservoir . . . . . . . . . . . . . . . . 66
12.6 Phase modulator . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
12.7 Training phase modulators . . . . . . . . . . . . . . . . . . . . . . 69
III Non classical states 71
13 Introduction 73
13.1 The generalized symplectic operator . . . . . . . . . . . . . . . . 73
14 Squeezing 75
14.1 Single Mode Squeezed state . . . . . . . . . . . . . . . . . . . . . 75
14.1.1 Symplectic representation for the squeezing . . . . . . . . 75
14.2 Multi-mode squeezed vacuum NN model . . . . . . . . . . . . . . 76
14.3 Covariance matrix and squeezing . . . . . . . . . . . . . . . . . . 78
14.4 Squeezed coherent states . . . . . . . . . . . . . . . . . . . . . . . 79
14.4.1 Displacing the squeezed vacuum . . . . . . . . . . . . . . 79
14.4.2 Squeezing the displaced vacuum . . . . . . . . . . . . . . 80
14.5 Two-mode squeezing layer . . . . . . . . . . . . . . . . . . . . . . 82
15 Beam splitters and detection 87
15.1 Beam splitter layer . . . . . . . . . . . . . . . . . . . . . . . . . . 87
15.2 Photon counter layer . . . . . . . . . . . . . . . . . . . . . . . . . 90
15.3 Homodyne detection . . . . . . . . . . . . . . . . . . . . . . . . . 94
15.4 Measuring the expected value of the quadrature operator . . . . 96
16 Uncertainties 99
16.1 The Heisenberg layer . . . . . . . . . . . . . . . . . . . . . . . . . 99
16.2 Heisenberg layer for general states . . . . . . . . . . . . . . . . . 100
16.2.1 The LaplacianLayer . . . . . . . . . . . . . . . . . . . . 100
16.2.2 The BiharmonicLayer . . . . . . . . . . . . . . . . . . . . 102
16.2.3 Using the BiharmonicLayer in the HeisenbergLayer . . 104
16.3 Heisenberg layer for Gaussian states . . . . . . . . . . . . . . . . 106
16.4 Testing the HeinsenbergLayer with a squeezed state . . . . . . . 108
16.4.1 Proof of equations (16.4) and (16.5)* and (16.9)* . . . . . 109
17 The DifferentialGaussianLayer 113
17.1 Uncertainties in Homodyne detection . . . . . . . . . . . . . . . . 113
17.2 Testing the DifferentialGaussianLayer on coherent state . . . 117
17.3 Using DifferentialGaussianLayer in homodyne detection . . . 119
17.3.1 Proof of Eqs. (17.3) and (17.5) . . . . . . . . . . . . . . . 120
18 Entanglement 121
18.1 Using beam splitters as entangler . . . . . . . . . . . . . . . . . . 121
18.2 Two squeezed states in a beam splitter . . . . . . . . . . . . . . . 121
18.3 Computing the entanglement . . . . . . . . . . . . . . . . . . . . 122
18.4 Training the model to maximize the entanglement . . . . . . . . 125
IV Gaussian Boson Sampling 127
19 Boson sampling introduction 129
20 Boson sampling 131
20.1 Boson sampling in a single mode . . . . . . . . . . . . . . . . . . 131
20.2 Boson sampling with many modes . . . . . . . . . . . . . . . . . 132
21 Simple cases 135
21.1 Using the Hafnian to compute . . . . . . . . . . . . . . . . . . . . 135
22 Machine learning implementation with functional approach 137
22.1 The Q-transform function . . . . . . . . . . . . . . . . . . . . . . 137
22.2 The multiderivative operator . . . . . . . . . . . . . . . . . . . . 140
22.3 Single mode coherent state . . . . . . . . . . . . . . . . . . . . . 142
22.4 Single mode squeezed vacuum state . . . . . . . . . . . . . . . . 144
22.5 Multimode coherent case . . . . . . . . . . . . . . . . . . . . . . . 144
22.6 A coherent mode and a squeezed mode . . . . . . . . . . . . . . . 148
22.7 A squeezed mode and a coherent mode in a random interferometer151
22.8 Using the functional approach to evaluate the derivatives . . . . 151
23 Testing the Boson sample protocol with Haar unitary 155
23.1 The Haar random layer . . . . . . . . . . . . . . . . . . . . . . . 155
23.2 A model with a varying number of layers . . . . . . . . . . . . . 157
23.3 Generating the sampling patterns . . . . . . . . . . . . . . . . . . 157
23.4 Computing the pattern probability . . . . . . . . . . . . . . . . . 158
24 Training a complex medium to enhance multiparticle events 163
24.1 Training by squeezing parameters . . . . . . . . . . . . . . . . . . 173
24.2 Training by linear interferometer . . . . . . . . . . . . . . . . . . 173
24.3 Training by displacing operators . . . . . . . . . . . . . . . . . . 173
V Programming a real quantum computer* 175
25 Introduction 177
26 Xanadu X8 hardware 179
27 Xanadu X8 model 181
28 Xanadu X8 training 183
VI Using NN to minimize many-body Hamiltonians* 185
VII Conclusions and future work 187
VIII Appendices* 189
Lisainfo e-raamatute kohta