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 . 21 6 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 Ï . 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)â and (16.9)â . 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.