applications of tensor networks machine learning quantum
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Applications of Tensor Networks: Machine Learning & Quantum - PowerPoint PPT Presentation

Nov 30, 2022 •355 likes •1.78k views

Applications of Tensor Networks: Machine Learning & Quantum Computing E.M. Stoudenmire Aug 2018 - ICTP Trieste Outline: Yesterday : intro to tensor networks, mainly matrix product states (MPS) computations with MPS intro to

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sha1_base64="NmkH4WbWmUEotzsNTq+cMw9UKIA=">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</latexit> General Philosophy of Machine Learning • Solution to problem just some function y ( x ) • Parameterize very flexible functions f ( x ) (prefer convenient over "correct") • Of all that come closest to for training data, f y prefer the simplest f

  2. Model Architectures

  3. Let's discuss the 3 most used types of models (increasing complexity) • The linear model • Kernel learning / support vector machines • Neural networks

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sha1_base64="93C+N7l2Qo/7ig64G7fC2C4Kjk4=">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</latexit> Example: Linear Supervised Learning Cost function for linear model: 1 X ( W · x ( j ) − y ( j ) ) 2 C = 2 N T j Gradient with respect to n th weight component 1 ∂ C X ( W · x ( j ) − y ( j ) ) x ( j ) = n ∂ W n N T j Update Wn with negative gradient times small step α W n ← W n − α ∂ C ∂ W n

  7. Kernel learning Want to separate classes, say f ( x ) Linear classifier f ( x ) = W · x may be insufficient ? ?

  8. Kernel learning Apply non-linear "feature map" x → Φ ( x )

  9. Kernel learning Apply non-linear "feature map" x → Φ ( x ) Φ

  10. Kernel learning Apply non-linear "feature map" x → Φ ( x ) Φ Decision function f ( x ) = W · Φ ( x )

  11. Kernel learning Φ Decision function f ( x ) = W · Φ ( x ) Linear classifier in feature space

  12. Kernel learning Example of feature map Φ x = ( x 1 , x 2 , x 3 ) Φ ( x ) = (1 , x 1 , x 2 , x 3 , x 1 x 2 , x 1 x 3 , x 2 x 3 ) is "lifted" to feature space x

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sha1_base64="7hTjoVUVOkmqY9Pj8DvZo+ngn10=">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</latexit><latexit sha1_base64="7hTjoVUVOkmqY9Pj8DvZo+ngn10=">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</latexit> Kernel learning Technical notes: • Also called "support vector machine" when using a particular choice of cost function • Name "kernel learning" comes from idea that Φ ( x ) may be too high dimensional, yet K ij = Φ ( x i ) · Φ ( x j ) may be efficiently computable, enough to optimize • Very generally, optimal weights have the form X W = α j Φ( x j ) j a result known as the "representer theorem"

  14. Kernel learning Kernel learning still popular among academics & for certain applications (e.g. life sciences) But "kernelization" approach scales as N 3 where N is size of training set – very costly! Thus kernel methods not popular with engineers Tomorrow : learning kernel models with tensor network weights

  15. Neural networks Current favorite of M.L. engineers Often notated diagrammatically (not a tensor diagram!)

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  18. Neural networks Many successful neural nets include "convolutional layers" These have sparser weight layers with few parameters. Recent upsurge of neural nets since 2012 (ImageNet paper) "Deep learning" often associated with 3 researchers: Yann LeCun (Facebook) Geoff Hinton (Vector/Google) Yoshua Bengio (Montreal)

  19. Other model types Graphical models very similar to tensor networks, except - always interpreted as probability - non-negative parameters only Boltzmann machines identical to random-bond classical Ising (T=1) J ij values learnable parameters generate data by sampling subset of spins Decision trees make decisions about input by taking forking paths

  20. Machine Learning Research Culture One sub-community is academic: papers often involve theorems Another community is engineering-oriented: papers focus on results, developments are intuitive/faddish Conference talks/posters valued above journal articles Strong industry ties: Google, Microsoft, etc. have booths at conferences, grad students poached often

  21. Recommended Resources • Online book by Michael Nielsen (quant. computing author) http://neuralnetworksanddeeplearning.com • Caltech Lectures by Yaser Abu-Mostafa CS 156 Available on YouTube. Companion book "Learning from Data" • M.L. review article by Pankaj Mehta, David Schwab aimed at physicists • TensorFlow examples (MNIST demo) • Blogs of Chris Olah and Andrej Karpathy

  22. Tensor Network Machine Learning

  23. Tensor Network Machine Learning Stoudenmire, Schwab, Advanced in Neural Information Processing Systems (NIPS) , 29 , 4799 [arxiv:1605.05775]

  24. Tensor network methods admit powerful optimization techniques, giving high precision results 10 0 bulk t=0.1 t=0.01 10 -2 t -1 * ( 〈 S + -r/2 S - r/2 〉 - 〈 S + -r/2 〉 〈 S - r/2 〉 ) power-UV 10 -1 power-IR 10 -3 t -2 (-1) r * ( 〈 S z -r/2 S z r/2 〉 - 〈 S z -r/2 〉 〈 S z r/2 〉 ) 10 -2 10 -4 10 -6 t -2 * ( 〈 J -r/2 J r/2 〉 - 〈 J -r/2 〉 〈 J r/2 〉 ) 1 10 100 r Lo, Fukusumi, Oshikawa, Kao, Chen, arxiv:1805.05006 Long-distance properties due to impurities in Luttinger liquids

  25. Tensor networks are highly interpretable, due to linear structure Q Q’ = e i 2 π h i i i Ground state degeneracy Topological spin of anyons i (b) i ∗ Ş ahino ğ lu et al., arxiv:1409.2150 MPO "pulling through" condition Williamson et al., arxiv:1412.5604 Bultinck et al., arxiv:1511.08090

  26. Applicable to classical systems too – tensor RG family of methods = exact TRG(64) TRG+env(64) TEFR(64) TNR(24) c 0.5 0.49982 0.49988 0.49942 0.50001 σ 0.125 0.12498 0.12498 0.12504 0.1250004 1 1.00055 1.00040 0.99996 1.00009 � 1.125 1.12615 1.12659 1.12256 1.12492 1.125 1.12635 1.12659 1.12403 1.12510 2 2.00243 2.00549 - 1.99922 2 2.00579 2.00557 - 1.99986 2 2.00750 2.00566 - 2.00006 2 2.01061 2.00567 - 2.00168 Levin, Nave, PRL 99 , 120601 (2007) Evenbly, Vidal, PRL 115 , 200401 (2015)

  27. ! Wavefunction, transfer matrix just large tensors Tensor network just a math technique Useful for more than physics?

  28. Machine learning has many connections to physics ↓ ↑ ↓ ↓ ↑ ↑ ↓ ↑ 1920s Boltzmann Disordered Machines Ising Model Deep Networks B Deep Architecture . . . . 1970s 2 (1) J (1) J J (1) (1) (1) (1) J J J r . . e . . . . y 1 a L J J J J J J J J J J . . . . . . 0 2 (n+1) = (n) tanh[ J ] = tanh [ J ] The "Renormalization Heirarchical PCA Group" Methods P . Mehta and D.J. Schwab, arxiv:1410.3831 S. Bradde and W. Bialek, arxiv:1610.09733 E.M. Stoudenmire, arxiv:1801.00315

  29. More recent ideas from physics useful for machine learning?

  30. "MERA" tensor network Convolutional neural network

  31. Analogy between wavefunctions & M.L. models machine learning – model functions Neural Nets physics – wavefunctions

  32. Analogy between wavefunctions & M.L. models machine learning – model functions Neural Nets Neural Quantum States physics – wavefunctions

  33. Analogy between wavefunctions & M.L. models machine learning – model functions Neural Nets Tensor Network Neural Quantum States States physics – wavefunctions

  34. Analogy between wavefunctions & M.L. models machine learning – model functions Tensor Network Neural Nets Weights Tensor Network Neural Quantum States States physics – wavefunctions

  35. Are tensor networks useful for machine learning? "MERA" tensor network Tensor networks can represent weights of useful and interesting machine learning models Realized benefits: Future benefits? • Linear scaling • Interpretability / theory • Adaptive weights • Better algorithms • Learning data "features" • Quantum computing

  36. Many proposals already to use tensor networks for machine learning Compressing weights of neural nets (& other models) Yu et al., Advances in Neural Information Processing (2017), arxiv:1711.00073 Izmailov et al., arxiv:1710.07324 (2017) Yang et al., arxiv:1707.01786 (2017) Garipov et al., arxiv:1611.03214 (2016) Novikov et al., Advances in Neural Information Processing (2015) (arxiv:1509.06569) Large scale PCA Lee, Cichocki, arxiv: 1410.6895 (2014) Gaussian Processes Izmailov, Novikov, Kropotov, arxiv:1710.07324 (2017) Feature extraction & tensor completion Bengua et al., arxiv:1606.01500, arxiv:1607.03967, arxiv:1609.04541 (2016) Phien et al., arxiv:1601.01083 (2016) Bengua et al., IEEE Congress on Big Data (2015)

  37. Example : compressing neural network weight layers } ≈ Novikov et al., Advances in Neural Information Processing (2015) (arxiv:1509.06569) Garipov, Podoprikhin, Novikov, arxiv:1611.03214 • Train very "wide" model: 262,144 hidden units • Achieve 80x compression, only 1% accuracy loss

  38. Framework where tensor network plays central role? W Motivation: • Can natural images be more complex than wavefunctions? • Import many ideas, algorithms from physics • Improve tensor network methods

  39. Raw data vectors x = ( x 1 , x 2 , x 3 , . . . , x N ) Example: grayscale images, components of are pixels x x j ∈ [0 , 1]

  40. Propose following model f ( x ) = W · Φ ( x ) X W s 1 s 2 s 3 ··· s N x s 1 1 x s 2 2 x s 3 3 · · · x s N s j = 0 , 1 = N s Weights are N-index tensor Like N-site wavefunction Cohen et al. arxiv:1509.05009 Novikov, Trofimov, Oseledets, arxiv:1605.03795 Stoudenmire, Schwab, arxiv:1605.05775

  41. N=3 example: X f ( x ) = W · Φ ( x ) W s 1 s 2 s 3 x s 1 1 x s 2 2 x s 3 = 3 s = W 000 + W 100 x 1 + W 010 x 2 + W 001 x 3 + W 110 x 1 x 2 + W 101 x 1 x 3 + W 011 x 2 x 3 + W 111 x 1 x 2 x 3 Contains linear classifier, plus other "feature maps"

  42. More generally, apply local "feature maps" φ s j ( x j ) f ( x ) = W · Φ ( x ) W s 1 s 2 s 3 ··· s N φ s 1 ( x 1 ) φ s 2 ( x 2 ) φ s 3 ( x 3 ) · · · φ s N ( x N ) X = s Highly expressive!

  43. x = input For example, following local feature map ⇣ π ⇣ π h ⌘ ⌘i φ ( x j ) = cos , sin x j ∈ [0 , 1] 2 x j 2 x j Picturesque idea of pixels as "spins"

  44. x = input φ = local feature map Total feature map Φ ( x ) Φ s 1 s 2 ··· s N ( x ) = φ s 1 ( x 1 ) ⊗ φ s 2 ( x 2 ) ⊗ · · · ⊗ φ s N ( x N ) • Tensor product of local feature maps / vectors • Just like product state wavefunction of spins • Vector in dimensional space 2 N

  45. x = input φ = local feature map Total feature map Φ ( x ) More detailed notation raw inputs x = [ x 1 , x N ] x 2 , x 3 , . . . , [ [ [ [ [ [ [ [ φ 1 ( ) φ 1 ( ) φ 1 ( ) φ 1 ( ) feature x 1 x 2 x 3 x N Φ ( x ) = ⊗ ⊗ ⊗ ⊗ . . . vector φ 2 ( ) φ 2 ( ) φ 2 ( ) φ 2 ( ) x 1 x 2 x 3 x N

  46. x = input φ = local feature map Total feature map Φ( x ) Tensor diagram notation raw inputs x = [ x 1 , x N ] x 2 , x 3 , . . . , s 3 s 4 s 5 s 6 s N s 1 s 2 feature vector Φ ( x ) = · · · φ s 1 φ s 2 φ s 3 φ s 4 φ s 5 φ s 6 φ s N

  47. Construct decision function f ( x ) = W · Φ ( x ) Φ ( x )

  48. Construct decision function f ( x ) = W · Φ ( x ) W Φ ( x )

  49. Construct decision function f ( x ) = W · Φ ( x ) W f ( x ) = Φ ( x )

  50. Construct decision function f ( x ) = W · Φ ( x ) W f ( x ) = Φ ( x ) W =

  51. Main approximation order-N tensor W = matrix ≈ product state (MPS)

  52. Main approximation order-N tensor W = matrix ≈ product state (MPS) PEPS ) ) ≈

  53. Linear scaling Can use algorithm similar to DMRG to optimize N = size of input Scaling is N · N T · m 3 N T = size of training set m = MPS bond dimension W f ( x ) = Φ ( x )

  54. Linear scaling Can use algorithm similar to DMRG to optimize N = size of input Scaling is N · N T · m 3 N T = size of training set m = MPS bond dimension W f ( x ) = Φ ( x )

  55. Linear scaling Can use algorithm similar to DMRG to optimize N = size of input Scaling is N · N T · m 3 N T = size of training set m = MPS bond dimension W f ( x ) = Φ ( x )

  56. Linear scaling Can use algorithm similar to DMRG to optimize N = size of input Scaling is N · N T · m 3 N T = size of training set m = MPS bond dimension W f ( x ) = Φ ( x )

  57. Linear scaling Can use algorithm similar to DMRG to optimize N = size of input Scaling is N · N T · m 3 N T = size of training set m = MPS bond dimension W f ( x ) = Φ ( x )

  58. Linear scaling Can use algorithm similar to DMRG to optimize N = size of input Scaling is N · N T · m 3 N T = size of training set m = MPS bond dimension W f ( x ) = Φ ( x )

  59. Linear scaling Can use algorithm similar to DMRG to optimize N = size of input Scaling is N · N T · m 3 N T = size of training set m = MPS bond dimension W f ( x ) = Φ ( x )

  60. Linear scaling Can use algorithm similar to DMRG to optimize N = size of input Scaling is N · N T · m 3 N T = size of training set m = MPS bond dimension W f ( x ) = Φ ( x )

  61. Linear scaling Can use algorithm similar to DMRG to optimize N = size of input Scaling is N · N T · m 3 N T = size of training set m = MPS bond dimension W f ( x ) = Φ ( x )

  62. Linear scaling Can use algorithm similar to DMRG to optimize N = size of input Scaling is N · N T · m 3 N T = size of training set m = MPS bond dimension W f ( x ) = Φ ( x )

  63. Linear scaling Can use algorithm similar to DMRG to optimize N = size of input Scaling is N · N T · m 3 N T = size of training set m = MPS bond dimension W f ( x ) = Φ ( x )

  64. Gradient step: At each bond, update "bond tensor" by computing and applying the gradient W B f ( x ) = Φ ( x ) ∂ f ( x ) = = ∂ B = − α B ′ � B ′ �

  65. Why should this work at all? Linear classifier exactly m=2 MPS f ( x ) = V · x W = [ [ [ [ 1 [ 1 [ ˆ ˆ ˆ 0 0 0 1 1 1 · · · ˆ ˆ ˆ [ [ ˆ ˆ ˆ V 3 V 1 V 2 V 0 1 1 ˆ 1 = [1 0] f ( x ) = W · Φ ( x ) ˆ φ s j ( x j ) = [1 , x j ] V j = [0 V j ] Novikov, Trofimov, Oseledets, arxiv:1605.03795

  66. Extendable to multiple outputs l W f l ( x ) = Φ ( x ) Output is a vector over the index l Models exhibit "feature sharing" – only differ in center tensor

  67. Experiment : handwriting classification (MNIST) Train to 99.95% accuracy on 60,000 training images Obtain 99.03% accuracy on 10,000 test images (only 97 incorrect) Stoudenmire, Schwab, arxiv:1605.05775

  68. Papers using tensor network machine learning Expressivity & priors of TN based models • Levine et al., " Deep Learning and Quantum Entanglement: Fundamental Connections with Implications to Network Design " arxiv:1704.01552 • Cohen, Shashua, " Inductive Bias of Deep Convolutional Networks through Pooling Geometry " arxiv:1605.06743 • Cohen et al., " On the Expressive Power of Deep Learning: A Tensor Analysis " arxiv: 1509.05009 Generative Models • Han et al., " Unsupervised Generative Modeling Using Matrix Product States " arxiv: 1709.01662 • Sharir et al., " Tractable Generative Convolutional Arithmetic Circuits " arxiv: 1610.04167 Supervised Learning • Novikov et al., " Expressive power of recurrent neural networks ", arxiv:1711.00811 • Liu et al., " Machine Learning by Two-Dimensional Hierarchical Tensor Networks: A Quantum Information Theoretic Perspective on Deep Architectures ", arxiv: 1710.04833 • Stoudenmire, Schwab, " Supervised Learning with Quantum-Inspired Tensor Networks ", arxiv:1605.05775 • Novikov et al., " Exponential Machines ", arxiv: 1605.03795

  69. Even startups getting into the game! Tunnel Tech, New York City John Terilla Generative Tensorial Networks (GTN), London

  70. Tensor Network Machine Learning Studies

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sha1_base64="W0a/NhSG5s1nLxQWD+he2qxFls0=">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</latexit> Unsupervised Generative Modeling Using MPS Zhao-Yu Han, Jun Wang, Heng Fan, Lei Wang, Pan Zhang • Map data to product state, tensor network weights • Squared output is probability – "Born machine" • "Perfect" sampling (no autocorrelation) 2 x 1 x 5 x 2 x 4 x 3 x 6 p ( x ) = 70 60 50 40 L 30 20 10 ln( |T | ) 0 50 100 150 200 250 300 350 D max (d) row reconstruction on testing Negative Log-Likelihood Reconstructing Testing Images

  72. Machine Learning By Hierarchical Tensor Networks... Ding Liu, Shi-Ju Ran, Peter Wittek, Cheng Peng, Raul Blazquez Garcia, Gang Su, Maciej Lewenstein • Supervised learning with tree tensor networks • Tests on MNIST, CIFAR-10 • Studied properties of the trained model (feature representations, entanglement) (b) (c) (a) (b) (c) (d) (d) (e) (f) Model Architecture Data Representation at Different Scales

  73. Deep Learning and Quantum Entanglement... Yoav Levine, David Yakira, Nadav Cohen, Amnon Shashua • "ConvAC" deep neural net = tree tensor network • Tensor network rank as capacity of model • Experiment on "inductive bias" of model architecture Tree Network as a Inductive Bias Experiment Deep Neural Net

  74. Matrix Product Operators for Sequence to Sequence... Guo, Jie, Lu, Poletti, arxiv:1803.10908 • Product-state input processed by an MPO model • Output is an MPS, approximated as another product state • Capabilities like recurrent neural nets; better results than LSTM! time series prediction errors 0 0.1 bidirectional-LSTM 0.05 0 1 25 50 75 100 MPO

  75. Learning Relevant Features of Data... E.M. Stoudenmire • Unsupervised determination of tree tensor network (compress data) • Supervised training of top layer • Excellent performance with "features" determined by tree tensors supervised top layer unsupervised tree data input ρ µ = X (1 − µ ) + µ j 89% accuracy on mixed training Fashion MNIST data set supervised / unsupervised

  76. Supervised Learning with Generalized Tensor Networks I. Glasser, N. Pancotti, J.I. Cirac, arxiv:1806.05964 • Models where inputs are copied, then processed by multiple tensor networks • Hybrid CNN / string bond architecture gives 92.3% on fashion MNIST test set! fashion MNIST entangled plaquette states string-bond states learning local features: (a) (b)

  77. Quantum Machine Learning with Tensor Networks Huggins, Patil, Whaley, Stoudenmire, arxiv:1803.11537 Grant, Benedetti, et al., arxiv:1804.03680 Bill Huggins

  78. What is a quantum computer? A set of coherent qubits for which one can: • efficiently prepare certain initial states • apply unitary operations (usually 1- and 2-qubit) • perform measurements

  79. <latexit sha1_base64="I/K6ahtHFVW/ESfgRWn+LDloxCU=">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</latexit><latexit sha1_base64="I/K6ahtHFVW/ESfgRWn+LDloxCU=">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</latexit><latexit sha1_base64="I/K6ahtHFVW/ESfgRWn+LDloxCU=">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</latexit><latexit sha1_base64="I/K6ahtHFVW/ESfgRWn+LDloxCU=">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</latexit> What is a quantum computer? A set of coherent qubits for which one can: • efficiently prepare certain initial states • apply unitary operations (usually 1- and 2-qubit) • perform measurements → 0 , 1

  80. <latexit sha1_base64="I/K6ahtHFVW/ESfgRWn+LDloxCU=">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</latexit><latexit sha1_base64="I/K6ahtHFVW/ESfgRWn+LDloxCU=">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</latexit><latexit sha1_base64="I/K6ahtHFVW/ESfgRWn+LDloxCU=">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</latexit><latexit sha1_base64="I/K6ahtHFVW/ESfgRWn+LDloxCU=">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</latexit> What is a quantum computer? A set of coherent qubits for which one can: • efficiently prepare certain initial states • apply unitary operations (usually 1- and 2-qubit) • perform measurements → 0 , 1 <latexit sha1_base64="I/K6ahtHFVW/ESfgRWn+LDloxCU=">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</latexit><latexit sha1_base64="I/K6ahtHFVW/ESfgRWn+LDloxCU=">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</latexit><latexit sha1_base64="I/K6ahtHFVW/ESfgRWn+LDloxCU=">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</latexit><latexit sha1_base64="I/K6ahtHFVW/ESfgRWn+LDloxCU=">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</latexit> → 0 , 1 <latexit sha1_base64="I/K6ahtHFVW/ESfgRWn+LDloxCU=">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</latexit><latexit sha1_base64="I/K6ahtHFVW/ESfgRWn+LDloxCU=">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</latexit><latexit sha1_base64="I/K6ahtHFVW/ESfgRWn+LDloxCU=">ACx3icdVFNTxsxEHUW2tL0g1COXKyukHqonVUqRyRuJQblQgZVfRrDPZWPHyvYGoiWH/o5eOJQf1X+Ds+QAJIxk6enNe35jT15K4XyS/G9FW9tv3r7bed/+8PHT593O3pcLZyrLsc+NPYqB4dSaOx74SVelRZB5RIv8+nJsn85Q+uE0ed+XmKmoNBiLDj4QA07u6kVxcSDteaJt/ZsBMn3aQpug7YCsRkVWfDvdZdOjK8Uqg9l+DcgCWlz2qwXnCJi3ZaOSyBT6HAQYAaFLqsbiZf0MPAjOjY2HC0pw371FGDcm6u8qBU4CfuZW9JbuoNKj8+ymqhy8qj5o9B40pSb+jyG+hIWORezgMAbkWYlfIJWOA+fFY71XjNjVKgR3U6Rb8YsKxOUbvK4jKrvo1ZakEX4YGL5+rcwpo6lY0Zrcb1M31vY0Ghxj76SBLPitaRgfOIKO2UvN7gOLnpdlnTZ7x/x8clquzvkgHwl3wgjP8kx+UXOSJ9wUpG/5B+5j04jE82im0dp1Fp59smziv48AMEa4Xo=</latexit><latexit sha1_base64="I/K6ahtHFVW/ESfgRWn+LDloxCU=">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</latexit> → 0 , 1 <latexit 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  81. Two recent ideas for machine learning with a quantum computer 1. supervised / discriminative learning Farhi, Neven, arxiv:1802.0600 Schuld, Killoran, arxiv:1803.07128

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sha1_base64="d5LGu7QT6qVk3LZkYdjrGkldXMg=">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</latexit> Two recent ideas for machine learning with a quantum computer 1. supervised / discriminative learning U D • prepare data as product state • apply gates to prepared state x → Farhi, Neven, arxiv:1802.0600 Schuld, Killoran, arxiv:1803.07128

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sha1_base64="9jLyt9okHWTekNW/5amNrwLzQIc=">ACuXicdVHBThsxEHW2LdBAKbTHXqyukDhF6wgJpF6Q4NAjiIZEyq6iWeyMbG9K9sLipZ8Atf2t/ib3CWHAhJRrL09OY9v7EnLaSwLoqeG8GHj5+2tnc+N3f3vux/PTj8dmvz0nDs8FzmpeCRSk0dpxwEnuFQVCpxG46uZj3u/dorMj1HzctMFGQaTESHJynbjqDy8FBGLWiugqYAsQkVdDQ4b/+NhzkuF2nEJ1vZVLikAuMElzhrxqXFAvgEMux7qEGhTap61hk98syQjnLj3a0Zt86KlDWTlXqlQrc2L7vzcl1vX7pRmdJXROtT8NWhUSupyOn84HQqD3MmpB8CN8LNSPgYD3PnvacYaH3iuFOhFU/QzfosqWLUtjQ4z6oeQxYb0Jl/4GxZnRpYUceylobscY26vr691kC9I2zTDUlwn21K8sY3Lr9T9n6Dq+C23WJRi12fhOcXi+3ukB/kJzkmjJySc/KbXJEO4SQjT+Qv+Rf8CiAYB3ev0qCx8HwnSxXYFyRP3Hc=</latexit><latexit sha1_base64="9jLyt9okHWTekNW/5amNrwLzQIc=">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</latexit> Two recent ideas for machine learning with a quantum computer 1. supervised / discriminative learning U D • prepare data as product state • apply gates to prepared state x → • measure output qubit → 0 , 1 Farhi, Neven, arxiv:1802.0600 Schuld, Killoran, arxiv:1803.07128

  85. Two recent ideas for machine learning with a quantum computer 2. generative modeling Gao, Zhang, Duan, arxiv:1711.02038 Benedetti, Garcia-Pintos, Nam, Perdomo-Ortiz, arxiv:1801.07686 


  86. Two recent ideas for machine learning with a quantum computer 2. generative modeling • prepare reference state Gao, Zhang, Duan, arxiv:1711.02038 Benedetti, Garcia-Pintos, Nam, Perdomo-Ortiz, arxiv:1801.07686 


  87. <latexit sha1_base64="PXscFMw4nvYexrqeqsh7vZqGrDw=">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</latexit><latexit sha1_base64="PXscFMw4nvYexrqeqsh7vZqGrDw=">ACuXicdVHBThsxEHW2LdBAKbTHXqyukDhF6wgJpF6QONAjiIZEyq6iWeyMbG9K9sLipZ8Atf2t/ib3CWHAhJRrL09OY9v7EnLaSwLoqeG8GHj5+2tnc+N3f3vux/PTj8dmvz0nDs8FzmpeCRSk0dpxwEnuFQVCpxG46uZj3u/dorMj1HzctMFGQaTESHJynbjqDy8FBGLWiugqYAsQkVdDQ4b/+NhzkuF2nEJ1vZVLikAuMElzhrxqXFAvgEMux7qEGhTap61hk98syQjnLj3a0Zt86KlDWTlXqlQrc2L7vzcl1vX7pRmdJXROtT8NWhUSupyOn84HQqD3MmpB8CN8LNSPgYD3PnvacYaH3iuFOhFU/QzfosqWLUtjQ4z6oeQxYb0Jl/4GxZnRpYUceylobscY26vr691kC9I2zTDUlwn21K8sY3Lr9T9n6Dq+C23WJRi12fhOcXi+3ukB/kJzkmjJySc/KbXJEO4SQjT+Qv+Rf8CiAYB3ev0qCx8HwnSxXYFytI3Ho=</latexit><latexit sha1_base64="PXscFMw4nvYexrqeqsh7vZqGrDw=">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</latexit><latexit sha1_base64="PXscFMw4nvYexrqeqsh7vZqGrDw=">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</latexit> Two recent ideas for machine learning with a quantum computer 2. generative modeling U G • prepare reference state • apply gates to qubit Gao, Zhang, Duan, arxiv:1711.02038 Benedetti, Garcia-Pintos, Nam, Perdomo-Ortiz, arxiv:1801.07686 


  88. <latexit sha1_base64="PXscFMw4nvYexrqeqsh7vZqGrDw=">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</latexit><latexit 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sha1_base64="2nIHmcpuqNasfc+yVKSjsqJZf+8=">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</latexit><latexit sha1_base64="2nIHmcpuqNasfc+yVKSjsqJZf+8=">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</latexit> Two recent ideas for machine learning with a quantum computer 2. generative modeling U G • prepare reference state • apply gates to qubit → x • measure all qubits Gao, Zhang, Duan, arxiv:1711.02038 Benedetti, Garcia-Pintos, Nam, Perdomo-Ortiz, arxiv:1801.07686 


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sha1_base64="Jt93F4dYNDzyEPAyamQ54qlMrI=">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</latexit> h 0 | Tensor networks are equivalent to quantum circuits h 0 | h 0 | h 0 | h 0 | h 0 | h 0 | h 0 | = D = 4

  93. Quantum circuit for matrix product state ( m = 4) =

  94. Quantum circuit for matrix product state ( m = 4) =

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