Classical and Quantum Machine Learning with Tensor Networks

Transcription

1 Classical and Quantum Machine Learning with Tensor Networks E.M. Stoudenmire M INTRIQ

2 Flatiron Institute The mission of the Flatiron Institute is to advance scientific research tough computational methods, including data analysis, modeling and simulation. CCA: Center for Computational Astrophysics CCB: Center for Computational Biology CCQ: Center for Computational Quantum Physics Plus fourth center to be decided

3 Overview Tensor networks = powerful model of many-body wavefunctions Tensor networks can parameterize machine learning models too Resulting learning architecture has interesting capabilities, connections to quantum computing

4 Machine learning galvanizing industry & science Language Processing Self-driving cars Medicine Materials Science / Chemistry

5 Google rebranded a "machine learning first company" Neural nets replace linguistic approach to Google Translate Quantum machine learning

6 Doing tasks thought impossible... Under review as a conference paper at ICLR 2016 U NSUPERVISED R EPRESENTATION L EARNING WITH D EEP C ONVOLUTIONAL G ENERATIVE A DVERSARIAL N ETWORKS Alec Radford & Luke Metz indico Research Vector arithmetic for visual concepts. For each column, the Z vectors of samples are Boston, MA Arithmetic was then performed on the mean vectors creating a new vector Y. The center {alec,luke}@indico.io 7 Jan 2016 Figure 7: averaged. sample on the right hand side is produce by feeding Y as input to the generator. To demonstrate the interpolation capabilities of the generator, uniform noise sampled with scale was added Chintala to Y to produce the Soumith 8 other samples. Applying arithmetic in the input space (bottom two examples) Facebook AI Research results in noisy overlap misalignment. Newdue York,toNY soumith@fb.com Further work is needed to tackle this from of instability. We think that extending this framework A BSTRACT

7 Machine learning has physics in its DNA # " # # " " # " Boltzmann Machines Disordered Ising Model Deep Belief Networks The "Renormalization Group" P. Mehta and D.J. Schwab, arxiv: S. Bradde and W. Bialek, arxiv:

8 Convolutional neural network "MERA" tensor network

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

10 What is machine learning?

11 What is machine learning? Data driven problem solving Any system that, given more data, performs increasingly better at some task Framework / philosophy, not single method

12 Goal: train model f(x) mapping input x to target f(x) 1 0 "dog"

13 Goal: train model f(x) mapping input x to target f(x) 0 1 "cat"

14 Philosophy of Machine Learning Map from images to labels is just a function Parameterize a set of very flexible functions (prefer convenient functions over "correct" ones) Prevent overfitting by regularization (prefer simple functions) = training data

15 Philosophy of Machine Learning Map from images to labels is just a function Parameterize a set of very flexible functions (prefer convenient functions over "correct" ones) Prevent overfitting by regularization (prefer simple functions) = training data = testing data

16 Supervised Learning Given labeled training data (labels A and B) Find decision function f(x) > 0 f(x) < 0 f(x) x 2 A x 2 B Example: identify photos of alligators and bears

17 Unsupervised Learning Given unlabeled training data {x j } Find function Find function f(x) such that f(x j ) ' p(x j ) f(x) such that f(x j ) 2 ' p(x j ) Find data clusters and which data belongs to each cluster Discover reduced representations of data for other learning tasks (e.g. supervised)

18 What are Tensor Networks?

19 Original setting is quantum mechanics Spin model (transverse field Ising model): z z h x Ĥ = X j ( z j z j+1 h x j ) " " " " " " " h<<1 " " " " " " "h>>1

20 Wavefunction just a rule to map spin configurations to numbers s 1 s 2 s 3 s 4 s 5 s 6 s 7 s 8 " # " " " " " " " # " " " " " " " # " # " " " " " " " " # " # " " # " # " " " " " " " " # " # " " # # # # " " " " # # # # " " " Simplest rule: store every amplitude separately

21 [ Let's make a different rule Introduce matrices, one for each spin " M " = [ # M # = [ [ Östlund, Rommer, Phys. Rev. Lett. 75, 3537 (1995)

22 Compute wavefunction by multiplying matrices together " # " " # M " 1 M # 2 M " 3 M " 4 M # 5 " " # # # M " 1 M " 2 M # 3 M # 4 M # 5 " # # " " M " 1 M # 2 M # 3 M " 4 M " 5

23 This rule is called a matrix product state (MPS) s 1 s 2 s 3 s 4 s 5 = M s 1 1 M s 2 2 M s 3 3 M s 4 4 M s 5 5 Size of matrices called m (the "bond dimension") For m = 2 N/2 can represent any state of N spins Just a w to compress a big tensor Represents 2 N amplitudes using only (2 N m 2 ) parameters

24 Tensor Diagrams

25 Helpful to draw N-index tensor as blob with N lines s 1 s 2 s 3 s 4 s N s 1 s 2 s 3 s N = No symmetries or transformation properties assumed

26 Diagrams for simple tensors j v j i j M ij i j k T ijk

27 Joining lines implies contraction, can omit names X i j j M ij v j A ij B jk = AB A ij B ji =Tr[AB]

28 Matrix product state in diagram notation s 1 s 2 s 3 s 4 s 5 s 6 = X M s 1 1 M s M s M s M s M s 6 5 s 1 s 2 s 3 s 4 s 5 s 6 = Can suppress index names, very convenient

29 Matrix product state in diagram notation s 1 s 2 s 3 s 4 s 5 s 6 = X M s 1 1 M s M s M s M s M s 6 5 s 1 s 2 s 3 s 4 s 5 s 6 = Can suppress index names, very convenient

30 Matrix product state in diagram notation s 1 s 2 s 3 s 4 s 5 s 6 = X M s 1 1 M s M s M s M s M s 6 5 = Can suppress index names, very convenient

31 Besides MPS, other successful tensor are PEPS and MERA PEPS (2D systems) MERA (critical systems) Evenbly, Vidal, PRB 79, (2009) Verstraete, Cirac, cond-mat/ (2004) Orus, Ann. Phys. 349, 117 (2014)

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

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

34 Propose following model f(x) =W (x) = X s W s1 s 2 s 3 s N x s 1 1 xs 2 2 xs 3 3 xs N N s j =0, 1 Weights are N-index tensor Like N-site wavefunction Novikov, Trofimov, Oseledets, arxiv:

35 N=3 example: f(x) =W (x) = X s W s1 s 2 s x s xs 2 2 xs 3 3 = W 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"

36 More generally, apply local "feature maps" s j (x j ) f(x) =W (x) = X s W s1 s 2 s 3 s N s 1 (x 1 ) s 2 (x 2 ) s 3 (x 3 ) s N (x N ) Highly expressive! Cohen et al. arxiv: Novikov, Trofimov, Oseledets, arxiv: Stoudenmire, Schwab, arxiv:

37 x = input For example, following local feature map (x j )= h cos 2 x j i, sin 2 x j x j 2 [0, 1] Picturesque idea of pixels as "spins" or "qubits"

38 x = input = local feature map Total feature map (x) s 1 s 2 s N (x) = s 1 (x 1 ) s 2 (x 2 ) s N (x N ) Tensor product of local feature maps / vectors Just like product state wavefunction of spins Vector in dimensional space 2 N

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

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

41 Construct decision function f(x) =W (x) (x)

42 Construct decision function f(x) =W (x) W (x)

43 Construct decision function f(x) =W (x) f(x) = W (x) W =

44 Main approximation W = order-n tensor matrix product state (MPS)

45 ) Main approximation W = order-n tensor matrix product state (MPS) PEPS )

46 Can use algorithm similar to DMRG to optimize Number of parameters is adaptive Scaling is N N T m 3 N = size of input N T = size of training set m = MPS bond dimension f(x) = W (x)

47 Can use algorithm similar to DMRG to optimize Number of parameters is adaptive Scaling is N N T m 3 N = size of input N T = size of training set m = MPS bond dimension f(x) = W (x)

48 Can use algorithm similar to DMRG to optimize Number of parameters is adaptive Scaling is N N T m 3 N = size of input N T = size of training set m = MPS bond dimension f(x) = W (x)

49 Can use algorithm similar to DMRG to optimize Number of parameters is adaptive Scaling is N N T m 3 N = size of input N T = size of training set m = MPS bond dimension f(x) = W (x)

50 Can use algorithm similar to DMRG to optimize Number of parameters is adaptive Scaling is N N T m 3 N = size of input N T = size of training set m = MPS bond dimension f(x) = W (x)

51 Can use algorithm similar to DMRG to optimize Number of parameters is adaptive Scaling is N N T m 3 N = size of input N T = size of training set m = MPS bond dimension f(x) = W (x)

52 [ Why should this work at all? Linear classifier f(x) =V x exactly m=2 MPS W = [ V 0 1 [ [ [ ˆ1 0 ˆV 1 ˆ1 [ [ ˆ1 0 ˆV 2 ˆ1 [ ˆ1 0 ˆV3 ˆ1 ˆ1 = [1 0] f(x) =W (x) ˆV j =[0 V j ] s j (x j )=[1,x j ] Novikov, Trofimov, Oseledets, arxiv:

53 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:

54 Tensor Network Machine Learning Studies

55 Deep Learning and Quantum Entanglement... Yoav Levine, David Yakira, Nadav Cohen, Amnon Shashua (arxiv: ) "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 Deep Neural Net Inductive Bias Experiment

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57 )). Compared with Refs. (Stoudenmire & Schwab, 2016; Han et al., 2017) where a onel (1D) TN (called matrix product state (MPS) (O stlund & Rommer, 1995)) is used, TTN he two-dimensional (2D) nature of images. The algorithm is inspired by the multipartite nt renormalization ansatz (MERA) approach (Vidal, 2007; 2008; Cincio et al., 2008; Vidal,Ding 2009),Liu, where the tensors the TN Wittek, are kept tocheng be unitary duringraul the training. We Garcia, Gang Su, Maciej Lewenstein Shi-Ju Ran,in Peter Peng, Blazquez orithm on both the MNIST (handwriting recognition with binary images) and CIFAR (arxiv: ) n of color images) databases and obtain accuracies comparable to the performance of al neural networks. More importantly, the TN states can then be defined that optimally ch class of images as a quantum many-body state, which is akin to the study of a duality obabilistic graphical modelslearning and TNs (Robeva Seigal,tensor 2017). Wenetworks contrast the bond Supervised with&tree and model complexity, with results indicating that a growing bond dimension overfits the udy the representation in the different lers in the hierarchical TN with t-sne (Van der Tests on MNIST, CIFAR-10 Hinton, 2008), and find that the level of abstraction changes the same w as in a deep al neural network (Krizhevsky et al., 2012) or a deep belief network (Hinton et al., 2006), Under review as a conference paper at ICLR 2018 Studied properties of the trained model (feature entanglement) hest level of the hierarchy allows for a clear separation of the classes. Finally, we showrepresentations, lities between each two TN states from the two different image classes are low, and we e entanglement entropy of each TN state, which gives an indication of the difficulty of Machine Learning By Hierarchical Tensor Networks... (b) (c) (a) (b) (c) (d) (e) (f) (d) Figure 3: Embedding of data instances of CIFAR-10 by t-sne corresponding to each ler in the TTN: (a) original data distribution and (b) the 1st, (c) 2nd, (d) 3rd, (e) 4th, and (f) 5th ler. Color online) The left figure (a) shows the configuration of TTN. The squares at the esent the vectors obtainedmodel from thearchitecture pixels of one image tough the feature map. The Data Representation at B ENCHMARK ON CIFAR-10 e top represents the label. The right figures are (b) the illustrations of 4.1 the environment Different Scales To verify the representation power of TTNs, we used the CIFAR-10 dataset (Krizhevsky & Hinton, he schematic diagram of fidelity and (d) entanglement entropy calculation. 2009). The dataset consists of 60, RGB images in 10 classes, with 6,000 instances per class. There are 50,000 training images and 10,000 test images. Each RGB image was originally pixels: we transformed them to grscale. Working with gr-scale images reduced the complexity of training, with the trade-off being that less information was available for learning.

58 Learning Relevant Features of Data... E.M. Stoudenmire (Quant. Sci. Tech. 3, [2018]) Unsupervised determination of tree tensor network (compress data) Supervised training of top ler Excellent performance with "features" determined by tree tensors supervised top ler unsupervised tree data input µ = (1 µ) X j + µ 89% accuracy on Fashion MNIST data set mixed training supervised / unsupervised

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

60 Two recent proposals for machine learning with a quantum computer 1. supervised / discriminative Farhi, Neven, arxiv: Schuld, Killoran, arxiv:

61 Two recent proposals for machine learning with a quantum computer 1. supervised / discriminative x! Farhi, Neven, arxiv: Schuld, Killoran, arxiv:

62 Two recent proposals for machine learning with a quantum computer 1. supervised / discriminative U D x! Farhi, Neven, arxiv: Schuld, Killoran, arxiv:

63 Two recent proposals for machine learning with a quantum computer 1. supervised / discriminative U D x!! 0, 1 Farhi, Neven, arxiv: Schuld, Killoran, arxiv:

64 Two recent proposals for machine learning with a quantum computer 2. unsupervised / generative Gao, Zhang, Duan, arxiv: Benedetti, Garcia-Pintos, Nam, Perdomo-Ortiz, arxiv:

65 Two recent proposals for machine learning with a quantum computer 2. unsupervised / generative Gao, Zhang, Duan, arxiv: Benedetti, Garcia-Pintos, Nam, Perdomo-Ortiz, arxiv:

66 Two recent proposals for machine learning with a quantum computer 2. unsupervised / generative U G Gao, Zhang, Duan, arxiv: Benedetti, Garcia-Pintos, Nam, Perdomo-Ortiz, arxiv:

67 <latexit sha1_base64="2nihmcpuqnasfc+yvksjsqjzf+8=">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</latexit> <latexit sha1_base64="2nihmcpuqnasfc+yvksjsqjzf+8=">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</latexit> <latexit sha1_base64="2nihmcpuqnasfc+yvksjsqjzf+8=">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</latexit> <latexit sha1_base64="2nihmcpuqnasfc+yvksjsqjzf+8=">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</latexit> Two recent proposals for machine learning with a quantum computer 2. unsupervised / generative U G! x Gao, Zhang, Duan, arxiv: Benedetti, Garcia-Pintos, Nam, Perdomo-Ortiz, arxiv:

68 <latexit sha1_base64="9jlyt9okhwteknw/5amnrwlzqic=">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</latexit> <latexit sha1_base64="9jlyt9okhwteknw/5amnrwlzqic=">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</latexit> <latexit sha1_base64="9jlyt9okhwteknw/5amnrwlzqic=">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</latexit> <latexit sha1_base64="9jlyt9okhwteknw/5amnrwlzqic=">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</latexit> <latexit sha1_base64="pxscfmw4nvyexrqeqsh7vzqgrdw=">aaacuxicdvhbthsxehw2ldbakbthxqyukd6wgjpf6qonajiizeyq6iwweymbg9k9ipz8atf22t/ib3cwhahjrrl09oy9v7enlaswloqeg8ghj5+2tnc+n3f3vux/ptj8dmvz0nds8fzmppecrsk0dpxwenufqvcpxg46uzj3u/dormj1hzctmfgqateshjynbjqdy8fbglwiuugqyasqkkvddq4b/+nhzkuf2nej1vzzvlikaumelzxqxfavgemux7qeghtap61hk98syqjnljj3a0zt86kldwtlxqlqrc2l7vzcl1vx7prmdjjxrrott8nwhusupyon84hqqd3mmpb8cn8lnspgyd3pnvacyah3iufohu/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> <latexit sha1_base64="pxscfmw4nvyexrqeqsh7vzqgrdw=">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</latexit> Two issues with these proposals U D U G discriminative generative

69 <latexit sha1_base64="9jlyt9okhwteknw/5amnrwlzqic=">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</latexit> <latexit sha1_base64="9jlyt9okhwteknw/5amnrwlzqic=">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</latexit> <latexit sha1_base64="9jlyt9okhwteknw/5amnrwlzqic=">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</latexit> <latexit sha1_base64="9jlyt9okhwteknw/5amnrwlzqic=">aaacuxicdvhbthsxehw2ldbakbthxqyukd6wgjpf6q4najiizeyq6iwweymbg9k9ipz8atf22t/ib3cwhahjrrl09oy9v7enlaswloqeg8ghj5+2tnc+n3f3vux/ptj8dmvz0nds8fzmppecrsk0dpxwenufqvcpxg46uzj3u/dormj1hzctmfgqateshjynbjqdy8fbglwiuugqyasqkkvddq4b/+nhzkuf2nej1vzzvlikaumelzxqxfavgemux7qeghtap61hk98syqjnljj3a0zt86kldwtlxqlqrc2l7vzcl1vx7prmdjjxrrott8nwhusupyon84hqqd3mmpb8cn8lnspgyd3pnvacyah3iufohu/qzfosqwlutjq4z6oeqxyb0jl/4gxznrpyuceylobscy26vr691kc9i2ztdulwn21k8sy3lr9t9n6dq+c23wjri12fhocxi+3ukb/kjzkmjjysc/kbxjeo4sqjt+qv+rf8ciayb3ev0qcx8hwnsxxyfyrp3hc=</latexit> <latexit sha1_base64="pxscfmw4nvyexrqeqsh7vzqgrdw=">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</latexit> <latexit sha1_base64="pxscfmw4nvyexrqeqsh7vzqgrdw=">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</latexit> <latexit sha1_base64="pxscfmw4nvyexrqeqsh7vzqgrdw=">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</latexit> <latexit sha1_base64="pxscfmw4nvyexrqeqsh7vzqgrdw=">aaacuxicdvhbthsxehw2ldbakbthxqyukd6wgjpf6qonajiizeyq6iwweymbg9k9ipz8atf22t/ib3cwhahjrrl09oy9v7enlaswloqeg8ghj5+2tnc+n3f3vux/ptj8dmvz0nds8fzmppecrsk0dpxwenufqvcpxg46uzj3u/dormj1hzctmfgqateshjynbjqdy8fbglwiuugqyasqkkvddq4b/+nhzkuf2nej1vzzvlikaumelzxqxfavgemux7qeghtap61hk98syqjnljj3a0zt86kldwtlxqlqrc2l7vzcl1vx7prmdjjxrrott8nwhusupyon84hqqd3mmpb8cn8lnspgyd3pnvacyah3iufohu/qzfosqwlutjq4z6oeqxyb0jl/4gxznrpyuceylobscy26vr691kc9i2ztdulwn21k8sy3lr9t9n6dq+c23wjri12fhocxi+3ukb/kjzkmjjysc/kbxjeo4sqjt+qv+rf8ciayb3ev0qcx8hwnsxxyfyti3ho=</latexit> Two issues with these proposals U D U G discriminative generative 1. efficient parameterization of N-qubit circuit? (vanishing gradient?*) * McClean, Boixo, et al., arxiv:

70 <latexit sha1_base64="9jlyt9okhwteknw/5amnrwlzqic=">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</latexit> <latexit sha1_base64="9jlyt9okhwteknw/5amnrwlzqic=">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</latexit> <latexit sha1_base64="9jlyt9okhwteknw/5amnrwlzqic=">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</latexit> <latexit sha1_base64="9jlyt9okhwteknw/5amnrwlzqic=">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</latexit> <latexit sha1_base64="pxscfmw4nvyexrqeqsh7vzqgrdw=">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</latexit> <latexit sha1_base64="pxscfmw4nvyexrqeqsh7vzqgrdw=">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</latexit> <latexit sha1_base64="pxscfmw4nvyexrqeqsh7vzqgrdw=">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</latexit> <latexit sha1_base64="pxscfmw4nvyexrqeqsh7vzqgrdw=">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</latexit> Two issues with these proposals U D U G discriminative generative 1. efficient parameterization of N-qubit circuit? (vanishing gradient?*) 2. require too many qubits for realistic data sizes * McClean, Boixo, et al., arxiv:

71 D =4 <latexit sha1_base64="jfp3kcp4tz8dri6gelidalutbak=">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</latexit> <latexit sha1_base64="jfp3kcp4tz8dri6gelidalutbak=">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</latexit> <latexit sha1_base64="jfp3kcp4tz8dri6gelidalutbak=">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</latexit> h0 yz EE AP Qm 45 lq rm =I >" AA CA Hv ci Vd LH AN JF G3 3d r3 s0 g1 FE Vy Ap qj NG ws rv Sh Ib fz 2S pd Zm Jh Jl cp 8R Fh dv F+ +v dj dh 6p ey wb j7 zl 0X uk Lu N8 wn Xa ll Wd 51 rr 5G bt z2 7u /e mc zk DT sd Et hp S8 CM 4i ow l7 DB kf ri bc Fp Te Hf Ee v2 MB d3 xl rj EG eo DI sz 6A ls sg gq te xz 1g QZ F0 R4 4T BZ 3o 7j Nu 6j fi ik ks Ay bm w0 Gy c1 rg ca WU we Cb HE Os pk ob r4 ul 6t HJ uj Tn aq db 5U Wb E7 H9 dc YK cy ou 9J nm z9 3E ej lr Yf Vw cl pc XU 7X jl oq LB ZU Qn Px 1+ jw 2s DA ND ax U+ GE tz K2 pm CH 5Z JZ fp ir an Gf YZ KT no ct Xo 4P 0w Kq be Ps tq md cu UU ie vl xy 1x X/ 71 or 6E 9h m+ 1/ CS f0 /C 47 +w vg 96 C4 oz Ft nv er 2D Lx 8+ hm 35 Rl Qy 3Q MJ nq KJ 1k Su de kw Kh sx bk fe uc PP SR 3T L5 cv Ub 0s +9 Th 6f va II j3 =M /< al et ix >t yz EE AP Qm 45 lq rm =I >" AA CA Hv ci Vd LH AN JF G3 3d r3 s0 g1 FE Vy Ap qj NG ws rv Sh Ib fz 2S pd Zm Jh Jl cp 8R Fh dv F+ +v dj dh 6p ey wb j7 zl 0X uk Lu N8 wn Xa ll Wd 51 rr 5G bt z2 7u /e mc zk DT sd Et hp S8 CM 4i ow l7 DB kf ri bc Fp Te Hf Ee v2 MB d3 xl rj EG eo DI sz 6A ls sg gq te xz 1g QZ F0 R4 4T BZ 3o 7j Nu 6j fi ik ks Ay bm w0 Gy c1 rg ca WU we Cb HE Os pk ob r4 ul 6t HJ uj Tn aq db 5U Wb E7 H9 dc YK cy ou 9J nm z9 3E ej lr Yf Vw cl pc XU 7X jl oq LB ZU Qn Px 1+ jw 2s DA ND ax U+ GE tz K2 pm CH 5Z JZ fp ir an Gf YZ KT no ct Xo 4P 0w Kq be Ps tq md cu UU ie vl xy 1x X/ 71 or 6E 9h m+ 1/ CS f0 /C 47 +w vg 96 C4 oz Ft nv er 2D Lx 8+ hm 35 Rl Qy 3Q MJ nq KJ 1k Su de kw Kh sx bk fe uc PP SR 3T L5 cv Ub 0s +9 Th 6f va II j3 =M /< al et ix >t yz EE AP Qm 45 lq rm =I >" AA CA Hv ci Vd LH AN JF G3 3d r3 s0 g1 FE Vy Ap qj NG ws rv Sh Ib fz 2S pd Zm Jh Jl cp 8R Fh dv F+ +v dj dh 6p ey wb j7 zl 0X uk Lu N8 wn Xa ll Wd 51 rr 5G bt z2 7u /e mc zk DT sd Et hp S8 CM 4i ow l7 DB kf ri bc Fp Te Hf Ee v2 MB d3 xl rj EG eo DI sz 6A ls sg gq te xz 1g QZ F0 R4 4T BZ 3o 7j Nu 6j fi ik ks Ay bm w0 Gy c1 rg ca WU we Cb HE Os pk ob r4 ul 6t HJ uj Tn aq db 5U Wb E7 H9 dc YK cy ou 9J nm z9 3E ej lr Yf Vw cl pc XU 7X jl oq LB ZU Qn Px 1+ jw 2s DA ND ax U+ GE tz K2 pm CH 5Z JZ fp ir an Gf YZ KT no ct Xo 4P 0w Kq be Ps tq md cu UU ie vl xy 1x X/ 71 or 6E 9h m+ 1/ CS f0 /C 47 +w vg 96 C4 oz Ft nv er 2D Lx 8+ hm 35 Rl Qy 3Q MJ nq KJ 1k Su de kw Kh sx bk fe uc PP SR 3T L5 cv Ub 0s +9 Th 6f va II j3 =M /< al et ix >t h0 yz EE AP Qm 45 lq rm =I >" AA CA Hv ci Vd LH AN JF G3 3d r3 s0 g1 FE Vy Ap qj NG ws rv Sh Ib fz 2S pd Zm Jh Jl cp 8R Fh dv F+ +v dj dh 6p ey wb j7 zl 0X uk Lu N8 wn Xa ll Wd 51 rr 5G bt z2 7u /e mc zk DT sd Et hp S8 CM 4i ow l7 DB kf ri bc Fp Te Hf Ee v2 MB d3 xl rj EG eo DI sz 6A ls sg gq te xz 1g QZ F0 R4 4T BZ 3o 7j Nu 6j fi ik ks Ay bm w0 Gy c1 rg ca WU we Cb HE Os pk ob r4 ul 6t HJ uj Tn aq db 5U Wb E7 H9 dc YK cy ou 9J nm z9 3E ej lr Yf Vw cl pc XU 7X jl oq LB ZU Qn Px 1+ jw 2s DA ND ax U+ GE tz K2 pm CH 5Z JZ fp ir an Gf YZ KT no ct Xo 4P 0w Kq be Ps tq md cu UU ie vl xy 1x X/ 71 or 6E 9h m+ 1/ CS f0 /C 47 +w vg 96 C4 oz Ft nv er 2D Lx 8+ hm 35 Rl Qy 3Q MJ nq KJ 1k Su de kw Kh sx bk fe uc PP SR 3T L5 cv Ub 0s +9 Th 6f va II j3 =M /< al et ix >t yz EE AP Qm 45 lq rm =I >" AA CA Hv ci Vd LH AN JF G3 3d r3 s0 g1 FE Vy Ap qj NG ws rv Sh Ib fz 2S pd Zm Jh Jl cp 8R Fh dv F+ +v dj dh 6p ey wb j7 zl 0X uk Lu N8 wn Xa ll Wd 51 rr 5G bt z2 7u /e mc zk DT sd Et hp S8 CM 4i ow l7 DB kf ri bc Fp Te Hf Ee v2 MB d3 xl rj EG eo DI sz 6A ls sg gq te xz 1g QZ F0 R4 4T BZ 3o 7j Nu 6j fi ik ks Ay bm w0 Gy c1 rg ca WU we Cb HE Os pk ob r4 ul 6t HJ uj Tn aq db 5U Wb E7 H9 dc YK cy ou 9J nm z9 3E ej lr Yf Vw cl pc XU 7X jl oq LB ZU Qn Px 1+ jw 2s DA ND ax U+ GE tz K2 pm CH 5Z JZ fp ir an Gf YZ KT no ct Xo 4P 0w Kq be Ps tq md cu UU ie vl xy 1x X/ 71 or 6E 9h m+ 1/ CS f0 /C 47 +w vg 96 C4 oz Ft nv er 2D Lx 8+ hm 35 Rl Qy 3Q MJ nq KJ 1k Su de kw Kh sx bk fe uc PP SR 3T L5 cv Ub 0s +9 Th 6f va II j3 =M /< al et ix >t yz EE AP Qm 45 lq rm =I >" AA CA Hv ci Vd LH AN JF G3 3d r3 s0 g1 FE Vy Ap qj NG ws rv Sh Ib fz 2S pd Zm Jh Jl cp 8R Fh dv F+ +v dj dh 6p ey wb j7 zl 0X uk Lu N8 wn Xa ll Wd 51 rr 5G bt z2 7u /e mc zk DT sd Et hp S8 CM 4i ow l7 DB kf ri bc Fp Te Hf Ee v2 MB d3 xl rj EG eo DI sz 6A ls sg gq te xz 1g QZ F0 R4 4T BZ 3o 7j Nu 6j fi ik ks Ay bm w0 Gy c1 rg ca WU we Cb HE Os pk ob r4 ul 6t HJ uj Tn aq db 5U Wb E7 H9 dc YK cy ou 9J nm z9 3E ej lr Yf Vw cl pc XU 7X jl oq LB ZU Qn Px 1+ jw 2s DA ND ax U+ GE tz K2 pm CH 5Z JZ fp ir an Gf YZ KT no ct Xo 4P 0w Kq be Ps tq md cu UU ie vl xy 1x X/ 71 or 6E 9h m+ 1/ CS f0 /C 47 +w vg 96 C4 oz Ft nv er 2D Lx 8+ hm 35 Rl Qy 3Q MJ nq KJ 1k Su de kw Kh sx bk fe uc PP SR 3T L5 cv Ub 0s +9 Th 6f va II j3 =M /< al et ix >t h0 yz EE AP Qm 45 lq rm =I >" AA CA Hv ci Vd LH AN JF G3 3d r3 s0 g1 FE Vy Ap qj NG ws rv Sh Ib fz 2S pd Zm Jh Jl cp 8R Fh dv F+ +v dj dh 6p ey wb j7 zl 0X uk Lu N8 wn Xa ll Wd 51 rr 5G bt z2 7u /e mc zk DT sd Et hp S8 CM 4i ow l7 DB kf ri bc Fp Te Hf Ee v2 MB d3 xl rj EG eo DI sz 6A ls sg gq te xz 1g QZ F0 R4 4T BZ 3o 7j Nu 6j fi ik ks Ay bm w0 Gy c1 rg ca WU we Cb HE Os pk ob r4 ul 6t HJ uj Tn aq db 5U Wb E7 H9 dc YK cy ou 9J nm z9 3E ej lr Yf Vw cl pc XU 7X jl oq LB ZU Qn Px 1+ jw 2s DA ND ax U+ GE tz K2 pm CH 5Z JZ fp ir an Gf YZ KT no ct Xo 4P 0w Kq be Ps tq md cu UU ie vl xy 1x X/ 71 or 6E 9h m+ 1/ CS f0 /C 47 +w vg 96 C4 oz Ft nv er 2D Lx 8+ hm 35 Rl Qy 3Q MJ nq KJ 1k Su de kw Kh sx bk fe uc PP SR 3T L5 cv Ub 0s +9 Th 6f va II j3 =M /< al et ix >t yz EE AP Qm 45 lq rm =I >" AA CA Hv ci Vd LH AN JF G3 3d r3 s0 g1 FE Vy Ap qj NG ws rv Sh Ib fz 2S pd Zm Jh Jl cp 8R Fh dv F+ +v dj dh 6p ey wb j7 zl 0X uk Lu N8 wn Xa ll Wd 51 rr 5G bt z2 7u /e mc zk DT sd Et hp S8 CM 4i ow l7 DB kf ri bc Fp Te Hf Ee v2 MB d3 xl rj EG eo DI sz 6A ls sg gq te xz 1g QZ F0 R4 4T BZ 3o 7j Nu 6j fi ik ks Ay bm w0 Gy c1 rg ca WU we Cb HE Os pk ob r4 ul 6t HJ uj Tn aq db 5U Wb E7 H9 dc YK cy ou 9J nm z9 3E ej lr Yf Vw cl pc XU 7X jl oq LB ZU Qn Px 1+ jw 2s DA ND ax U+ GE tz K2 pm CH 5Z JZ fp ir an Gf YZ KT no ct Xo 4P 0w Kq be Ps tq md cu UU ie vl xy 1x X/ 71 or 6E 9h m+ 1/ CS f0 /C 47 +w vg 96 C4 oz Ft nv er 2D Lx 8+ hm 35 Rl Qy 3Q MJ nq KJ 1k Su de kw Kh sx bk fe uc PP SR 3T L5 cv Ub 0s +9 Th 6f va II j3 =M /< al et ix >t yz EE AP Qm 45 lq rm =I >" AA CA Hv ci Vd LH AN JF G3 3d r3 s0 g1 FE Vy Ap qj NG ws rv Sh Ib fz 2S pd Zm Jh Jl cp 8R Fh dv F+ +v dj dh 6p ey wb j7 zl 0X uk Lu N8 wn Xa ll Wd 51 rr 5G bt z2 7u /e mc zk DT sd Et hp S8 CM 4i ow l7 DB kf ri bc Fp Te Hf Ee v2 MB d3 xl rj EG eo DI sz 6A ls sg gq te xz 1g QZ F0 R4 4T BZ 3o 7j Nu 6j fi ik ks Ay bm w0 Gy c1 rg ca WU we Cb HE Os pk ob r4 ul 6t HJ uj Tn aq db 5U Wb E7 H9 dc YK cy ou 9J nm z9 3E ej lr Yf Vw cl pc XU 7X jl oq LB ZU Qn Px 1+ jw 2s DA ND ax U+ GE tz K2 pm CH 5Z JZ fp ir an Gf YZ KT no ct Xo 4P 0w Kq be Ps tq md cu UU ie vl xy 1x X/ 71 or 6E 9h m+ 1/ CS f0 /C 47 +w vg 96 C4 oz Ft nv er 2D Lx 8+ hm 35 Rl Qy 3Q MJ nq KJ 1k Su de kw Kh sx bk fe uc PP SR 3T L5 cv Ub 0s +9 Th 6f va II j3 =M /< al et ix >t h0 yz EE AP Qm 45 lq rm =I >" AA CA Hv ci Vd LH AN JF G3 3d r3 s0 g1 FE Vy Ap qj NG ws rv Sh Ib fz 2S pd Zm Jh Jl cp 8R Fh dv F+ +v dj dh 6p ey wb j7 zl 0X uk Lu N8 wn Xa ll Wd 51 rr 5G bt z2 7u /e mc zk DT sd Et hp S8 CM 4i ow l7 DB kf ri bc Fp Te Hf Ee v2 MB d3 xl rj EG eo DI sz 6A ls sg gq te xz 1g QZ F0 R4 4T BZ 3o 7j Nu 6j fi ik ks Ay bm w0 Gy c1 rg ca WU we Cb HE Os pk ob r4 ul 6t HJ uj Tn aq db 5U Wb E7 H9 dc YK cy ou 9J nm z9 3E ej lr Yf Vw cl pc XU 7X jl oq LB ZU Qn Px 1+ jw 2s DA ND ax U+ GE tz K2 pm CH 5Z JZ fp ir an Gf YZ KT no ct Xo 4P 0w Kq be Ps tq md cu UU ie vl xy 1x X/ 71 or 6E 9h m+ 1/ CS f0 /C 47 +w vg 96 C4 oz Ft nv er 2D Lx 8+ hm 35 Rl Qy 3Q MJ nq KJ 1k Su de kw Kh sx bk fe uc PP SR 3T L5 cv Ub 0s +9 Th 6f va II j3 =M /< al et ix >t yz EE AP Qm 45 lq rm =I >" AA CA Hv ci Vd LH AN JF G3 3d r3 s0 g1 FE Vy Ap qj NG ws rv Sh Ib fz 2S pd Zm Jh Jl cp 8R Fh dv F+ +v dj dh 6p ey wb j7 zl 0X uk Lu N8 wn Xa ll Wd 51 rr 5G bt z2 7u /e mc zk DT sd Et hp S8 CM 4i ow l7 DB kf ri bc Fp Te Hf Ee v2 MB d3 xl rj EG eo DI sz 6A ls sg gq te xz 1g QZ F0 R4 4T BZ 3o 7j Nu 6j fi ik ks Ay bm w0 Gy c1 rg ca WU we Cb HE Os pk ob r4 ul 6t HJ uj Tn aq db 5U Wb E7 H9 dc YK cy ou 9J nm z9 3E ej lr Yf Vw cl pc XU 7X jl oq LB ZU Qn Px 1+ jw 2s DA ND ax U+ GE tz K2 pm CH 5Z JZ fp ir an Gf YZ KT no ct Xo 4P 0w Kq be Ps tq md cu UU ie vl xy 1x X/ 71 or 6E 9h m+ 1/ CS f0 /C 47 +w vg 96 C4 oz Ft nv er 2D Lx 8+ hm 35 Rl Qy 3Q MJ nq KJ 1k Su de kw Kh sx bk fe uc PP SR 3T L5 cv Ub 0s +9 Th 6f va II j3 =M /< al et ix >t yz EE AP Qm 45 lq rm =I >" AA CA Hv ci Vd LH AN JF G3 3d r3 s0 g1 FE Vy Ap qj NG ws rv Sh Ib fz 2S pd Zm Jh Jl cp 8R Fh dv F+ +v dj dh 6p ey wb j7 zl 0X uk Lu N8 wn Xa ll Wd 51 rr 5G bt z2 7u /e mc zk DT sd Et hp S8 CM 4i ow l7 DB kf ri bc Fp Te Hf Ee v2 MB d3 xl rj EG eo DI sz 6A ls sg gq te xz 1g QZ F0 R4 4T BZ 3o 7j Nu 6j fi ik ks Ay bm w0 Gy c1 rg ca WU we Cb HE Os pk ob r4 ul 6t HJ uj Tn aq db 5U Wb E7 H9 dc YK cy ou 9J nm z9 3E ej lr Yf Vw cl pc XU 7X jl oq LB ZU Qn Px 1+ jw 2s DA ND ax U+ GE tz K2 pm CH 5Z JZ fp ir an Gf YZ KT no ct Xo 4P 0w Kq be Ps tq md cu UU ie vl xy 1x X/ 71 or 6E 9h m+ 1/ CS f0 /C 47 +w vg 96 C4 oz Ft nv er 2D Lx 8+ hm 35 Rl Qy 3Q MJ nq KJ 1k Su de kw Kh sx bk fe uc PP SR 3T L5 cv Ub 0s +9 Th 6f va II j3 =M /< al et ix >t h0 yz EE AP Qm 45 lq rm =I >" AA CA Hv ci Vd LH AN JF G3 3d r3 s0 g1 FE Vy Ap qj NG ws rv Sh Ib fz 2S pd Zm Jh Jl cp 8R Fh dv F+ +v dj dh 6p ey wb j7 zl 0X uk Lu N8 wn Xa ll Wd 51 rr 5G bt z2 7u /e mc zk DT sd Et hp S8 CM 4i ow l7 DB kf ri bc Fp Te Hf Ee v2 MB d3 xl rj EG eo DI sz 6A ls sg gq te xz 1g QZ F0 R4 4T BZ 3o 7j Nu 6j fi ik ks Ay bm w0 Gy c1 rg ca WU we Cb HE Os pk ob r4 ul 6t HJ uj Tn aq db 5U Wb E7 H9 dc YK cy ou 9J nm z9 3E ej lr Yf Vw cl pc XU 7X jl oq LB ZU Qn Px 1+ jw 2s DA ND ax U+ GE tz K2 pm CH 5Z JZ fp ir an Gf YZ KT no ct Xo 4P 0w Kq be Ps tq md cu UU ie vl xy 1x X/ 71 or 6E 9h m+ 1/ CS f0 /C 47 +w vg 96 C4 oz Ft nv er 2D Lx 8+ hm 35 Rl Qy 3Q MJ nq KJ 1k Su de kw Kh sx bk fe uc PP SR 3T L5 cv Ub 0s +9 Th 6f va II j3 =M /< al et ix >t yz EE AP Qm 45 lq rm =I >" AA CA Hv ci Vd LH AN JF G3 3d r3 s0 g1 FE Vy Ap qj NG ws rv Sh Ib fz 2S pd Zm Jh Jl cp 8R Fh dv F+ +v dj dh 6p ey wb j7 zl 0X uk Lu N8 wn Xa ll Wd 51 rr 5G bt z2 7u /e mc zk DT sd Et hp S8 CM 4i ow l7 DB kf ri bc Fp Te Hf Ee v2 MB d3 xl rj EG eo DI sz 6A ls sg gq te xz 1g QZ F0 R4 4T BZ 3o 7j Nu 6j fi ik ks Ay bm w0 Gy c1 rg ca WU we Cb HE Os pk ob r4 ul 6t HJ uj Tn aq db 5U Wb E7 H9 dc YK cy ou 9J nm z9 3E ej lr Yf Vw cl pc XU 7X jl oq LB ZU Qn Px 1+ jw 2s DA ND ax U+ GE tz K2 pm CH 5Z JZ fp ir an Gf YZ KT no ct Xo 4P 0w Kq be Ps tq md cu UU ie vl xy 1x X/ 71 or 6E 9h m+ 1/ CS f0 /C 47 +w vg 96 C4 oz Ft nv er 2D Lx 8+ hm 35 Rl Qy 3Q MJ nq KJ 1k Su de kw Kh sx bk fe uc PP SR 3T L5 cv Ub 0s +9 Th 6f va II j3 =M /< al et ix >t yz EE AP Qm 45 lq rm =I >" AA CA Hv ci Vd LH AN JF G3 3d r3 s0 g1 FE Vy Ap qj NG ws rv Sh Ib fz 2S pd Zm Jh Jl cp 8R Fh dv F+ +v dj dh 6p ey wb j7 zl 0X uk Lu N8 wn Xa ll Wd 51 rr 5G bt z2 7u /e mc zk DT sd Et hp S8 CM 4i ow l7 DB kf ri bc Fp Te Hf Ee v2 MB d3 xl rj EG eo DI sz 6A ls sg gq te xz 1g QZ F0 R4 4T BZ 3o 7j Nu 6j fi ik ks Ay bm w0 Gy c1 rg ca WU we Cb HE Os pk ob r4 ul 6t HJ uj Tn aq db 5U Wb E7 H9 dc YK cy ou 9J nm z9 3E ej lr Yf Vw cl pc XU 7X jl oq LB ZU Qn Px 1+ jw 2s DA ND ax U+ GE tz K2 pm CH 5Z JZ fp ir an Gf YZ KT no ct Xo 4P 0w Kq be Ps tq md cu UU ie vl xy 1x X/ 71 or 6E 9h m+ 1/ CS f0 /C 47 +w vg 96 C4 oz Ft nv er 2D Lx 8+ hm 35 Rl Qy 3Q MJ nq KJ 1k Su de kw Kh sx bk fe uc PP SR 3T L5 cv Ub 0s +9 Th 6f va II j3 =M /< al et ix >t h0 yz EE AP Qm 45 lq rm =I >" AA CA Hv ci Vd LH AN JF G3 3d r3 s0 g1 FE Vy Ap qj NG ws rv Sh Ib fz 2S pd Zm Jh Jl cp 8R Fh dv F+ +v dj dh 6p ey wb j7 zl 0X uk Lu N8 wn Xa ll Wd 51 rr 5G bt z2 7u /e mc zk DT sd Et hp S8 CM 4i ow l7 DB kf ri bc Fp Te Hf Ee v2 MB d3 xl rj EG eo DI sz 6A ls sg gq te xz 1g QZ F0 R4 4T BZ 3o 7j Nu 6j fi ik ks Ay bm w0 Gy c1 rg ca WU we Cb HE Os pk ob r4 ul 6t HJ uj Tn aq db 5U Wb E7 H9 dc YK cy ou 9J nm z9 3E ej lr Yf Vw cl pc XU 7X jl oq LB ZU Qn Px 1+ jw 2s DA ND ax U+ GE tz K2 pm CH 5Z JZ fp ir an Gf YZ KT no ct Xo 4P 0w Kq be Ps tq md cu UU ie vl xy 1x X/ 71 or 6E 9h m+ 1/ CS f0 /C 47 +w vg 96 C4 oz Ft nv er 2D Lx 8+ hm 35 Rl Qy 3Q MJ nq KJ 1k Su de kw Kh sx bk fe uc PP SR 3T L5 cv Ub 0s +9 Th 6f va II j3 =M /< al et ix >t yz EE AP Qm 45 lq rm =I >" AA CA Hv ci Vd LH AN JF G3 3d r3 s0 g1 FE Vy Ap qj NG ws rv Sh Ib fz 2S pd Zm Jh Jl cp 8R Fh dv F+ +v dj dh 6p ey wb j7 zl 0X uk Lu N8 wn Xa ll Wd 51 rr 5G bt z2 7u /e mc zk DT sd Et hp S8 CM 4i ow l7 DB kf ri bc Fp Te Hf Ee v2 MB d3 xl rj EG eo DI sz 6A ls sg gq te xz 1g QZ F0 R4 4T BZ 3o 7j Nu 6j fi ik ks Ay bm w0 Gy c1 rg ca WU we Cb HE Os pk ob r4 ul 6t HJ uj Tn aq db 5U Wb E7 H9 dc YK cy ou 9J nm z9 3E ej lr Yf Vw cl pc XU 7X jl oq LB ZU Qn Px 1+ jw 2s DA ND ax U+ GE tz K2 pm CH 5Z JZ fp ir an Gf YZ KT no ct Xo 4P 0w Kq be Ps tq md cu UU ie vl xy 1x X/ 71 or 6E 9h m+ 1/ CS f0 /C 47 +w vg 96 C4 oz Ft nv er 2D Lx 8+ hm 35 Rl Qy 3Q MJ nq KJ 1k Su de kw Kh sx bk fe uc PP SR 3T L5 cv Ub 0s +9 Th 6f va II j3 =M /< al et ix >t yz EE AP Qm 45 lq rm =I >" AA CA Hv ci Vd LH AN JF G3 3d r3 s0 g1 FE Vy Ap qj NG ws rv Sh Ib fz 2S pd Zm Jh Jl cp 8R Fh dv F+ +v dj dh 6p ey wb j7 zl 0X uk Lu N8 wn Xa ll Wd 51 rr 5G bt z2 7u /e mc zk DT sd Et hp S8 CM 4i ow l7 DB kf ri bc Fp Te Hf Ee v2 MB d3 xl rj EG eo DI sz 6A ls sg gq te xz 1g QZ F0 R4 4T BZ 3o 7j Nu 6j fi ik ks Ay bm w0 Gy c1 rg ca WU we Cb HE Os pk ob r4 ul 6t HJ uj Tn aq db 5U Wb E7 H9 dc YK cy ou 9J nm z9 3E ej lr Yf Vw cl pc XU 7X jl oq LB ZU Qn Px 1+ jw 2s DA ND ax U+ GE tz K2 pm CH 5Z JZ fp ir an Gf YZ KT no ct Xo 4P 0w Kq be Ps tq md cu UU ie vl xy 1x X/ 71 or 6E 9h m+ 1/ CS f0 /C 47 +w vg 96 C4 oz Ft nv er 2D Lx 8+ hm 35 Rl Qy 3Q MJ nq KJ 1k Su de kw Kh sx bk fe uc PP SR 3T L5 cv Ub 0s +9 Th 6f va II j3 =M /< al et ix >t h0 yz EE AP Qm 45 lq rm =I >" AA CA Hv ci Vd LH AN JF G3 3d r3 s0 g1 FE Vy Ap qj NG ws rv Sh Ib fz 2S pd Zm Jh Jl cp 8R Fh dv F+ +v dj dh 6p ey wb j7 zl 0X uk Lu N8 wn Xa ll Wd 51 rr 5G bt z2 7u /e mc zk DT sd Et hp S8 CM 4i ow l7 DB kf ri bc Fp Te Hf Ee v2 MB d3 xl rj EG eo DI sz 6A ls sg gq te xz 1g QZ F0 R4 4T BZ 3o 7j Nu 6j fi ik ks Ay bm w0 Gy c1 rg ca WU we Cb HE Os pk ob r4 ul 6t HJ uj Tn aq db 5U Wb E7 H9 dc YK cy ou 9J nm z9 3E ej lr Yf Vw cl pc XU 7X jl oq LB ZU Qn Px 1+ jw 2s DA ND ax U+ GE tz K2 pm CH 5Z JZ fp ir an Gf YZ KT no ct Xo 4P 0w Kq be Ps tq md cu UU ie vl xy 1x X/ 71 or 6E 9h m+ 1/ CS f0 /C 47 +w vg 96 C4 oz Ft nv er 2D Lx 8+ hm 35 Rl Qy 3Q MJ nq KJ 1k Su de kw Kh sx bk fe uc PP SR 3T L5 cv Ub 0s +9 Th 6f va II j3 =M /< al et ix >t yz EE AP Qm 45 lq rm =I >" AA CA Hv ci Vd LH AN JF G3 3d r3 s0 g1 FE Vy Ap qj NG ws rv Sh Ib fz 2S pd Zm Jh Jl cp 8R Fh dv F+ +v dj dh 6p ey wb j7 zl 0X uk Lu N8 wn Xa ll Wd 51 rr 5G bt z2 7u /e mc zk DT sd Et hp S8 CM 4i ow l7 DB kf ri bc Fp Te Hf Ee v2 MB d3 xl rj EG eo DI sz 6A ls sg gq te xz 1g QZ F0 R4 4T BZ 3o 7j Nu 6j fi ik ks Ay bm w0 Gy c1 rg ca WU we Cb HE Os pk ob r4 ul 6t HJ uj Tn aq db 5U Wb E7 H9 dc YK cy ou 9J nm z9 3E ej lr Yf Vw cl pc XU 7X jl oq LB ZU Qn Px 1+ jw 2s DA ND ax U+ GE tz K2 pm CH 5Z JZ fp ir an Gf YZ KT no ct Xo 4P 0w Kq be Ps tq md cu UU ie vl xy 1x X/ 71 or 6E 9h m+ 1/ CS f0 /C 47 +w vg 96 C4 oz Ft nv er 2D Lx 8+ hm 35 Rl Qy 3Q MJ nq KJ 1k Su de kw Kh sx bk fe uc PP SR 3T L5 cv Ub 0s +9 Th 6f va II j3 =M /< al et ix >t yz EE AP Qm 45 lq rm =I >" AA CA Hv ci Vd LH AN JF G3 3d r3 s0 g1 FE Vy Ap qj NG ws rv Sh Ib fz 2S pd Zm Jh Jl cp 8R Fh dv F+ +v dj dh 6p ey wb j7 zl 0X uk Lu N8 wn Xa ll Wd 51 rr 5G bt z2 7u /e mc zk DT sd Et hp S8 CM 4i ow l7 DB kf ri bc Fp Te Hf Ee v2 MB d3 xl rj EG eo DI sz 6A ls sg gq te xz 1g QZ F0 R4 4T BZ 3o 7j Nu 6j fi ik ks Ay bm w0 Gy c1 rg ca WU we Cb HE Os pk ob r4 ul 6t HJ uj Tn aq db 5U Wb E7 H9 dc YK cy ou 9J nm z9 3E ej lr Yf Vw cl pc XU 7X jl oq LB ZU Qn Px 1+ jw 2s DA ND ax U+ GE tz K2 pm CH 5Z JZ fp ir an Gf YZ KT no ct Xo 4P 0w Kq be Ps tq md cu UU ie vl xy 1x X/ 71 or 6E 9h m+ 1/ CS f0 /C 47 +w vg 96 C4 oz Ft nv er 2D Lx 8+ hm 35 Rl Qy 3Q MJ nq KJ 1k Su de kw Kh sx bk fe uc PP SR 3T L5 cv Ub 0s +9 Th 6f va II j3 =M /< al et ix >t h0 yz EE AP Qm 45 lq rm =I >" AA CA Hv ci Vd LH AN JF G3 3d r3 s0 g1 FE Vy Ap qj NG ws rv Sh Ib fz 2S pd Zm Jh Jl cp 8R Fh dv F+ +v dj dh 6p ey wb j7 zl 0X uk Lu N8 wn Xa ll Wd 51 rr 5G bt z2 7u /e mc zk DT sd Et hp S8 CM 4i ow l7 DB kf ri bc Fp Te Hf Ee v2 MB d3 xl rj EG eo DI sz 6A ls sg gq te xz 1g QZ F0 R4 4T BZ 3o 7j Nu 6j fi ik ks Ay bm w0 Gy c1 rg ca WU we Cb HE Os pk ob r4 ul 6t HJ uj Tn aq db 5U Wb E7 H9 dc YK cy ou 9J nm z9 3E ej lr Yf Vw cl pc XU 7X jl oq LB ZU Qn Px 1+ jw 2s DA ND ax U+ GE tz K2 pm CH 5Z JZ fp ir an Gf YZ KT no ct Xo 4P 0w Kq be Ps tq md cu UU ie vl xy 1x X/ 71 or 6E 9h m+ 1/ CS f0 /C 47 +w vg 96 C4 oz Ft nv er 2D Lx 8+ hm 35 Rl Qy 3Q MJ nq KJ 1k Su de kw Kh sx bk fe uc PP SR 3T L5 cv Ub 0s +9 Th 6f va II j3 =M /< al et ix >t yz EE AP Qm 45 lq rm =I >" AA CA Hv ci Vd LH AN JF G3 3d r3 s0 g1 FE Vy Ap qj NG ws rv Sh Ib fz 2S pd Zm Jh Jl cp 8R Fh dv F+ +v dj dh 6p ey wb j7 zl 0X uk Lu N8 wn Xa ll Wd 51 rr 5G bt z2 7u /e mc zk DT sd Et hp S8 CM 4i ow l7 DB kf ri bc Fp Te Hf Ee v2 MB d3 xl rj EG eo DI sz 6A ls sg gq te xz 1g QZ F0 R4 4T BZ 3o 7j Nu 6j fi ik ks Ay bm w0 Gy c1 rg ca WU we Cb HE Os pk ob r4 ul 6t HJ uj Tn aq db 5U Wb E7 H9 dc YK cy ou 9J nm z9 3E ej lr Yf Vw cl pc XU 7X jl oq LB ZU Qn Px 1+ jw 2s DA ND ax U+ GE tz K2 pm CH 5Z JZ fp ir an Gf YZ KT no ct Xo 4P 0w Kq be Ps tq md cu UU ie vl xy 1x X/ 71 or 6E 9h m+ 1/ CS f0 /C 47 +w vg 96 C4 oz Ft nv er 2D Lx 8+ hm 35 Rl Qy 3Q MJ nq KJ 1k Su de kw Kh sx bk fe uc PP SR 3T L5 cv Ub 0s +9 Th 6f va II j3 =M /< al et ix >t yz EE AP Qm 45 lq rm =I >" AA CA Hv ci Vd LH AN JF G3 3d r3 s0 g1 FE Vy Ap qj NG ws rv Sh Ib fz 2S pd Zm Jh Jl cp 8R Fh dv F+ +v dj dh 6p ey wb j7 zl 0X uk Lu N8 wn Xa ll Wd 51 rr 5G bt z2 7u /e mc zk DT sd Et hp S8 CM 4i ow l7 DB kf ri bc Fp Te Hf Ee v2 MB d3 xl rj EG eo DI sz 6A ls sg gq te xz 1g QZ F0 R4 4T BZ 3o 7j Nu 6j fi ik ks Ay bm w0 Gy c1 rg ca WU we Cb HE Os pk ob r4 ul 6t HJ uj Tn aq db 5U Wb E7 H9 dc YK cy ou 9J nm z9 3E ej lr Yf Vw cl pc XU 7X jl oq LB ZU Qn Px 1+ jw 2s DA ND ax U+ GE tz K2 pm CH 5Z JZ fp ir an Gf YZ KT no ct Xo 4P 0w Kq be Ps tq md cu UU ie vl xy 1x X/ 71 or 6E 9h m+ 1/ CS f0 /C 47 +w vg 96 C4 oz Ft nv er 2D Lx 8+ hm 35 Rl Qy 3Q MJ nq KJ 1k Su de kw Kh sx bk fe uc PP SR 3T L5 cv Ub 0s +9 Th 6f va II j3 =M /< al et ix >t =<latexit sha1_base64="ufbkrtbq720kdpn34ecddtho5ne=">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</latexit> <latexit sha1_base64="ufbkrtbq720kdpn34ecddtho5ne=">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</latexit> <latexit sha1_base64="ufbkrtbq720kdpn34ecddtho5ne=">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</latexit> Tensor networks are equivalent to quantum circuits

72 Quantum circuit for matrix product state (m = 4) =

73 Quantum circuit for matrix product state (m = 4) =

74 <latexit sha1_base64="2nihmcpuqnasfc+yvksjsqjzf+8=">aaacznicdvhlattafb0rfatuy0mhm26gikjxrmmc7tkqtzco1enaeuzqfcupnhmjmzfti4tu+xml2/sh+jcdkv4ksxnh4hduoxnfasmfdvh0rxccphn67pnhi/7lv6/fvb0chv/yojicj7yqhblkwaiugidooilxpufqqctldhnw5i9xakwo9he3ljfrkgurcq7ou7pb+9iifohamokaxgrcis3qh81seebdqau6c9gwhgqb49lr73c8l3iludsuwdopi0qx1gcc4bkbflxzlievicephxou2qtubmjoj8/mavyy/7sjhxvxuyoydq1sr2xbta9zlbkvn61c9jwphs4rh5rffsoqsv1b223qutdinvx7anwi3yvlczdand9zp9z4zqulqm/reimumbkkjlhbymbbq96eldagcz9gc1+dgthrx7kthmyzr919p9pron4rjugjlwcvp1bjg++4/e3zwwvugovrkevddn4snp5tr3tippcp5dnh5as5jd/imewijxvyh9yqv8e4wavn8pnwgvs2nnfkxgs//gosyevx</latexit> <latexit sha1_base64="2nihmcpuqnasfc+yvksjsqjzf+8=">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</latexit> <latexit sha1_base64="2nihmcpuqnasfc+yvksjsqjzf+8=">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</latexit> <latexit sha1_base64="2nihmcpuqnasfc+yvksjsqjzf+8=">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</latexit> Suggests MPS parameterization of generative quantum model! x much fewer paramaters than arbitrary circuit can initialize with classically optimized MPS

75 Discriminative model basically the reverse

76 Test discriminative idea, using only operations available to quantum hardware: Bill Huggins 8x8 images (MNIST) distinguish 0's from 1's Obtain 99% accuracy training & test