Quantum fields as deep learning Jae-Weon Lee (Jungwon univ.) arXiv:1708.07408

Brief History of Complexity 1920: Hilbert's program Decision problem=asking for an algorithm to decide whether a given statement is provable from the axioms using the rules of logic 1931:Kurt Gödel's incompleteness theorem 1936:Turing machine  computational complexity theory Easy problem = in P, Hard problem = not in P Church–Turing Thesis = “All physically computable functions are Turing-computable” 1981: Feynman proposes Q. Computer (N spins need 2N coefficients) Quantum Church–Turing thesis =“A quantum Turing machine can efficiently simulate any realistic model of computation.” Church–Turing -Dutch principle  The universe is equivalent to a Turing machine digital physics.

Machine learning “A computer program is said to learn from experience E with respect to some class of tasks T and performance measure P, if its performance at tasks in T, as measured by P, improves with experience E” -Tom M. Mitchell (x) Supervised Learning (data x-label y) Classification Regression Unsupervised Learning (data x) Clustering Underlying Probability Density Estimation Semisupervised learning Reinforcement Learning – delayed reward problem 0 6 4 7 (y) (x) Algorithms: Gradient descent, Regression, Bayesian, HMM, SVM, ANN

Artificial neural networks (ANN) O(1011) cells with O(104) synapses 가중치 역치 Activation function perceptron 1958 Rosenblatt

How DLs recognize a dog Roger Parloff

WHY DOES DEEP LEARNING WORK? RG? Combination? Simple physical laws arXiv:1410.3831 Pankaj Mehta, David J. Schwab Lin, Tegmark 1608.08225

Hopfield network (energy based, deterministic) Training a Hopfield net = lowering the energy of states that the net should "remember" 1 3 1)wij = wji, 2) asynchronous  associative memory

Boltzmann machine (BM)  Stochastic(Monte Carlo), generative counterpart of Hopfield nets Similar to Spin glass Unsupervised learning impractical

Restricted Boltzmann machine (RBM) no two nodes of the same layer are linked interaction

차원감소 Autoencoder

Renormalization Kadanoff h v Boltzmann distribution Variational RG

An exact mapping between the Variational Renormalization Group and Deep Learning arXiv:1410.3831 Pankaj Mehta, David J. Schwab

An exact mapping between the Variational Renormalization Group and Deep Learning

AdS/MERA gA CFTd A AdSd+1 Ryu-Takayanagi formula 2006 s steps L~ 2s Swingle 2009 Ryu-Takayanagi formula 2006 A gA CFTd AdSd+1 s steps L~ 2s E.E ~ # of tracing out indices ~# of cuts ~ s ~ log(L)  geodesic in AdS  Redundancy, QEC, Holography…

Holography as deep learning Wen-Cong Ganand Fu-Wen Shu, 1705.05750

Gravity from entanglement Lee, Lee, Kim, 1001.5445 JKPS 2013 Vacuum in Euclidean L R t x dE Boost H. a with proper time Unruh effect BH horizon too

Quantum fields as deep RBM arXiv:1708.07408 Higgs?

Conclusion Deep learning has physics in it Quantum fields, and spacetime, could be deep learning [Q] Can nature think? If yes, then why doesn’t it speak to us??

Machine Learning Spatial Geometry from Entanglement Hayden etal 1601.01694 the random tensor network (RTN) states satisfy the Ryu-Takayanagi formula Yi-Zhuang You etal 1709.01223

Other ideas 1) Euclidean quantum fieeld theory in d+1 dimensional at spacetime and the statistical mechanics in d+1 dimensional at space using an imaginary time 2) QNN

Entanglement entropy of a state How broadly spread over, or Information in a density matrix A Ex) Vacuum or ground Partial trace For pure states Entanglement Entropy (vNE of subsystem) More correlated, less we know about the subsystem alone.

Vanishing gradient problem Multi-layer perceptron 기울기<1/4 출력 입력 Backpropagation Backward:Hinton et al. (1986) Nature Using chainrule

Tensor network Orus arXiv:1306.2164 Tensor network: to find the ground states of critical quantum systems in an efficient manner. (MPS, PEPS, MERA…) (Feynman’s original motivation for QC.) Ex) N spin wave fn ansatz Ground state Vidal

aXiv:1112.4101

MERA ( Multiscale entanglement renormalization ansatz) Vidal 2007 time RG Removal UV DOF (preserves the inner product) Removal UV entanglement

Jacobson’s Great idea Padmanabhan 1st law (assumption) R Covariant form where Raychaudhuri equation For all null using contracted Bianchi identity Einstein equation is related to 1st law for local Rindler horizons!

Entanglement 1st law

Holographic Q. error correcting code HaPPY Perfect tensor with 2n indices: Isometry Maximally entangled n to n 2n-1 qubit EC for 1 qubit The code can tolerate loss of 2 qubit 5 qubit code Logical op. preserving code space reside in the bulk Geodesic line = epr pairs Quantum code on boundary

Activation function needs to be nonlinear (Logistic) 미분에 유리 (Logistic) Rectified linear unit Moujahid

Entanglement as spacetime glue Raamsdonk 1005.3035 A B L Minimal surface The smaller entanglement, the less ST connected gA CFT CFT Two CFTs in HH states =dual to eternal AdS black hole spacetime (EE is equal to thermal entropy of BH) Quantum superposition of disconnected spacetime  Emergent spacetime Maldacena, hepth/0106112

Modern interpretation L R t x a