Langevin dynamics in a deep belief network Stefanie Czischek Cold - PowerPoint PPT Presentation

Violating Bell’s inequality with Langevin dynamics in a deep belief network Stefanie Czischek Cold Quantum Coffee L. Kades, J. M. Pawlowski, M. Gärttner, T. Gasenzer 20.11.2018

Spin-½ Systems Two spins: Three spins: four configurations eight configurations One spin: two configurations 20.11.2018 Cold Quantum Coffee Stefanie Czischek

Spin-½ Systems 20.11.2018 Cold Quantum Coffee Stefanie Czischek

Spin-½ Systems • Wave function as weighted sum over all product states | ۧ ۧ Ψ = ෍ 𝑑 𝑗 1 …𝑗 𝑂 | 𝑗 1 … 𝑗 𝑂 𝑗 1 …𝑗 𝑂 • Look for functional Hilbert space 0,1 𝑂 → ℂ 𝑗 1 … 𝑗 𝑂 → 𝑑 𝑗 1 …𝑗 𝑂 = 𝑔 𝑗 1 … 𝑗 𝑂 ; 𝑋 Physical States Analogy to Machine Learning! 20.11.2018 Cold Quantum Coffee Stefanie Czischek

Machine Learning Netflix Prize Pattern Recognition ➢ Predict user ratings for films based on ➢ Is there a cat in the picture? previous ratings Machine Color Machine Films rated Rating for a Cat or not Learning value for Learning by user given film System each pixel System 20.11.2018 Cold Quantum Coffee Stefanie Czischek

Quantum Dynamics and Machine Learning [Carleo and Troyer, Science 2017] • Represent Spin-½ system using artificial neural networks • Use unsupervised learning to find ground states and calculate dynamics • Where is the simulation method efficient? • Where does it struggle? 20.11.2018 Cold Quantum Coffee Stefanie Czischek

Neural-network quantum states [Carleo and Troyer, Science 2017] 𝒘 𝒜 | ۧ Ψ = ෍ 𝑑 𝑗 1 …𝑗 𝑂 | 𝑗 1 … 𝑗 𝑂 = ෍ ۧ 𝑑 𝒘 𝑨 | ۧ 𝒘 𝑨 𝑗 1 …𝑗 𝑂 Restricted Boltzmann machine : 𝑓 −𝐹 𝒘 𝑨 ,𝒊 𝑑 𝒘 𝑨 = ෍ • 𝑗 𝑂 visible variables 𝑤 𝑨 𝒊 • M = 𝛽𝑂 hidden variables ℎ 𝑘 𝑨 − ෍ 𝑨 𝑋 𝐹 𝒘 𝑨 , 𝒊 = − ෍ 𝑤 𝑗 𝑗,𝑘 ℎ 𝑘 − ෍ 𝑏 𝑗 𝑤 𝑗 𝑐 𝑘 ℎ 𝑘 • Biases 𝑏 𝑗 , 𝑐 𝑘 and weights 𝑋 𝑗,𝑘 as 𝑗,𝑘 𝑗 𝑘 variational parameters 𝑨 ෑ 𝑑 𝒘 𝑨 = 𝑓 σ 𝑗 𝑏 𝑗 𝑤 𝑗 𝑨 𝑋 2 cosh 𝑐 𝑘 + ෍ 𝑤 𝑗 𝑗,𝑘 𝑘 𝑗 𝑨 , ℎ 𝑘 𝜗 −1,1 𝑤 𝑗 20.11.2018 Cold Quantum Coffee Stefanie Czischek

Neural-network quantum states [Carleo and Troyer, Science 2017] 𝑙 𝑞 − 𝛿 𝜖𝐹 𝑋 𝑙 𝑞 + 1 = 𝑋 𝜖𝑋 𝑙 Imaginary time evolution Ground state search Random initial weights Learn to represent ground state If necessary: Monte Carlo Markov Chain Stochastic gradient descent: Sample states from 𝑑 𝒘 2 Wave function optimize weights Initial ground state weights Represent time evolution Time evolution Calculating expectation values: 𝑙 𝑢 − 𝑗Δ𝑢 𝜖𝐹 𝑄 𝑋 𝑙 𝑢 + Δ𝑢 = 𝑋 𝒫 = 1 𝒫 𝒘 𝑨 𝑑 𝒘 𝑨 2 ≈ 1 𝜖𝑋 ෠ 𝑨 𝑙 𝑎 ෍ 𝑄 ෍ 𝒫 𝒘 𝑞 Real time evolution 𝒘 𝑨 𝑞=1 20.11.2018 Cold Quantum Coffee Stefanie Czischek

Transverse Field Ising Model (TFIM) • Quantum critical point at ℎ 𝑑 = 1 𝑨 ො ෡ 𝑨 𝑦 𝐼 = − ෍ 𝜏 𝑗 ො 𝜏 𝑗+1 − ℎ 𝑦 ෍ 𝜏 𝑗 ො T 𝑗 𝑗 • Analytical solutions available [Lieb , Calabrese,…] • Quenches studied in detail [Karl, Cakir et al. PRE 2017] Quantum critical • Hard for MPS based methods 𝜗 = ℎ 𝑦 − ℎ 𝑑 LOW T LOW T ℎ 𝑑 Magnetic long-range order Quantum paramagnet ℎ 𝑦 0 ℎ 𝑑 = 1 Sudden quench Picture from: S. Sachdev, Quantum Phase Transitions 20.11.2018 Cold Quantum Coffee Stefanie Czischek

Neural-network quantum states [Carleo and Troyer, Science 2017] 𝑙 𝑞 − 𝛿 𝜖𝐹 𝑋 𝑙 𝑞 + 1 = 𝑋 𝜖𝑋 𝑙 Imaginary time evolution Ground state search Random initial weights Learn to represent ground state If necessary: Monte Carlo Markov Chain Stochastic gradient descent: Sample states from 𝑑 𝒘 2 Wave function optimize weights Initial ground state weights Represent time evolution Time evolution Calculating expectation values: 𝑙 𝑢 − 𝑗Δ𝑢 𝜖𝐹 𝑄 𝑋 𝑙 𝑢 + Δ𝑢 = 𝑋 𝒫 = 1 𝒫 𝒘 𝑨 𝑑 𝒘 𝑨 2 ≈ 1 𝜖𝑋 ෠ 𝑨 𝑙 𝑎 ෍ 𝑄 ෍ 𝒫 𝒘 𝑞 Real time evolution 𝒘 𝑨 𝑞=1 20.11.2018 Cold Quantum Coffee Stefanie Czischek

Quenches in the TFIM Correlation length 𝑨𝑨 𝑢 = 𝑨 ො 𝑨 𝐷 𝑒 𝜏 𝑗 ො 𝜏 𝑗+𝑒 𝑨𝑨 𝑢 = 𝑓 − Τ 𝑒 𝜊 𝑢 𝐷 𝑒 -1 ℎ 𝑦 0 1 𝑂 visible, 𝑁 hidden neurons 20.11.2018 Cold Quantum Coffee Stefanie Czischek

TFIM in a longitudinal field ℎ 𝑨 ℎ 𝑦,𝑗 = 100 𝑨 ො 𝑦 − ℎ 𝑨 ෍ ෡ 𝑨 𝑨 𝐼 = − ෍ 𝜏 𝑗 ො 𝜏 𝑗+1 − ℎ 𝑦 ෍ 𝜏 𝑗 ො 𝜏 𝑗 ො ℎ 𝑨,𝑗 = 0 𝑗 𝑗 𝑗 𝑂 = 10 𝑨𝑨 𝑢 = 𝑓 − Τ 𝑒 𝜊 𝑢 𝐷 𝑒 ℎ 𝑦 QCP 0 QCP ℎ 𝑨,𝑔 = 2 Δ𝜊 𝑢 = 𝜊 − 𝜊exact 20.11.2018 Cold Quantum Coffee Stefanie Czischek

Simulating large spin systems ℎ 𝑦,𝑔 = 0.5, ℎ 𝑨,𝑔 = 1 𝑂 = 42 𝑨 ො 𝑦 − ℎ 𝑨 ෍ ෡ 𝑨 𝑨 𝐼 = − ෍ 𝜏 𝑗 ො 𝜏 𝑗+1 − ℎ 𝑦 ෍ 𝜏 𝑗 ො 𝜏 𝑗 ො 𝑗 𝑗 𝑗 d=1 Half chain entanglement entropy ANN, M=N tDMRG, D=5 ANN, M=2N tDMRG, D=64 tDMRG, D=5 tDMRG, D=96 tDMRG, D=128 tDMRG, D=128 d=2 Bond dimension D ND 2 vs. NM 20.11.2018 Cold Quantum Coffee Stefanie Czischek

Simulating large spin systems ℎ 𝑦,𝑔 = 0.5, ℎ 𝑨,𝑔 = 1 𝑂 = 42 𝑨 ො 𝑦 − ℎ 𝑨 ෍ ෡ 𝑨 𝑨 𝐼 = − ෍ 𝜏 𝑗 ො 𝜏 𝑗+1 − ℎ 𝑦 ෍ 𝜏 𝑗 ො 𝜏 𝑗 ො 𝑗 𝑗 𝑗 d=1 Half chain entanglement entropy ANN, M=N tDMRG, D=5 ANN, M=2N tDMRG, D=64 tDMRG, D=5 tDMRG, D=96 tDMRG, D=128 tDMRG, D=128 d=2 Bond dimension D ND 2 vs. NM 20.11.2018 Cold Quantum Coffee Stefanie Czischek

Neural-network quantum states [Carleo and Troyer, Science 2017] 𝑙 𝑞 − 𝛿 𝜖𝐹 𝑋 𝑙 𝑞 + 1 = 𝑋 𝜖𝑋 𝑙 Imaginary time evolution Ground state search Random initial weights Learn to represent ground state If necessary: Monte Carlo Markov Chain Stochastic gradient descent: Sample states from 𝑑 𝒘 2 Wave function optimize weights Initial ground state weights Represent time evolution Time evolution Calculating expectation values: 𝑙 𝑢 − 𝑗Δ𝑢 𝜖𝐹 𝑄 𝑋 𝑙 𝑢 + Δ𝑢 = 𝑋 𝒫 = 1 𝒫 𝒘 𝑨 𝑑 𝒘 𝑨 2 ≈ 1 𝜖𝑋 ෠ 𝑨 𝑙 𝑎 ෍ 𝑄 ෍ 𝒫 𝒘 𝑞 Real time evolution 𝒘 𝑨 𝑞=1 20.11.2018 Cold Quantum Coffee Stefanie Czischek

Going to Langevin Dynamics Brownian motion: Particle in a fluid Sampling spin states: Walk around in Hilbert space Hilbert space Physical States Analogy to sampling spin states Equation of motion: Langevin equation Equation of motion: Langevin equation 𝒘 𝑨 𝑢 = −𝜇𝒘 𝑨 𝑢 + 𝜽 𝑨 𝑢 𝑛 ሷ 𝑦 𝑢 = −𝜇 ሶ 𝑦 𝑢 + 𝜽 𝑢 𝑛 ሶ Friction coefficient Noise term (representing collisions) What is the force? Gaussian white noise: Gaussian white noise: 𝑨 𝑢 𝜃 𝑗 𝑢 = 0 𝜃 𝑗 = 0 𝑨 𝑢 𝜃 𝑘 𝑨 𝑢 ′ 𝜃 𝑗 𝑢 𝜃 𝑘 𝑢 ′ = 2𝜇𝑙 𝐶 𝑈𝜀 𝑗,𝑘 𝜀 𝑢 − 𝑢 ′ = 2𝜀 𝑗,𝑘 𝜀 𝑢 − 𝑢 ′ 𝜃 𝑗 20.11.2018 Cold Quantum Coffee Stefanie Czischek

ሶ ሶ Sampling with the Langevin Equation 𝑨 +σ 𝑗,𝑘 𝑤 𝑗 𝑨 𝑋 𝑗,𝑘 ℎ 𝑘 +σ 𝑘 𝑐 𝑘 ℎ 𝑘 =: 𝑓 −𝑇 𝑑 𝒘 𝑨 ,𝒊 = 𝑓 σ 𝑗 𝑏 𝑗 𝑤 𝑗 𝑐 1 𝑐 2 𝑐 3 𝑐 𝑁 𝑨 − ෍ ⋯ ℎ 1 ℎ 2 ℎ 3 ℎ 𝑁 𝑨 𝑋 Action: 𝑇 = − ෍ 𝑏 𝑗 𝑤 𝑗 𝑤 𝑗 𝑗,𝑘 ℎ 𝑘 − ෍ 𝑐 𝑘 ℎ 𝑘 𝑗 𝑗,𝑘 𝑘 𝑋 𝑋 1,1 𝑂,𝑁 𝑨 = − 𝜖𝑇 𝑨 = 𝑏 𝑗 + ෍ 𝑨 𝑤 𝑗 𝑨 + 𝜃 𝑗 𝑋 𝑗,𝑘 ℎ 𝑘 + 𝜃 𝑗 𝜖𝑤 𝑗 𝑨 𝑨 𝑨 𝑘 𝑤 1 𝑤 2 𝑤 𝑂 ⋯ Langevin 𝑏 1 𝑏 2 𝑏 𝑂 Equations: ℎ 𝑘 = − 𝜖𝑇 ℎ = ෍ 𝑨 𝑋 ℎ + 𝜃 𝑘 𝑤 𝑗 𝑗,𝑘 + 𝑐 𝑘 + 𝜃 𝑘 𝜖ℎ 𝑘 𝑗 Complex Real 20.11.2018 Cold Quantum Coffee Stefanie Czischek

ሶ ሶ Sampling with the Langevin Equation Idea: Complex Langevin equations can be applied to complex actions 𝑨 − ෍ 𝑨 𝑋 Action: 𝑇 = − ෍ 𝑏 𝑗 𝑤 𝑗 𝑤 𝑗 𝑗,𝑘 ℎ 𝑘 − ෍ 𝑐 𝑘 ℎ 𝑘 𝑗 𝑗,𝑘 𝑘 Complexification: 𝑨 → ෤ 𝑨 = 𝑤 𝑗 𝑨 + 𝑗𝑤 𝑗 𝑨,𝐽 𝑤 𝑗 𝑤 𝑗 𝑨,𝐽 𝑗𝑤 𝑗 ℎ 𝑘 → ෨ 𝐽 ℎ 𝑘 = ℎ 𝑘 + 𝑗ℎ 𝑘 Complexified equations of motion: 𝑗 𝑨 → ෤ 𝑨 𝑨 = − 𝜖𝑇 𝑤 𝑗 𝑤 𝑗 𝑨 = 𝑏 𝑗 + ෍ 𝑤 𝑗 ෤ + 𝜃 ෤ 𝑋 𝑗,𝑘 ℎ 𝑘 + 𝜃 ෤ 𝑨 𝑨 𝑤 𝑗 𝑤 𝑗 𝜖 ෤ 𝑤 𝑗 𝑨 𝑨 𝑘 𝑤 𝑗 𝑤 𝑗 −1 1 −1 1 ℎ 𝑘 = − 𝜖𝑇 ℎ 𝑘 → ෨ ℎ 𝑘 ෨ 𝑨 𝑋 −𝑗 + 𝜃 ෨ ℎ 𝑘 = 𝑐 𝑘 + ෍ 𝑤 𝑗 𝑗,𝑘 + 𝜃 ෨ 𝑨 = 𝜃 𝑤 𝑗 𝑨 + 𝑗𝜃 𝑤 𝑗 𝜃 ෤ ℎ 𝑘 𝜖෨ ℎ 𝑘 𝑨,𝐽 𝑤 𝑗 𝑗 𝜃 ෨ ℎ 𝑗 = 𝜃 ℎ 𝑗 + 𝑗𝜃 ℎ 𝑗 𝐽 20.11.2018 Cold Quantum Coffee Stefanie Czischek