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

Stefanie Czischek

• L. Kades, J. M. Pawlowski, M. Gärttner, T. Gasenzer

Cold Quantum Coffee 20.11.2018

Violating Bell’s inequality with Langevin dynamics in a deep belief network

20.11.2018 Cold Quantum Coffee

Spin-½ Systems

One spin: two configurations Two spins: four configurations Three spins: eight configurations

Stefanie Czischek

20.11.2018 Cold Quantum Coffee

Spin-½ Systems

Stefanie Czischek

20.11.2018 Cold Quantum Coffee

Spin-½ Systems

Physical States

• Wave function as weighted sum over all product states

| ۧ Ψ = ෍

𝑗1…𝑗𝑂

𝑑𝑗1…𝑗𝑂| ۧ 𝑗1 … 𝑗𝑂

• Look for functional

0,1 𝑂 → ℂ 𝑗1 … 𝑗𝑂 → 𝑑𝑗1…𝑗𝑂 = 𝑔 𝑗1 … 𝑗𝑂; 𝑋 Analogy to Machine Learning! Hilbert space

Stefanie Czischek

20.11.2018 Cold Quantum Coffee

Machine Learning

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

Stefanie Czischek

20.11.2018 Cold Quantum Coffee

Quantum Dynamics and Machine Learning

• 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?

[Carleo and Troyer, Science 2017]

Stefanie Czischek

20.11.2018 Cold Quantum Coffee

Neural-network quantum states

Restricted Boltzmann machine:

• 𝑂 visible variables 𝑤𝑨

𝑗

• M= 𝛽𝑂 hidden variables ℎ𝑘
• Biases 𝑏𝑗, 𝑐

𝑘 and weights 𝑋 𝑗,𝑘 as

variational parameters

[Carleo and Troyer, Science 2017]

| ۧ Ψ = ෍

𝑗1…𝑗𝑂

𝑑𝑗1…𝑗𝑂| ۧ 𝑗1 … 𝑗𝑂 = ෍

𝒘𝑨

𝑑𝒘𝑨| ۧ 𝒘𝒜 𝑑𝒘𝑨 = ෍

𝒊

𝑓−𝐹 𝒘𝑨,𝒊 𝐹 𝒘𝑨, 𝒊 = − ෍

𝑗,𝑘

𝑤𝑗

𝑨𝑋 𝑗,𝑘ℎ𝑘 − ෍ 𝑗

𝑏𝑗 𝑤𝑗

𝑨 − ෍ 𝑘

𝑐

𝑘ℎ𝑘

𝑑𝒘𝑨 = 𝑓σ𝑗 𝑏𝑗𝑤𝑗

𝑨 ෑ

𝑘

2 cosh 𝑐

𝑘 + ෍ 𝑗

𝑤𝑗

𝑨𝑋 𝑗,𝑘

Stefanie Czischek

𝑤𝑗

𝑨, ℎ𝑘𝜗 −1,1

20.11.2018 Cold Quantum Coffee

Neural-network quantum states

[Carleo and Troyer, Science 2017]

Ground state search Random initial weights Wave function Sample states from 𝑑𝒘 2 Learn to represent ground state 𝑋

𝑙 𝑞 + 1 = 𝑋 𝑙 𝑞 − 𝛿 𝜖𝐹

𝜖𝑋

𝑙

Imaginary time evolution Stochastic gradient descent:

• ptimize weights

Initial ground state weights If necessary: Monte Carlo Markov Chain Represent time evolution 𝑋

𝑙 𝑢 + Δ𝑢 = 𝑋 𝑙 𝑢 − 𝑗Δ𝑢 𝜖𝐹

𝜖𝑋

𝑙

Real time evolution Time evolution ෠ 𝒫 = 1 𝑎 ෍

𝒘𝑨

𝒫 𝒘𝑨 𝑑𝒘𝑨 2 ≈ 1 𝑄 ෍

𝑞=1 𝑄

𝒫 𝒘𝑞

𝑨

Calculating expectation values:

Stefanie Czischek

20.11.2018 Cold Quantum Coffee

Transverse Field Ising Model (TFIM)

෡ 𝐼 = − ෍

𝑗

ො 𝜏𝑗

𝑨 ො

𝜏𝑗+1

𝑨

− ℎ𝑦 ෍

𝑗

ො 𝜏𝑗

𝑦

T ℎ𝑑 = 1 ℎ𝑦 LOW T Magnetic long-range order LOW T Quantum paramagnet Quantum critical Sudden quench

• Quantum critical point at ℎ𝑑 = 1
• Analytical solutions available

[Lieb, Calabrese,…]

• Quenches studied in detail

[Karl, Cakir et al. PRE 2017]

• Hard for MPS based methods

𝜗 = ℎ𝑦 − ℎ𝑑 ℎ𝑑

Picture from: S. Sachdev, Quantum Phase Transitions

Stefanie Czischek

20.11.2018 Cold Quantum Coffee

Neural-network quantum states

[Carleo and Troyer, Science 2017]

Ground state search Random initial weights Wave function Sample states from 𝑑𝒘 2 Learn to represent ground state 𝑋

𝑙 𝑞 + 1 = 𝑋 𝑙 𝑞 − 𝛿 𝜖𝐹

𝜖𝑋

𝑙

Imaginary time evolution Stochastic gradient descent:

• ptimize weights

Initial ground state weights If necessary: Monte Carlo Markov Chain Represent time evolution 𝑋

𝑙 𝑢 + Δ𝑢 = 𝑋 𝑙 𝑢 − 𝑗Δ𝑢 𝜖𝐹

𝜖𝑋

𝑙

Real time evolution Time evolution ෠ 𝒫 = 1 𝑎 ෍

𝒘𝑨

𝒫 𝒘𝑨 𝑑𝒘𝑨 2 ≈ 1 𝑄 ෍

𝑞=1 𝑄

𝒫 𝒘𝑞

𝑨

Calculating expectation values:

Stefanie Czischek

20.11.2018 Cold Quantum Coffee

Quenches in the TFIM

ℎ𝑦 1

• 1

𝐷𝑒

𝑨𝑨 𝑢 =

ො 𝜏𝑗

𝑨 ො

𝜏𝑗+𝑒

𝑨

𝐷𝑒

𝑨𝑨 𝑢

= 𝑓− Τ

𝑒 𝜊 𝑢

𝑂 visible, 𝑁 hidden neurons Correlation length

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20.11.2018 Cold Quantum Coffee

TFIM in a longitudinal field

෡ 𝐼 = − ෍

𝑗

ො 𝜏𝑗

𝑨 ො

𝜏𝑗+1

𝑨

− ℎ𝑦 ෍

𝑗

ො 𝜏𝑗

𝑦 − ℎ𝑨 ෍ 𝑗

ො 𝜏𝑗

𝑨

ℎ𝑦,𝑗 = 100 ℎ𝑨,𝑗 = 0 𝑂 = 10 ℎ𝑨,𝑔 = 2 ℎ𝑦 ℎ𝑨 QCP QCP 𝐷𝑒

𝑨𝑨 𝑢

= 𝑓− Τ

𝑒 𝜊 𝑢

Δ𝜊 𝑢 = 𝜊 − 𝜊exact

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20.11.2018 Cold Quantum Coffee

Simulating large spin systems

෡ 𝐼 = − ෍

𝑗

ො 𝜏𝑗

𝑨 ො

𝜏𝑗+1

𝑨

− ℎ𝑦 ෍

𝑗

ො 𝜏𝑗

𝑦 − ℎ𝑨 ෍ 𝑗

ො 𝜏𝑗

𝑨

𝑂 = 42 ANN, M=N ANN, M=2N tDMRG, D=128 tDMRG, D=5 tDMRG, D=5 tDMRG, D=64 tDMRG, D=128 tDMRG, D=96 ℎ𝑦,𝑔 = 0.5, ℎ𝑨,𝑔 = 1 d=1 d=2 Half chain entanglement entropy Bond dimension D ND2 vs. NM

Stefanie Czischek

20.11.2018 Cold Quantum Coffee

Simulating large spin systems

෡ 𝐼 = − ෍

𝑗

ො 𝜏𝑗

𝑨 ො

𝜏𝑗+1

𝑨

− ℎ𝑦 ෍

𝑗

ො 𝜏𝑗

𝑦 − ℎ𝑨 ෍ 𝑗

ො 𝜏𝑗

𝑨

𝑂 = 42 ANN, M=N ANN, M=2N tDMRG, D=128 tDMRG, D=5 tDMRG, D=5 tDMRG, D=64 tDMRG, D=128 tDMRG, D=96 ℎ𝑦,𝑔 = 0.5, ℎ𝑨,𝑔 = 1 d=1 d=2 Half chain entanglement entropy Bond dimension D ND2 vs. NM

Stefanie Czischek

20.11.2018 Cold Quantum Coffee

Neural-network quantum states

[Carleo and Troyer, Science 2017]

Ground state search Random initial weights Wave function Sample states from 𝑑𝒘 2 Learn to represent ground state 𝑋

𝑙 𝑞 + 1 = 𝑋 𝑙 𝑞 − 𝛿 𝜖𝐹

𝜖𝑋

𝑙

Imaginary time evolution Stochastic gradient descent:

• ptimize weights

Initial ground state weights If necessary: Monte Carlo Markov Chain Represent time evolution 𝑋

𝑙 𝑢 + Δ𝑢 = 𝑋 𝑙 𝑢 − 𝑗Δ𝑢 𝜖𝐹

𝜖𝑋

𝑙

Real time evolution Time evolution ෠ 𝒫 = 1 𝑎 ෍

𝒘𝑨

𝒫 𝒘𝑨 𝑑𝒘𝑨 2 ≈ 1 𝑄 ෍

𝑞=1 𝑄

𝒫 𝒘𝑞

𝑨

Calculating expectation values:

Stefanie Czischek

20.11.2018 Cold Quantum Coffee

Going to Langevin Dynamics

Brownian motion: Particle in a fluid Equation of motion: Langevin equation 𝑛 ሷ 𝑦 𝑢 = −𝜇 ሶ 𝑦 𝑢 + 𝜽 𝑢 Friction coefficient Noise term (representing collisions) Gaussian white noise: 𝜃𝑗 𝑢 = 0 𝜃𝑗 𝑢 𝜃𝑘 𝑢′ = 2𝜇𝑙𝐶𝑈𝜀𝑗,𝑘𝜀 𝑢 − 𝑢′ Sampling spin states: Walk around in Hilbert space

Physical States Hilbert space

Equation of motion: Langevin equation 𝑛 ሶ 𝒘𝑨 𝑢 = −𝜇𝒘𝑨 𝑢 + 𝜽𝑨 𝑢 What is the force? Gaussian white noise: 𝜃𝑗

𝑨 𝑢

= 0 𝜃𝑗

𝑨 𝑢 𝜃𝑘 𝑨 𝑢′

= 2𝜀𝑗,𝑘𝜀 𝑢 − 𝑢′ Analogy to sampling spin states

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20.11.2018 Cold Quantum Coffee

Sampling with the Langevin Equation

𝑇 = − ෍

𝑗

𝑏𝑗𝑤𝑗

𝑨 − ෍ 𝑗,𝑘

𝑤𝑗

𝑨𝑋 𝑗,𝑘ℎ𝑘 − ෍ 𝑘

𝑐

𝑘ℎ𝑘

ሶ 𝑤𝑗

𝑨 = − 𝜖𝑇

𝜖𝑤𝑗

𝑨 + 𝜃𝑗 𝑨 = 𝑏𝑗 + ෍ 𝑘

𝑋

𝑗,𝑘ℎ𝑘 + 𝜃𝑗 𝑨

ሶ ℎ𝑘 = − 𝜖𝑇 𝜖ℎ𝑘 + 𝜃𝑘

ℎ = ෍ 𝑗

𝑤𝑗

𝑨𝑋 𝑗,𝑘 + 𝑐 𝑘 + 𝜃𝑘 ℎ

Real Complex

Action: Langevin Equations:

ℎ1 ℎ2 ℎ3 ℎ𝑁 𝑤𝑂

𝑨

𝑤2

𝑨

𝑤1

𝑨

𝑐1 𝑐2 𝑐3 𝑐𝑁 𝑏𝑂 𝑏2 𝑏1 𝑋

1,1

𝑋

𝑂,𝑁

⋯ ⋯ 𝑑𝒘𝑨,𝒊 = 𝑓σ𝑗 𝑏𝑗𝑤𝑗

𝑨+σ𝑗,𝑘 𝑤𝑗 𝑨𝑋𝑗,𝑘ℎ𝑘+σ𝑘 𝑐𝑘ℎ𝑘 =: 𝑓−𝑇

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20.11.2018 Cold Quantum Coffee

Sampling with the Langevin Equation

Idea: Complex Langevin equations can be applied to complex actions

𝑤𝑗

𝑨 → ෤

𝑤𝑗

𝑨 = 𝑤𝑗 𝑨 + 𝑗𝑤𝑗 𝑨,𝐽

ℎ𝑘 → ෨ ℎ𝑘 = ℎ𝑘 + 𝑗ℎ𝑘

𝐽

𝜃෤

𝑤𝑗

𝑨 = 𝜃𝑤𝑗 𝑨 + 𝑗𝜃𝑤𝑗 𝑨,𝐽

𝜃෨

ℎ𝑗 = 𝜃ℎ𝑗 + 𝑗𝜃ℎ𝑗

𝐽

𝑇 = − ෍

𝑗

𝑏𝑗𝑤𝑗

𝑨 − ෍ 𝑗,𝑘

𝑤𝑗

𝑨𝑋 𝑗,𝑘ℎ𝑘 − ෍ 𝑘

𝑐

𝑘ℎ𝑘

Action: Complexification:

Stefanie Czischek

1 𝑤𝑗

𝑨

−1 1 𝑤𝑗

𝑨

−1 𝑗 −𝑗 𝑗𝑤𝑗

𝑨,𝐽

ሶ ෤ 𝑤𝑗

𝑨 = − 𝜖𝑇 𝑤𝑗 𝑨 → ෤

𝑤𝑗

𝑨

𝜖 ෤ 𝑤𝑗

𝑨

+ 𝜃෤

𝑤𝑗

𝑨 = 𝑏𝑗 + ෍

𝑘

𝑋

𝑗,𝑘 ℎ𝑘 + 𝜃෤ 𝑤𝑗

𝑨

ሶ ෨ ℎ𝑘 = − 𝜖𝑇 ℎ𝑘 → ෨ ℎ𝑘 𝜖෨ ℎ𝑘 + 𝜃෨

ℎ𝑘 = 𝑐 𝑘 + ෍ 𝑗

𝑤𝑗

𝑨𝑋 𝑗,𝑘 + 𝜃෨ ℎ𝑘

Complexified equations of motion:

20.11.2018 Cold Quantum Coffee

Sampling with the Langevin Equation

Idea: Complex Langevin equations can be applied to complex actions

𝑤𝑗

𝑨 → ෤

𝑤𝑗

𝑨 = 𝑤𝑗 𝑨 + 𝑗𝑤𝑗 𝑨,𝐽

ℎ𝑘 → ෨ ℎ𝑘 = ℎ𝑘 + 𝑗ℎ𝑘

𝐽

𝜃෤

𝑤𝑗

𝑨 = 𝜃𝑤𝑗 𝑨 + 𝑗𝜃𝑤𝑗 𝑨,𝐽

𝜃෨

ℎ𝑗 = 𝜃ℎ𝑗 + 𝑗𝜃ℎ𝑗

𝐽

𝑇 = − ෍

𝑗

𝑏𝑗𝑤𝑗

𝑨 − ෍ 𝑗,𝑘

𝑤𝑗

𝑨𝑋 𝑗,𝑘ℎ𝑘 − ෍ 𝑘

𝑐

𝑘ℎ𝑘

Action: Complexification:

Stefanie Czischek

1 𝑤𝑗

𝑨

−1 1 𝑤𝑗

𝑨

−1 𝑗 −𝑗 𝑗𝑤𝑗

𝑨,𝐽

ሶ ෤ 𝑤𝑗

𝑨 = − 𝜖𝑇 𝑤𝑗 𝑨 → ෤

𝑤𝑗

𝑨

𝜖 ෤ 𝑤𝑗

𝑨

+ 𝜃෤

𝑤𝑗

𝑨 = 𝑏𝑗 + ෍

𝑘

𝑋

𝑗,𝑘 ℎ𝑘 + 𝜃෤ 𝑤𝑗

𝑨

ሶ ෨ ℎ𝑘 = − 𝜖𝑇 ℎ𝑘 → ෨ ℎ𝑘 𝜖෨ ℎ𝑘 + 𝜃෨

ℎ𝑘 = 𝑐 𝑘 + ෍ 𝑗

𝑤𝑗

𝑨𝑋 𝑗,𝑘 + 𝜃෨ ℎ𝑘

Complexified equations of motion:

20.11.2018 Cold Quantum Coffee

Use Langevin Dynamics for the real part

𝑇 = − ෍

𝑗

𝑏𝑗𝑤𝑗

𝑨 − ෍ 𝑗,𝑘

𝑤𝑗

𝑨𝑋 𝑗,𝑘ℎ𝑘 − ෍ 𝑘

𝑐

𝑘ℎ𝑘 =: 𝑇𝑆 Re 𝒃 , Re 𝒄 , Re 𝑋

+ 𝑗𝑇𝐽 Im 𝒃 , Im 𝒄 , Im 𝑋

Action:

ሶ 𝑤𝑗

𝑨 = Re 𝑏𝑗 + ෍ 𝑘

Re 𝑋

𝑗,𝑘 ℎ𝑘 + 𝜃𝑤𝑗

𝑨

ሶ ℎ𝑘 = Re 𝑐

𝑘 + ෍ 𝑗

𝑤𝑗

𝑨Re 𝑋 𝑗,𝑘 + 𝜃ℎ𝑘

Real Langevin: Calculate observables: ෠ 𝒫 = 1 𝑎 ෍

𝒘𝑨

෍

𝒘𝑨′

𝒫 𝒘𝑨 𝑑𝒘𝑨𝑑𝒘𝑨′

∗

𝜀𝒘𝑨,𝒘𝑨′ ≈ 1 ෨ 𝑄 ෍

𝑞=1 𝑄

෍

𝑟=1 𝑄

𝒫 𝒘𝑞

𝑨 𝑓𝑗𝑇𝐽 𝒘𝑞

𝑨,𝒊𝑞 −𝑗𝑇𝐽 𝒘𝑟 𝑨,𝒊𝑟 𝜀𝒘𝑞 𝑨,𝒘𝑟 𝑨 ,

(for diagonal operators) ෨ 𝑄 = ෍

𝑞=1 𝑄

෍

𝑟=1 𝑄

𝑓𝑗𝑇𝐽 𝒘𝑞

𝑨,𝒊𝑞 −𝑗𝑇𝐽 𝒘𝑟 𝑨,𝒊𝑟 𝜀𝒘𝑞 𝑨,𝒘𝑟 𝑨

Stefanie Czischek

Include phase in expectation values!

20.11.2018 Cold Quantum Coffee

Measuring in different bases

ො 𝜏𝑗

𝑨 = ෍ 𝒘

𝑤𝑗

𝑨 𝑑𝒘𝑨 2 ≈ 1

෨ 𝑄 ෍

𝑞=1 𝑄

෍

𝑟=1 𝑄

𝑤𝑗,𝑞

𝑨 𝑓𝑗𝑇𝐽 𝒘𝑞

𝑨,𝒊𝑞 −𝑗𝑇𝐽 𝒘𝑟 𝑨,𝒊𝑟 𝜀𝒘𝑞 𝑨,𝒘𝑟 𝑨

Expectation values of diagonal operators: But what about operators in other bases, such as ො 𝜏𝑗

𝑦 or ො

𝜏𝑗

𝑧 ?

 Locally rotate quantum state: 𝑑𝒘𝑦 = ෍

𝒘𝑨,𝒊

𝑓𝒘𝑨𝑋𝒊+𝒃𝒘𝑨+𝒄𝒊+𝑗𝜌

4 𝒘𝑦𝒘𝑨−𝒘𝑦−𝒘𝑨+1

𝑑𝒘𝑧 = ෍

𝒘𝑨,𝒊

𝑓𝒘𝑨𝑋𝒊+𝒃𝒘𝑨+𝒄𝒊+𝑗𝜌

4 𝒘𝑧𝒘𝑨−1

Stefanie Czischek

෨ 𝑄 = ෍

𝑞=1 𝑄

෍

𝑟=1 𝑄

𝑓𝑗𝑇𝐽 𝒘𝑞

𝑨,𝒊𝑞 −𝑗𝑇𝐽 𝒘𝑟 𝑨,𝒊𝑟 𝜀𝒘𝑞 𝑨,𝒘𝑟 𝑨

z y x z y x z y x

20.11.2018 Cold Quantum Coffee

Setting up a deep belief network (DBN)

𝑑𝒘𝑦 = ෍

𝒘𝑨,𝒊

𝑓𝒘𝑨𝑋𝒊+𝒃𝒘𝑨+𝒄𝒊+𝑗𝜌

4 𝒘𝑦𝒘𝑨−𝒘𝑦−𝒘𝑨+1

𝑑𝒘𝑧 = ෍

𝒘𝑨,𝒊

𝑓𝒘𝑨𝑋𝒊+𝒃𝒘𝑨+𝒄𝒊+𝑗𝜌

4 𝒘𝑧𝒘𝑨−1

𝑑𝒘𝑨 = ෍

𝒊

𝑓𝒘𝑨𝑋𝒊+𝒃𝒘𝑨+𝒄𝒊 ℎ1 ℎ2 ℎ3 ℎ𝑁 𝑤𝑂

𝑨

𝑤2

𝑨

𝑤1

𝑨

𝑐1 𝑐2 𝑐3 𝑐𝑁 𝑏𝑂 + 𝑗 𝜌 4 𝑏1 + 𝑗 𝜌 4 𝑋

1,1

𝑋

𝑂,𝑁

⋯ ⋯ 𝑤1

𝑦

𝑤2

𝑦

𝑤𝑂

𝑦

𝑤1

𝑧

𝑤𝑂

𝑧

𝑤2

𝑧

⋯

• 𝑗

𝜌 4

• 𝑗

𝜌 4

• 𝑗 𝜌

4

𝑗 𝜌 4 𝑗 𝜌 4 𝑗 𝜌 4 𝑗 𝜌 4 𝑗 𝜌 4 𝑗 𝜌 4 ሶ 𝒘𝑦 = ሶ 𝒘𝑧 = 0 + 𝜽𝑦/𝑧 (only complex phase) ሶ 𝒘𝑨 = Re 𝑋 𝒊 + Re 𝒃 + 𝜽𝑨 ሶ 𝒊 = 𝒘𝑨Re 𝑋 + Re 𝒄 + 𝜽ℎ Equations of motion:

Stefanie Czischek

20.11.2018 Cold Quantum Coffee

The Bell-pair state

ۧ Ψ = 1 2 | ۧ ↑↓ + | ۧ ↓↑ = ෍

𝒘𝑨

𝑑𝒘𝑨 ۧ 𝑤1

𝑨, 𝑤2 𝑨

⇒ 𝑑𝒘𝑨 = ቐ 1 2 if 𝑤1

𝑨 = −𝑤2 𝑨

else 𝑋

1,1 = −𝜕𝑓 + 𝑗 𝜌

2 , 𝑋

2,1 = 𝜕𝑓 + 𝑗 𝜌

2 , 𝑏1 = 𝑗 𝜌 2 , 𝑏2 = 0, 𝑐1 = 𝑗 𝜌 2 , 𝜕𝑓 = 1 2 sinh−1 1 8 𝑋

1,1 = 𝑗 −𝜕𝑝 + 𝜌

4 , 𝑋

2,1 = 𝑗 𝜕𝑝 + 𝜌

4 , 𝑏1 = 0, 𝑏2 = 0, 𝑐1 = 0, 𝜕𝑝 = 1 2 cos−1 1 8 ℎ1 𝑤2

𝑨

𝑤1

𝑨

𝑐1 𝑏2 + 𝑗 𝜌 4 𝑏1 + 𝑗 𝜌 4 𝑋

1,1

𝑤1

𝑦

𝑤2

𝑦

𝑤1

𝑧

𝑤2

𝑧

• 𝑗

𝜌 4

• 𝑗 𝜌

4

𝑗 𝜌 4 𝑗 𝜌 4 𝑗 𝜌 4 𝑗 𝜌 4 𝑋

2,1

Choice for complex weights: Choice for imaginary weights: There are many possible choices for the weights!

Stefanie Czischek

20.11.2018 Cold Quantum Coffee

Quantum entanglement: Bell’s inequality

CHSH-inequality (J. Clauser, M. Horne, A. Shimony, R. Holt): ℬCHSH = መ 𝐵1⨂ ෠ 𝐶1 + መ 𝐵1⨂ ෠ 𝐶2 + መ 𝐵2⨂ ෠ 𝐶1 − መ 𝐵2⨂ ෠ 𝐶2 ≤ 2 in a classical system Choose: መ 𝐵1 = ො 𝜏1

𝑦

መ 𝐵2 = ො 𝜏1

𝑨

෠ 𝐶1 =

1 2 ො

𝜏2

𝑦 − ො

𝜏2

𝑨

෠ 𝐶2 =

1 2 ො

𝜏2

𝑦 + ො

𝜏2

𝑨

⇒ ℬCHSH = 2 ො 𝜏1

𝑦 ො

𝜏2

𝑦 − ො

𝜏1

𝑨 ො

𝜏2

𝑨

= 2 2 > 2 for the Bell-pair state

Complex weights Imaginary weights

Re( ) Re( ) Im( ) Im( )

Stefanie Czischek

20.11.2018 Cold Quantum Coffee

Violating Bell’s inequality with a DBN

Complex Weights Imaginary Weights

z basis z basis x basis x basis Magnetization Correlation Magnetization Correlation Magnetization Correlation

Exact solution Sampled results Absolute error

Stefanie Czischek

20.11.2018 Cold Quantum Coffee

Violating Bell’s inequality with a DBN

Stefanie Czischek

Complex Weights Imaginary Weights

z basis z basis x basis x basis Magnetization Correlation

Exact solution

Magnetization Correlation

Sampled results

Magnetization Correlation

Absolute error

20.11.2018 Cold Quantum Coffee

Violating Bell’s inequality with a DBN

ℬCHSH = 2 ො 𝜏1

𝑦 ො

𝜏2

𝑦 − ො

𝜏1

𝑨 ො

𝜏2

𝑨

Complex Weights Imaginary Weights Exact solution Sampled results Absolute error

2 4 2 4

Stefanie Czischek

20.11.2018 Cold Quantum Coffee

Violating Bell’s inequality with a DBN

ℬCHSH = 2 ො 𝜏1

𝑦 ො

𝜏2

𝑦 − ො

𝜏1

𝑨 ො

𝜏2

𝑨

Complex Weights Imaginary Weights Exact solution Sampled results Absolute error

2 4 2 4

Stefanie Czischek

20.11.2018 Cold Quantum Coffee

Outlook: Connection to the BrainScaleS group

BUT: ➢ We still need to evaluate the phase 𝑓−𝑗𝑇𝐽 to calculate expectation values ➢ This needs to be done on a classical computer ➢ Can the evaluation be done in an efficient way such that the sampling is the bottleneck? ➢ Sampling with Leaky-Integrate-and-Fire (LIF) neurons can be described via Langevin dynamics ➢ Sampling can be done in a very efficient and fast way on the neuromorphic hardware ➢ Huge sample sets can be created in short times (help with sign problem?)

Stefanie Czischek

[s] [s]

Sampling, 1 hidden neuron Evaluation, 1 hidden neuron Sampling, 3 hidden neurons Evaluation, 3 hidden neurons Sampling, 6 hidden neurons Evaluation, 6 hidden neurons Sampling, 20 hidden neurons Evaluation, 20 hidden neurons Sampling, 100 hidden neurons Evaluation, 100 hidden neurons

20.11.2018 Cold Quantum Coffee

Conclusion

➢ We can use Langevin dynamics in a deep belief network to sample spin states which show qantum entanglement ➢ With this ansatz we can go to deeper and more complex networks ➢ We can perform measurements in different bases ➢ We still need to evaluate the phase separately, this can end up in the sign problem ➢ We can create many samples efficiently with the neuromorphic hardware, but for large networks the evaluation becomes expensive

Stefanie Czischek