Neural Simulated Classical and Quantum Annealing

Mohamed Hibat-Allah

Neural Simulated Classical and

Quantum Annealing

AQC, June 2021

2

Joint work with

RoelandWiersema

JuanCarrasquilla

EstelleInack

Roger Melko

Mohamed Hibat-Allah, Estelle Inack, Roeland Wiersema, Roger G. Melko, Juan Carrasquilla, Variational Neural Annealing, arXiv:2101.10154.

Neural Simulated Classical and Quantum Annealing

Question?

3

Can we solve real-world Optimization Problems using

< Neural Networks | Annealing >?


Question?

4

Can we solve real-world Optimization Problems using

Recurrent Neural Networks by taking advantage of Simulated

(Classical and Quantum) Annealing?


5

Outline

o 1 - Recurrent Neural Networks (RNN).

o 2 - Variational Classical Annealing with RNNs.

o 3 - Variational Quantum Annealing with RNNs.

o 4 - Promising results on prototypical models.


6

1 - Recurrent Neural Networks


Recurrent Neural Networks (RNNs)

7

Very powerful at generating sequential data.

• Speech recognition, machine translation, times series reconstruction.


Recurrent Neural Networks (RNNs)

8

Very powerful at generating sequential data.

• Speech recognition, machine translation, times series reconstruction.

-Juan Carrasquilla, Giacomo Torlai, Roger G. Melko, Leandro Aolita

Reconstructing quantum states with generative models, Nature Machine Intelligence, 2019.

-Mohamed Hibat-Allah, Martin Ganahl, Lauren E. Hayward, Roger G. Melko, Juan Carrasquilla, Recurrent Neural Network Wave Functions,

PRReasearch, 2020.

-Christopher Roth, Iterative Retraining of Quantum Spin Models using Recurrent Neural Networks,

2020.

RNNs and Quantum-many body physics


RNN Wave Functions

9


Recurrent Neural Network Wave Functions,

PRReasearch, Jun 2020.

10

RNN Wave Functions




11

RNN Wave Functions




12

RNN Wave Functions




13

RNN Wave Functions

An approximation of the ground state is found!




14

2 - Variational Classical Annealing (VCA)

with RNNs


15

-Goal: Find ground state of a target (classical) Hamiltonian:

Simulated Classical Annealing


16



-Examples:

• Spin glass models.

• Traveling salesman problems.

• Protein folding problems,…


17



Entropy term (thermal fluctuations)

-Free Energy:



18Neural Simulated Classical and Quantum Annealing

19

High-Temperature Regime


20

Warmup Phase

𝐅𝐑𝐍𝐍 from a randomly initialized RNN


21

Warmup Phase

Gradient descent


22

Warmup Phase

Gradient descent


23

Warmup Phase

Gradient descent


24

Warmup Phase


25

Annealing Step


26

Training Step

Gradient descent


27

Close to zero temperature

Gradient descent

Global minimum


28

Optimization at T = 0

Gradient descent


29

Gradient descent



30

Gradient descent


Local minimum


31

3 - Variational Quantum Annealing (VQA)

with RNN Wave Functions


32

RNNwave function

Variational Quantum Annealing with RNNs

Initial Hamiltonian:

“Ground state is simple”


33

Gradient descent

Warmup phase


34

Gradient descent

Warmup phase


35

Gradient descent

Warmup phase


36

Annealing Step


37

Annealing Step


38

Training Step

Gradient descent


39

Training Step

Gradient descent


40

Annealing Step


41

Training Step


42

Variational Quantum Annealing (VQA)


43



44



45


Ground state of target Hamiltonian is found!


The Variational Adiabatic Theorem

46

is sufficient to guarantee:

Under a set of assumptions [1], a total number of gradient steps:

[1] Variational Neural Annealing, 2101.10154


The Variational Adiabatic Theorem

47

Conclusions:

[1] Variational Neural Annealing, 2101.10154


48

4 - Results on spin-glass models


49

Spin Ising glass in 2D

Edwards-Anderson model:

The couplings are drawn from random uniform distribution in

the range [-1,1)


50

Spin Ising glass in 2D (N = 10x10 spins)


Residual energy:

Variational Classical Annealing (VCA)

• 𝐓(𝐭) = 𝐓𝟎 𝟏 − 𝐭/𝐍𝐚𝐧𝐧𝐞𝐚𝐥𝐢𝐧𝐠


• 𝐁𝐱(𝐭) = 𝐁𝐱𝟎 𝟏 − 𝐭/𝐍𝐚𝐧𝐧𝐞𝐚𝐥𝐢𝐧𝐠


51


Variational Classical Annealing (VCA)

• 𝐓(𝐭) = 𝐓𝟎 𝟏 − 𝐭/𝐍𝐚𝐧𝐧𝐞𝐚𝐥𝐢𝐧𝐠


• 𝐁𝐱(𝒕) = 𝐁𝐱𝟎 𝟏 − 𝐭/𝐍𝐚𝐧𝐧𝐞𝐚𝐥𝐢𝐧𝐠

Classical-Quantum Optimization (CQO)

• 𝐓 = 𝟎, 𝐁𝐱 = 𝟎



52


SA: Simulated Annealing.

SQA: Path-Integral Quantum Monte Carlo.



Sampling advantage of RNNs

53Neural Simulated Classical and Quantum Annealing

SA and SQAMetropolis sampling

VCA (RNNs)Autoregressive (exact) sampling

Fully-connected spin glasses (N = 100 spins)

54

The couplings are drawn from a gaussian distribution with

mean 0 and variance 1.

Sherrington-Kirkpatrick model:

Residual energy:


55

Takeaways

o We can use neural networks to emulate Classical and Quantum Annealing to

solve optimization problems.

o Neural Machine Translation (RNNs) meets Classical and Quantum Annealing.

o VCA is superior compared to VQA.

o VCA is superior compared to SA and SQA.


More results: Variational Neural Annealing, arXiv:2101.10154.

56

Thank you for your attention!


o We can use neural networks to emulate Classical and Quantum Annealing to

solve optimization problems.

o Neural Machine Translation (RNNs) meets Classical and Quantum Annealing.

o VCA is superior compared to VQA.

o VCA is superior compared to SA and SQA.

More results: Variational Neural Annealing, arXiv:2101.10154.

Documents

Neural Simulated Classical and Quantum Annealing