Quantum neurons Yudong Cao with Gian Giacomo Guerreschi, Aln - - PowerPoint PPT Presentation

Quantum neurons

Yudong Cao

with Gian Giacomo Guerreschi, Alán Aspuru-Guzik

Quantum Techniques in Machine Learning 2017, Verona, Italy.

The quest for quantum neural nets

• Parametrized quantum system that can be trained

to accomplish tasks such as classification

• In many cases, it is not easy to identify what is the

fundamental building block with which one could describe the quantum system as a learning algorithm

• This work can be seen as a conceptual attempt at

addressing this issue

Nonlinear and parallel Builds up its own rules through experience Neural network

a machine that is designed to mimic the way in which the brain performs a particular task or function of interest

Basic requirements for quantum NN

• 1. Initial state encodes

any N-bit binary string

• 2. Reflects one or more

basic neural computing mechanisms

• 3. The evolution is based
• n quantum effects

e.g. attractor dynamics, synaptic connections, integrate & fire, training rules, structure of a NN 01001 01001 Schuld, M., Sinayskiy, I. & Petruccione, F. Quantum Inf Process (2014) 13: 2567 Superposition and entanglement

(artificial) Neuron

𝜄 𝜄 =

𝑗

𝑥𝑗𝑦𝑗 + 𝑐 𝑐 1 𝜄 𝜏 𝜄

Can we reali lize art rtif ific icia ial l neurons on a quantum computer?

QM + NN: an unlikely match ?

• Unitary evolution
• Rotation in Hilbert space

Quantum Mechanics (QM) Neural Networks (NN)

• Lossy transformations
• Clustering, classification,

compression etc

Challenges

• Sigmoid / step function activation

How to realize on quantum computers, whose dynamics is lin linear?

• Measurement? Open system?

May collapse the state / reduce to classical probabilistic algorithms

Dissipative dynamics

Story of quantum error correction

Reversible circuits Cost scaling?

Our proposal

Neuron Qubit Activation Rotation angle

rest active Activation 𝑧 = 𝜏(𝜄)

rest active 1

𝑆𝑧(𝜒) 𝜒

𝜄 =

𝑗

𝑥𝑗𝑦𝑗 + 𝑐 … 𝑦1 𝑦2 𝑦𝑜 𝑥1 𝑥2 𝑥𝑜 1 𝜄 𝜏 𝜄 Information from previous layer

𝜒 = 𝛿𝜄 + 𝜌 4

Introduce nonlinearity

Repeat-until-success (RUS) circuits: Given ability to realize 𝑆𝑧 2𝑦 One could use RUS to realize 𝑆𝑧(2𝑔(𝑦))

𝑔 𝑦 = arctan tan2 𝑦

𝑦 Measure 0: 𝑆𝑧(𝑔(𝑦)) 𝜔 Measure 1: 𝑆𝑧(𝜌/4) 𝜔 Su Success Fail ail bu but eas easil ily cor

• rrectable

Nonlinear!

Repeat until success

𝑆𝑧 𝜄 = cos 𝜄 2 −sin 𝜄 2 sin 𝜄 2 cos 𝜄 2

𝑔 𝑔 … 𝑔 𝑦 … = 𝑔°𝑙(𝑦)

𝑙 times

𝑆𝑧 𝑦 𝑆𝑧 𝑔(𝑦 ) 𝑆𝑧 𝑔°𝑙(𝑦)

… …

𝑆𝑧(2𝑔(𝜄)) 1 𝑆𝑧(𝑔∘𝑙(𝜄)) 𝑆𝑧(ߠ) 0 𝜄

• Prev. layer

|010…> RUS x k Controlled rotations by angle 𝑥𝑗, 𝑐 … 𝑦1 = 0 𝑦2 = 1 𝑦3 = 0 Close to either 0 or 1 due to nonlinear function 𝑔 Weighted sum Nonlinear

• utput

𝜄 =

𝑗

𝑥𝑗𝑦𝑗 + 𝑐 … 𝑦1 𝑦2 𝑦3 𝑥1 𝑥2 𝑥3 𝜄 =

𝑗

𝑥𝑗𝑦𝑗 + 𝑐 𝑧 = 𝜏(𝜄) Prev. layer Weighted sum Nonlinear

• utput

• Size
• Neuron type
• Connectivity
• Activation function
• Weight/bias setting
• Training method
• …

Feedforward network

“cat”

XOR network

𝑦1 𝑦2

Train the network such that 𝑡 = 𝑦1⨁𝑦2

𝑡 1 2 00 1 + 01 0 + 10 0 + 11 1 𝒚𝟐 𝒚𝟐 𝒕 1 1 1 1 1 1 Input Correct output 𝑎𝑎

Accuracy:

1+ 𝑎𝑎 2

1 2 00 1 + 01 0 + 10 0 + 11 1 Solid: training on Dashed: testing on 00 10 01 11 average

8-bit parity network

𝑦1 𝑦2

Train the network such that 𝑡 = 𝑦1⨁ … ⨁𝑦8

𝑡 𝑎𝑎

Accuracy:

1+ 𝑎𝑎 2

𝑦3 𝑦4 𝑦5 𝑦6 𝑦7 𝑦8

⋮

8

1 28

𝑗=0 28−1

𝑗 Parity(𝑗) Solid: training on Dashed: testing on 28=256 states 00000000 00000001 ⋯ 11111111 average

Hopfield network

Initial state Update Repeat Final state (attractor)

𝑡𝑗

𝑡𝑗 = 1 𝜄𝑗 > 0 −1 𝜄𝑗 < 0

𝑡

𝑘

𝑥𝑗𝑘

𝜄𝑗 =

𝑘≠𝑗

𝑥𝑗𝑘𝑡

𝑘 + 𝑐𝑗

Hopfield net of quantum neurons

𝑡1

𝑡𝑗 = 1 𝜄𝑗 > 0 −1 𝜄𝑗 < 0 𝜄𝑗 =

𝑘≠𝑗

𝑥𝑗𝑘𝑡

𝑘 + 𝑐𝑗

𝑡2 𝑡3 𝑡4 𝑟1

(0)

𝑟2

(0)

𝑟3

(0)

𝑟4

(0)

𝑟3

(1)

RUS x k 𝑟4

(2)

RUS x k 𝑟2

(3)

RUS x k

…

𝑜 + 𝑢 + 𝑙 qubits for Hopfield network of 𝑜 neurons and 𝑢 updates

Numerical example

attractors: letters C and Y 3x3 grid

1

+ + +

1 1 1 initial input after 1 update after 2 updates after 3 updates

Summary

• Building block for quantum neural network satisfying
• Initial state encoding n-bit strings

Neuron <-> Qubit

• One or more neural computing mechanisms

Sigmoid/step function, attractor

• Evolution based on quantum effects

Train with superposition of examples

• Application and extensions
• Superposition of weights (networks) ?
• Different forms of networks
• Different activation functions

Acknowledgements

Gian Giacomo Guerreschi Alán Aspuru-Guzik

Post

• stdo

docs cs Peter Johnson Jonathan Olson Gr Grad adua uate te stu stude dent nts Jhonathan Romero Fontalvo Hannah (Sukin) Sim Tim Menke Florian Hase

Thanks!