Nathan WiebeWork with Maria Kieferova

Quantum Neural Networks

Machine learning

• Fast scalable quantum computers will fundamentally change machine learning.

• Quadratic (or greater) speedups.

• Better models for data.

• Improved privacy and security.

Quantum Machine Learning

⋮

• Fast scalable quantum computers will fundamentally change machine learning.

• Quadratic (or greater) speedups.

• Better models for data.

• Improved privacy and security.

Quantum Machine Learning

Quantum Boltzmann Machine Training.

⋮

Boltzmann Machines

Class of recurrent neural network that is strongly related to physics.

𝑃 𝑥 ∝ 𝑒−𝐸(𝑥)

1 = cat

0 = Not cat

Energy

, 1

, 0

, 1

, 0

𝐸 𝑥 =

<𝑖,𝑗>

𝑤𝑖,𝑗𝑛𝑖𝑛𝑗

𝐸 𝑥 → 𝐻 =

<𝑖,𝑗>

𝑤𝑖,𝑗 1 − 𝑍𝑖 1 − 𝑍𝑗4

𝑤 = 𝑎𝑟𝑔𝑚𝑖𝑛𝑤(𝐷𝐾𝐿(𝑃 𝑥 |𝑃𝑡𝑟𝑎𝑖𝑛(𝑥)))

Quantizing the Boltzmann Machine

𝜌 ∝ 𝑒−𝐻

𝐻 =

𝑗

𝑤𝑗𝐻𝑗

𝑤 = 𝑎𝑟𝑔𝑚𝑖𝑛𝑤 𝑆(𝜌|𝜌𝑡𝑟𝑎𝑖𝑛)

Quantum Boltzmann machineslearn directly from quantum states.

They also generate quantum statesthat closely resemble the training data.

𝐻𝑗Do not commute

Energy

, 1

, 0

, 1

, 0

Classically we have only energy penalties.

Quantumly, we controlboth the energiesand the eigenbasis theenergies are applied in.

Classical Quantum

Training quantum Boltzmann Machines

We train using gradient descent on the quantum relative entropy (quantum KL divergence).

𝜎 =𝑒− σ𝑗𝑤𝑗𝐻𝑗

𝑇𝑟(𝑒−σ𝑗𝑤𝑗𝐻𝑗)

log 𝜎 = −σ𝑗𝑤𝑗𝐻𝑗 − log 𝑇𝑟(𝑒− σ𝑗𝑤𝑗𝐻𝑗

𝜕𝑤𝑗log 𝜎 = −𝐻𝑗 + 𝑇𝑟 𝜎𝐻𝑗

𝜕𝑤𝑗𝑆 𝜌 𝜎 = −Tr 𝜌𝐻𝑗 + 𝑇𝑟 𝜎𝐻𝑗 = 𝐻𝑗 𝑚𝑜𝑑𝑒𝑙

− 𝐻𝑗 𝑑𝑎𝑡𝑎

The number of times the trainingdata is accessed and Gibbs states

are prepared scales as 𝑂𝑀2

𝜖2.

𝑀 = number of weights.

Relative entropy training

Train

ing

ob

jectiv

e fu

nctio

n

Learning a model for a quantum system

Assume a quantum Hamiltonian thatis a transverse Ising model:

Find, 𝛼, 𝛽 and 𝛾 thatbest represent the Hamiltonian giventraining data

𝜌 = 𝑒−෩𝐻/𝑍

Want to minimize Δ 𝐻 = || 𝐻 − ෩𝐻||

Building a quantum associative memory

Quantum Training vectors: 𝑣𝑗 = 𝜓𝑗 ⊗𝑒𝑗 Data qubits

0

0

Address qubits

Pick 𝐻𝑗 from a universal set of Hamiltonians on 6 qubits. 1000 epochs, learning rate=0.2

Relative entropy ≈ 10−3.

Conclusions

Quantum computers enable new forms of machine learning.

Quantum Neural networks can learn directly from quantum data.

This opens up much richer models of data that cannot be efficiently trained using classical computers.

What is machine learning?Machine learning uses computers to find patterns in data,classify data, or perform tasks.

Computers are not usually told explicitly how to do this, or what “features” to use.

4 0 5

⋯

Training Data

???

4

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