Towards Quantum-Assisted Artificial Intelligence Peter Wittek - - PowerPoint PPT Presentation

Towards Quantum-Assisted Artificial Intelligence

Peter Wittek

Research Fellow, Quantum Information Theory Group ICFO-The Institute of Photonic Sciences Barcelona Institute of Science and Technology & Academic Director, Quantum Machine Learning Initiative Creative Destruction Lab University of Toronto

November 2017

Max Tegmark (2017). Life 3.0.

Introduction GOFAI Probabilistic Graphical Models Statistical Relational Learning Quantum Gibbs Sampling Summary

The Virtuous Cycle!

Quantum information processing and quantum computing Machine learning and artificial intelligence

• Reinforcement learning in control problems
• Deep learning and neural networks as representation
• Quantum algorithms in machine learning
• Improved sample and computational complexity

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Introduction GOFAI Probabilistic Graphical Models Statistical Relational Learning Quantum Gibbs Sampling Summary

Diversity is key

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Statistical learning theory Marco Loog Gael Sent´ ıs Quantum data Jonathan Olson Antonio Gentile Nana Liu John Calsamiglia Deep architectures & Many-body physics William Huggins Shi-Ju Ran Alexandre Dauphin Agency Vedran Dunjko (Discrete) Optimization Davide Venturelli William Santos Sampling Alejandro Perdomo-Ortiz Causal networks, kangaroos, and cockroaches Christina Giarmatzi Andreas Winter

Introduction GOFAI Probabilistic Graphical Models Statistical Relational Learning Quantum Gibbs Sampling Summary

So let’s add one more component. . .

Good Old-Fashioned AI Formalize causal relations in higher order logic. Classical data in, classical data out. Complexity of entailment is in NP. Fragile and largely dead since the 1980s. Add uncertainty Bump complexity to #P. Sampling helps. Took off in 2006, still a niche.

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Introduction GOFAI Probabilistic Graphical Models Statistical Relational Learning Quantum Gibbs Sampling Summary

First-order logic

Basic components (abbreviated): Constant: representing objects in the domain. E.g., Alice, Bob. Variable: taking values in a domain, e.g., people. Predicate: representing relations among objects, e.g., Flies(x), Physicist(y), Coauthors(x,y). Formulas: Atom: predicate applied to a tuple of objects. E.g., Coauthors(x, Bob). Ground atom: atom with no variable. E.g., Coauthors(Alice, Bob). Formula: atoms with logical connectives and quantifiers. E.g., ∀x (Flies(x)⇒Flies(MotherOf(x))).

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Introduction GOFAI Probabilistic Graphical Models Statistical Relational Learning Quantum Gibbs Sampling Summary

Knowledge base

Knowledge base (KB): conjunctive set of formulas. Every referee is competent: ∀x,y (Referees(x,y)⇒Competent(x)) Referees of physicists are physicists: ∀x,y (Referees(x,y)∧Physicist(y)⇒Physicist(x))

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Introduction GOFAI Probabilistic Graphical Models Statistical Relational Learning Quantum Gibbs Sampling Summary

Grounding out and Herbrand interpretation

Finite domain: {Alice, Bob} Functions are not relevant, they serve as substitutions. Grounding out the atoms grow exponentially: Referees(Alice,Bob), Referees(Bob,Alice), Referees(Alice,Alice), Referees(Bob,Bob). Competent(Alice), Competent(Bob). Physicist(Alice), Physicist(Bob). Grounding out the knowledge base: Referees(Alice,Bob)⇒Competent(Alice) Referees(Bob,Alice)⇒Competent(Bob) Referees(Alice,Alice)⇒Competent(Alice) . . . Herbrand interpretation (possible world): assign a truth value to each ground atom.

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Introduction GOFAI Probabilistic Graphical Models Statistical Relational Learning Quantum Gibbs Sampling Summary

Deduction

Let us have a formula from outside the KB: F: Bob referees Alice. Bob is not competent. Referees(Bob, Alice)∧¬Competent(Bob) Problem of entailment: KBF. What about contradictions?

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Introduction GOFAI Probabilistic Graphical Models Statistical Relational Learning Quantum Gibbs Sampling Summary

Restricted Boltzmann machines

E(v, h) = −

• i

aivi −

• j

bjhj −

• i
• j

viwi,jhj Obtain a probability distribution: P(v, h) = 1 Z e−E(v,h) Trace out over the hidden nodes to approximate a target probability distribution. This is a generative probabilistic model.

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Introduction GOFAI Probabilistic Graphical Models Statistical Relational Learning Quantum Gibbs Sampling Summary

Probabilistic Graphical Models

Uncertainty (probabilities) and logical structure (independence constraints). Goal: compact representation of a joint probability distribution.

For {X1, . . . , XN} binary random variables, there are 2N assignments.

Complexity is dealt through graph theory. Factorization: compactness. Inference: reassembling factors. Conditional independence (X ⊥ Y |Z): P(X = x, Y = y|Z = z) = P(X = x|Z = z)P(Y = y|Z = z) ∀x ∈ X, y ∈ Y , z ∈ Z

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Introduction GOFAI Probabilistic Graphical Models Statistical Relational Learning Quantum Gibbs Sampling Summary

Markov random fields

Ising model generalized to hypergraphs. A distribution factorizes over G if:

P(X1, . . . , XN) = 1

Z P′(X1, . . . , XN),, where

P′(X1, . . . , XN) = exp(−

i ǫ[Ck]) and

Ci is a clique in G.

Connection to Boltzmann machines

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Introduction GOFAI Probabilistic Graphical Models Statistical Relational Learning Quantum Gibbs Sampling Summary

Probabilistic inference and learning in Markov networks

How to apply the learned model? Complexity is in #P. Two types of queries:

Conditional probability: P(Y |E = e) = P(Y ,e)

P(e) .

Maximum a posteriori: argmaxyP(y|e) = argmaxy

• Z P(y, Z|e).

Generic case for approximation: Markov chain Monte Carlo Gibbs sampling. What prevents from accelerating this with quantum-enhanced sampling?

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Introduction GOFAI Probabilistic Graphical Models Statistical Relational Learning Quantum Gibbs Sampling Summary

Statistical relational learning

Uncertainty + relational structure. Combine (first-order) logic and probabilistic graphical models. Non-IID data. Markov logic networks are a type of statistical relational learning.

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Introduction GOFAI Probabilistic Graphical Models Statistical Relational Learning Quantum Gibbs Sampling Summary

Markov logic networks

Real world can never match a KB. Weight each formula in a KB: high weight indicates high probability. Markov Logic Network Apply a KB {Fi} with matching weights {wi} to a finite set of constants C to define a Markov network: Add a binary node for each possible grounding for each atom. Add a binary feature for each possible grounding of each formula Fi. It is like a template to generate Markov networks.

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Introduction GOFAI Probabilistic Graphical Models Statistical Relational Learning Quantum Gibbs Sampling Summary

An example

∀x,y (Referees(x,y)⇒Competent(x)) ∀x,y (Referees(x,y)∧Physicist(y)⇒Physicist(x)) C={Alice, Bob}

Physicist(Alice) Physicist(Bob) Referees(Alice,Bob) Competent(Alice) Referees(Bob,Alice) Competent(Bob) Referees(Alice,Alice) Referees(Bob,Bob)

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Introduction GOFAI Probabilistic Graphical Models Statistical Relational Learning Quantum Gibbs Sampling Summary

What do we gain?

First-order logic is recovered in the limit of uniform weights. Unlikely statements will be assigned a low probability. Cross-over between formal reasoning and probabilistic inference.

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Introduction GOFAI Probabilistic Graphical Models Statistical Relational Learning Quantum Gibbs Sampling Summary

Include evidence

Physicist(Alice) Physicist(Bob) Referees(Alice,Bob) Competent(Alice) Referees(Bob,Alice) Competent(Bob) Referees(Alice,Alice) Referees(Bob,Bob)

We have true evidence for these: Physicist(Bob) Referees(Alice,Bob)

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Introduction GOFAI Probabilistic Graphical Models Statistical Relational Learning Quantum Gibbs Sampling Summary

Assignment 1

Physicist(Alice) Physicist(Bob) Referees(Alice,Bob) Competent(Alice) Referees(Bob,Alice) Competent(Bob) Referees(Alice,Alice) Referees(Bob,Bob)

With this assignment, we actually violate two formulas: Referees(Alice,Bob)⇒Competent(Alice) Referees(Alice,Bob)∧Physicist(Bob)⇒Physicist(Alice)

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Introduction GOFAI Probabilistic Graphical Models Statistical Relational Learning Quantum Gibbs Sampling Summary

Assignment 2

Physicist(Alice) Physicist(Bob) Referees(Alice,Bob) Competent(Alice) Referees(Bob,Alice) Competent(Bob) Referees(Alice,Alice) Referees(Bob,Bob)

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Introduction GOFAI Probabilistic Graphical Models Statistical Relational Learning Quantum Gibbs Sampling Summary

Assignment 3

Physicist(Alice) Physicist(Bob) Referees(Alice,Bob) Competent(Alice) Referees(Bob,Alice) Competent(Bob) Referees(Alice,Alice) Referees(Bob,Bob)

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Introduction GOFAI Probabilistic Graphical Models Statistical Relational Learning Quantum Gibbs Sampling Summary

Clique size

Markov logic networks are essentially templates to generate Markov networks. Size of the largest clique: longest formula in KB. Restricted formula length: controlled clique structure.

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Introduction GOFAI Probabilistic Graphical Models Statistical Relational Learning Quantum Gibbs Sampling Summary

Lifted inference

Do not operate on a propositional level: exploit symmetries. Variable elimination (2001-2003). Belief propagation (2008). Domain lifting (2011). Weighted first-order model counting (2011). Massive reduction in maximum degree of nodes. Source: Kersting (2012): Lifted Probabilistic Inference. Proceedings of ECAI

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Introduction GOFAI Probabilistic Graphical Models Statistical Relational Learning Quantum Gibbs Sampling Summary

Example: domain-lifted inference

A domain is like a type.

E.g., a variable x ranges over people, {Alice, Bob,. . . }.

Coarse-grain the part of the network that we are not grounding out by the observation.

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Introduction GOFAI Probabilistic Graphical Models Statistical Relational Learning Quantum Gibbs Sampling Summary

Quantum annealing and Gibbs sampling

Real-world quantum annealing is noisy. So what can we do?

A B

thermal annealing thermal state quantum annealing

Sampling is the most useful quantum-enhanced routine for ML today.

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Introduction GOFAI Probabilistic Graphical Models Statistical Relational Learning Quantum Gibbs Sampling Summary

Probabilistic inference with quantum Gibbs sampling

Restricted Boltzmann machine: P(v, h) = 1

Z e−E(v,h).

Idea: representative Boltzmann statistics with fewer samples than MCMC Gibbs sampling. N >> 1 proposals for quantum BMs:

1

Quantum annealing to do sampling.

2

Prepare a Gibbs state starting from a mean-field approximation.

3

Prepare a Gibbs state starting from an arbitrary state.

4

. . .

The actual topology of the nodes is secondary. Instead of training a BM, we can use the same ideas for inference in an MLN.

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Introduction GOFAI Probabilistic Graphical Models Statistical Relational Learning Quantum Gibbs Sampling Summary

Sampling by annealing

Quantum sampling: use quantum annealing to draw representative samples from a Boltzmann distribution. Why does it work?

Actual implementation of quantum annealing has > 0 probability of ending in excited state. Distribution of excited states follows a Boltzmann distribution: P(x) = 1

Z exp (−βeff Hf (x)).

Use the Hamiltonian describing a Markov network as Hf .

Problems? Many:

Connectivity: hardware qubit connectivity may have nothing to do with the structure of the Markov network.

But with Markov logic networks:

Locality is guaranteed.

• Max. degree is reduced by lifting.

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Introduction GOFAI Probabilistic Graphical Models Statistical Relational Learning Quantum Gibbs Sampling Summary

Gate-based model

Prepare a thermal state and sample it. ‘Asymptotic’ is not good enough if you want a thermal state. From Riera et al., 2012:

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Introduction GOFAI Probabilistic Graphical Models Statistical Relational Learning Quantum Gibbs Sampling Summary

Overview of matching concepts

Probability distribution of an MLN: PM(ω) :=

1 Z(M) exp

• j wj N(fj, ω)
• Formula

Weight

Friends(A,B) Friends(A,A) Friends(B,B) Friends(B,A) Smokes(B) Smokes(A) Cancer(A) Cancer(B) Local space Local dimension d=2

Total number of nodes State vector

Max degree frequency

• f atom in formulas
• Max. clique size: c. For every

clique, there is a term in the Hamiltonian.

Domain: {A,B} Domain size: D = |{A,B}| =2 Inverse temperature:

x,y Friends(x,y) (Smokes(x) Smokes(y)) x Smokes(x) Cancer(x)

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Introduction GOFAI Probabilistic Graphical Models Statistical Relational Learning Quantum Gibbs Sampling Summary

Prepare a thermal state

Best known state preparation protocol (arXiv:1603.02940):

Almost linear in

• dNβ/Z

Polynomial in log(1/ǫ)

Where: N is number of subsystems and d local Hilbert space dimension. The good: Huge improvement over ∼ dN/ǫ run time of classical simulated annealing. The bad: Still exponential in N. The ugly: Requires essentially a universal quantum computer.

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Introduction GOFAI Probabilistic Graphical Models Statistical Relational Learning Quantum Gibbs Sampling Summary

Can we hope for something better?

Not really. . . Finding and sampling from thermal states of (quantum) many-body problems is usually hard even though they have much more structure:

Very small cliques Sites on a simple lattice Cliques geometrically local

For ground states we even have rigorous hardness results:

Classical 3-SAT is NP-complete. Estimating quantum ground state energy is QMA-hard.

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Introduction GOFAI Probabilistic Graphical Models Statistical Relational Learning Quantum Gibbs Sampling Summary

How real is all this?

We started the first incubator program in the world for QML startups. Close to 30 ventures as of today. Includes:

One month of intensive technical training in September. Business guidance and investor access. Pre-seed investment up to $80k from three Silicon Valley venture capital firms.

Creative Destruction Lab at the University of Toronto. Also host of the largest ML accelerator in the world. Sick of academia? Apply in the next intake: https://creativedestructionlab.com/quantum

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Introduction GOFAI Probabilistic Graphical Models Statistical Relational Learning Quantum Gibbs Sampling Summary

Summary

Deep architectures are good at artificial intuition. Probabilistic graphical models are better at capturing causation. Quantum-enhanced sampling can enable these models the same way GPUs enabled deep learning. arXiv:1611.08104. If you want some entertainment, read the acknowledgement of the published version.

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