Towards Quantum-Assisted Artificial...

Towards Quantum-Assisted Artificial Intelligence

Peter Wittek

Research Fellow, Quantum Information Theory GroupICFO-The Institute of Photonic Sciences

Barcelona Institute of Science and Technology&

Academic Director, Quantum Machine Learning InitiativeCreative Destruction LabUniversity 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 processingand

quantum computing

Machine learningand

artificial intelligence

•Reinforcement learning in control problems•Deep learning and neural networks as representation

•Quantum algorithms in machine learning•Improved sample and computational complexity

Towards Quantum-Assisted AI November 2017 5 / 35


Diversity is key


Statistical learning theoryMarco LoogGael Sent́ıs

Quantum dataJonathan OlsonAntonio GentileNana LiuJohn Calsamiglia

Deep architectures & Many-body physicsWilliam HugginsShi-Ju RanAlexandre Dauphin

AgencyVedran Dunjko

(Discrete) OptimizationDavide VenturelliWilliam Santos

SamplingAlejandro Perdomo-Ortiz

Causal networks, kangaroos,and cockroachesChristina GiarmatziAndreas Winter


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.



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))).



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))



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.Towards Quantum-Assisted AI November 2017 10 / 35


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: KB�F.

What about contradictions?



Restricted Boltzmann machines

E (v , h) = −∑

i

aivi −∑

j

bjhj −∑

i

∑j

viwi ,jhj

Obtain a probability distribution:

P(v , h) =1

Ze−E(v ,h)

Trace out over the hidden nodes to approximate a target probability distribution.This is a generative probabilistic model.



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



Markov random fields

Ising model generalized to hypergraphs.

A distribution factorizes over G if:

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

′(X1, . . . ,XN ),, whereP ′(X1, . . . ,XN ) = exp(−

∑i ε[Ck ]) and

Ci is a clique in G .

Connection to Boltzmann machines



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?



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.



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 Markovnetwork:

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.



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)



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.



Include evidence

Physicist(Alice)

Physicist(Bob)



Referees(Bob,Bob)

We have true evidence for these:Physicist(Bob)

Referees(Alice,Bob)



Assignment 1

Physicist(Alice)

Physicist(Bob)



Referees(Bob,Bob)

With this assignment, we actually violate two formulas:

Referees(Alice,Bob)⇒Competent(Alice)

Referees(Alice,Bob)∧Physicist(Bob)⇒Physicist(Alice)



Assignment 2

Physicist(Alice)

Physicist(Bob)



Referees(Bob,Bob)



Assignment 3

Physicist(Alice)

Physicist(Bob)



Referees(Bob,Bob)



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.



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 ECAITowards Quantum-Assisted AI November 2017 25 / 35

https://dx.doi.org/10.3233/978-1-61499-098-7-33


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.



Quantum annealing and Gibbs sampling

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

A B

thermal annealing

thermal state

quantumannealing

Sampling is the most useful quantum-enhanced routine for ML today.Towards Quantum-Assisted AI November 2017 27 / 35


Probabilistic inference with quantum Gibbs sampling

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



Sampling by annealing

Quantum sampling: use quantum annealing to draw representative samples from aBoltzmann 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 theMarkov network.

But with Markov logic networks:

Locality is guaranteed.Max. degree is reduced by lifting.



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:

This is not the optimal protocol.


https://arxiv.org/abs/1102.2389


Overview of matching concepts

Probability distribution of an MLN: PM(ω) := 1Z(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 spaceLocal dimension d=2

Total number of nodesState vector

Max degree frequency of atom in formulas

Max. clique size: c. For everyclique, there is a term inthe Hamiltonian.

Domain: {A,B}Domain size: D = |{A,B}| =2Inverse temperature: x,y Friends(x,y) (Smokes(x) Smokes(y))

x Smokes(x) Cancer(x)



Prepare a thermal state

Best known state preparation protocol (arXiv:1603.02940):

Almost linear in√dNβ/Z

Polynomial in log(1/ε)

Where: N is number of subsystems andd local Hilbert space dimension.

The good: Huge improvement over ∼ dN/ε run time of classical simulated annealing.

The bad: Still exponential in N.

The ugly: Requires essentially a universal quantum computer.


http://arxiv.org/abs/1603.02940


Can we hope for something better?

Not really. . .

Finding and sampling from thermal states of (quantum) many-body problems is usuallyhard even though they have much more structure:

Very small cliquesSites on a simple latticeCliques 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.



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


https://creativedestructionlab.com/quantum


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 deeplearning.

arXiv:1611.08104. If you want some entertainment, read the acknowledgement of thepublished version.


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