Artificial Intelligence in a Quantum...

Artificial Intelligence in a Quantum World

Alexander Hentschel

19 June 2008

supervisor: Dr. Barry Sandersco-supervisor: Dr. Gilad Gour

Acknowledgements:

Content

1 Introduction to Artificial Intelligence

2 Quantum Learning

3 Application: Gravitational Wave Detection

4 Comparison and Conclusions

Alexander Hentschel Artificial Intelligence in a Quantum World 1/12

Motivation

Artificial Intelligence (AI)

award ability to computers for:

independent accumulationof knowledge

reasoning

communication

learning

perception

Quantum Information

(AI)

extend computing model

Qubits

|ψ〉 = α|

0

〉+ β|

1

〉

QC proven more powerful

searching tasks


Motivation




reasoning

communication

learning

perception

Quantum Information

(AI)


Qubits

|ψ〉 = α|0〉+ β|1〉


searching tasks


Motivation




reasoning

communication

learning

perception

Quantum Information

(AI)


Qubits

|ψ〉 = α|0〉+ β|1〉


searching tasks


Motivation

Quantum Machine Intelligence




reasoning

communication

learning

perception

Quantum Information

(AI)


Qubits

|ψ〉 = α|0〉+ β|1〉


searching tasks

How classical AI applies to quantum information?

Make use of increased efficiency of quantum computing for machineintelligence?


Motivation





reasoning

communication

learning

perception

Quantum Information

(AI)


Qubits

|ψ〉 = α|0〉+ β|1〉


searching tasks




Motivation





reasoning

communication

learning

perception

Quantum Information

(AI)


Qubits

|ψ〉 = α|0〉+ β|1〉


searching tasks




Introduction to Artificial Intelligence

Applications of artificial intelligence:

object recognition & robot locomotion: Mars rovers Spirit, Opportunity

medical diagnostics: adaptive medical therapies

stock market analysis: Stratego-Funds

search engines: Google

spam filtering: SpamAssassin

computer games

Approach to classical artificial intelligence:

Artificial Neural Networks: object recognition & robot locomotion

search algorithms: chess

statistical approach: spam filtering









computer games













computer games













computer games






Bayesian Inference

Bayesian Inference

Idea:

define: set of hypotheses H1,H2, . . .

define:

prior beliefs in hypotheses: P(H1),P(H2), . . .adapting degree of belief in hypothesis depending on results of ongoingobservations


Bayesian Inference

Bayesian Inference

Idea:


define:



Bayesian Inference

Bayesian Inference

Idea:


define:


10 theories predicting different hours of rain per day

10%

50%

100%

hours ofrain per day

0 12 24

belief

global warming


Bayesian Inference

Bayesian Inference

Idea:


define:



10%

50%

100%


0 12 24

belief

-12 hours

of rain


0 12 24

belief

10%

50%

100%

global warming


Bayesian Inference

Bayesian Inference

Idea:


define:



10%

50%

100%


0 12 24

belief

-30 days of

observations


0 12 24

belief

10%

50%

100%

global warming


Bayesian Inference

Bayesian Inference

Idea:


define:



10%

50%

100%


0 12 24

belief

-30 days of

observations


0 12 24

belief

10%

50%

100%

-ongoing

observations


0 12 24

belief

10%

50%

100%

global warming


Bayesian Inference

Bayesian Inference

Idea:


define:


Mathematical formulation:

observe event ε

Posterior probability: P(Hi |ε) =P(ε|Hi)P(Hi)

P(ε)

P(ε|Hi) conditional probability for observing event ε given Hi

P(ε) a priori probability of witnessing evidence ε under all possible hypotheses

P(ε) =∑j

P(Hj )P(ε|Hj )


Bayesian Inference

Bayesian Inference

Idea:


define:


Mathematical formulation:

observe event ε

Posterior probability: P(Hi |ε) =P(ε|Hi)P(Hi)

P(ε)

P(ε|Hi) conditional probability for observing event ε given Hi

P(ε) a priori probability of witnessing evidence ε under all possible hypotheses

P(ε) =∑j

P(Hj )P(ε|Hj )


Quantum Learning

In quantum World:

system described by quantum state |Ψ〉 = α1|ψ1〉+ · · ·+ αn |ψn〉measurement affects quantum state |Ψ〉

Questions:

full control over quantum system |Ψ〉,i.e. prepare and measure |Ψ〉 repeatedly:

How much can classical Bayesian Learner infer about|Ψ〉 via quantum tomography?

|Ψ〉

?quantum tomographyα1, . . . , αk , . . . , αn


Quantum Learning

In quantum World:


Questions:



|Ψ〉



Quantum Learning

In quantum World:


Questions:



like classical system: all system parameters known

|Ψ〉


ClassicalBayesian Learner

HHHj?��

??


Quantum Learning

In quantum World:


Questions:



partial control over quantum system:measurement affects |Ψ〉

How much can classical Bayesian Learner infer about|Ψ〉 before system is dominated by measurement?

implement learning on quantum level

|Ψ〉

?random outputα1, . . . , αk , . . . , αn


?

??


Quantum Learning

In quantum World:


Questions:






|Ψ〉

?random outputα1, . . . , αk , . . . , αn


?

??


Quantum Learning

In quantum World:


Questions:






|Ψ〉

QuantumBayesian Learner

?6


Application: Gravitational Wave Detection

Possible application:

Detection of Gravitational Waves

move with speed of lightresource: time ⇔ N pulses

extremely weak: ∼ 10−20

valuable resource: sensitivity

LIGO, Washington, USA

Detection: Mach-Zehnder Interferometer



Possible application:

Detection of Gravitational Waves

move with speed of lightresource: time ⇔ N pulses

extremely weak: ∼ 10−20

valuable resource: sensitivity

LIGO, Washington, USA

Detection: Mach-Zehnder Interferometer



gravitational wave deforms optical path length

⇔ phase shift ΦN photons: phase sensitivity ∆Φ ∝ 1√

N

use entangled state |Ψ〉: ∆Φ ∝ 1N

required POVM experimentally very difficult

increase sensitivity by compensating for Φ[D. Berry, H. Wiseman, J. Breslin, Phys. Rev. A 63, 53804 (2001)]

include controllable phase shifter ϕ

adjust ϕ by feedback control mechanism

using general Bayesian Learner

Mach-Zehnder Interferometer

Beam Splitter

DetectorPhase Shifter





N








Beam Splitter






N








Beam Splitter


Φ





N








Beam Splitter


Φ





N








Beam Splitter


|Ψ〉Φ





N








Beam Splitter


|Ψ〉Φ

ϕ





N








Beam Splitter


|Ψ〉Φ

ϕ



-



N








Beam Splitter


|Ψ〉Φ

ϕ



-�



N








Beam Splitter


|Ψ〉Φ

ϕ



-�



N








Beam Splitter


|Ψ〉Φ

ϕ

BayesianLearner

6


Feedback Algorithms

A simple feedback strategy

input: N photons in horizontal arm

tune phaseshifter ϕ untiloutput: only photons on horizontal arm

disadvantages: sensitivity limited by∆Φ ∝ 1√

N

requires large N ⇔ long time


Beam Splitter


Φ

ϕ

BayesianLearner

6


Feedback Algorithms





N



Beam Splitter


Φ

ϕ

BayesianLearner

6


Feedback Algorithms





N



Beam Splitter


Φ

ϕ

BayesianLearner

6


Feedback Algorithms

A high sensitivity feedback strategy

input: fixed state |Ψ〉 of

input:

N entangled photons

1234N-1N . . . . .

no prior information about Φ:select first phase ϕ(0) at random

adjust phase ϕ(0) → ϕ(1)

according to measurement outcome...

final phase estimates Φ̃ determined bymeasurements

learning

evaluate fitness of decision tree with Bayes Theoremvary decision tree using evolutionary algorithm

��

��1


Detector 0

Detector 1

|Ψ〉Φ

ϕ


Feedback Algorithms



input:

N entangled photons

1234N-1N . . . . .





learning


��

��1


Detector 0

Detector 1

|Ψ〉Φ

ϕ

ϕ(0)

:


Feedback Algorithms



input:

N entangled photons

234N-1N . . . . .





learning


��

��1


Detector 0

Detector 1

|Ψ〉Φ

ϕ

ϕ(0)

:

��0 @@R

1


Feedback Algorithms



input:

N entangled photons

234N-1N . . . . .





learning


��

��1


Detector 0

Detector 1

|Ψ〉Φ

ϕ

ϕ(0)

:

��0 @@R

1

ϕ(1)1 ϕ

(1)2


Feedback Algorithms



input:

N entangled photons

34N-1N . . . . .





learning


��

��1


Detector 0

Detector 1

|Ψ〉Φ

ϕ

ϕ(0)

:

��0 @@R

1

ϕ(1)1 ϕ

(1)2

��0 @@R

1 ��0 @@R

1


Feedback Algorithms



input:

N entangled photons





learning


��

��1


Detector 0

Detector 1

|Ψ〉Φ

ϕ

ϕ(0)

:

��0 @@R

1

ϕ(1)1 ϕ

(1)2

��0 @@R

1 ��0 @@R

1

final phase estimates eΦ:? ? ? ? ? ?


Feedback Algorithms



input:

N entangled photons





learning


��

��1


Detector 0

Detector 1

|Ψ〉Φ

ϕ

ϕ(0)

:

��0 @@R

1

ϕ(1)1 ϕ

(1)2

��0 @@R

1 ��0 @@R

1



Feedback Algorithms



input:

N entangled photons





learning


��

��1


Detector 0

Detector 1

|Ψ〉Φ

ϕ

ϕ(0)

:

��0 @@R

1

ϕ(1)1 ϕ

(1)2

��0 @@R

1 ��0 @@R

1



Comparison

Shut Noise Limit

photons feed into single arm

Canonical Measurement[1]

scaling: ∼ Heisenberg limit

achieves maximal phase sensitivity

realization: unknown

realization:

not by photon counting

Adaptive Measurement Scheme[2]

scaling: & Heisenberg limit

realization: photon counting

noise not included in analysis

not optimal for N > 4requires infinitely fast detectors

Learning Feedback Algorithm

expected scaling: worse than [2]much more flexible

log(N )

∆Φ

∼ 1N

[1]

[2]

∼ 1√N

[1] B. Sanders, G. Milburn, Phys. Rev. Lett. 75, 2944 (1995)

[2] D. Berry, H. Wiseman, J. Breslin, Phys. Rev. A 63, 53804 (2001)


Comparison

Shut Noise Limit






realization:









log(N )

∆Φ

∼ 1N

[1]

[2]

∼ 1√N




Comparison

Shut Noise Limit






realization:









log(N )

∆Φ

∼ 1N

[1]

[2]

∼ 1√N




Comparison

Shut Noise Limit






realization:









log(N )

∆Φ

∼ 1N

[1]

[2]

∼ 1√N




Comparison

Shut Noise Limit






realization:









log(N )

∆Φ

∼ 1N

[1]

[2]

∼ 1√N




Comparison

Shut Noise Limit






realization:









log(N )

∆Φ

∼ 1N

[1]

[2]

∼ 1√N




Comparison

Shut Noise Limit






realization:









log(N )

∆Φ

∼ 1N

[1]

[2]

∼ 1√N




Comparison

Advantages of Learning

noise tolerantwithout knowledge of specific noise model

potential to do better than adaptivemeasurement scheme (depending on training)

works for any prior distribution of phase Φ

optimal input state:

|Ψopt 〉 ∝N/2X

µ=−N/2

αµ

˛̨̨N2, µE

only via post selection

same learning algorithm works for differentinput states |Ψ〉

Future work:

input |Ψ〉 chosen by learning algorithm


Beam Splitter


|Ψ〉Φ

ϕ

BayesianLearner

6


Comparison






|Ψopt 〉 ∝N/2X

µ=−N/2

αµ

˛̨̨N2, µE



Future work:



Beam Splitter


|Ψ〉Φ

ϕ

BayesianLearner

6

log(N )

∆Φ

∼ 1N

[1]

[2]

∼ 1√N


Comparison






|Ψopt 〉 ∝N/2X

µ=−N/2

αµ

˛̨̨N2, µE



Future work:



Beam Splitter


|Ψ〉Φ

ϕ

BayesianLearner

6

log(N )

∆Φ

∼ 1N

[1]

[2]

∼ 1√N


Comparison






|Ψopt 〉 ∝N/2X

µ=−N/2

αµ

˛̨̨N2, µE



Future work:



Beam Splitter


|Ψ〉Φ

ϕ

BayesianLearner

6

log(N )

∆Φ

∼ 1N

[1]

[2]

∼ 1√N


Comparison






|Ψopt 〉 ∝N/2X

µ=−N/2

αµ

˛̨̨N2, µE



Future work:



Beam Splitter


|Ψ〉Φ

ϕ

BayesianLearner

6

log(N )

∆Φ

∼ 1N

[1]

[2]

∼ 1√N


Comparison






|Ψopt 〉 ∝N/2X

µ=−N/2

αµ

˛̨̨N2, µE



Future work:



Beam Splitter


|Ψ〉Φ

ϕ

BayesianLearner

6

log(N )

∆Φ

∼ 1N

[1]

[2]

∼ 1√N


Conclusions

classical learning techniques

heuristical methods

can be applied to quantum systems

much more flexible than analytic methodsbut usually perform worse

potential to work on extremely complex systems, for which

good analytic methods do not exist

system is too complex for humans to understand

classical engineered systems: aircrafts, Internet . . .autonomous feedback control systems maintain stability & predictability

no certainty about quantum state of system⇒ involved integrals over probability distributions of possible system states

automated learning: stabilize complex engineered quantum systems


Conclusions


heuristical methods










Conclusions


heuristical methods










Conclusions


heuristical methods










Conclusions


heuristical methods










Conclusions


heuristical methods










Thank you for your attention.

Questions?

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