Raffaele SantagatiRaffaele.Santagati@Bristol.ac.uk

Learning quantum physics from experiments with Bayesian inference and quantum computers

Outline

• Quantum Computers a (very) short introduction

• Quantum Hamiltonian Learning (QHL) -> Learning parameters of a model.

• From learning parameters to learning models…

• Wiebe et al., Hamiltonian Learning and Certification Using Quantum Resources. Phys. Rev. Lett. 112, (2014)

• Wang et al., Experimental quantum Hamiltonian learning - Nature Physics 1, 149 (2017)

• Santagati et al., Magnetic-field-learning using a single electronic spin in diamond… - Phys. Rev. X (2019)

• Gentile et al., Characterising quantum systems with Bayesian inference – manuscript in preparation (2019)

• Flynn et al., Exploring acyclic graphs for the study of quantum systems – manuscript in preparation (2019)

Quantum physics

EINSTEIN, N. ROSEN and B. PODOLSKY, Phys. Rev. 47, 777 (1935) D. R. INGLIS, Rev. Mod. Phys. 33, 1 (1961)

Entanglement

Quantum mechanics although counterintuitive is correct in its predictions

Defines probability

Schrödinger’s cat

PROBABILISTIC MEASUREMENT OUTCOMES

Superposition

Quantum superposition of 2 states:

Quantum Simulation

Quantum systems are hard to simulate.

If we study a system comprising N particles with two basis states and wewant to store their quantum state on a classical computer using an 8 bits (1 byte) resolution.

2 p 4 bytes

1 p 2 bytes

3 p 8 bytes

200 p ~ 1.6 1060 bytes

The total amount of memory required grows exponentially with the value of N.

Mem

ory

Number of particles

Total memory available on earth today is ~ 5.21 1021 bytes

Quantum Simulation

Quantum systems are hard to simulate.

Feynman - Simulating Physics with Computers International Journal of Theoretical Physics, VoL 21, 6-7, 1982

Mem

ory

Number of particles

We need a new kind of computer to simulatequantum systems efficiently.

0

1

Bit Quantum bit

Measurement outcomes are probabilistic

Probability of 0

Probability of 1

Quantum computers and what they are good at

Factoring Numbers (Shor’s algorithm) Quantum Machine Learning

Quantum Simulation Optimization

Exponential speed-upPoly and exponential speed-up

Exponential speed-ups Poly speed-ups

Ashley Montanaro Quantum Information 2, 15023, 2016

Quantum algorithm zoohttps://math.nist.gov/quantum/zoo/

Quantum computing funding landscape

Topological• UK -> 400 million £

• EU -> 1.6 billion €

• GE -> 600 million €

And many more….

Investments inQuantum technologies

Other Solid state

Photonics

Superconducting

Exponential scaling of

required classical

resources

31

Bayesian approach

(similar to Bayesian phase estim.)

Ø Choose experiment (i.e. evol. time ) given .

Ø Obtain data from the system.

Ø Use the quantum simulator to calculate the

likelihood functions .

Ø Update the prior distribution using Bayes’ rule

The information is encoded in a prior distribution

• Near-optimal choice: .

• Simulated Monte-Carlo approximation used to make the Bayesian update practical.

Quantum Hamiltonian LearningWhat is a Hamiltonian operator?

Quantum System Model of the system (H)

with parameters

Quantum Hamiltonian Learning

QHL aims to find efficiently the set of parameters which best describe the dynamic of the system.

Mohseni et al. Phys. Rev. A 77, 032322 (2008)Hentschel and Sanders Phys. Rev. Lett. 104, 063603 (2010)Da Silva et al. Phys. Rev. Lett. 107, 210404 (2011)Wiebe et al. Phys. Rev. Lett. 112, 190501 (2014)Wiebe et al. Phys. Rev. A 89, 042314 (2014)

1. From choose an

experiment (e.g. )

2. Perform experiment on

System and obtain

outcome E

Phys. Rev. Lett. 112, 190501 (2014)

3. Calculate likelihoods

using quantum simulator

4. Update the prior distribution

using Bayes’ rule:

Nature Physics 13, 551-555 (2017)

31

Bayesian approach

(similar to Bayesian phase estim.)

Ø Choose experiment (i.e. evol. time ) given .

Ø Obtain data from the system.

Ø Use the quantum simulator to calculate the

likelihood functions .

Ø Update the prior distribution using Bayes’ rule

The information is encoded in a prior distribution

• Near-optimal choice: .

• Simulated Monte-Carlo approximation used to make the Bayesian update practical.

Quantum Likelihood Estimation

Experimental set-up

Nature Physics 13, 551-555 (2017)

Hamiltonian: Where we want to

estimate the frequency

of oscillations

c

Experimental results

frequency inferred with QLE:

Rescaling

Nature Physics 13, 551-555 (2017)

First part: QHL Take home message

• Finding the parameter of H in a scalable way

• We learned only parameters not models

• Can we use similar techniques to help us

improve models?

31

Bayesian approach

(similar to Bayesian phase estim.)

Ø Choose experiment (i.e. evol. time ) given .

Ø Obtain data from the system.

Ø Use the quantum simulator to calculate the

likelihood functions .

Ø Update the prior distribution using Bayes’ rule

The information is encoded in a prior distribution

• Near-optimal choice: .

• Simulated Monte-Carlo approximation used to make the Bayesian update practical.

• Gentile et al., Experimental Quantum Model Learning –manuscript in preparation (2019)

• Flynn et al., Exploring acyclic graphs for the study of quantum systems – manuscript in preparation (2019)

Methods to model physical systems with ML

Crutchfield and McNamara, Complex Syst. 1, 417 (1987)Schmidt and Lipson, Science 324, 81 (2009)Torlai and Melko Phys Rev B 94, 165134 (2016)Bairey, Arad, and Lindner PRL 122, 020504 (2019)Iten, et al., arXiv preprint arXiv:1807.10300 (2018)

Devise an automatic strategy to generate and select models by performing a search on a tree for the Hamiltonian model which best approximate the system dynamics.

QMLA prior

Quantum Model Learning

Explore and prune

for in DAG:

QHL update Consolidate

..

.

C. Granade, C. Ferrie, I. Hincks, S. Casagrande, T. Alexander, J. Gross, M. Kononenko, and Y. Sanders, Qinfer: Statistical Inference Software for Quantum Applications, Quantum 1, 5 (2017).

The protocol compares in pairs (Hi,Hj) all models within the

same layer using the Bayes factor

where

Learning models of quantum system:NV centre in diamond

Electron spin interacting with a Carbon nuclear spin

We want to learn the basics of the

interaction between the spin in the NV centre in diamond and

the closest 13C nuclear spin through Hahn echo measurements.

Hyperfine gyromagnetic tensors

Electron gyromagnetic tensors

cc

Results – NV-centre model learning

Conclusions

Wang et al. Nature Physics 13, 551-555 (2017)Santagati et al. Phys. Rev. X 9, 021019 (2019)Gentile et al., man. in prep. (2019)Flynn et al., man. in prep. (2019)

• Quantum computers• Statistical inference (QHL) for parameters in H• Quantum model learning to learn the models of a quantum system• From the experiments we find models compatible with theoretical

developments.

Raffaele.Santagati@Bristol.ac.uk

Backup slides

Results – NV-centre model learning

Other models performance

Experiment measurements

c

We measure the traced-out state i.e. Measure only the electron and not theother interacting particles.

It is a bit like studying the property of an iceberg by looking only at the tip.

electron

nucleus

What quantum computers are good at and what they are not?

Quantum computers don’t compute all the answers in parallel

Quantum computers are not faster than classical computers at every task

Quantum computers can give exponential and polynomial speed ups at specific tasks.

G Wendin 2017 Rep. Prog. Phys. 80 106001

https://www.smbc-comics.com/comic/the-talk-3

Experiment – NV-centre model learning

Experimental results: Variance

Variance:

SATURATION = Bad quantum simulator or bad model!

Wang et al. Nature Physics 13, 551-555 (2017)

Model I:

Model II:

From QHL to comparing models

Bayes Factor:

Updated value (for model II, including

chirping) of Norm of covariance matrix:

Old Variance with model I:

Model I

Model II

0 10 20 30 40 5010- 6

10- 5

10- 4

10- 3

10- 2

10- 1

1

10- 6

10- 5

10- 4

10- 3

10- 2

10- 1

1

Experiment Number

2(

)

From quantum simulation to qubits

A new theory of information based on quantum mechanics.

0

1

Bit Quantum bit

Measurement

Measurement outcomes are probabilistic

Probability of 0

Probability of 1

Quantum Computation and Quantum Information - M. Nielsen and I.L. Chuang - Cambridge

Experiment measurements

c

electron

nucleus

Model1 Model2

BF(H2; H1) = 1010

Estimating the number of spins

Gali, Fyta, and Kaxiras Ab initio supercell calculations on nitrogen-vacancy center in diamond:Electronic structure and hyperfine tensors. PHYSICAL REVIEW B 77, 155206 (2008)

Galli - Identification of individual 13C isotopes of nitrogen-vacancy center in diamond by combining thepolarization studies of nuclear spins and first-principles calculations. PHYSICAL REVIEW B 80, 241204 (2009)

Sequential Monte Carlo

Phys. Rev. Lett. 112, 190501 (2014)

2p W

2p W

P(W

)P

(W)

2 spins model – Expectation values

Simulation – NV-centre model learning

Measuring the traced out state state

QMLA prior

Model Learning

for in DAG:

QHL update

Learn from data

each model

Consolidate

The protocol compares in pairs (Hi,Hj) all models within the

same layer using the Bayes factor

where

Explore and prune

Generate new models till

n qubits allows or generate new layer.

.

.

The problem is NP Hard… but

A Bayesian analysis of decision-tree learning has shown that near-optimal solutions can be found in polynomial time.

Muggleton, S. Page, D.: A learnability model of universal representations and its application to top-down induction of decision trees. In K. Furukawa, D. Michie, & S. Muggleton(eds.) Machine Intelligence 15, Oxford University Press, page 16 (1999)

Dynamics examples with BF and PGH times