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Hybrid Quantum-Classical Graph Neural Networks for Track Reconstruction

Cenk Tüysüz

Middle East Technical University, Ankara, Turkey

Quantum Optics and Information Meeting 5

22-23 April 2021

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Outline

The particle track reconstruction problem
Quantum Computing and Machine Learning
Hybrid Graph Neural Network approach
Results
Comments on future improvements

Cenk Tüysüz

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Large Hadron Collider (LHC)

and particle track reconstruction

High Luminosity upgrade of LHC brings many computational challenges.

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https://atlas.cern/updates/atlas-news/counting-collisions

ATLAS computing model

projections for Phase-2

Number of tracks is expected to be increase by 12-15 times

μ: Average number of interactions per bunch crossing

H. Gray, Track reconstruction in the ATLAS experiment, 2016.

	Run 1	Run 2	Run 3
μ	21	40	150-200?
Tracks	~280	~600	~7-10k

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High Luminosity LHC

High Luminosity upgrade of LHC brings many computational challenges.

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ATLAS computing model projections for Phase-2

Number of tracks is expected to be increase by 12-15 times

μ: Average number of interactions per bunch crossing

H. Gray, Track reconstruction in the ATLAS experiment, 2016.

	Run 1	Run 2	Run 3
μ	21	40	150-200?
Tracks	~280	~600	~7-10k

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TrackML Dataset

https://www.kaggle.com/c/trackml-particle-identification/overview

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Contains: 10k collision events (200 soft QCD interactions)

(arXiv: 1904.06778)

Retrieved from: Farrell et al. 2018 (arXiv: 1810.06111)

endcaps produce a lot of ambiguity and therefore many track candidates, we omit endcaps as we want to limit our model to simpler cases.

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Hep.TrkX GNN

Segment Classification

arXiv: 1810.06111

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Model Scores (with 0.5 threshold):

Purity: 99.5%, Efficiency: 98.7%

Overall Accuracy: 99.5%

The project is extended with the name Exa.TrkX to continue investigating use of GNNs in track reconstruction.

https://exatrkx.github.io

arXiv:2007.00149

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Quantum Graph Neural Network

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Node information

(3D cylindrical coordinates)

(Graph connectivity matrix)

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Preprocessing

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Use only the barrel region to avoid track ambiguity.

Only 100 events are used!

Use hits of particles

with pT > 1 GeV

pT distribution of an event

The preprocessing is used to reduce both the track ambiguity and the size of the dataset. Quantum Machine Learning simulations can not handle large datasets at the moment!

~15% of hits survive

\|Δr/Δ𝜙\|	< 0.0006
\|z₀\|	< 200 mm
\|𝜂\|	< 5

apply cuts to segments

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Preprocessing

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After preprocessing (100 events):

Total edges: 880k (true: 450k, fake: 430k)

Total nodes: 560k

edges per graph: 8783.7 +/- 1877.3

nodes per graph: 5583.1 +/- 804.4

Distribution of 100 graphs

Single Event

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Edge Network

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Edge

Features

Edge Network:

Input: Node information of each edge

Learn edge features

Node i (v_i)

Node j (v_j)

v_i= [r_i, ϕ_i, z_i, (h_i¹, … , h_i^N)]

v_j= [r_j, ϕ_j, z_j, (h_j¹, … , h_j^N)]

N = hidden dimension size

Edge Network

[v_i, v_j]

e_ij

= 0 if edge is fake

= 1 if edge is true

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Node Network

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Target Node (v_t)

Neighbour Nodes

(v_i, v_j, v_k )

Node Network:

Input: Triplet node information

Learns hidden node features

v_input = v_j. e_jt+ v_k. e_kt

combine nodes using the output of the edge network

v_output = v_i. e_i_t

Node Network

[v_input, v_output, v_target]

h_target (update hidden node features)

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Q.C. for Machine Learning

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We can use parameterized gates to embed data in the Hilbert Space.

Then, we can use other parametrized gates that we can optimize to do tasks such as classification.

Train

Classify

Parameterized Gates

Adapted from: Sim et al. 2019 (arXiv:1905.10876)

Adapted from: Lloyd et al. 2020 (arXiv:2001.03622)

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Quantum Classification

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arXiv: 1804.03680

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Hybrid Neural Network

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Single Fully Connected Layer

IQC (Information Encoding Quantum Circuit): Encodes the Classical Information to Quantum States

PQC (Parametrized Quantum Circuit): Contains trainable parameters that does operations to the Quantum States on the Hilbert Space

Single Fully Connected Layer

Variational Quantum Circuit

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Parametrized Quantum Circuits

Circuits are taken from (Sim et al. 2019, arXiv:1905.10876)

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a layer

Circuit 19:

Circuit 10:

a layer

Layers are repeated blocks of Quantum Circuits. (They can have the same or different parameters)

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Parametrized Quantum Circuits

How do we choose a Quantum Circuit?

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There are metrics in the literature to assess the capacity of Quantum Circuits.

However, they haven’t been shown to have correlation with their learning capacity (yet)!

(Sim et al. 2019 (arXiv:1905.10876))

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Training Results

Number of Layers (N_iteration= 3, qc = 19, N_hid= 4, N_qubits= 4)

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AUC: Area Under ROC, a measure of accuracy for different thresholds.

AUC = 1.0 means perfect score.

Training set: 50 graphs, Test set: 50 graphs, using ADAM, binary cross entropy, lr = 0.01, analytic results.

N_Layers has a positive effect on the performance as expected!

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Training Results

Hidden Dimension Size (N_qubits= N_hid, N_layers= 1, N_iteration= 3)

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AUC: Area Under ROC, a measure of accuracy for different thresholds.

AUC = 1.0 means perfect score.

Training set: 50 graphs, Test set: 50 graphs, using ADAM, binary cross entropy, lr = 0.01, analytic results.

Hidden Dimension size has a positive effect on the performance as expected.

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Training Results

Number of Iterations (N_qubits= 4, N_hid= 4, N_layers= 1)

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AUC: Area Under ROC, a measure of accuracy for different thresholds.

AUC = 1.0 means perfect score.

Training set: 50 graphs, Test set: 50 graphs, using ADAM, binary cross entropy, lr = 0.01, analytic results.

N_iterations has a positive effect on the performance as expected!

(Exa.TrkX team reported 8 as the optimal amount.)

https://exatrkx.github.io

very small

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Training Results

Comparing Results with Hep.TrkX (N_iteration= 3, qc = 10 with 1 layer)

Farrell et al. 2018 (arXiv: 1810.06111)

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Training set: 50 graphs, Test set: 50 graphs, using ADAM, binary cross entropy, lr = 0.01, analytic results.

AUC: Area Under ROC, a measure of accuracy for different thresholds.

AUC = 1.0 means perfect score.

Our approach shows similar characteristics.

But, it can achieve better AUC and loss with better circuits and more layers!

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Conclusion

QGNN results are promising.

They can achieve similar performance compared to a novel classical model. However, there are still challenges to use this algorithm on a Quantum Computer.

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How to improve?

Use more layers.

Explore different circuits.

Explore different architectures.

Use more events.

Challenges

Simulation times are long. Quantum models are hard to simulate. (Training takes 1-2 days depending on model complexity)

Things to explore

Effects of hardware and shot noise.

Complete overview with more layers and iterations.

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Contributors

Cenk Tüysüz

C. Tüysüz¹, C. Rieger², K. Novotny³, B. Demirköz¹, D. Dobos³^,⁴, K. Potamianos³^,5, S. Vallecorsa⁶, J.R. Vlimant⁷, Richard Forster³

¹Middle East Technical University, Ankara, Turkey, ²ETH Zurich, Zurich, Switzerland,

³gluoNNet, Geneva, Switzerland,

⁴Lancaster University, Lancaster, UK,

⁵Oxford University, Oxford, UK,

⁶CERN, Geneva, Switzerland,

⁷California Institute of Technology, Pasadena, California, USA,

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Contributors

Cenk Tüysüz, Carla Rieger

C. Tüysüz^1,2, C. Rieger⁸, K. Novotny⁴, B. Demirköz¹, D. Dobos^4,6,

K. Potamianos^4,5, S. Vallecorsa³, J.R. Vlimant⁷

¹Middle East Technical University, Ankara, Turkey, ²STB Research, Ankara, Turkey, ³CERN, Geneva, Switzerland, ⁴gluoNNet, Geneva, Switzerland, ⁵Oxford University, Oxford, UK, ⁶Lancaster University, Lancaster, UK, ⁷California Institute of Technology, Pasadena, California, USA, ⁸ETH Zurich, Zurich, Switzerland

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Thank you.

Email: ctuysuz@cern.ch

Twitter: @cenk_tuysuz

Cenk Tüysüz

Results shown here will be published soon, with a complete overview.

You can refer to our recent conference paper for previous results: arXiv:2012.01379

The current code base will be public with the release of the paper.

You can refer to our old codebase: https://github.com/cnktysz/HepTrkX-quantum

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Backup Slides

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Information Encoding Quantum Circuit

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Simple Angle Encoding Circuit:

Requires N_qubits = Size of the input

Single Qubit Bloch Sphere Representation

We limit the use of full bloch sphere for two reasons:

Full circle prevents a 1-1 relation between data and measurements
Use of full sphere requires complex PQCs (on our radar for future improvements)

x_i ∈ [0, 𝜋]