Charged Particle Tracking as a QUBO Problem Solved with ...

Connecting the Dots 2019, Valencia, April 2-5 2019

Jean-Roch Vlimant on behalf of the QMLQCF project teamSpecial credits to

Abhishek Anand and Alexander Zlokapa

Charged Particle Tracking as a QUBOProblem Solved with Quantum Annealing-

inspired Optimization

04/02/19 2CTD19, QA-Tracking, J-R Vlimant

Outline

Forewords on charged particletracking and dataset

Introduction to D-Wave andquantum annealing

Hopfield Network and segmentclassification as a QUBOproblem

Results and outlooks


The Challenge of ChargedParticle Tracking at the HL-LHC


Tracking in a Nutshell

Seeding Kalman Filter

● Particle trajectory bended in asolenoid magnetic field

● Curvature is a proxy tomomentum

● Particle ionize silicon pixel and strip throughout several concentric layers

● Thousands of sparse hits● Lots of hit pollution from low

momentum, secondary particles

● Explosion of hit combinatorics in both seeding and stepping pattern recognition● Highly time consuming task in extracting physics content from LHC data


Cost of Tracking● CPU time consumption in HL-LHC era surpasses computing budget

➔ Need for faster algorithms● Charged particle track reconstruction is one of the most CPU consuming

task in event reconstruction➔ Optimizations mostly saturated

● Large fraction of CPU required in the HLT. Cannot perform trackinginclusively

➔ Approximation allowed in the trigger

@HL-LHC <μ> >200


Fast Hardware Tracking● Track trigger implementation for Trigger

upgrades development on-going● Several approaches investigated● Dedicated hardware is the key to fast

computation.● Not applicable for offline processing unless

through adopting heterogeneous computing.Tracklets

Hough

Transform

Kalman

Filter


Deep Learning Approaches

Seq-to-seq track finding

https://tinyurl.com/yb3v93y9

End-to-end hit assignment

Track following with RNNhttps://heptrkx.github.io/

https://indico.cern.ch/event/587955/contributions/2937570/


See Xiangyang later today https://indico.cern.ch/event/742793/contributions/3274328/


https://heptrkx.github.io/





Charged Particle Tracking Dataset

https://www.kaggle.com/c/trackml-particle-identificationhttps://competitions.codalab.org/competitions/20112

● This work uses the publicdataset of the TrackMLParticle Tracking Challenge(Kaggle, codalab).

● Simulating the denseenvironment expected forHL-HLC. Average of 200proton-proton interaction perbunch crossing.

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

https://competitions.codalab.org/competitions/20112


Motivation

● Classical charged particle tracking algorithmssuffer from combinatorial explosion

● Embrace the combinatorics considering allpossible branches of track candidates, andsolve the complex optimization problem withquantum annealing


The D-Wave Computing System


D-Wave 2XTM

1098 qubitsOperates at 15mKAnneals in 5-20 μs


qubit and qubit

Quantum CircuitsSeries of quantum gates

operating on a set ofquantum states.

Quantum AnnealingEvolution of a quantum

system to a low T Gibbs stateThat's D-Wave !


Quantum Annealing


Adiabatic Quantum Annealing ➢ System setup with trivial hamiltonian H(0) and ground state➢ Evolve adiabatically the hamiltonian towards the desired

Hamiltonian Hp

➢ Adiabatic theorem : with a slow evolution of the system, thestate stays in the ground state.

https://arxiv.org/abs/quant-ph/0001106 https://arxiv.org/abs/quant-ph/0104129

https://arxiv.org/abs/quant-ph/0001106

https://arxiv.org/abs/quant-ph/0104129


Space of Hamiltonian

Externalmagnetic field

Interactions

Runs overadjacent quBits

Runs over all quBit pairs


Interactions

Chimera graphembedding

➢ ~1000 qubits on chimera graph canonly encode ~40 qubits full IsingHamiltonian

➢ Quadratic Unconstrained BinaryOptimization (QUBO) can bemapped to an Ising Hamiltonianwith change of variable {0,1}↔{-1,1}


Ising Model Heuristic Solution● Monte-Carlo based method to find ground state

of energy functions● Random walk across phase space

➔ accepting descent➔ accepting ascent with probability e-ΔE/kT

● Decrease T with time

Applied to the QUBO problem, and finds theground state. SA in the legends.


Charged Particle Tracking usingAdiabatic Quantum Annealing

See also L. Linder https://indico.cern.ch/event/742793/contributions/3274780/



Hopfield Network Approach


Framing the Problem● Segment ≡ pair of hits on

consecutive layers of the detector● Assign a boolean to each segment

representing whether the segment iswithin a track or not

● Limits the number of hits/segments➔ Separating the hits in 16 sector in φ➔ pre-filtering the segments on Δφ and

Δz to reduce the number of spuriousbad segments

● Segment opening in r-phi-z plane inwhich helical segments are aligned

● Azimuthal angle in cartesiancoordinate in which high pT trackssegments are straight

θabc

rab

rbc


Segment QUBO

∑a , b ,c ( cos

λθabc+ρcosλϕabcrab+rbc

+η(zc−r c zc−zar c−ra )ζ

) sab sbc+α( ∑a , b≠c sab sac+ ∑

a≠b , csac sbc)+∑

a , b(β+γ P (sab)) sab

Helix TermSegments along

an helix

High pT TermAligned pair of segments

Beam spot TermSegment pointing at

the origin

Bifurcation TermNo shared hits

on valid segment

Inhibition TermReduced number

of segments

GP TermUse the quality

of segment


Resolving Sub-Group

● Train a gaussian kernel densityestimator on true single segment

● Aiming at reducing the number offalse segments, retaining

● Force segment off based on cosθabc

➢ cosθabc

= 0 if θabc

> θ0

● 5 best neighbors● Solvable in polynomial time

● Full all-to-all QUBO problem cannot fit on dWave. Aim at identifyingsub-groups of segments that can fit on the hardware


QA-Tracking Workflow

ResolvingSub-group (θ

0)

SegmentQUBO

ResolvingSub-group (θ

1)

SegmentQUBO


Performance● Simulated annealing and

Quantum annealing are inperfect agreement at 200 tracks

● Simulated annealing solves theexact problem at all multiplicity

● Limitation on number of qubitsprevents from solving eventsbeyond 200 tracks on Dwave ;solving a contrive problem

● Purity and Efficiency aremeasured with respect to truetracks with at least three hits

➢ Promising tracking efficiency forthe algorithm up to 2000 tracksper event

<P

U>

~16

<P

U>

~33

<P

U>

~8


Conclusion

● QMLQCF Scouting for applications of quantumannealing (among others) in HEP

● Charged particle tracking interpreted as asegment classification can be expressed in aQUBO problem

● Experimentation on dWave imposes somestringent algorithmic restrictions

● Limited hardware size limits the complexity of theproblem that can be solved


This project is supported in part by the United States Department ofEnergy, Office of High Energy Physics Research TechnologyComputational HEP and Fermi Research Alliance, LLC underContract No. DE-AC02-07CH11359. The project is also supported inpart under ARO grant number W911NF-12-1-0523 and NSF grantnumber INSPIRE-1551064. The work is supported in part by theAT&T Foundry Innovation Centers through INQNET, a program foraccelerating quantum technologies. We wish to thank the AdvancedScientific Computing Research program of the DOE for theopportunity to first present and discuss this work at the ASCRworkshop on Quantum Computing for Science (2015). AwardNumber: DE-SC0019227, Quantum Machine Learning and QuantumComputation Frameworks for HEP (QMLQCF), California Institute ofTechnology, Pasadena, CA

Part of this work was conducted at "iBanks", the AI GPU cluster atCaltech. We acknowledge NVIDIA, SuperMicro and the KavliFoundation for their support of "iBanks".

Acknowledgments


Problem Parameters Optimization

● Parameters of the hamiltonian are tuned usingbayesian optimization, modeling the figure of meritwith gaussian processes.

● Accuracy (# of properly labeled / # of segments)use as f.o.m

● Global inhibition model : α=3.E-3, β=2.63E-8, λ=7

● Threshold model : α=5.E-3, β=1.E-6, λ=7


Edge Aff inity

• Helical bias: tracks are straight in cylindrical coordinates• Momentum bias: high-PT tracks are straight in rectangular

coordinates• Short-edge bias: long tracks of short edge segments

Momentum bias(rectangular angle)

Helical bias(cylindrical angle)

Short-edge bias Ising variables (1 or 0)

θabc

rab

rbc


Cross-Term Penalties• Beam spot penalty: penalize tracks that originate further from the

origin

Z-intercept penalty

Ising variables (1 or 0)


• Global inhibition: limits total number of edges turned on• Prior probability: Bayesian prior based on edge position in rz-plane

• Computed using Gaussian kernel density estimation

Global inhibition

Ising variable (1 or 0)

Single-Edge Bias

Bayesian prior


Extra Material


References


https://www.dwavesys.com/ 1999 Founded 2011 D-Wave One : 128 qubits2013 D-Wave Two : 512 qubits2015 D-Wave 2X : 1000 qubits2017 D-Wave 2000Q : 2000 qubits2019? 5000 qubits ?

The D-Wave Company

https://www.dwavesys.com/


D-Wave HamiltonianAnd

Chimera Graph



Interactions

D-Wave Hamiltonian

Runs overadjacent quBits


D-Wave qubit Adjacency

Active qubits in greenCoupling to 5-6 qubitsInactive qubits in redNot a fully connected graph


Model Embedding


Full Ising Model

https://arxiv.org/abs/1210.8395

● Create chains of spins throughthe chimera graph

● Split local fields across allqubits in the chain

● Tightly couple (JF=6)

● Non-unique embedding.Heuristic approach.

● Suppressing spin flip withinchain as error correction.

● Use majority vote

➔ Approximately full Ising Modelwith ~<40 spins

https://arxiv.org/abs/1210.8395


Ising Hamiltonian

Runs over all quBit pairs


Interactions


High Luminosity LHCThe Challenge


HL-LHC Challenge

<PU>=7 <PU>=21

<PU>=140-20010x more hits

Circa 2025

● CPU time extrapolation into HL-LHC era far surpasses growth incomputing budget

● Need for faster algorithms

● Approximation allowed in the trigger


Complexity and Ambiguity

Shown trajectories are reconstructed objects

The future holds much more hits


HEP.TrkX Approaches

Seq-to-seq track finding

https://tinyurl.com/y87saehf


End-to-end hit assignment

Track following with RNN https://heptrkx.github.io/

https://tinyurl.com/y87saehf


https://heptrkx.github.io/


Tracking Not In a Nutshell

● Hits preparation

● Seeding

● Pattern recognition

● Track fitting

● Track cleaning

Sev

eral

Tim

es


Hit Preparation

● Calculate the hit position from barycenter of chargedeposits

● Use of neural net classifier to split cluster in ATLAS

● Access to trajectory local parameter from clustershape

● Remove hits from previous tracking iterations

● HL-LHC design include double layers giving moreconstraints on the local trajectory parameters


Seeding

● Combinatorics of 2 or 3 hitswith tight/loose constraintsto the beam spot or vertex

● Seed cleaning/purity playsin an important in reducingthe CPU requirements ofsub-sequent steps➔ Consider pixel cluster

shape and charge toremove incompatibleseeds

● Initial track parameters fromhelix fit


Pattern Recognition● Use of the Kalman filter

formalism with weight matrix

● Identify possible next layersfrom geometrical considerations

● Combinatorics with compatibleshits, retain N best candidates

● No smoothing procedure

● Resilient to missing modules

● Hits are mostly belonging to onetrack and one track only

● Hit sharing can happen in denseevents, in the innermost part

● Lots of hits from low momentumparticles


Kalman Filter● Trajectory state propagation

done either✔ Analytical (helix, fastest)✔ Stepping helix (fast)✔ Runge-Kutta (slow)

● Material effect added totrajectory state covariance

● Projection matrix of local helixparameters onto module surface➔ Trivial expression due to local

helix parametrisation● Hits covariance matrix for pixel

and stereo hits properly formed✗ Issue with strip hits and

longitudinal error being nongaussian (square)


Track Fitting

● Use of the Kalman filterformalism with weightmatrix

● Use of smoothingprocedure to identifyoutliers

● Field non uniformity aretaken into account

● Detector alignmenttaken into account


Cleaning, Selection

● Track qualityestimated usingranking orclassification method➔Use of MVA

● Hits from high qualitytracks are removefor the next iterationswhere applicable


A Charged Particle Journey


First order effect : electromagnetic elasticinteraction of the charge particle with nuclei (heavy

and multiply charged) and electrons (light andsingle charged)

Second order effect : inelastic interaction withnuclei.


Magnetic Field● Magnetic fieldB acts on charged particles

in motion : Lorentz Force

● The solution in uniform magnetic field isan helix along the field : 5 parameters

● Helix radius proportional to the componentof momentum perpendicular to B

● Separate particles in dense environment

➔ Bending induces radiation :bremsstrahlung

➔ The magnetic field has to be known to agood precision for accurate tracking ofparticle


Multiple Scattering● Deflection on nuclei (effect from

electron are negligible)

● Addition of scattering processes

● Gaussian approximation valid forsubstantial material traversed

Gaussian Approximation


Bremsstrahlung● Electromagnetic radiation of

charged particles under accelerationdue to nuclei charge

● Significant at low mass or highenergy

● Discontinuity in energy lossspectrum due to photon emissionand track curvature

➔ Can be observed as kink in thetrajectory or presence of collinearenergetic photons


Energy Loss● Momentum transfer to electrons when

traversing material (effect of nuclei isnegligible

● Energy loss at low momentumdepends on mass : can be used asmass spectrometer

ALICE Experiment


Summary on Material Effects

● Collective effects can be estimatedstatistically and taken into account in how theymodify the trajectory

● Bremstrahlung and nuclear interactionssignificantly distort trajectories

Charged Particle Tracking as a QUBO Problem Solved with ...

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