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
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The Challenge of ChargedParticle Tracking at the HL-LHC
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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
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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
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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
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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/
https://indico.cern.ch/event/587955/contributions/2937540/
See Xiangyang later today https://indico.cern.ch/event/742793/contributions/3274328/
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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.
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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
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The D-Wave Computing System
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D-Wave 2XTM
1098 qubitsOperates at 15mKAnneals in 5-20 μs
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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 !
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Quantum Annealing
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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
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Space of Hamiltonian
Externalmagnetic field
Interactions
Runs overadjacent quBits
Runs over all quBit pairs
Externalmagnetic field
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}
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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.
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Charged Particle Tracking usingAdiabatic Quantum Annealing
See also L. Linder https://indico.cern.ch/event/742793/contributions/3274780/
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Hopfield Network Approach
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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
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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
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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
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QA-Tracking Workflow
ResolvingSub-group (θ
0)
SegmentQUBO
ResolvingSub-group (θ
1)
SegmentQUBO
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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
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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
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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
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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
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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
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Cross-Term Penalties• Beam spot penalty: penalize tracks that originate further from the
origin
Z-intercept penalty
Ising variables (1 or 0)
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• 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
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Extra Material
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References
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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
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D-Wave HamiltonianAnd
Chimera Graph
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Externalmagnetic field
Interactions
D-Wave Hamiltonian
Runs overadjacent quBits
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D-Wave qubit Adjacency
Active qubits in greenCoupling to 5-6 qubitsInactive qubits in redNot a fully connected graph
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Model Embedding
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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
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Ising Hamiltonian
Runs over all quBit pairs
Externalmagnetic field
Interactions
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High Luminosity LHCThe Challenge
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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
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Complexity and Ambiguity
Shown trajectories are reconstructed objects
The future holds much more hits
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HEP.TrkX Approaches
Seq-to-seq track finding
https://tinyurl.com/y87saehf
https://tinyurl.com/yb3v93y9
End-to-end hit assignment
Track following with RNN https://heptrkx.github.io/
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Tracking Not In a Nutshell
● Hits preparation
● Seeding
● Pattern recognition
● Track fitting
● Track cleaning
Sev
eral
Tim
es
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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
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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
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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
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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)
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Track Fitting
● Use of the Kalman filterformalism with weightmatrix
● Use of smoothingprocedure to identifyoutliers
● Field non uniformity aretaken into account
● Detector alignmenttaken into account
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Cleaning, Selection
● Track qualityestimated usingranking orclassification method➔Use of MVA
● Hits from high qualitytracks are removefor the next iterationswhere applicable
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A Charged Particle Journey
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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.
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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
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Multiple Scattering● Deflection on nuclei (effect from
electron are negligible)
● Addition of scattering processes
● Gaussian approximation valid forsubstantial material traversed
Gaussian Approximation
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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
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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
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Summary on Material Effects
● Collective effects can be estimatedstatistically and taken into account in how theymodify the trajectory
● Bremstrahlung and nuclear interactionssignificantly distort trajectories
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