novel machine learning algorithms for quantum annealing ...novel machine learning algorithms for...
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Quantum Machine Learning and Quantum Computation Frameworks for HEP (QMLQCF)
Novel machine learning algorithms for quantum annealing with applications in high energy physics
Alexander Zlokapa California Institute of Technology
A. Anand Harvard University
A. Mott DeepMind
J. Job Lockheed Martin
J.-R. Vlimant Caltech
J. M. Duarte Fermilab/UC San Diego
D. Lidar University of Southern California
M. Spiropulu Caltech
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Overview
Higgs boson classification (QAML-Z):
• Phrase error minimization in an Ising model• Use multiple anneals to zoom into the energy surface
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Overview
Higgs boson classification (QAML-Z):
• Phrase error minimization in an Ising model• Use multiple anneals to zoom into the energy surface
Charged particle tracking:
• Adapt large-scale computations to NISQ hardware• Match state-of-the-art classical tracking algorithms
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QAML-Z: Higgs boson classification
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QAML algorithm
“Quantum annealing for machine learning” (QAML)
Rationale: minimize squared error
Method: create strong classifier from sum of weak classifiers
A. Mott, J. Job, J.-R. Vlimant, D. Lidar, M. Spiropulu. "Solving a Higgs optimization problem with quantum annealing for machine
learning." Nature 550.7676 (2017): 375.
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QAML algorithm
Rationale: minimize squared error
Method: create strong classifier from sum of weak classifiers
argminsi
S
∑τ=1
yτ −N
∑i=1
si ci(xτ)
2
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QAML algorithm
Rationale: minimize squared error
Method: create strong classifier from sum of weak classifiers
argminsi
S
∑τ=1
yτ −N
∑i=1
si ci(xτ)
2Training set
Training label
Training input
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QAML algorithm
Rationale: minimize squared error
Method: create strong classifier from sum of weak classifiers
argminsi
S
∑τ=1
yτ −N
∑i=1
si ci(xτ)
2Training set
Training label
Training input
Weak classifier = ±1/N
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QAML algorithm
Rationale: minimize squared error
Method: create strong classifier from sum of weak classifiers
argminsi
S
∑τ=1
yτ −N
∑i=1
si ci(xτ)
2Training set
Training label
Training input
Weak classifier = ±1/N
Classifier weight
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QAML algorithm
Rationale: minimize squared error
Method: create strong classifier from sum of weak classifiers
HIsing =N
∑i=1
N
∑j>i
S
∑τ=1
si ci(xτ) sj cj(xτ) −N
∑i=1
S
∑τ=1
si ci(xτ) yτ
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Higgs problem construction
Can we “rediscover” the Higgs boson with QAML?
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Higgs problem construction
Can we “rediscover” the Higgs boson with QAML?
Diphoton pair
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Higgs problem construction
Higgs boson
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Higgs problem construction
Higgs boson Other Standard Model (SM) processes
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Higgs problem construction
Eight kinematic observables assembled from decay photons:
p1T/mγγ , p2
T/mγγ , (p1T + p2
T)2/mγγ , (p1T − p2
T)2/mγγ , pγγT /mγγ , Δη , ΔR , |ηγγ |
Transverse momentum + diphoton mass
Diphoton angle
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Higgs problem construction
Thirty-six weak classifiers constructed from division and multiplication of eight observables
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Higgs problem construction
Thirty-six weak classifiers constructed from division and multiplication of eight observables
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Higgs classification results
Optimize simulated annealing, deep neural network, and XGBoost hyperparameters
Measure area under ROC curve on 200,000 simulated events
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Higgs classification results
Optimize simulated annealing, deep neural network, and XGBoost hyperparameters
Measure area under ROC curve on 200,000 simulated events
Bette
r
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Higgs classification results
Optimize simulated annealing, deep neural network, and XGBoost hyperparameters
Measure area under ROC curve on 200,000 simulated events
D-Wave 2X
Bette
r
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QAML-Z algorithmA. Zlokapa, A. Mott, J. Job, J.-R. Vlimant, D. Lidar, M. Spiropulu.
“Quantum adiabatic machine learning with zooming." arXiv:1908.04480 [quant-ph] (2019).
Two improvements:• Zoom into the energy surface — continuous optimization• Augment the set of classifiers — stronger ensemble
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QAML-Z algorithm: Zooming
Zooming: perform a binary search on continuous classifier weights by running multiple quantum anneals
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QAML-Z algorithm: Zooming
Zooming: perform a binary search on continuous classifier weights by running multiple quantum anneals
Ener
gy
µ0 = 1s0 = 1
1-1
QAML: take discrete values ±1
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QAML-Z algorithm: Zooming
Zooming: perform a binary search on continuous classifier weights by running multiple quantum anneals
Ener
gy
µ0 = 1s0 = 1
1-1
QAML: take discrete values ±1
Classifier weight
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QAML-Z algorithm: Zooming
Zooming: perform a binary search on continuous classifier weights by running multiple quantum anneals
Ener
gy
µ0 = 1s0 = 1
1-1
QAML: take discrete values ±1
Classifier weight Ising model spin
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QAML-Z algorithm: Zooming
Zooming: perform a binary search on continuous classifier weights by running multiple quantum anneals
Ener
gy
µ0(0) = 0 1-1
QAML-Z: search for weights in [-1, 1]
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QAML-Z algorithm: Zooming
Zooming: perform a binary search on continuous classifier weights by running multiple quantum anneals
µ0(0) = 0En
ergy
µ0(1) = 0.5s0 = 1
1-1
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QAML-Z algorithm: Zooming
Zooming: perform a binary search on continuous classifier weights by running multiple quantum anneals
µ0(0) = 0 µ0(1) = 0.5
Ener
gy
µ0(2) = 0.25s0 = -1
1-1
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QAML-Z algorithm: Augmentation
Augmentation: create multiple classifiers from the same combination of physical variables by offsetting distribution cut
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QAML-Z algorithm: Augmentation
Augmentation: create multiple classifiers from the same combination of physical variables by offsetting distribution cut
QAML
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QAML-Z algorithm: Augmentation
Augmentation: create multiple classifiers from the same combination of physical variables by offsetting distribution cut
QAML
Kinematic variables
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QAML-Z algorithm: Augmentation
Augmentation: create multiple classifiers from the same combination of physical variables by offsetting distribution cut
QAML
Kinematic variables
Weak classifier
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QAML-Z algorithm: Augmentation
Augmentation: create multiple classifiers from the same combination of physical variables by offsetting distribution cut
QAML
Background
Higgs
Background Higgs
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QAML-Z algorithm: Augmentation
Augmentation: create multiple classifiers from the same combination of physical variables by offsetting distribution cut
QAML QAML-Z
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Higgs classification results
QAML-Z vs. QAML
Improves advantage over DNN by ~40% for small training sets
Shrinks disadvantage to DNN by ~50% for large training sets
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Higgs classification resultsQAML-Z
(D-Wave 2X)
QAML(D-Wave 2X)
QAML-Z vs. QAML
Improves advantage over DNN by ~40% for small training sets
Shrinks disadvantage to DNN by ~50% for large training sets
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Higgs classification results
Deep neural network
QAML-Z(D-Wave 2X)
QAML(D-Wave 2X)
QAML-Z vs. QAML
Improves advantage over DNN by ~40% for small training sets
Shrinks disadvantage to DNN by ~50% for large training sets
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Higgs classification results
Both zooming and augmentation improve
performance
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Higgs classification results
Both zooming and augmentation improve
performanceAugmentation
Zooming
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Charged particle tracking
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Track reconstruction
Cluster “hits” in a detector by particle instance
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Track reconstruction
Cluster “hits” in a detector by particle instance
B
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Track reconstruction
Cluster “hits” in a detector by particle instance
B
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Track reconstruction
Cluster “hits” in a detector by particle instance
Strandlie, Are, and Rudolf Frühwirth. "Track and vertex reconstruction: From classical to adaptive methods." Reviews of Modern Physics 82.2 (2010): 1419.
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Track reconstruction
Cluster “hits” in a detector by particle instance
Strandlie, Are, and Rudolf Frühwirth. "Track and vertex reconstruction: From classical to adaptive methods." Reviews of Modern Physics 82.2 (2010): 1419.
Low momentum
High momentum
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Classical methods
Upgrade of LHC to high luminosity increases the number of hits per event by a factor of 5
Current tracking (Kalman filter) is thought to scale exponentially with the number of hits
Possibility of quantum speedup?
CMS Collaboration. "CMS Tracking POG Performance Plots For 2017 with PhaseI pixel detector." (2017).
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Ising model formulation
Make each edge a binary variable: turn edge “on” or “off”
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Ising model formulation
H1 = − ∑i
∑j>i
Jijsisj − ∑i
hisi
Affinity between edges i and j
Prior expectation on edge i
A. Zlokapa, A. Anand, J.-R. Vlimant, J. M. Duarte, J. Job, D. Lidar, M. Spiropulu. “Charged particle tracking with quantum annealing-
inspired optimization." arXiv:1908.04475 [quant-ph] (2019).
1 if edge is on; 0 if edge is off
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Expect helical tracks due to a charged particle moving in a uniform magnetic field
Ising model formulation
a
b c
rabrbc
θabc
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Expect helical tracks due to a charged particle moving in a uniform magnetic field
Ising model formulation
a
b c
rabrbc
θabc−( cosλ(θabc)rab + rbc ) sabsbc
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Ising model formulation
E = − ∑a,b,c ( cosλ(θabc) + ρ cosλ(ϕabc)
rab + rbc ) sabsbc + η∑a,bc (zc −
zc − za
rc − rarc)
ζ
sabsbc + α ∑b≠c
sabsac + ∑a≠c
sabscb + ∑a,b
(γ − βP(sab)) sab
Helical tracks High-momentum bias
Beam spot geometry
Track bifurcation penalty Global edge penalty
Edge orientation probability(Gaussian kernel density estimation)
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Dimension challengeHiggs event at LHC: 103 to 104 detector hits ~107 edges
• Divide into 16 sectors: ~105 edges• Remove edges with Gaussian KDE: ~103 edges
⟹
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Dimension challengeHiggs event at LHC: 103 to 104 detector hits ~107 edges
• Divide into 16 sectors: ~105 edges• Remove edges with Gaussian KDE: ~103 edges
D-Wave 2X: 33 fully-connected qubits
• Sparse Ising model weights: ~102 qubits• Split into disjoint sub-graphs: ~10 problems per sector
⟹
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Disjoint sub-graphs: prune and divide
Dimension challenge
Initial graph True graph
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Disjoint sub-graphs: prune and divide
Dimension challenge
Pruned graph True graph
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Disjoint sub-graphs: prune and divide
Dimension challenge
Disjoint sub-graphs True graph
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Dimension challengeHiggs event at LHC: 103 to 104 detector hits ~107 edges
• Divide into 16 sectors: ~105 edges• Remove edges with Gaussian KDE: ~103 edges
D-Wave 2X: 33 fully-connected qubits
• Sparse Ising model weights: ~102 qubits• Split into disjoint sub-graphs: ~10 problems per sector
⟹
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Higgs event at LHC: 103 to 104 detector hits ~107 edges
• Divide into 16 sectors: ~105 edges• Remove edges with Gaussian KDE: ~103 edges
D-Wave 2X: 33 fully-connected qubits
• Sparse Ising model weights: ~102 qubits• Split into disjoint sub-graphs: ~10 problems per sector
Result: ~100 Ising model variables on ~100 qubits
⟹
Dimension challenge
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ResultsPerformance metrics: efficiency (recall) and purity (precision) measured on the TrackML dataset
purity =# true tracks reconstructed
# tracks reconstructed
efficiency =# true tracks reconstructed
# true tracks
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Results
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Results
Maximum efficiency(after pre-processing)
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Results
Maximum efficiency(after pre-processing)
D-Wave 2XSimulated annealing
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!63
Results
Maximum efficiency(after pre-processing)
Higgs in 2011-2012
D-Wave 2XSimulated annealing
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Results
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Results
Efficiency
Purity
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Results
97%
99%
Efficiency
Purity
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Conclusion
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Substantial improvement demonstrated by QAML-Z
• Widespread applicability of successive anneals on iteratively refined problems
Beyond HEP: What’s new in QML?
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Substantial improvement demonstrated by QAML-Z
• Widespread applicability of successive anneals on iteratively refined problems
Successful encoding of big data in the era of NISQ
• General methodology of pruning Ising models with a successful outcome
Beyond HEP: What’s new in QML?
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Substantial improvement demonstrated by QAML-Z
• Widespread applicability of successive anneals on iteratively refined problems
Successful encoding of big data in the era of NISQ
• General methodology of pruning Ising models with a successful outcome
Competitive results with state-of-the-art classical algorithms
Beyond HEP: What’s new in QML?
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Thank you
!71
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Supplementary slides
!72
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Higgs problem construction
Can we “rediscover” the Higgs boson with QAML?
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!74
Higgs problem construction
Can we “rediscover” the Higgs boson with QAML?
B
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!75
Higgs problem construction
Can we “rediscover” the Higgs boson with QAML?
BpT
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!76
QAML-Z algorithm: Hamiltonian
Zooming: replace each binary weight � with continuous weight � governed with search breadth � .
Augmentation: generate multiple shifted classifiers � for each original classifier � .
Anneal for iterations � .
siμi(t) σ(t) = 1/2t
cil(xτ)ci(xτ)
t = 0, 1, 2, …, T − 1
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QAML-Z algorithm: Hamiltonian
H(t) =A
∑l=−A [
N
∑i=1
−Cil +N
∑j>i
μjl(t)Cijl σ(t)sil +N
∑i=1
N
∑j>i
Cijlσ2(t)silsjl]
Each iteration � , anneal:
where we have defined:
t
Cijl =S
∑τ=1
cil(xτ)cjl(xτ)Cil =S
∑τ=1
cil(xτ)yτ
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!78
QAML-Z algorithm: Hamiltonian
H(t) =A
∑l=−A [
N
∑i=1
−Cil +N
∑j>i
μjl(t)Cijl σ(t)sil +N
∑i=1
N
∑j>i
Cijlσ2(t)silsjl]
Each iteration � , anneal:
and update continuous weights from spins:
t
μil(t + 1) = μil(t) + silσ(t + 1)
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!79
ROC curve
Metric of performance: area under receiver operating
characteristic (ROC) curve
True
pos
itive
rate
(sig
nal e
ffici
ency
)
False positive rate(background rejection)
Random
clas
sifier
0 10
1
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!80
ROC curve
Metric of performance: area under receiver operating
characteristic (ROC) curve
True
pos
itive
rate
(sig
nal e
ffici
ency
)
False positive rate(background rejection)
Random
clas
sifier
0 10
1
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!81
ROC curve
Metric of performance: area under receiver operating
characteristic (ROC) curve
True
pos
itive
rate
(sig
nal e
ffici
ency
)
False positive rate(background rejection)
Random
clas
sifier
0 10
1Perfect classifier
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!82
ROC curve
Metric of performance: area under receiver operating
characteristic (ROC) curve
True
pos
itive
rate
(sig
nal e
ffici
ency
)
False positive rate(background rejection)
Random
clas
sifier
0 10
1
Typica
l clas
sifier
Perfect classifier
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Higgs classification results
Dashes indicate test set, solid line indicates training set
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DatasetTrackML Challenge: top quark events with 15% noise
Amrouche, Sabrina, et al. "The Tracking Machine Learning challenge: Accuracy phase." arXiv preprint arXiv:1904.06778 (2019).
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Bias towards high momentum tracks that are more important
Ising model formulation
−( cosλ(ϕabc)rab + rbc ) sabsbc
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!86
Bias towards high momentum tracks that are more important
Ising model formulation
−( cosλ(ϕabc)rab + rbc ) sabsbc
High �pT
Low �pTϕ
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!87
Tracks should point towards the beam spot at the origin
Ising model formulation
(zc −zc − za
rc − ra ) sabsbc
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!88
In general, tracks shouldn’t split at or merge into a single hit
Ising model formulation
a
b
ca
b
csabsac + sabscb
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!89
Use Gaussian kernel density estimation to provide a prior on an edge being “on” or “off” based on orientation and position
Dimension challenge
(γ − P(sab))sab
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!90
ResultsGround state energy True solution energy
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Inconclusive results for quantum speedup
Quantum speedup?
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!92
Inconclusive results for quantum speedup
• Classical: �
• Quantum: preprocess at � , inconclusive QUBO-solving time
O(exp(# of hits))
O ((# of hits)2)
Quantum speedup?
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!93
Inconclusive results for quantum speedup
• Classical: �
• Quantum: preprocess at � , inconclusive QUBO-solving time
Could use specialized classical hardware for particle tracking at the trigger level: 1 PB/s reduced to 1 GB/s
O(exp(# of hits))
O ((# of hits)2)
Quantum speedup?
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!94
Quantum speedup?
Pre-processing to construct the Ising model scales like � where an event has � hitsO(h2) h
Annealing Classical: O(exp(ch2))
Quantum: O(?)
Disjoint sub-graph flood-fill search
O(h2)
Ising model construction O(h2)
Singlet selection with Gaussian KDE
O(h2)
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!95
Quantum speedup?
Pre-processing reduces the simulated annealing solving time from � where an event has � hitsO(exp(ch2)) h
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!96
Quantum speedup?
The problem remains NP-hard after pre-processing, so SA is exponential in the number of Ising model variables
For an event divided into � sub-graphs with � edges each, we expect solving time
Kmi
O (K
∑i=1
exp (cmi))
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!97
Quantum annealing is expected to reduce the size of � but leave the problem exponential
�
Can’t measure a single time to solution, but can change annealing time and measure change in performance
c
O (K
∑i=1
exp (cmi))
Quantum speedup?
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!98
Quantum speedup?