Introduction to Quantum Computing - CERN
Transcript of Introduction to Quantum Computing - CERN
1
Introduction to Quantum
Computing
Summer Student Lecture
Tuesday, July 9th, 2019
Federico Carminati
2Driving innovation in HEP computing
3Driving innovation in HEP computing
The Higgs Boson
Elementary particles0,2%
Atoms, stars, diffused gas4%
Exotic dark matter(neutrinos, neutralinos,β¦) 30%
Dark energy(Vacuum energy,β¦) 66%
Whe ignore most things about the 4% of the Universe
But we do not even know of what the remaining 96% is made of
What is the place of matter in the universe
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From ridiculously difficultβ¦
...to almost impossibleDriving innovation in HEP computing
6Driving innovation in HEP computing
Worldwide LHC Computing Grid
Tier-0 (CERN):
β’Data recording
β’Initial data reconstruction
β’Data distribution
Tier-1 (14 centres):
β’Permanent storage
β’Re-processing
β’Analysis
Tier-2 (72
Federations, ~149
centres):
β’ Simulation
β’ End-user analysis
β’760,000 cores
β’700 PB
6Driving innovation in HEP computing
Worldwide LHC Computing Grid
Tier-0 (CERN):
β’Data recording
β’Initial data reconstruction
β’Data distribution
Tier-1 (14 centres):
β’Permanent storage
β’Re-processing
β’Analysis
Tier-2 (72 Federations, ~149 centres):
β’ Simulationβ’ End-user analysis
β’760,000 coresβ’700 PB
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β’ Raw data volume increases exponentially Processing and analysis load
β’ Technology at ~20%/year will bring x6-10 in ~10 years Estimates of resource needs x10 above what is realistic to expect
Driving innovation in HEP computing
HL-LHC: data volume
https://arxiv.org/pdf/1712.06982.pdf
Flat budget Flat budget
CMS
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29 December 1959, Pasadena, California
β’ Why cannot we write the entire 24 volumes of the Encyclopedia Britannica on the head of a pin?
β’ The principles of physics, as far as I can see, do not speak against the possibility of maneuvering things atom by atom
β’ Miniaturizing the computer... the wires should be 10 or 100 atoms in diameter, and the circuits should be a few thousand angstroms across
Driving innovation in HEP computing
There is plenty of room
at the bottom
9Driving innovation in HEP computing
Quantum Computing?
"Nature is quantum, goddamn it! So if we want to simulate it, we need a quantum computer.βR.Feynman, 1981, Endicott House, MIT
`Blochβs sphere
Use qubits instead of bitsβ¦e.g. bits that exhibit quantum
behavior
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Size of an atom
The three frontiersShort distance -> High Energy Physics
Long distance -> Cosmology
Entanglement (i.e. complexity) -> Quantum Information Technology
Since Turing it was believed that the βhardnessβ of a problem was intrinsic to it
Quantum Computing is now challenging this
Driving innovation in HEP computing
Quantum Computing in
perspective
We could argue that Quantum Computingis a natural consequence of Mooreβs law
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Can we control complex quantum systems and if we can, so what? (J.Preskill, 2012)
β’ Can quantum computers outperform classical computers on all algorithms?
β’ Can quantum computers do things that cannot be done by classical computers (quantum supremacy)?
β’ The golden apple is βsuperpolynomial speedupβ Reducing to polynomial time what in classic computing is exponential or more
Theoretically achieved with some algorithms
β’ But polynomial speedup can be very appealing Particularly for large problems
β’ For the moment it is debatable whether Quantum Supremacy has been demonstratedor not
Driving innovation in HEP computing
Quantum supremacy
β¦or is the gain worth the pain?
Tim
e
Problem size
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EU Quantum Flagship β large-scale initiative
funded at the 1b β¬ level on a 10 years timescale.
US-DoE Quantum Information Science Enabled Discovery (QuantISED) for High Energy Physics
Up to $13M total of awards in FY 2018 (FOA+LAB)
US-DoE Quantum Information Science in FY 2019 HEP
Presidentβs Budget Request: $27.5M
Driving innovation in HEP computing
β¦ and money is flowing inβ¦
What people
laugh aboutWhat matters
at the end
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I think there can be a world market for maybe five computers. (Thomas Watson, CEO of IBM, 1943)
There is no reason for an individual to have a computer at home . (Ken Olsen , president, director and founder of Digital Equipment Corp., 1977)
I think that this thing that Tim (Berners-Lee) has shown me has no future (F.Carminati, 1989)
Driving innovation in HEP computing
Just for the skeptical
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β’ QC can be used to solve directly Quantum Many Body and Quantum Field Theory problems
β’ In chemistry we already have variational calculations of atomic orbital configurations Complex molecules are the βkilling appβ here
β’ Similarly for Nuclear Physics the challenge will be to describe nucleiand their scattering and interactions
β’ This is well beyond exascale computingand current theoretical understanding
Quantum Computing for Theoretical Particle Physics
Quantum on Quantum
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β’ Classical computing is based on discrete elements (transistors)
taking the value of either 0 or 1
β’ Operations are performed by gates i.e. electronic circuits that
act as operators
β’ This is a NAND gate
β’ This gate is βirreversibleβ and it
increases world entropy by
π₯S=k ln 2 (~3 10-21 joules@70Β°C)
Driving innovation in HEP computing
Principles of QC
See http://theory.caltech.edu/~preskill/ph229
INPUT OUTPUT
A B A NAND B
0 0 1
0 1 1
1 0 1
1 1 0
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β’ Quantum Computers use quantum objects to store information
(see later for practical implementation)
β’ The simplest representation is the Bloch Sphere
β’ Or more generally
Driving innovation in HEP computing
Principles of Quantum
Computing
ΰ΅ΏΘπ = cos(π/2)Θ0 + ΰ΅Ώπβππsin(π/2)Θ1
ΰ΅ΏΘπ = Ξ±Θ0 + Ξ²Θ1
Ξ± 2 + Ξ² 2=1
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β’ What if we have 2 qubits?
β’ And in general
Driving innovation in HEP computing
Principles of quantum
computing
Θπ = π1Θ0 + π1Θ1 π2Θ0 + π2Θ1 =
πΌ Θ0 Θ0 + π½ Θ0 Θ1 + πΎ Θ1 Θ0 + πΏ Θ1 Θ1 =
πΌ Θ00 + π½ Θ01 + πΎ Θ10 + πΏ Θ11
Θπ =
0
2πβ1
ππ₯Θπ₯
Θπ₯ = Θ10010010β¦ .111101001
Θπ = πΌ
1000
+π½
0100
+ πΎ
0010
+ πΏ
0001
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β’ In general, a state may be such that
β’ Take for instance the Bell states
β’ If I measure the first qubit as 0 in a separable state I have
Driving innovation in HEP computing
Separable and entangled states
Θπ = πΌ Θ00 + π½ Θ01 + πΎ Θ10 + πΏ Θ11 β π1Θ0 + π1Θ1 π2Θ0 + π2Θ1
ΰ΅ΏΰΈ«π1 =1
2Θ00 + Θ11 ; ΰ΅ΏΰΈ«π2 =
1
2Θ00 β Θ11
Θπ = πΌ Θ00 + π½ Θ01 + πΎ Θ10 + πΏ Θ11measure 1
π 2 + π½ 2πΌ Θ0 + π½ Θ1
= k π2Θ0 + π2Θ1
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β’ But if I do the same in the Bell states, I get
β’ The measure of one qubit entirely determines the measure of
the other
β’ This is the basic concept of quantum entanglement
Driving innovation in HEP computing
Separable and entangled states
ΰ΅ΏΰΈ«π1 =1
2Θ00 + Θ11 ; ΰ΅ΏΰΈ«π2 =
1
2Θ00 β Θ11
measure 01Θ0
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EPR state
β’ Bob knows what Alice measured (or will measure!) At the precise moment of its measurement
β’ The effect of the act of measurement is not local β’ It is again in contradiction with our intuition
β’ But the theory is consistent (it is not in contradiction with itself)
β’ It is said that e1 and e2 are "entangled"
Some light year
If Bob finds that, then Alice must find
1
2+
1
2
e1
1
2+
1
2
e2
Driving innovation in HEP computing
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Small reminder
Diachronic
Oldcausa efficiens
Newcausa finalis
Everything
causa formalis
Constituentsca
usa
mat
eria
lis
event
Determinism Fatalism, vitalism,
idealism, religion
Holism
Riductionism
Driving innovation in HEP computing
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β’ We need 8 coefficients
β’ Which may not be reducible to 6!
β’ In this case the state is βentangledβ, we do need all the 8
coefficients
Driving innovation in HEP computing
And with 3?
The trouble is that 2+2 = 22
Θπ = π1Θ0 + π1Θ1 π2Θ0 + π2Θ1 π3Θ0 + π3Θ1 =
πΌ0 Θ000 +πΌ1 Θ001 +πΌ2 Θ010 +πΌ3 Θ011 + πΌ4 Θ100 +πΌ5 Θ101 +πΌ6 Θ110 +πΌ7 Θ111
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Driving innovation in HEP computing
The power of Quantum
Computing
1 Θ0 , Θ1 21=2
2 Θ00 , Θ01 , Θ00 , Θ01 22=4
3 Θ000 , Θ001 , β¦ , Θ111 23=8
10 Θ00β¦0 , Θ00β¦1 ,β¦ , Θ11β¦1 210=1kβ¦
20 Θ00β¦0 , Θ00β¦1 ,β¦ , Θ11β¦1 220=1Mβ¦
30 Θ00β¦0 , Θ00β¦1 ,β¦ , Θ11β¦1 230=1Gβ¦
40 Θ00β¦0 , Θ00β¦1 ,β¦ , Θ11β¦1 240=1Tβ¦
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β’ A classical computer can do everything a quantum computer can do
β’ But to describe the status of a quantum computer you need 2N
complex numbers
β’ For a 100 qubit quantum computer you need ~ 1030 complex numbers!
β’ But for this we need to preserve the status of entanglement and this is possible only with reversible, norm preserving operations
β’ In QM these are called Unitary operations
Driving innovation in HEP computing
Quantum vs Classical
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β’ We have seen the NAND gate, which is non-reversible
β’ To implement a NAND on a Quantum Computer we must render this operation reversible, adding one qubit
β’ This is the so-called Toffoli gate
β’ Note that this is an isoentropictransformation
Driving innovation in HEP computing
Unitary operations
π, π, π β (π, π, πβ¨πβπ)
INPUT OUTPUT
a b c a b c
0 0 0 0 0 0
0 1 0 0 1 0
1 0 0 1 0 0
1 1 0 1 1 1
0 0 1 0 0 1
0 1 1 0 1 1
1 0 1 1 0 1
1 1 1 1 1 0
β
a b c
0 0 0
0 1 1
1 0 1
1 1 0
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β’ Let be π: Θ0 , Θ1 β 0,1 . Suppose we need to know whether it
is constant i.e. π Θ0 = π( Θ1 )
β’ Classically we should calculate it twice
⒠If we want to use a QC, since it might not be invertible⦠we
use a unitary transformation
Driving innovation in HEP computing
Quantum parallelism
Deutsch (1985) β Preskill example
ππ: Θπ₯ Θπ¦ = Θπ₯ Θπ¦β¨π(π₯)
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β’ If we prepare the second qubit in a superposition state, we
have
β’ And now letβs prepare the first qubit in the orthogonal state
Driving innovation in HEP computing
Quantum parallelism
Deutsch (1985) β Preskill example
ππ: Θπ₯1
2Θ0 β Θ1 = Θπ₯
1
2Θ0β¨π(π₯) β Θ1β¨π(π₯) =
Θπ₯ β1 π(π₯)1
2Θ0 β Θ1
ππ:1
2Θ0 + Θ1
1
2Θ0 β Θ1 =
Θπ₯1
2β1 π(0) Θ0 + β1 π(1) Θ1
1
2Θ0 β Θ1
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β’ Now we project the first qubit on the basis
β’ If the function is constant we have Β±Θ+ and if it is not Β±Θβ
β’ We have achieved our result with one calculation instead of two
Driving innovation in HEP computing
Quantum parallelism
Deutsch (1985) β Preskill example
ΘΒ± =1
2Θ0 Β± Θ1
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β’ Not gate X
β’ Generator of rotation around Y
β’ Generator of rotation around Z
β’ Hadamard gate
β’ Ο/8 gate
Driving innovation in HEP computing
Basic one-qubit gates
π =0 11 0
; X Θ0 = Θ1 ; X Θ1 = Θ0
Y=0 βππ 0
; y Θ0 = i Θ1 ; X Θ1 = -i Θ0
Z=1 00 β1
; Z Θ0 = Θ1 ; Z Θ1 = - Θ0
H= Ξ€(π + π) 2;
H Θ0 =1
2Θ0 + Θ1 ; H Θ1 =
1
2Θ0 β Θ1 ;
T=1 0
0 π ΰ΅ππ4
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β’ CNOT
β’ CNOT, H and T are a universal set
Driving innovation in HEP computing
C-not two qubit gate
INPUT OUTPUT
a b aβ bβ
0 0 0 0
0 1 0 1
1 0 1 1
1 1 1 0
CNOT=
1000
0100
0001
0010
;
CNOT Θ00 = Θ00CNOT Θ01 = Θ01CNOT Θ10 = Θ11CNOT Θ11 = Θ10
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β’ Start with a catβ¦
β’ The state Θπππ‘ is possible, in principle, but is rarely seen because it is extremely unstable.
Driving innovation in HEP computing
Errors and environment
The world is not perfect
Θπππ‘ =1
2Θππππ +
1
2Θππππ£π
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β’ Even if isolation from the environment is possible our
transformations may be faulty
β’ There is an extensive theory of (classical) error correcting
codes
β’ If the first bit flips, we can still use majority voting to determine
the right bit
Driving innovation in HEP computing
Errors errors errors
π = π0(1 + π π )
0 β 000 ; 1 β 111
000 β 100 ; 111 β 011
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β’ Of course more than one bit may flip, but if the
probability of a flip is π <1
2, the error probability is
π = 3π2 β 2π3 < π
β’ And in general with N correction bits
β’ Where πππππ =1
2β ν
Driving innovation in HEP computing
Errors errors errors
ππππππ~πβππ2
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β’ Flip & phase errors
β¦ phase errors are serious!
β’ Small errors
β’ Measurements collapse the status and are not reversible
β’ Cannot βcloneβ a state without errors
Driving innovation in HEP computing
Quantum errors β not so simple
ππππ: Θ0 β Θ1 ; Θ1 β Θ0 ; πβππ π: Θ0 β Θ0 ; Θ1 β βΘ1
πΘ0 + π Θ1 β (π + ν)Θ0 + (π β ν) Θ1
1
2Θ0 + Θ1 β
1
2Θ0 β Θ1
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β’ We encode each qubit as three qubits
β’ Measuring wonβt do, as we will simply βprepareβ the state and
break any possible entanglement
β’ But we can create a reversible diagnostic measure (a
syndrome)
Driving innovation in HEP computing
Do we give up?
Never!
Θ0 β ΰ΅ΏΰΈ«ΰ΄€0 = Θ000 ; Θ1 β ΰ΅ΏΰΈ«ΰ΄€1 = Θ111 ;
π = π2β¨π3, π1β¨π3
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β’ It is easy to see that if a bit flips, the syndrome is the binary
position of the bit to flip back
Driving innovation in HEP computing
The error syndrome
ΰ΅ Θ000 β Θ100Θ111 β Θ011
β π = 0,1 = 1
ΰ΅ Θ000 β Θ010Θ111 β Θ101
β π = 1,0 = 2
ΰ΅ Θ000 β Θ001Θ111 β Θ110
β π = 1,1 = 3
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β’ Flip & phase errors β we have shown how to fix flip
errors
β’ Measurements collapse the status and are not
reversible β we can do it in a reversible way
β’ Cannot βcloneβ a state without errors β no need to
clone
Driving innovation in HEP computing
What we have achieved
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β’ Small errors β we can have small errors instead of large bit flips
β’ But in this case for 1 β ν 2 of the cases we have π = 0 so nothing to do
β’ And in ν 2of the cases we have π β 0 and we know which bit to flip
β’ For the phase anomaly we would need 9 qubits, but this is for another lecture ;-)
Driving innovation in HEP computing
What we have achieved
Θ000 β Θ000 + ν Θ100Θ111 β Θ111 + ν Θ011
39Driving innovation in HEP computing
Quantum Algorithms
Driving innovation in HEP computing
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β’ Use interactions between quantum elements to simulate the continuous-time evolution governed by a given Hamiltonian.
β’ Same equations - same physics
β’ Direct implementation of SchrΓΆdinger's equation.
β’ Usually special purpose systems
Driving innovation in HEP computing
Two approaches to QoQ
Analog quantum simulations
42Driving innovation in HEP computing
Analog Quantum Simulation
One important example
Ultracold atoms in optical lattices to describe many-body physics & high-temperature superconductivity Hart et al., Nature 519:211 2015
β’ Study of quantum phase transitionsβ’ Quantum magnetismβ’ High-temperature superconductorsβ’ Quantum Hall effect β’ Address problems in quantum filed theory
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β’ Digital Quantum Simulation which can solve the Schrodingerequation using a discretized approximation of the time-evolution operator.
β’ Use efficient methods for constructing the system Hamiltonian and then decompose the time-evolution operator into a sequence of well-defined instructions
β’ These instructions are applied to the register in order to carry out a specific simulation sequence
β’ All this in a βgenericβ quantum computer
Driving innovation in HEP computing
Two approaches to QoQ
Digital Quantum Simulation
44Driving innovation in HEP computing
Recalling a bit of notation
Q
qubitquantum circuit
timemany qubits
n
45Driving innovation in HEP computing
Recall β the Hadamard gate
HΘ01
2Θ0 + Θ1
HΘ11
2Θ0 β Θ1
π» =1
2
1 11 β1
0 π
1 π
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β’ Remember the c-not gate?
Driving innovation in HEP computing
Recall -- C-not gate
1000
0100
0001
0010
00 π
01 π
10 π
11 πA C
B D A B C D
0 0 0 0
0 1 0 1
1 0 1 1
1 1 1 0
47Driving innovation in HEP computing
Producing entangled states
HΘ1
Θ1
1
2Θ0 β Θ1 Θ1
1
2Θ01 β Θ10
A c-not gate is a unitary operatorjust like the time evolution operator
π π‘ Θπ = πβππ» π‘
β Θπ
48Driving innovation in HEP computing
Controlled Unitary
Evolution
Θ0
Θπ U
1000
0100
00π11π21
00π12π22
00 π
01 π
10 π
11 π
n
H
1
2Θ0 + Θ1 Θπ
π π‘ Θπ = πβπππ‘ Θπ
1
2Θ0 Θπ + ΰ΅Ώπβπππ‘Θ1 Θπ
If Θπ is an eigenstate of π 1
2Θ0 + ΰ΅Ώπβπππ‘Θ1
Θπ
Eigenstate does not change but the control bit oscillates!
49Driving innovation in HEP computing
How to measure the phase?
H1
2Θ0 + ΰ΅Ώπβπππ‘Θ1 πβπππ‘/2 cos ππ‘/2 Θ0 + sin ππ‘/2 Θ1
β¦and voila, the phase is an amplitudeβ¦
β¦but we still need π(π‘)β¦
Θ0
Θπ U
n
H
Θπ
Θ0 H
Θ0 H
U2 U4
QFT-1 Digits of the energy}Courtesy of Peter Love, Department of Physics, Tufts University
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β’ For high-energy processes in small volumes of space- time, QCD can be solved by expansions
β’ Conversely, the only technique for solving QCD in the intermediate regime is Lattice QCD (LQCD), in which space-time is discretized on a grid and the theory is solved numerically
β’ But these calculations are affected by the βsign problemβ Which also affect the weights of path integral solutions!
β’ Real-time evolution of strongly interacting quarks and gluons cannot be determined with current computers and algorithms Fragmentation, QGP, matter in extreme conditions and the origin of the universe, star
structures, supernovae
Driving innovation in HEP computing
Getting serious about it
Simulating QCD processes
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β’ Quantum computer can naturally manipulate complex
amplitudes and thus does not suffer from sign or complex
weight problems
β’ New approaches such as the Tensor Networks representation
of the wave function in Lattice Gauge Theories and Quantum
Link Model formulation of LGT are particularly suited for
Quantum Computers
Driving innovation in HEP computing
Simulating QED
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β’ Dashed line is single meson moving through the lattice
β’ Colored lines are cuts of the entanglement entropy at different times
β’ A singlet state has been created between the two indistinguishable mesons
β’ The entropy has increased by one ebit because the information of the fate of the two mesons (bouncing back or continue traveling) is lost due to the superposition state
β’ This kind of calculations are particularly suited for digital or analog quantum computers
Driving innovation in HEP computing
One example
Entanglement entropy in the scattering of two mesons in the Schwinger model calculated using tensor networks.
T Pichler, et al. Phys. Rev. X., vol. 6, p. 011023, 2016.
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β’ Problem: distinguish signal from background
Driving innovation in HEP computing
QC and Higgs Analysis
Mott A et al. Nature 2017, 550:175
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β’ The D-Wave system
Driving innovation in HEP computing
Take 1 β Quantum Annealing
Quantum Circuit
Quantum Annealer
1098 qubitsOperates @ 15mKAnneals in 5-20Β΅s
D-Wave 2XTM
55Driving innovation in HEP computing
Take 1 β Quantum Annealing
How does it work
β’ Setup with trivial H0 and evolve to target Hp in the ground state
https://arxiv.org/abs/quant-ph/0001106 https://arxiv.org/abs/quant-ph/0104129
T.Caneva et al. PRA (2014)
π» π‘ = π΄ π‘ π»0 + π΅ π‘ π»π
56Driving innovation in HEP computing
D-Wave qubit connectivity
Not fully connected
π»πΌπ πππ =
π
βππππ§ +
ππ
π½πππππ§ππ
π§
External FieldInteractions
Ising Hamiltonian
But what if we do not have all connections?
57Driving innovation in HEP computing
D-Wave Chimera network
https://arxiv.org/abs/1210.8395
β’ Realize full Ising via spin chains by the Chimera graph
β’ Split local fields across all qubits in the chain β’ Tightly intra-chain coupling (JF up to 6) β’ Non-unique, heuristic embedding β’ Post-process to correct broken chainsβ’ Majority vote β’ Approximately 40 spins full Ising Model
58Driving innovation in HEP computing
Now letβs do thisβ¦!
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β’ hi(π) β [-1,1] are functions
of the variables such that
β’ P(S|hi>0) > P(B|hi>0)
β’ P(B|hi<0) > P(S|hi<0)
i.e.
β’ hi>0 probably Signal
β’ hi<0 probably Background
Driving innovation in HEP computing
Weak β Strong classifier
How to obtain a strong classifier
h1
h2
h3
hN
O
π π₯ =
π
π€πβπ π₯β¦
β
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β’ Since we have a MC, we can define a precise target
Driving innovation in HEP computing
The gory detailsβ¦
π¦ π₯ = α+1, ππ π₯ β πβ1, ππ π₯ β π΅
β’ So the error per event is
πΈπ = πΈ π₯π = π¦ π₯π β
π=1
π
π€πβπ π₯π
β’ And the total error is
πΏ π₯ =
π
πΈπ 2 = π¦π
2 +
π,π=1
π
πΆπππ€ππ€π β 2
π=1
π
πΆπ¦ππ€π πΆππ =
π
βπ π₯π βπ π₯π πΆπ¦π =
π
βπ π₯π π¦π
πΏβ² π₯ =
π,π=1
π
πΆπππ€ππ€π + 2
π=1
π
π β πΆπ¦π π€πβ½ + sparsity penalty (Ξ», Hamming weight)β constant ( π¦π
2)
61Driving innovation in HEP computing
So here we are!
πΏβ² π₯ =
π=1
π
2 π β πΆπ¦π π€π +
π,π=1
π
πΆπππ€ππ€π β½ π»πΌπ πππ =
π
βππππ§ +
ππ
π½πππππ§ππ
π§
Training on 20k events
DNN & XGBβ’ Classical ML
DW & XGBβ’ D-Wave annealing
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β’ XGBoost (XGB)
Extremely efficient library for training decision trees (http://xgboost.readthedocs.io)
Discovered during the higgs-ml challenge (https://www.kaggle.com/c/higgs-boson)
Moderately optimize the hyper-parameters
β’ Deep Neural Network (DNN)
Simple fully connected model 2 layers 1000 nodes
https://keras.io/ http://deeplearning.net/software/theano/
Moderately optimize the hyper-parameters
Driving innovation in HEP computing
For reference
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β’ Same problem β different take
β’ Analysis done with Support Vector Machine
β’ Separate two sets of points with the widest possible margin
β’ The decision function is fully specified by a (usually very small) subset of training samples, the support vectors.
β’ The solution is fully specified by a (usually small) subset of training samples, the support vectors.
β’ If there is an hyperplane that divides the points it is a simple quadratic optimization
Driving innovation in HEP computing
Take 2 β Quantum Circuits
The IBM Q-machine
Support Vectors
Maximize marginSupport Vectors: vectors that βsupportβ the dividing planes
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β’ Input: set of training pair samples with a result function π¦ π₯π ββ1,1 ;
β’ Output: set of π€π whose linear combination predicts the value of
π¦ π₯π
β’ Important difference: optimization has two objectives: maximize
the margin (βstreet widthβ) and reduce the number of weights to
the (usually few) support vectors
Driving innovation in HEP computing
Almost a DNN
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β’ Distance from support point to centerline
Driving innovation in HEP computing
One word on how SVM works
H0
H1
H2
π€ Τ¦π₯ + π = +1
π€ Τ¦π₯ + π = β1
π€ Τ¦π₯ + π = 0
π₯0, π¦0 π+
πβ
π = Ξ€π€ Τ¦π₯ + π π€ = Ξ€1 π€
β’ We have to minimize π€ and
impose no points βin betweenβ π¦π π€ Τ¦π₯π + π β₯ 1
β’ Well defined quadratic minimization problem with linear constraint solved with Lagrangian multipliers
minπ€ ,π
β Τ¦π₯, Τ¦π = minπ€ ,π
ΰ΅1 2 π€ β
π
ππ π¦π π€ Τ¦π₯π + π β 1
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β’ What about this?
Driving innovation in HEP computing
This is great butβ¦
π₯πβ² = π₯π
π¦πβ² = π₯π
2 + π¦π2
β½
β’ With the bonus of the Kernel Trick
We do not need π₯β² = Ξ¦ Τ¦π₯ but just K π₯π , π₯π = Ξ¦ π₯π β Ξ¦ π₯π !
67
β’ Step 0: Build a classifier like before
β’ Step 1: Feature-map the data to a much larger dimensional
space
β’ Step 2: Train a the weights
β’ Step 3: Apply Quantum Classification
Driving innovation in HEP computing
Now on Quantum
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β’ Feature-map to a high-dimensional space (with entanglement)
Driving innovation in HEP computing
Step 1
Single qubit mapping with
phase gate πΞ¦ π₯ =1 00 πππ₯
π°Ξ¦ Τ¦π₯ = πΞ¦ Τ¦π₯ π»β¨ππΞ¦ Τ¦π₯ π»
β¨π
πΞ¦ Τ¦π₯ = exp π
πβ π
ππ Τ¦π₯ ΰ·
πβπ
ππ
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β’ Define the training network as a short-depth quantum circuit
made of layers of single-qubit unitaries and entangling gates
Driving innovation in HEP computing
Step 2a
π Τ¦π = πππππππ ππππ‘β―ππππ
2π2 ππππ‘ππππ
1π1
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β’ Apply a binary measurement ππ¦ to get the classifier and
measure the probability of the foreseen outcome
Driving innovation in HEP computing
Step 2b
ππ¦ Τ¦π₯ = Ξ¦ Τ¦π₯ πβ Τ¦π ππ¦π Τ¦π Ξ¦ Τ¦π₯
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β’ Train the network
β’ Obtain the empirical distribution ΰ·ππ¦
β’ Assign label π Τ¦π₯ = π¦ πππ ΰ·ππ¦ Τ¦π₯ > ΰ·πβπ¦ Τ¦π₯ β π¦π
β’ Use cost π πππ =1
πΟ Τ¦π₯βπππ π Τ¦π₯ β π Τ¦π₯ on training set
β’ Optimize for Τ¦π, π
Driving innovation in HEP computing
Step 3
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β’ Results
Driving innovation in HEP computing
Apply to data -- simulator
ttH(H->πΈπΈ)
accuracy 200 800 3200
QSVM 0.775 0.798 0.774
BDT 0.810 0.796 0.781
ttH(H->πΈπΈ)
auc 200 800 3200
QSVM 0.849 0.834 0.826
BDT 0.880 0.867 0.869
73Driving innovation in HEP computing
Apply to data -- simulator
ROC = Receiver Operating Characteristic
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β’ Accuracy and AUC with different number of
iterations.
β’ QSVM accuracy increases with iterations
β’ QSVM AUC increases rapidly with iterations
β’ We plan to run the test with many more
iterations if possible
Driving innovation in HEP computing
Apply to data -- hardware
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β’ Origin of Matter
β’ Unification of forces
β’ Black hole formation
Driving innovation in HEP computing
Six simple pieces β 1
DUNE experiment
β’ Supervised Quantum Learning to
reconstruct neutrino interactions
with a Quantum Computer
β’ Unsupervised learning to analyze the simulated and real event structures
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β’ Simulate refugee camp tents with GAN
β’ Train DNN to count simulated tents
Driving innovation in HEP computing
Six simple pieces β 2
Refugee camp evaluation
β’ Generate a reality to train upon!
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β’ Simulate any kind of calorimeter
β’ Hyper-parameter scan & DNN
training fast & in one go
Driving innovation in HEP computing
Six simple pieces β 3
Super-fast training
β’ Classify the calorimeter via a DNN & chose the closest DNN for the job
β’ Keep a set of βwarmβ weights ready
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β’ ALICE Grid 70 computing centres in 40 countries
150,000 CPU cores and 120 PB of storage
~140.000 jobs running 24 x 7 x 365
Driving innovation in HEP computing
Six simple pieces β 4
Optimize Grid workflow
β’ Optimize storage location and job workflow
β’ Use Quantum Computing algorithms to find best distribution in a dynamic environment
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β’ Track candidates are identified via combinatorial search
β’ And then βfollowedβ via Kalmnanfilters
β’ The track is no better than its seeds!
Driving innovation in HEP computing
Six simple pieces β 5
Track reconstruction in dense environments
β’ Use Quantum Computing to
speed up combinatorial searches
β’ And Genetic Algorithms to quickly
optimize the search
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β’ Use extensive HIS from SNUBH (Korea)
β’ Analyze with DL
β’ Explore supervised and non supervised
learning
Driving innovation in HEP computing
Six simple pieces β 6
Explore wide-range medical data
β’ Relate DNN classification to
existing diagnoses
β’ Explore symptoms correlations
β’ Include medical imagery via
DNN classification
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CERN openlab is a unique public β private infrastructure fostering collaboration between research and ICT industry
We have presented two specific fields of investigation that have a high relevance both for fundamental research and for society at large
While still not a ready for prime-time production, Quantum Computing holds the promise to herald a revolution in ICT
CERN openlab intends to investigate the opportunities offered by these and other advanced ICT fields, fostering collaborations between scientists and industry
Driving innovation in HEP computing
Conclusions