introduction to quantum computing - cern

1

Introduction to Quantum

Computing

Summer Student Lecture

Tuesday, July 9th, 2019

Federico Carminati

2Driving innovation in HEP computing


The Higgs Boson

Elementary particles0,2%

Atoms, stars, diffused gas4%

Exotic dark matter(neutrinos, neutralinos,…) 30%

Dark energy(Vacuum energy,…) 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

5

From ridiculously difficult…

...to almost impossibleDriving innovation in HEP computing


Worldwide LHC Computing Grid

Tier-0 (CERN):

•Data recording

•Initial data reconstruction

•Data distribution

Tier-1 (14 centres):

•Permanent storage

•Re-processing

•Analysis

Tier-2 (72

Federations, ~149

centres):

• Simulation

• End-user analysis

•760,000 cores

•700 PB


Worldwide LHC Computing Grid

Tier-0 (CERN):

•Data recording

•Initial data reconstruction

•Data distribution

Tier-1 (14 centres):

•Permanent storage

•Re-processing

•Analysis

Tier-2 (72 Federations, ~149 centres):

• Simulation• End-user analysis

•760,000 cores•700 PB

7

• Raw data volume increases exponentially Processing and analysis load

• 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

8

29 December 1959, Pasadena, California

• Why cannot we write the entire 24 volumes of the Encyclopedia Britannica on the head of a pin?

• The principles of physics, as far as I can see, do not speak against the possibility of maneuvering things atom by atom

• Miniaturizing the computer... the wires should be 10 or 100 atoms in diameter, and the circuits should be a few thousand angstroms across


There is plenty of room

at the bottom


Quantum Computing?

"Nature is quantum, goddamn it! So if we want to simulate it, we need a quantum computer.”R.Feynman, 1981, Endicott House, MIT

`Bloch’s sphere

Use qubits instead of bits…e.g. bits that exhibit quantum

behavior

10

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 “hardness” of a problem was intrinsic to it

Quantum Computing is now challenging this


Quantum Computing in

perspective

We could argue that Quantum Computingis a natural consequence of Moore’s law

11

Can we control complex quantum systems and if we can, so what? (J.Preskill, 2012)

• Can quantum computers outperform classical computers on all algorithms?

• Can quantum computers do things that cannot be done by classical computers (quantum supremacy)?

• The golden apple is “superpolynomial speedup” Reducing to polynomial time what in classic computing is exponential or more

Theoretically achieved with some algorithms

• But polynomial speedup can be very appealing Particularly for large problems

• For the moment it is debatable whether Quantum Supremacy has been demonstratedor not


Quantum supremacy

…or is the gain worth the pain?

Tim

e

Problem size

12

EU Quantum Flagship – large-scale initiative

funded at the 1b € 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

President’s Budget Request: $27.5M


… and money is flowing in…

What people

laugh aboutWhat matters

at the end

13

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)


Just for the skeptical

14

• QC can be used to solve directly Quantum Many Body and Quantum Field Theory problems

• In chemistry we already have variational calculations of atomic orbital configurations Complex molecules are the ”killing app” here

• Similarly for Nuclear Physics the challenge will be to describe nucleiand their scattering and interactions

• This is well beyond exascale computingand current theoretical understanding

Quantum Computing for Theoretical Particle Physics

Quantum on Quantum

15

• Classical computing is based on discrete elements (transistors)

taking the value of either 0 or 1

• Operations are performed by gates i.e. electronic circuits that

act as operators

• This is a NAND gate

• This gate is “irreversible” and it

increases world entropy by

𝛥S=k ln 2 (~3 10-21 joules@70°C)


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

16

• Quantum Computers use quantum objects to store information

(see later for practical implementation)

• The simplest representation is the Bloch Sphere

• Or more generally


Principles of Quantum

Computing

ൿȁ𝜓 = cos(𝜃/2)ȁ0 + ൿ𝑒−𝑖𝜙sin(𝜃/2)ȁ1

ൿȁ𝜓 = αȁ0 + βȁ1

α 2 + β 2=1

17

• What if we have 2 qubits?

• And in general


Principles of quantum

computing

ȁ𝜓 = 𝑎1ȁ0 + 𝑏1ȁ1 𝑎2ȁ0 + 𝑏2ȁ1 =

𝛼 ȁ0 ȁ0 + 𝛽 ȁ0 ȁ1 + 𝛾 ȁ1 ȁ0 + 𝛿 ȁ1 ȁ1 =

𝛼 ȁ00 + 𝛽 ȁ01 + 𝛾 ȁ10 + 𝛿 ȁ11

ȁ𝜓 =

0

2𝑁−1

𝑎𝑥ȁ𝑥

ȁ𝑥 = ȁ10010010… .111101001

ȁ𝜓 = 𝛼

1000

+𝛽

0100

+ 𝛾

0010

+ 𝛿

0001

18

• In general, a state may be such that

• Take for instance the Bell states

• If I measure the first qubit as 0 in a separable state I have


Separable and entangled states

ȁ𝜓 = 𝛼 ȁ00 + 𝛽 ȁ01 + 𝛾 ȁ10 + 𝛿 ȁ11 ≠ 𝑎1ȁ0 + 𝑏1ȁ1 𝑎2ȁ0 + 𝑏2ȁ1

ൿห𝜓1 =1

2ȁ00 + ȁ11 ; ൿห𝜓2 =

1

2ȁ00 − ȁ11

ȁ𝜓 = 𝛼 ȁ00 + 𝛽 ȁ01 + 𝛾 ȁ10 + 𝛿 ȁ11measure 1

𝑎 2 + 𝛽 2𝛼 ȁ0 + 𝛽 ȁ1

= k 𝑎2ȁ0 + 𝑏2ȁ1

19

• But if I do the same in the Bell states, I get

• The measure of one qubit entirely determines the measure of

the other

• This is the basic concept of quantum entanglement


Separable and entangled states

ൿห𝜓1 =1

2ȁ00 + ȁ11 ; ൿห𝜓2 =

1

2ȁ00 − ȁ11

measure 01ȁ0

20

EPR state

• Bob knows what Alice measured (or will measure!) At the precise moment of its measurement

• The effect of the act of measurement is not local • It is again in contradiction with our intuition

• But the theory is consistent (it is not in contradiction with itself)

• 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


21

Small reminder

Diachronic

Oldcausa efficiens

Newcausa finalis

Everything

causa formalis

Constituentsca

usa

mat

eria

lis

event

Determinism Fatalism, vitalism,

idealism, religion

Holism

Riductionism


22

• We need 8 coefficients

• Which may not be reducible to 6!

• In this case the state is “entangled”, we do need all the 8

coefficients


And with 3?

The trouble is that 2+2 = 22

ȁ𝜓 = 𝑎1ȁ0 + 𝑏1ȁ1 𝑎2ȁ0 + 𝑏2ȁ1 𝑎3ȁ0 + 𝑏3ȁ1 =

𝛼0 ȁ000 +𝛼1 ȁ001 +𝛼2 ȁ010 +𝛼3 ȁ011 + 𝛼4 ȁ100 +𝛼5 ȁ101 +𝛼6 ȁ110 +𝛼7 ȁ111

23


The power of Quantum

Computing

1 ȁ0 , ȁ1 21=2

2 ȁ00 , ȁ01 , ȁ00 , ȁ01 22=4

3 ȁ000 , ȁ001 , … , ȁ111 23=8

10 ȁ00…0 , ȁ00…1 ,… , ȁ11…1 210=1k…

20 ȁ00…0 , ȁ00…1 ,… , ȁ11…1 220=1M…

30 ȁ00…0 , ȁ00…1 ,… , ȁ11…1 230=1G…

40 ȁ00…0 , ȁ00…1 ,… , ȁ11…1 240=1T…

24

• A classical computer can do everything a quantum computer can do

• But to describe the status of a quantum computer you need 2N

complex numbers

• For a 100 qubit quantum computer you need ~ 1030 complex numbers!

• But for this we need to preserve the status of entanglement and this is possible only with reversible, norm preserving operations

• In QM these are called Unitary operations


Quantum vs Classical

25

• We have seen the NAND gate, which is non-reversible

• To implement a NAND on a Quantum Computer we must render this operation reversible, adding one qubit

• This is the so-called Toffoli gate

• Note that this is an isoentropictransformation


Unitary operations

𝑎, 𝑏, 𝑐 → (𝑎, 𝑏, 𝑐⨁𝑎⋀𝑏)

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

⊕

a b c

0 0 0

0 1 1

1 0 1

1 1 0

26

• Let be 𝑓: ȁ0 , ȁ1 → 0,1 . Suppose we need to know whether it

is constant i.e. 𝑓 ȁ0 = 𝑓( ȁ1 )

• Classically we should calculate it twice

• If we want to use a QC, since it might not be invertible… we

use a unitary transformation


Quantum parallelism

Deutsch (1985) – Preskill example

𝑈𝑓: ȁ𝑥 ȁ𝑦 = ȁ𝑥 ȁ𝑦⨁𝑓(𝑥)

27

• If we prepare the second qubit in a superposition state, we

have

• And now let’s prepare the first qubit in the orthogonal state


Quantum parallelism


𝑈𝑓: ȁ𝑥1

2ȁ0 − ȁ1 = ȁ𝑥

1

2ȁ0⨁𝑓(𝑥) − ȁ1⨁𝑓(𝑥) =

ȁ𝑥 −1 𝑓(𝑥)1

2ȁ0 − ȁ1

𝑈𝑓:1

2ȁ0 + ȁ1

1

2ȁ0 − ȁ1 =

ȁ𝑥1

2−1 𝑓(0) ȁ0 + −1 𝑓(1) ȁ1

1

2ȁ0 − ȁ1

28

• Now we project the first qubit on the basis

• If the function is constant we have ±ȁ+ and if it is not ±ȁ−

• We have achieved our result with one calculation instead of two


Quantum parallelism


ȁ± =1

2ȁ0 ± ȁ1

29

• Not gate X

• Generator of rotation around Y

• Generator of rotation around Z

• Hadamard gate

• π/8 gate


Basic one-qubit gates

𝑋 =0 11 0

; X ȁ0 = ȁ1 ; X ȁ1 = ȁ0

Y=0 −𝑖𝑖 0

; y ȁ0 = i ȁ1 ; X ȁ1 = -i ȁ0

Z=1 00 −1

; Z ȁ0 = ȁ1 ; Z ȁ1 = - ȁ0

H= Τ(𝑍 + 𝑋) 2;

H ȁ0 =1

2ȁ0 + ȁ1 ; H ȁ1 =

1

2ȁ0 − ȁ1 ;

T=1 0

0 𝑒 ൗ𝑖𝜋4

30

• CNOT

• CNOT, H and T are a universal set


C-not two qubit gate

INPUT OUTPUT

a b a’ b’

0 0 0 0

0 1 0 1

1 0 1 1

1 1 1 0

CNOT=

1000

0100

0001

0010

;

CNOT ȁ00 = ȁ00CNOT ȁ01 = ȁ01CNOT ȁ10 = ȁ11CNOT ȁ11 = ȁ10

31

• Start with a cat…

• The state ȁ𝑐𝑎𝑡 is possible, in principle, but is rarely seen because it is extremely unstable.


Errors and environment

The world is not perfect

ȁ𝑐𝑎𝑡 =1

2ȁ𝑑𝑒𝑎𝑑 +

1

2ȁ𝑎𝑙𝑖𝑣𝑒

32

• Even if isolation from the environment is possible our

transformations may be faulty

• There is an extensive theory of (classical) error correcting

codes

• If the first bit flips, we can still use majority voting to determine

the right bit


Errors errors errors

𝑈 = 𝑈0(1 + 𝑂 𝜖 )

0 → 000 ; 1 → 111

000 → 100 ; 111 → 011

33

• Of course more than one bit may flip, but if the

probability of a flip is 𝑝 <1

2, the error probability is

𝑃 = 3𝑝2 − 2𝑝3 < 𝑝

• And in general with N correction bits

• Where 𝑝𝑓𝑙𝑖𝑝 =1

2− 휀


Errors errors errors

𝑃𝑒𝑟𝑟𝑜𝑟~𝑒−𝑁𝜀2

34

• Flip & phase errors

… phase errors are serious!

• Small errors

• Measurements collapse the status and are not reversible

• Cannot “clone” a state without errors


Quantum errors – not so simple

𝑓𝑙𝑖𝑝: ȁ0 → ȁ1 ; ȁ1 → ȁ0 ; 𝑝ℎ𝑎𝑠𝑒: ȁ0 → ȁ0 ; ȁ1 → −ȁ1

𝑎ȁ0 + 𝑏 ȁ1 → (𝑎 + 휀)ȁ0 + (𝑏 − 휀) ȁ1

1

2ȁ0 + ȁ1 →

1

2ȁ0 − ȁ1

35

• We encode each qubit as three qubits

• Measuring won’t do, as we will simply “prepare” the state and

break any possible entanglement

• But we can create a reversible diagnostic measure (a

syndrome)


Do we give up?

Never!

ȁ0 → ൿหത0 = ȁ000 ; ȁ1 → ൿหത1 = ȁ111 ;

𝑆 = 𝑏2⨁𝑏3, 𝑏1⨁𝑏3

36

• It is easy to see that if a bit flips, the syndrome is the binary

position of the bit to flip back


The error syndrome

ൠȁ000 → ȁ100ȁ111 → ȁ011

→ 𝑆 = 0,1 = 1

ൠȁ000 → ȁ010ȁ111 → ȁ101

→ 𝑆 = 1,0 = 2

ൠȁ000 → ȁ001ȁ111 → ȁ110

→ 𝑆 = 1,1 = 3

37

• Flip & phase errors – we have shown how to fix flip

errors

• Measurements collapse the status and are not

reversible – we can do it in a reversible way

• Cannot “clone” a state without errors – no need to

clone


What we have achieved

38

• Small errors – we can have small errors instead of large bit flips

• But in this case for 1 − 휀 2 of the cases we have 𝑆 = 0 so nothing to do

• And in 휀 2of the cases we have 𝑆 ≠ 0 and we know which bit to flip

• For the phase anomaly we would need 9 qubits, but this is for another lecture ;-)


What we have achieved

ȁ000 → ȁ000 + 휀 ȁ100ȁ111 → ȁ111 + 휀 ȁ011


Quantum Algorithms

41

• Use interactions between quantum elements to simulate the continuous-time evolution governed by a given Hamiltonian.

• Same equations - same physics

• Direct implementation of Schrödinger's equation.

• Usually special purpose systems


Two approaches to QoQ

Analog quantum simulations


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

• Study of quantum phase transitions• Quantum magnetism• High-temperature superconductors• Quantum Hall effect • Address problems in quantum filed theory

43

• Digital Quantum Simulation which can solve the Schrodingerequation using a discretized approximation of the time-evolution operator.

• Use efficient methods for constructing the system Hamiltonian and then decompose the time-evolution operator into a sequence of well-defined instructions

• These instructions are applied to the register in order to carry out a specific simulation sequence

• All this in a “generic” quantum computer


Two approaches to QoQ

Digital Quantum Simulation


Recalling a bit of notation

Q

qubitquantum circuit

timemany qubits

n


Recall – the Hadamard gate

Hȁ01

2ȁ0 + ȁ1

Hȁ11

2ȁ0 − ȁ1

𝐻 =1

2

1 11 −1

0 𝜓

1 𝜓

46

• Remember the c-not gate?


Recall -- C-not gate

1000

0100

0001

0010

00 𝜓

01 𝜓

10 𝜓

11 𝜓A C

B D A B C D

0 0 0 0

0 1 0 1

1 0 1 1

1 1 1 0


Producing entangled states

Hȁ1

ȁ1

1

2ȁ0 − ȁ1 ȁ1

1

2ȁ01 − ȁ10

A c-not gate is a unitary operatorjust like the time evolution operator

𝑈 𝑡 ȁ𝜓 = 𝑒−𝑖𝐻 𝑡

ℏ ȁ𝜓


Controlled Unitary

Evolution

ȁ0

ȁ𝜓 U

1000

0100

00𝑈11𝑈21

00𝑈12𝑈22

00 𝜓

01 𝜓

10 𝜓

11 𝜓

n

H

1

2ȁ0 + ȁ1 ȁ𝜓

𝑈 𝑡 ȁ𝜓 = 𝑒−𝑖𝜔𝑡 ȁ𝜓

1

2ȁ0 ȁ𝜓 + ൿ𝑒−𝑖𝜔𝑡ȁ1 ȁ𝜓

If ȁ𝜓 is an eigenstate of 𝑈 1

2ȁ0 + ൿ𝑒−𝑖𝜔𝑡ȁ1

ȁ𝜓

Eigenstate does not change but the control bit oscillates!


How to measure the phase?

H1

2ȁ0 + ൿ𝑒−𝑖𝜔𝑡ȁ1 𝑒−𝑖𝜔𝑡/2 cos 𝜔𝑡/2 ȁ0 + sin 𝜔𝑡/2 ȁ1

…and voila, the phase is an amplitude…

…but we still need 𝑈(𝑡)…

ȁ0

ȁ𝜓 U

n

H

ȁ𝜓

ȁ0 H

ȁ0 H

U2 U4

QFT-1 Digits of the energy}Courtesy of Peter Love, Department of Physics, Tufts University

50

• For high-energy processes in small volumes of space- time, QCD can be solved by expansions

• 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

• But these calculations are affected by the “sign problem” Which also affect the weights of path integral solutions!

• 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


Getting serious about it

Simulating QCD processes

51

• Quantum computer can naturally manipulate complex

amplitudes and thus does not suffer from sign or complex

weight problems

• 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


Simulating QED

52

• Dashed line is single meson moving through the lattice

• Colored lines are cuts of the entanglement entropy at different times

• A singlet state has been created between the two indistinguishable mesons

• 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

• This kind of calculations are particularly suited for digital or analog quantum computers


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.

53

• Problem: distinguish signal from background


QC and Higgs Analysis

Mott A et al. Nature 2017, 550:175

54

• The D-Wave system


Take 1 – Quantum Annealing

Quantum Circuit

Quantum Annealer

1098 qubitsOperates @ 15mKAnneals in 5-20µs

D-Wave 2XTM


Take 1 – Quantum Annealing

How does it work

• 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)

𝐻 𝑡 = 𝐴 𝑡 𝐻0 + 𝐵 𝑡 𝐻𝑝


D-Wave qubit connectivity

Not fully connected

𝐻𝐼𝑠𝑖𝑛𝑔 =

𝑖

ℎ𝑖𝜎𝑖𝑧 +

𝑖𝑗

𝐽𝑖𝑗𝜎𝑖𝑧𝜎𝑗

𝑧

External FieldInteractions

Ising Hamiltonian

But what if we do not have all connections?


D-Wave Chimera network

https://arxiv.org/abs/1210.8395

• Realize full Ising via spin chains by the Chimera graph

• Split local fields across all qubits in the chain • Tightly intra-chain coupling (JF up to 6) • Non-unique, heuristic embedding • Post-process to correct broken chains• Majority vote • Approximately 40 spins full Ising Model


Now let’s do this…!

59

• hi(𝒙) ∈ [-1,1] are functions

of the variables such that

• P(S|hi>0) > P(B|hi>0)

• P(B|hi<0) > P(S|hi<0)

i.e.

• hi>0 probably Signal

• hi<0 probably Background


Weak ➙ Strong classifier

How to obtain a strong classifier

h1

h2

h3

hN

O

𝑂 𝑥 =

𝑖

𝑤𝑖ℎ𝑖 𝑥…

➠

60

• Since we have a MC, we can define a precise target


The gory details…

𝑦 𝑥 = ቊ+1, 𝑖𝑓 𝑥 ∈ 𝑆−1, 𝑖𝑓 𝑥 ∈ 𝐵

• So the error per event is

𝐸𝑠 = 𝐸 𝑥𝑠 = 𝑦 𝑥𝑠 −

𝑖=1

𝑁

𝑤𝑖ℎ𝑖 𝑥𝑠

• And the total error is

𝛿 𝑥 =

𝑠

𝐸𝑠2 = 𝑦𝑠

2 +

𝑖,𝑗=1

𝑁

𝐶𝑖𝑗𝑤𝑖𝑤𝑗 − 2

𝑖=1

𝑛

𝐶𝑦𝑖𝑤𝑖 𝐶𝑖𝑗 =

𝑠

ℎ𝑖 𝑥𝑠 ℎ𝑗 𝑥𝑠 𝐶𝑦𝑗 =

𝑠

ℎ𝑖 𝑥𝑠 𝑦𝑠

𝛿′ 𝑥 =

𝑖,𝑗=1

𝑁

𝐶𝑖𝑗𝑤𝑖𝑤𝑗 + 2

𝑖=1

𝑁

𝜆 − 𝐶𝑦𝑖 𝑤𝑖➽ + sparsity penalty (λ, Hamming weight)– constant ( 𝑦𝑠

2)


So here we are!

𝛿′ 𝑥 =

𝑖=1

𝑁

2 𝜆 − 𝐶𝑦𝑖 𝑤𝑖 +

𝑖,𝑗=1

𝑁

𝐶𝑖𝑗𝑤𝑖𝑤𝑗 ➽ 𝐻𝐼𝑠𝑖𝑛𝑔 =

𝑖

ℎ𝑖𝜎𝑖𝑧 +

𝑖𝑗

𝐽𝑖𝑗𝜎𝑖𝑧𝜎𝑗

𝑧

Training on 20k events

DNN & XGB• Classical ML

DW & XGB• D-Wave annealing

62

• 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

• 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


For reference

http://xgboost.readthedocs.io/

https://www.kaggle.com/c/higgs-boson

http://deeplearning.net/software/theano/

63

• Same problem – different take

• Analysis done with Support Vector Machine

• Separate two sets of points with the widest possible margin

• The decision function is fully specified by a (usually very small) subset of training samples, the support vectors.

• The solution is fully specified by a (usually small) subset of training samples, the support vectors.

• If there is an hyperplane that divides the points it is a simple quadratic optimization


Take 2 – Quantum Circuits

The IBM Q-machine

Support Vectors

Maximize marginSupport Vectors: vectors that “support” the dividing planes

64

• Input: set of training pair samples with a result function 𝑦 𝑥𝑖 ∈−1,1 ;

• Output: set of 𝑤𝑖 whose linear combination predicts the value of

𝑦 𝑥𝑖

• Important difference: optimization has two objectives: maximize

the margin (“street width”) and reduce the number of weights to

the (usually few) support vectors


Almost a DNN

65

• Distance from support point to centerline


One word on how SVM works

H0

H1

H2

𝑤 Ԧ𝑥 + 𝑏 = +1

𝑤 Ԧ𝑥 + 𝑏 = −1

𝑤 Ԧ𝑥 + 𝑏 = 0

𝑥0, 𝑦0 𝑑+

𝑑−

𝑑 = Τ𝑤 Ԧ𝑥 + 𝑏 𝑤 = Τ1 𝑤

• We have to minimize 𝑤 and

impose no points “in between” 𝑦𝑖 𝑤 Ԧ𝑥𝑖 + 𝑏 ≥ 1

• Well defined quadratic minimization problem with linear constraint solved with Lagrangian multipliers

min𝑤 ,𝑏

ℒ Ԧ𝑥, Ԧ𝑎 = min𝑤 ,𝑏

൘1 2 𝑤 −

𝑖

𝑎𝑖 𝑦𝑖 𝑤 Ԧ𝑥𝑖 + 𝑏 − 1

66

• What about this?


This is great but…

𝑥𝑖′ = 𝑥𝑖

𝑦𝑖′ = 𝑥𝑖

2 + 𝑦𝑖2

➽

• With the bonus of the Kernel Trick

We do not need 𝑥′ = Φ Ԧ𝑥 but just K 𝑥𝑖 , 𝑥𝑗 = Φ 𝑥𝑖 ⋅ Φ 𝑥𝑗 !

67

• Step 0: Build a classifier like before

• Step 1: Feature-map the data to a much larger dimensional

space

• Step 2: Train a the weights

• Step 3: Apply Quantum Classification


Now on Quantum

68

• Feature-map to a high-dimensional space (with entanglement)


Step 1

Single qubit mapping with

phase gate 𝑈Φ 𝑥 =1 00 𝑒𝑖𝑥

𝒰Φ Ԧ𝑥 = 𝑈Φ Ԧ𝑥 𝐻⨂𝑛𝑈Φ Ԧ𝑥 𝐻

⨂𝑛

𝑈Φ Ԧ𝑥 = exp 𝑖

𝑆⊆ 𝑛

𝜙𝑠 Ԧ𝑥 ෑ

𝑖∈𝑆

𝑍𝑖

69

• Define the training network as a short-depth quantum circuit

made of layers of single-qubit unitaries and entangling gates


Step 2a

𝑊 Ԧ𝜃 = 𝑈𝑙𝑜𝑐𝑙𝜃𝑙 𝑈𝑒𝑛𝑡⋯𝑈𝑙𝑜𝑐

2𝜃2 𝑈𝑒𝑛𝑡𝑈𝑙𝑜𝑐

1𝜃1

70

• Apply a binary measurement 𝑀𝑦 to get the classifier and

measure the probability of the foreseen outcome


Step 2b

𝑝𝑦 Ԧ𝑥 = Φ Ԧ𝑥 𝑊† Ԧ𝜃 𝑀𝑦𝑊 Ԧ𝜃 Φ Ԧ𝑥

71

• Train the network

• Obtain the empirical distribution ො𝑝𝑦

• Assign label 𝑚 Ԧ𝑥 = 𝑦 𝑖𝑓𝑓 ො𝑝𝑦 Ԧ𝑥 > ො𝑝−𝑦 Ԧ𝑥 − 𝑦𝑏

• Use cost 𝑅𝑒𝑚𝑝 =1

𝑇σ Ԧ𝑥∈𝑇𝑃𝑟 𝑚 Ԧ𝑥 ≠ 𝑚 Ԧ𝑥 on training set

• Optimize for Ԧ𝜃, 𝑏


Step 3

72

• Results


Apply to data -- simulator

ttH(H->𝜸𝜸)

accuracy 200 800 3200

QSVM 0.775 0.798 0.774

BDT 0.810 0.796 0.781

ttH(H->𝜸𝜸)

auc 200 800 3200

QSVM 0.849 0.834 0.826

BDT 0.880 0.867 0.869


Apply to data -- simulator

ROC = Receiver Operating Characteristic

74

• Accuracy and AUC with different number of

iterations.

• QSVM accuracy increases with iterations

• QSVM AUC increases rapidly with iterations

• We plan to run the test with many more

iterations if possible


Apply to data -- hardware

75

• Origin of Matter

• Unification of forces

• Black hole formation


Six simple pieces – 1

DUNE experiment

• Supervised Quantum Learning to

reconstruct neutrino interactions

with a Quantum Computer

• Unsupervised learning to analyze the simulated and real event structures

76

• Simulate refugee camp tents with GAN

• Train DNN to count simulated tents



Refugee camp evaluation

• Generate a reality to train upon!

77

• Simulate any kind of calorimeter

• Hyper-parameter scan & DNN

training fast & in one go



Super-fast training

• Classify the calorimeter via a DNN & chose the closest DNN for the job

• Keep a set of “warm” weights ready

78

• 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



Optimize Grid workflow

• Optimize storage location and job workflow

• Use Quantum Computing algorithms to find best distribution in a dynamic environment

79

• Track candidates are identified via combinatorial search

• And then “followed” via Kalmnanfilters

• The track is no better than its seeds!



Track reconstruction in dense environments

• Use Quantum Computing to

speed up combinatorial searches

• And Genetic Algorithms to quickly

optimize the search

80

• Use extensive HIS from SNUBH (Korea)

• Analyze with DL

• Explore supervised and non supervised

learning



Explore wide-range medical data

• Relate DNN classification to

existing diagnoses

• Explore symptoms correlations

• Include medical imagery via

DNN classification

81

CERN openlab is a unique public – 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


Conclusions

82

Thanks for your attention!

[email protected]


introduction to quantum computing - cern

Documents