An Introduction to Quantum Computing: From Basic Concepts ...Concepts to Hardware 1. Introduction 2. Basic principles of quantum computing 3. Quantum algorithms 4. Hardware implementations

Shuntaro Takeda1The University of Tokyo

2JST PRESTO

An Introduction to Quantum

Computing: From Basic

Concepts to Hardware

@shuntaro_takeda

~20 people in total

Self-introduction

Professor

A. Furusawa

Project Lecturer

S. Takeda

- Department of Applied Physics, The Univ. of Tokyo

- Research on optical quantum information processing

Oct. 1st, 2019

Became Associate Professor

and started my own group!

An Introduction to Quantum

Computing: From Basic

Concepts to Hardware

1. Introduction

2. Basic principles of quantum computing

3. Quantum algorithms

4. Hardware implementations

What is a quantum computer?

Quantum computers are quantum mechanical

devices that enable us to perform certain

computational tasks more efficiently than what

is possible on classical (existing) computers.

Simulation of

quantum systems

Optimization Machine learning

Applications:

IBM’s commercial QCs

QC is now an exceptionally hot topic!

Heavy investment in USA/Europe/China

Japan started a big project

Much attention

from industry

History of quantum computing

1980’s Proposal of quantum computingD. Deutsch

R. Feynman

2010’s Companies join quantum computer race

・Small-scale QCs with tens of qubits have emerged

1990’s Quantum algorithm discovered

・Integer factorization（Shor, 1994）・Quantum error correction（Shor,1995）・Database search（Grover, 1996）

2000’s Development of basic technologies

・Basic experiments with spins, ions, photons, etc.

There’s still a long way to go to build a practical QC

・Formalization of QC (Deutsch, 1985)

An Introduction to Quantum

Computing: From Basic

Concepts to Hardware

1. Introduction

2. Basic principles of quantum computing

3. Quantum algorithms

4. Hardware implementations

Double-slit experiment

Classical mechanics

Quantum mechanics

Two slits

Electron, atom,

photon, etc.

Interference

“Left” or “Right”

Superposition

RightLeft

Measurement

Wall

Introduce these principles

into computing!

Principle of quantum computing

Classical computer’s information unit

0 1or

Bit

Quantum computer’s information unit

01

Qubit

Superposition10 10 cc

　　 12

1

2

0 cc

0

1x

z

y

12

sin02

cos ie

“Bloch sphere”Real

Principle of quantum computing

Classical computer’s information unit

0 1or

Bit

Quantum computer’s information unit

01

Qubit

Superposition10 10 cc

　　 12

1

2

0 cc

0

1

2

00 cP

2

11 cP

Measure

Principle of quantum computing

Classical computer’s information unit

Quantum computer’s information unit

Two bits

or or or

01

Two qubits

0111100100 11100100 cccc

Superposition

　　 12

11

2

10

2

01

2

00 cccc

n qubits can be any superposition of 2n states

n bits can be only one of 2n states

Classical computer’s logic gates

Quantum computer’s logic gates

0 1

Principle of quantum computing

1

11

01 01

1-bit NOT 2-bit AND

R̂　

1-qubit rotation

1 0

10 10 cc 10 10 cc

otherwise 0

Classical computer’s logic gates

Quantum computer’s logic gates

0 1

Principle of quantum computing

1

11

1-bit NOT 2-bit AND

R̂　

1-qubit rotation 2-qubit Controlled-NOT

1 0

otherwise 0

0 01 1

0 11

11

R̂　

1 0

0 01 1

101 0

01 01

01 01

1-bit NOT 2-bit AND

Classical computer’s logic gates

2-qubit Controlled-NOT1-qubit rotation

Quantum computer’s logic gates

otherwise 0

Principle of quantum computing

11100100 11100100 cccc

10110100 11100100 cccc

R̂　 R̂　

R̂　

R̂　

Classical computer’s calculation

Quantum computer’s calculation

01

01

01

01

01

01

01

01

0

1

1

0 0

0

1

Perform multiple

calculations

in parallel

Perform only

one calculation

at a time

NOT

AND

C-NOT

Rotation

Principle of quantum computing

111100010000 111100010000 ccc

R̂　 R̂　

R̂　

R̂　

Classical computer’s calculation

Quantum computer’s calculation

01

01

01

01

0

1

1

0 0

0

1 Perform only

one calculation

at a time

NOT

AND

Principle of quantum computing

111100010000 111100010000 ccc

Quantum algorithm

needed

Meas.

Meas.

Meas.

Meas.Only one

outcome can

be read out

An Introduction to Quantum

Computing: From Basic

Concepts to Hardware

1. Introduction

2. Basic principles of quantum computing

3. Quantum algorithms

4. Hardware implementations

Quantum algorithm

- Quantum computers are only better than

classical computers at specific computational tasks

- What problems will we use quantum computers to

solve? What algorithms will solve them?

VS

・Database search

・Integer factorization

・Quantum simulation

Examples

Database search

https://slideplayer.com/slide/3169508/

Search space size N

Num

ber

of opera

tions

Grover’s algorithm (Grover, 1996)

- Searching through a database for a specific item

- Application: optimization, NP-complete problems, etc.

- Quadratic speedup over classical algorithm

Yamada 2359

Suzuki 8723

Hayashi 3850

Sato 1123

Tanaka 5678

Database

N

Enter PINQuantum

Database search

Classical algorithm

- Sequentially try all N possibilities

- Average search takes N/2 steps

Grover’s algorithm

- Simultaneously try all possibilities

- Average search takes N1/2 steps

Enter PIN

1 2 3 4Wrong!

Correct!or

×N/2

2 5 9 1Correct PIN

×N1/2

0 0 0 00 0 0 10 0 0 2

9 9 9 9

Superposition

01

01

01

Database searchGrover’s algorithm’s circuit (3-qubit case)

1. Equal superposition of all states

2. Selectively invert the solution state

Inter-

ference

3. Invert all states about the mean

111001000 111001000 ccc

Am

plit

ude

0mean

Inter-

ference

Database searchGrover’s algorithm’s circuit (3-qubit case)

01

01

01

Inter-

ference

111001000 111001000 ccc

Am

plit

ude

0

1. Equal superposition of all states

2. Selectively invert the solution state

3. Invert all states about the mean

Repeat

~N1/2

times

Inter-

ference

Database searchGrover’s algorithm’s circuit (3-qubit case)

01

01

01

Inter-

ference

111001000 111001000 ccc

Am

plit

ude

0

1. Equal superposition of all states

2. Selectively invert the solution state Repeat

~N1/2

times3. Invert all states about the mean

0

1

1

Meas.

Meas.

Meas.

Inter-

ference

Database searchGrover’s algorithm’s circuit (3-qubit case)

01

01

01

Inter-

ference

1. Equal superposition of all states

2. Selectively invert the solution state Repeat

~N1/2

times3. Invert all states about the mean

0

1

1

Meas.

Meas.

Meas.

Simultaneously investigate all patterns by superposition

Increase the amplitude of the solution by interference

Quadratic speedup!

Quantum algorithm

・Database search

・Integer factorization

・Quantum simulation

- Quantum computers are only better than

classical computers at specific computational tasks

- What problems will we use quantum computers to

solve? What algorithms will solve them?

VS

Examples

Integer factorizationShor’s Factoring algorithm（Shor, 1994）

- Factoring integers efficiently (e.g. 191207=367×521)

https://github.com/mett29/Shor-s-Algorithm

1010

1020

1030

Num

be

r of op

era

tion

s

100

0 50 100 150 200 250

Number of digits d

exp(const×d1/3)

Best classical

algorithm

const×d3

Shor’s algorithm

- Exponential speedup over classical algorithm

- Application: breaking RSA encryption

speedup

Quantum simulation

Quantum simulation

- Application: material & chemical engineering

“Nature isn't classical, dammit, and if you

want to make a simulation of nature, you'd

better make it quantum mechanical, and by

golly it's a wonderful problem, because it

doesn't look so easy.” (1982)

R. Feynman

Shown to be correct! Science 273, 1073 (1996)

- Classical simulation of quantum systems

(such as molecules) is hard

- QCs can simulate dynamics or find ground states of

a Hamiltonian Science 309, 5741 (2005)

Quantum algorithm

・Database search

・Integer factorization

・Quantum simulation

- Quantum computers are only better than

classical computers at specific computational tasks

- What problems will we use quantum computers to

solve? What algorithms will solve them?

VS

Examples

Quantum algorithm

About 60 kinds of quantum algorithms

An Introduction to Quantum

Computing: From Basic

Concepts to Hardware

1. Introduction

2. Basic principles of quantum computing

3. Quantum algorithms

4. Hardware implementations

0

1

1

0 0

0

1

Hardware of classical computer

Vth

Time

VoltageControl of

voltages by

transistors

Bit Logic gates

Hardware of quantum computer

R̂　 R̂　

R̂　

R̂　

01

01

01

01

Meas.

Meas.

Meas.

Meas.

Qubit

Any system that obeys

quantum mechanics

Quantum logic gates

Control of quantum states

of the system

atom

electron

photon

Hardware of quantum computer

R̂　 R̂　

R̂　

R̂　

01

01

01

01

Meas.

Meas.

Meas.

Meas.

atom

electron

photon

111100010000 111100010000 ccc

Small change = error

Error is accumulated

Qubit

Any system that obeys

quantum mechanics

Quantum logic gates

Control of quantum states

of the system

Hardware of quantum computer

Building a quantum computer is super difficult!

atom

electron

photon

Qubit

Any system that obeys

quantum mechanics

Quantum logic gates

Control of quantum states

of the system

Hardware of quantum computer

Various systems to build a quantum computer

- Superconducting circuits

- Trapped ions

- Semiconductor quantum dots

- Photonics

- Neutral atoms

- Diamond vacancies

- Topological qubits

・・・

Tens of qubits

>99% gate accuracy

A few qubits

Low gate accuracy

All approaches have pros & cons!

Superconducting circuits

Refrigerator

Superconducting

chip (10~20mK)

Superconducting chip

Device

Superconducting circuits

etc.

qubit

Electric

wiring

0 1

ElectrodesJosephson

junction

Qubit

High gate accuracy>99%

Scalable fabrication

×Short lifetime of qubits

×Requires cryogenic temperature

Device

Electrodes

Trapped ions

Vacuum

chamber

Ions

Ion

0

1

Qubit

High gate accuracy>99%

Long lifetime of qubits

×Low gate speed

×Requires vacuum

Semiconductor quantum dots

DeviceElectrodes

Electrons2DEGelectron

spin

0 1

Qubit

Small footprint

×Gate accuracy is still low

×Requires cryogenic temperature

Photonics

Optical

circuit

Device

V-pol H-pol

Photon

Qubit

0 1crystal Photon

Room temperature/ in Air

High gate speed

×Probabilistic operation

×Gate accuracy is still low My tomorrow’s talk!

100 101 102 103 104 105 106 107 108

10%

1%

0.1%

0.01%

0.001%

Number of qubits

Gate

err

or

Roadmap to practical QC

Practical

QC

Current state-of-the-art

Improve quantity & quality

Introduce error correction

Factoring/Database search

NISQ=Noisy Intermediate-Scale Quantum computer

Recent trend: Developing useful algorithms that run

on NISQ computers (without error correction)

An Introduction to Quantum

Computing: From Basic

Concepts to Hardware

1. Introduction

2. Basic principles of quantum computing

3. Quantum algorithms

4. Hardware implementations

Summary of quantum computing

Basic principles Algorithm

Hardware

・Database search

・Integer factorization

・Quantum simulation

- All approaches have pros & cons

- There’s still a long way to go to build a practical QC

Qubit & quantum logic gates

& measurement

etc.