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 Takeda 1The 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 computing D. 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

Superposition 10 10 cc 

 　　 12120  cc





0

1 x

z

y

1 2

sin0 2

cos  ie

“Bloch sphere”Real

Principle of quantum computing

Classical computer’s information unit

0 1or

Bit

Quantum computer’s information unit

01

Qubit

Superposition 10 10 cc 

 　　 12120  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

01 11100100 11100100 cccc 

Superposition

 　　 1211210201200  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

1 1

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

1 1

1-bit NOT 2-bit AND

R̂　

1-qubit rotation 2-qubit Controlled-NOT

1 0

otherwise 0

0 01 1

0 1 1

1 1

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

N u m

b e r

o f o p e ra

ti o n s

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 PIN Quantum

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 4 Wrong!

Correct! or

×N/2

2 5 9 1 Correct PIN

×N1/2

0 0 0 00 0 0 10 0 0 2

9 9 9 9

Superposition

01

01

01

Database search Grover’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  

A m

p lit

u d e

0 mean

Inter-

ference

Database search Grover’s algorithm’s circuit (3-qubit case)

01

01

01

Inter-

ference

111001000 111001000 ccc  

A m

p lit

u d e

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 search Grover’s algorithm’s circuit (3-qubit case)

01

01

01

Inter-

ference

111001000 111001000 ccc  

A m

p lit

u d e

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 search Grover’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