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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

Aug 10, 2020

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  • 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