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Lecture 1 - IAP2019

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Overview

Instructors: Dr. Jacques Carolan, Prof. Dirk Englund�TA’s: Mihika Prabhu, Eric Bersin, Hyeongrak ‘Chuck’ Choi, Ian Christen�Contact: carolanj@mit.edu

When: M/W/F this week @ 34-303

Goals of the course:

Familiarity with quantum computing
Hands-on experience with a QC programming language (Qiskit)
Resources and references to come up with your own quantum algorithm

Outline of the three classes:

Theory (qubit, operations, mmnt, entanglement) + programming
Hardware + Applications
Group presentations!

https://mybinder.org/v2/gh/jacquescarolan/H2PQC.git/master

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Resources

Field of quantum computing is very well documented!

Nielsen and Chuang�Quantum Computation and Information

Scott Aaronson�Quantum Computing Since Democritus

IBM quantum experience�

Algorithm Zoo

John Preskill Lecture Notes

David Mermin Lecture Notes

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Highly Interdisciplinary

Expertise required:

Physics
Mathematics
Computer Science
Information Theory
Electrical Engineering
Nano Fabrication

Applications

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Elise Booker

Colossus Mark 2 (1943)

Enigma Machine

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Miniaturizing the Colossus

Thermionic vacuum tubes in Colossus Mark 2 (1940s)

(Bletchley Park Museum)

First Transistor: 1947

1950s First Si transistor (TI)

First IC - 1958 - J. Kilby, TI

Today: Intel Phi with ~ 5 Billion transistors

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Moore’s law meets the atomic scale

Microns

Feature size (nm)

Source: Intel

Intel FinFET

gate electrode

Si

Quantum effects: electrons no longer confined, can tunnel through a wall, etc.

2 nm

Quantum effects:

electron spins

nuclear spins

qu. tunneling

…

Si

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Different quantum bits (qubits) for different uses

waveguide

Energy levels of a

trapped ion

Quantized charge in a SC circuit

Spin state of an atomic defect

Spin state of an optically trapped

atom

Position of a photon

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Classical Computing

programme

tape head

tape

Alan Turing

Brit of the Century Award, JC 2019

0

1

0

[Jacques] To get an intuition for quantum computing I think its helpful to understand what we mean by a classical computer.

To do that we just needn’t look further Alan Turing, and not just because he’s my favourite Brit, because what Turing showed (amongst other things) is that you don’t need to consider each computing machine on a case by case basis to understand what it is capable of, you can just consider the Turing Machine.

Briefly: the way it works, is that you have a piece of tape with symbols, say 0s and 1s, which encode the input to your computation.

Then you have a tapehead that moves across the tape, reads in the symbols and performs logical operations, and writes the output back on the tape.

So to say add two numbers together:

the tape will initially have the input numbers written in binary
the tapehead will move acros read in the numbers
performing logic according to the ‘multiply’ programme
Write the output on the tape!

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Computational Complexity

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Quantum “speed up”

T_classical

T_quantum

YAY!!

:)

Uh oh!!

:(

Quantum Computing Memes

For QMA complete teens - Twitter

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Mathematical Primer (It’s all Matrix Multiplication!)

matrix x vector = vector

Dot means ‘multiply’

Big matrix

Big vector

Big

vector

matrix x matrix = matrix

2 x 2 square matrix

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The Quantum Bit

Other useful states:

Pssst… its all just vectors!

Probability Amplitude

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More Quantum Bits

2 qubits = 4 possible states

first qubit

second qubit

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MORE Quantum Bits

3 qubits = 8 possible states

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MORE Quantum Bits

n qubits = 2ⁿ possible states
300 qubits = 2³⁰⁰ ~ 10⁹⁰ > # particles in the observable universe.
Nature somehow keeps track of all these numbers!

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Quantum simulation

Nature isn’t classical, dammit, and if you want to make a simulation of nature, you’d better make it quantum mechanical... - R. Feynman, 1982

Google Bristlecone 72 - qubit quantum computer

IBM 50-qubit

Intel 49 - qubit device

Greiner (Harvard); Lukin (Harvard) ;Vuletic (MIT)

Ultracold Atom Array Quantum Simulators

for 300, to write down all the correlations, it would require more numbers than there are atoms in the visible universe

The computational power required to even describe a quantum system scales exponentially with the number of its constituents. For instance, to describe the most general (pure) quantum state for N spin-1⁄2 particles, we have to store in our computer 2^Ncoefficients — a task that becomes practically impossible if N is greater than 50 or so. Moreover, to predict the value of a physical quantity, we will have to add, multiply or otherwise combine all of those coefficients, something that

will require a time that scales exponentially with N as well.

Feynman’s 1981 ‘Simulating physics with computers’ emphasized the complexity of simulating quantum systems using classical computers.

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Quantum Simulation

Don’t need full wavefunction, just certain properties
QC can efficiently simulate many-body quantum systems by evolving in small time steps

controlled qubits

Seth Lloyd

In 1996, Seth Lloyd proved3 that evolving in small time steps would allow efficient simulation of any many-body quantum Hamiltonian containing few- particle interactions; that is, the overall time needed for the simulation would not grow exponentially with the number of particles, but polynomially. The main idea is to approximate each time step by a sequence of simpler operations according to a so-called Trotter decomposition, resulting in a sequence of quantum gates. Importantly, a quantum simulation might allow us to determine not only the dynamical behaviour of a many body quantum system, but also (using adiabatic algorithms4) properties at zero temperature, corresponding to the ground state of a given Hamiltonian.

Those advances notwithstanding, a quantum computer is still a long-term goal, requiring the full control of a many-body system and, eventually, the implementation of sophisticated error-correction protocols to achieve fault tolerance. So, as we still have to wait for a fully fledged universal quantum- information processor, is it possible to build on the experimental advances made so far, and construct a device not quite at the level of complexity of a quantum computer, but one that still could perform some tasks that classical devices cannot?

Feynman proposed that to overcome these problems, ‘quantum simulators’ that operate according to the laws of quantum mechanics should be used.

If we had a system at our disposal, also composed of spin-1⁄2 particles, that

we could manipulate at will, then we would be able to engineer the interaction between those particles according to the one we want to simulate, and thus predict the value of physical quantities by simply performing the appropriate measurements on our system

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Quantum computing

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Quantum computing shifts boundaries between ‘impossible’ problems and ‘easy’ problems.

...

Quantum algorithm

Huge superposition ∝2^N

Code-breaking, search big data, machine learning, ..

1022-bit number: ~1000 years on classical computer, minutes on quantum computer (if we had one!)

months

Peter Shor

Example: Prime number factoring

compute time: μs

1230186684530117755130494958384962720772853569595334792197322452151726400507263657518745202199786469389956474942774063845925192557326303453731548268507917026122142913461670429214311602221240479274737794080665351419597459856902143413;

RSA record: 768-bit

33478071698956898786044169848212690817704794983713768568912431388982883793878002287614711652531743087737814467999489

× 36746043666799590428244633799627952632279158164343087642676032283815739666511279233373417143396810270092798736308917

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Quantum Technologies

Atomic magnetometers

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Investment in Quantum Technologies

E.U. Quantum Flagship: >$2B €

U.K. national quantum technologies programme $341M

China: > $10B

Strong industry interest, but much work remains in basic science to foster technology transition, etc.

Success depends on “quantum ecosystem”

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Quantum Gates

QUESTION:

In	Out
0	1
1	0

Classical NOT

In	Out

Quantum NOT

Transform vectors ∴ MATRICES!!

?

Useful gates

Quantum gates →unitary matrices!

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Measurement

Hey qubit…

are you ‘0’ or ‘1’???

Thanks for asking Tom!�

I’m a ‘0’

I’m a ‘1’

or

Prob |𝛼|²

Prob |𝛽|²

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Measurement

Oh no… I should probably read some foundations of physics

Thanks for asking Tom!�

I’m a ‘0’

I’m a ‘1’

or

Prob |𝛼|²

Prob |𝛽|²

Q3: What happens if you measure the first qubit and get ‘0’?

Q1: Measure in 0,1:

‘0’ with probability ½ → |0>
‘1’ with probability ½ → |1>�

Q2: Measure +,-:

‘+’ with probability 1 → |+>

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Quantum Cheat Sheet

INPUT: n qubits a 2ⁿ dimensional (normalised) complex vector.�
OPERATIONS: Gates are unitary matrices in 2ⁿ dimensions. �
OUTPUT: Measurements probabilistically collapse the state → | |² of amplitudes

Input state

Quantum Gates

Measurement

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Let’s get programing!

https://mybinder.org/v2/gh/jacquescarolan/H2PQC.git/master

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John Bell

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The GHZ Game

3 people get individually asked one of 2 meaningless questions:

‘Z’ or ‘H’

And they must answer:

‘+1’ or ‘-1’

They win if the following condition is met:

Z . Z . Z = -1
H . H . Z = +1
H. Z . H = +1
Z . H . H = +1

What proportion can they win?

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Projects!

�

In 5 groups will research one of the following topics:

Grover Search
Quantum Communication
Teleportation and Superdense Coding
Shor’s Factoring Algorithm
Quantum Machine Learning

Aim:

Understand how the protocol works
Try to implement it in the provided jupyter notebooks
Think up some cool applications
Look into implementations!