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MULT90063 Introduction to Quantum Computing

Welcome to MULT90063 “Introduction to Quantum Computing” University of Melbourne

MULT90063 Lecture 1, 2022

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MULT90063 Introduction to Quantum Computing

Subject summary 2022

24 Lectures:

Tuesday 11am (start 11.05am):

, 104 ( ) – dual delivery lecture

Wednesday 11am (start 11.05am):

, 104 ( ) – dual delivery lecture

12 Computer-based lab classes (commencing in week 1):

Friday, Baker Computer Lab, Level 6, Building or online (dual delivery)

Assessment (see Subject Outline on LMS for timing):

2 assignments (30% and 30%) , two-hour exam (40%)

Lecturers/coordinators: Dr. Prof.

LMS: Access lecture notes, lab notes, assignments (recommend bringing print-out of lab notes to labs)

Suggested reading:

E. Rieffel and W. Polak – “Quantum Computing: A Gentle Introduction”

P. Kaye, R. Laflamme and M. Mosca – “An Introduction to Quantum Computing” M. Nielsen & I. Chuang – “Quantum Computation and Quantum Information”

MULT90063 Introduction to Quantum Computing

Opportunities for interaction

Ask questions during lectures

Ask questions after lectures

Ask your lab class demonstrators Ask on the discussion board on LMS

MULT90063 Introduction to Quantum Computing

Assessment

Description

Project 2 Examination (2 hour)

Timing Percentage

Week 6 30% Week 12 30% During the examination period 40%

MULT90063 Introduction to Quantum Computing

Week by week

(1) Introduction to quantum computing

(2) Single qubit representation and operations

(3) Two and more qubits

(4) Simple quantum algorithms

(5) Quantum search (Grover’s algorithm)

(6) Quantum factorization (Shor’s algorithm)

(7) Quantum supremacy and noise

(8) Programming real quantum computers (IBM Q)

(9) Quantum error correction (QEC)

(10) QUBO problems and Adiabatic Quantum Computation (AQC)

(11) Variational/hybrid quantum algorithms (QAOA and VQE)

(12) Solving linear equations, QC computing hardware

MULT90063 Introduction to Quantum Computing

Classical mechanics and Quantum mechanics

“Classical” mechanics rules our medium scale world.

“Quantum” mechanics rules the world of atoms and molecules.

Images: Pixabay

MULT90063 Introduction to Quantum Computing

Information is physical

Images: Wikimedia commons

Your computer is a physical device.

It operates according to the laws of physics.

MULT90063 Introduction to Quantum Computing

Moore’s Law

Image: Wikimedia commons

“The number of transistors incorporated in a chip will approximately double every 24 months.”

MULT90063 Introduction to Quantum Computing

The limits of conventional computing

Conventional computers: some problems are just too hard to solve on even the biggest supercomputers -> More powerful CPUs?

Moore’s Law – since 1960’s transistor size halves every 1.5 years

As transistor size < 10nm we’re getting close to the limit (atoms are about 1nm or less).
The end-of-Moore’s law signals new computer technology is needed...
MULT90063 Introduction to Quantum Computing
Image: Wikimedia commons
I am not afraid to consider the final question as to whether, ultimately – in the great future – we can arrange the atoms the way we want; the very atoms, all the way down!
There’s plenty of room at the bottom —
MULT90063 Introduction to Quantum Computing
Quantum Mechanics
Classical mechanics
• Newton’s laws, currents, ‘classical’ computing....
Entanglement, tuneling, Quantum computing,...
ntum mechanics
MULT90063 Introduction to Quantum Computing
Quantum Computing gives an advantage!
Surprisingly, for some problems quantum mechanics allows more efficient solutions than are possible using classical mechanics.
Quantum computers based on the laws of quantum mechanics circumvent limitations of classical information processing
MULT90063 Introduction to Quantum Computing
MULT90063: Introduction to Quantum Computing
This subject is a introduction to quantum computing.
In this subject we will cover:
• The basic concepts of quantum information and quantum logic
• Programming quantum computers to perform computational tasks
• Quantum algorithms
• Quantum error correction
• The status of current quantum computer technology
Teaching system: UoM Quantum User Interface (QUI)
• Prior to Lab Class 1: access and sign in at quispace.org (using your unimelb email address)
• In Lab Class: registered users have expanded capabilities
MULT90063 Introduction to Quantum Computing
Bits and qubits
MULT90063 Introduction to Quantum Computing
“Classical” computing: Bits
Conventional computers obey the laws of “classical physics” (pre-quantum).
m Tri-Gate Transistor
A classical bit (e.g. in an Intel chip) is like a switch with two values:
on “1” off “0”
Only “0” or “1” at a time in any given physical representation (transistor)
Gates Fins
From classical bits ® we can construct binary numbers eg. 0000 or 1010 8
MULT90063 Introduction to Quantum Computing
Decimal notation
Our number system is decimal – based on powers of 10. Computers work on a binary number system – base 2.
We know instinctively that “110” is shorthand for:
1x100 + 1x10 + 0x1 = “one hundred and one ten and no ones”
If we remember we’re in base 10, we write it as:
1x102 + 1x101 + 0x100
= one of 102 plus one of 101 plus zero of 100
But, in base 2 the number “110” is actually “six”, i.e.
”one hundred and ten”
base 2 ”six”
c. 100,000 BC
102 = 100 101 = 10 100 = 1
MULT90063 Introduction to Quantum Computing
Decimal and binary notation
”one hundred and ten”
110 = 1x102 + 1x101 + 0x100 = one of 102 plus one of 101 plus zero of 100 = 100 + 10 + 0 = 110
102 = 100 101 = 10 100 = 1
base 2 ”six”
110=1x22 +1x21 +0x20 =oneof22 plusoneof21 pluszeroof20 =4+2+0=6
22 = 4 21 = 2 20 = 1
A number, n, with digits, dk, and base, b has a value of
𝑛 = #𝑑!𝑏! !
There are 10 types of people in this world...
– those who understand binary numbers...and those who don’t
MULT90063 Introduction to Quantum Computing
Some examples
103 = 1000 102 = 100 101 = 10 100 = 1
Base 10 “decimal”
Base 2 “binary”
25 = 32 24 = 16 23 = 8 22 = 4 21 = 2 20 = 1
The point: as computers naturally use binary numbers, so do quantum computers
® need to be familiar with bits and binary and decimals (see table)
57=5x101 +7x100 =fiveof101 plussevenof100=50+7=57 Note: the numbers in blue run over base 10 “digits” i.e. 0-9
111001 = 1x25 + 1x24 + 1x23 + 0x22 + 0x21 + 1x20 = 1x32 + 1x16 + 1x8 + 0x4 + 0x2 + 1x1
= 32 + 16 + 8 + 0 + 0 + 1 = 57
Note: the numbers in blue run over base 2 “bits” i.e. 0-1
MULT90063 Introduction to Quantum Computing
“Classical” computing: bits and binary numbers
Gates Fins
on “1” off “0”
Only “0” or “1” at a time in any given physical representation (transistor)
From classical bits ® we can construct binary numbers 0000 or 1010 etc
One binary number at one time in the physical representation
MULT90063 Introduction to Quantum Computing
Start with a (semi) familiar concept – the atom
Planck (1900): postulated that energy is quantised
Bohr (1913): constructed a simple “quantised” model of the hydrogen atom
m æ e2 ö2 1 En =-2!2 ç4pe ÷ n2
Consider lowest two levels
excited state ® ground state ®
Simply re-label states to bit notation
A quantum bit, or “qubit” is a two level quantum system
n=1 energy levels
MULT90063 Introduction to Quantum Computing
Quantum superposition and measurement
quantum superposition
One electron in quantum superposition.
measurement/observation
|1ñ Y |0ñ
|0ñ “and” |1ñ
state collapse ® random outcome
Bubble: Brocken commons.wikimedia.org Bursting bubble: wallpaperswide.com
MULT90063 Introduction to Quantum Computing
Quantum superposition and measurement
quantum superposition
One electron in quantum superposition.
measurement/observation
|1ñ Y |0ñ
|0ñ “and” |1ñ
state collapse ® random outcome
Bubble: Brocken commons.wikimedia.org Bursting bubble: wallpaperswide.com
MULT90063 Introduction to Quantum Computing
Quantum Bits = Qubits
Instead of bits we have qubits.
As we will see, qubits can be in superpositions:
|1ñ Y |0ñ where α and 𝛽 are called amplitudes, and are
↵|0i + |1i complex numbers.
If we were to measure we would randomly measure “0” with probability:
|↵|2 or “1” with probability,
Since probabilities must sum to 1,
|↵|2 + ||2 = 1
|1ñ measurement/observation
state collapse ® random outcome
MULT90063 Introduction to Quantum Computing
Combining qubits – binary notation
|0ñ, |1ñ
|0ñ, |1ñ
|0ñ, |1ñ
Binary combinations
|0ñ, |1ñ
binary representation of decimals 0 to 1
|00ñ, |01ñ, |10ñ, |11ñ
binary representation of decimals 0 to 3
|000ñ, |001ñ, |010ñ, |011ñ |100ñ, |101ñ, |110ñ, |111ñ
binary representation of decimals 0 to 7
|0000ñ, |0001ñ, |0010ñ, |0011ñ |0100ñ, |0101ñ, |0110ñ, |0111ñ |1000ñ, |1001ñ, |1010ñ, |1011ñ |1100ñ, |1101ñ, |1110ñ, |1111ñ
|0ñ, |1ñ
|0ñ, |1ñ
|0ñ, |1ñ
|0ñ, |1ñ
|0ñ, |1ñ
|0ñ, |1ñ
|0ñ, |1ñ
binary representation of decimals 0 to 15 Each of these states has an associated amplitude.
MULT90063 Introduction to Quantum Computing
A general quantum superposition
In a quantum computer, we can be in a superposition of many different states. The state of a quantum computer is a superposition over all binary strings, “k”,
amplitudes states (binary string)
| i=Xak|ki k
MULT90063 Introduction to Quantum Computing
Multiple qubits and quantum processing
Basic representation of binaries as quantum information:
N qubits Independent quantum superpositions ® superposition over
N-bit binaries |000...0ñ,..., |111...1ñ (and there are 2N of these)
Not very useful...measurement of qubits collapses to one random N-bit string
Quantum computation: qubits interact to create complex superpositions and entangled states
probability
Bubble: Brocken commons.wikimedia.org
QC Immersion Project
Quantum information processing
Digital quantum computing:
• logic gates between qubits perform mathematical operations on binary data
• complex states created ® binary data in a large superposition of many states
• quantum interference amplifies probability of desired output (answer)
|1010ñ |0010ñ
|0010ñ |0011ñ
|0110ñ |0111ñ
|1101ñ |0011ñ
|1101ñ |1011ñ
|1110ñ |1111ñ
quantum program
|0000ñ |0001ñ
|0100ñ |1000ñ
|1000ñ |0000ñ |1010ñ
|1001ñ |0101ñ |1100ñ
|0001ñ |01|00110ñ1ñ |1100ñ
|1011ñ |1110ñ
|0111ñ |1111ñ
probability
Bubble: Brocken commons.wikimedia.org Bursting bubble: wallpaperswide.com
MULT90063 Introduction to Quantum Computing
NISQ Devices
• Noisy Intermediate Scale Quantum (NISQ)
- medium-scale system (10 ~1000 qubits)
- simplified control and error correction
- potentially polynomial speed-up (e.g. √CPU) - optimisation, machine learning, chemistry,... - pathway to full-scale universal QC...
• Fault Tolerant Quantum Computing
- large-scale system (>>1000 qubits)

– high redundancy for quantum error correction – potentially exponential speed-up for some

problem size

polynomial speed-up

exponential speed-up

problems (e.g. Shor’s factoring algorithm)

pprorobblelemmssizizee

computational time

computational time

MULT90063 Introduction to Quantum Computing

Timeline and state of the art

Google “Quantum Supremacy”, random quantum circuit:

Quantum advantage 100-1000 qubits?

Universal QC: (millions of qubits) poly to exponential speed-up on various problems

increasing # qubits and quality (error rates)

Hardware race: IBM, Google, Intel, D-Wave, Rigetti, Microsoft, SQC,…

DWave: 2000 “qubits” “analogue QC”

IBM-Q: 65 qubits digital QC

In the meantime, the era of quantum software and app development has begun…

MULT90063 Introduction to Quantum Computing

Week by week

(1) Introduction to quantum computing

(2) Single qubit operations and representation

(3) Two and more qubits

(4) Simple quantum algorithms

(5) Quantum search (Grover’s algorithm)

(6) Quantum factorization (Shor’s algorithm)

(7) Quantum supremacy and noise

(8) Programming real quantum computers (IBM Q)

(9) Quantum error correction (QEC)

(10) QUBO problems and Adiabatic Quantum Computation (AQC)

(11) Variational/hybrid quantum algorithms (QAOA and VQE)

(12) Solving linear equations, QC computing hardware

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