Quantum Computation simplified.pptx

Quantum Computation
Simplified
Kathiresan S
Part 1 - Introduction

Introduction
For whom?
 Those who wanted to learn Quantum Computing but got
frustrated/ confused, looking at the notations and
mathematics given in textbooks & videos
 Those who are not aware and/ or want to learn Quantum
Computing.

Introduction
Why Quantum Computing?
 We are approaching physical limit of computer technology (Moore’s law - the number
of transistors on a microchip doubles every two years, though the cost of computers
is halved)
 It is expected that quantum computer will solve a real-world problem that can’t be
solved with classical computer at all or in shorter time
 IBM, Google, Microsoft, Honeywell and many other companies are spending a lot in
quantum computing – It is going to be big market

Real World Problems
 Delta Airlines - reschedule flights after hurricanes
 Daimler – developing advanced low cost, greener batteries
 JP Morgan - improving stock trading strategies
 Cambridge Quantum Computing - advance cryptography for device security (extending Shor
algorithm)
 Google and IBM - improving artificial intelligence
 Cleaner Fertilizer manufacturing
 Drug Development
 ..........
https://builtin.com/hardware/quantum-computing-applications

Cyber Security/ Data Encryption
 RSA (Rivest–Shamir–Adleman) is an algorithm used by modern computers to encrypt
and decrypt messages. The algorithm is based on the fact that finding the factors of a
large composite number is difficult (eg. 8633 has two prime factors – 89 and 97)
 Approximate time taken for factorization in classical computer - 90 s for a 40-digit
number, 10 min for a 50-digit number, 2 h for a 60-digit number, 20 h for a 70-digit
number and 100 h for a 75-digit number.
 Private key (kept with the bank) is a function of two prime numbers
 Public key (given to customer) is a function of product of these two prime numbers
https://simple.wikipedia.org/wiki/RSA_algorithm

Cleaner Fertilization
 Fertilizer is made by heating and pressurizing atmospheric nitrogen into
ammonia
 The so-called Haber process, consumes three percent (approx.) of annual
global energy output and accounts for more than one percent of greenhouse
gas emissions
 Some bacteria perform that process naturally. The active site of nitrogenase
(enzyme used by nitrogen fixing bacteria) is made up of transition metals (iron,
cobalt, and platinum) which have many electrons and the quantum mechanical
behavior of those electrons can’t be modelled on classical computers
 Sundar Pichai, Google’s CEO, recently told MIT that he thinks the quantum
improvement of Haber process is roughly a decade away.

Quantum basics
 "Anyone who claims to understand quantum theory is either
lying or crazy"
 "If you think you understand quantum mechanics, you don't
understand quantum mechanics."
- Richard Feynman

Quantum Basics ....contd.
 In classical computing, a bit has two possible states: either 0 or 1
 In quantum computing, a qubit (short for “quantum bit”) is a unit of quantum
information - the quantum analogue to a classical bit
 Superposition, which states that instead of holding one binary value (“0” or
“1”) like a classical bit, a qubit can hold a combination of “0” and “1”
simultaneously
 In classical computing for example if there are 4 bits, then the combination of 4
bits can represent 24
=16 values in total but one value a given instant
(0000, 0001,...., 0100,.....1111)
0, 1,......... 4,........... 15
 But in a combination of 4 qubits, all 16 values are possible at once.

Quantum Basics ....contd.
 Entanglement – It is a property of two or more qubits that allows a set of qubits to
express higher correlation than is possible in classical systems
 The qubits will remain linked even when they are separated by any distance
 When one of the qubits is manipulated, the manipulation happens instantly to its
entangled twin. Einstein euphemistically called “spooky action at a distance”
 Entanglement is also a point of weakness for quantum computers. Interference from
the outside world can break the fragile correlation (entanglement) between two or
more qubits
 Any number of Qubits can be entangled as described by Greenberger–Horne–
Zeilinger state (GHZ state).

Qubits ....Contd.
 To write the state of a classical bit, we express as 0 or 1
 State of a Qubit is expressed in a different way, even when it is
measured as 0 or 1
|0 and |1
 It is part of the bra-ket notation, introduced by Dirac
 It means that the states are expressed in terms of vectors. The symbols |
and are called ‘ket’ and tell us that it is a column vector. Similarly, row
vector is expressed as ⟨ and | and called ‘bra’

Qubits
Qbit/ qubit – Quantum Bit – Physical Implementation
 To create a qubit, scientists must find a spot in a material where they can access and control
these quantum properties. It can be made from molecules, atoms, electrons, ions, photons…
 Superconducting loops (current oscillates back-and-forth around a circuit loop)
 Trapped Ions (Lasers are used to cool and trap ions or atoms and put them in a
superposition state)
 Silicon Quantum Dots (Artificial atoms are created by adding an electron to pure silicon, and
then microwaves are used to control the superposition state)
 Diamond Vacancies (diamond lattice is combined with a nitrogen atom and vacancy, and a
superposition state is controlled by light)
 .......
https://en.wikipedia.org/wiki/Qubit

Quantum Computer (HW)
 Too early to have one in every home (in fact in every country)
 We need a classical computer to work with a quantum computer
 Quantum Computers are available for everyone at free of cost including simulators
through cloud service
 Hardware Manufacturers
 IBM (53, Transmon Qbit , Superconducting circuit, 18 computers)
 Google (72, Superconducting circuit, 5 computers)
 Honeywell (QV – 64, Ions excited by laser, 6 computers)
 Riggit (128, Superconducting circuit,?)
 Dwave (2048 Qbits Annealer, Superconducting loop, ?)
 IonQ (79/160, Trapped ion,?)

Quantum Computation
Simplified
Kathiresan S
Part 2 – Minimum Essential Mathematics

Essential Mathematics
 Vectors
 Complex Numbers
 Matrix Operations
 Tensor Product
 Inner product
 Unitary matrix
 Adjoint, Conjugate, Inverse, Identity Matrices
 Sum modulo 2 (0⊕0 =0, 0⊕1 =1, 1⊕0 =1, 1⊕1 =0) (XOR?)

Vectors & Complex numbers
Complex Numbers
 Complex Numbers have real part and imaginary part
a + ib, where a is real part, b is imaginary part and 𝑖2
= −1, 𝑖 = −1
 Complex conjugate of 2 + i is 2 – i and their product is a real number (5)
 Euler’s formula, 𝑒𝑖𝜑
= 𝑐𝑜𝑠𝜑 + 𝑖𝑠𝑖𝑛𝜑
Vectors
 Vectors are used to represent quantities that have both magnitude and
direction, and may be added, subtracted and scaled (i.e. multiplied by a
real number) for forming a vector space
 The Dot Product gives a scalar (ordinary number) as answer, and is
sometimes called the scalar product
a · b = |a| × |b| × cos(θ)
|a| is the magnitude of vector a
|b| is the magnitude of vector b
θ is the angle between vector a and b

Simple matrix operations
Multiplication & Division by scalar
2 ∗
1 2
3 4
=
2 4
6 8
, 2 ÷
1 2
3 4
=
0.5 1
1.5 2
,
3
2
5
2
3.5
−9
2
=
1
2
3 5
7 −9
Matrix Addition & Subtraction
1 2
3 4
+
5 6
7 8
=
6 8
10 12
,
5 6
7 8
−
1 2
3 4
=
4 4
4 4
Matrix multiplication
If A is an n × m matrix and B is an m × p matrix, then the product C = AB is defined to be n x p matrix

Tensor & Inner products
1
2
⊗
3
4
⊗
5
6
=
1
2
⊗
3
5
6
4
5
6
=
1
2
⊗
15
18
20
24
=
1
15
18
20
24
2
15
18
20
24
=
15
18
20
24
30
36
40
48
& |0 ⊗ |0 = |0 |0 = |00
Tensor Product
Inner Product
|𝑣1 =
𝑎1
.
.
𝑎𝑛
𝑎𝑛𝑑 |𝑣2 =
𝑏1
.
.
𝑏𝑛
, 𝑡ℎ𝑒𝑖𝑟 𝑖𝑛𝑛𝑒𝑟 𝑝𝑟𝑜𝑑𝑢𝑐𝑡 𝑖𝑠 𝑔𝑖𝑣𝑒𝑛 𝑏𝑦 𝑣1 𝑣2 =
𝑖=1
𝑛
𝑎𝑖
∗
𝑏𝑖 𝑤ℎ𝑖𝑐ℎ 𝑤𝑖𝑙𝑙 𝑏𝑒 𝑎 𝑠𝑐𝑎𝑙𝑎𝑟
𝑣1 =
1
0
𝑎𝑛𝑑 𝑣2 =
2
3
, 𝑖𝑛𝑛𝑒𝑟 𝑝𝑟𝑜𝑑𝑢𝑐𝑡 𝑣1 𝑣2 = 1 0
2
3
= 2
|𝑎 =
𝑎0
𝑎1
𝑎𝑛𝑑 |𝑏 =
𝑏0
𝑏1
|𝑏𝑎 = |𝑏 ⊗ |𝑎 =
𝑏0 x
𝑎0
𝑎1
𝑏1 x
𝑎0
𝑎1
=
𝑏0𝑎0
𝑏0𝑎1
𝑏1𝑎0
𝑏1𝑎1

Unitary Matrices
 A Unitary Matrix is a complex square matrix in which its conjugate
transpose is also its inverse
 Square matrix where number of rows and columns are equal
3 4
−5 1
,
1 + 𝑖 1 − 𝑖
1 − 𝑖 1 + 𝑖
,
1 0
0 1
,
5 −9 𝑒2
−𝑖
7
2
8
4 𝜋 2
 Complex Conjugate & Conjugate Transpose
A =
1 + 2𝑖 −5
𝑖 2
𝐴 =
1 − 2𝑖 −5
−𝑖 2
𝐴 𝑇
=
1 − 2𝑖 −𝑖
−5 2

Unitary Matrices
 A Unitary Matrix is a complex square matrix in which its conjugate
transpose is also its inverse
1 + 𝑖
2
1 − 𝑖
2
1 − 𝑖
2
1 + 𝑖
2
=
1
2
1 + 𝑖 1 − 𝑖
1 − 𝑖 1 + 𝑖
M𝑢𝑙𝑡𝑖𝑝𝑙𝑦𝑖𝑛𝑔 𝑤𝑖𝑡ℎ 𝑖𝑡𝑠 𝑐𝑜𝑛𝑔𝑢𝑔𝑎𝑡𝑒 transpose,
1
4
1 + 𝑖 1 − 𝑖
1 − 𝑖 1 + 𝑖
1 − 𝑖 1 + 𝑖
1 + 𝑖 1 − 𝑖
=
1
4
4 0
0 4
=
1 0
0 1
= 𝐼2
Therefore, the conjugate transpose is its inverse

Unitary Matrices
 All Quantum logic gates are represented by unitary matrices
and they are reversible
Applying Hadamard gate on |0 ,
𝐻|0 =
1
2
1 1
1 −1
1
0
=
1
2
1 + 0
1 − 0
=
1
2
1
1
=
|0 + |1
2
Applying Hadamard gate again on the result,
𝐻
|0 + |1
2
=
1
2
1 1
1 −1
1
2
1
1
=
1
2
1 + 1
1 − 1
=
1
2
2
0
=
1
0
= |0

Definitions
 Euclidean space is what we are all familiar with. It is just a mathematical space (set
of objects that conform to certain rules) that agrees with our own physical world.
 Hilbert space is a generalization of this idea to an infinite number (meaning more) of
dimensions having the property that it is complete or closed.
 A basis of a vector space (C, R) is a minimal collection of vectors such that every
vector in that space can be expressed as a linear combination of the basis vectors
(for C2 3
2
&
4
−𝑖
, and R2 ,
1
0
&
0
1
 The computational basis is simply the basis states composed by (any of) the distinct
quantum states that the qubit can be in physically and can be measured
 If a and b are vectors and if their dot product a * b (written as a . b = |a| |b| cos (𝜃)
where 𝜃 is angle between the two vectors) = 0, then a and b vectors are orthogonal
(perpendicular) vectors and it happens when 𝜃 = π/2. If orthogonal vectors have unit
length (magnitude), then they are orthonormal vectors (>2?)

Quantum Computation
Simplified
Kathiresan S
Part 3 – Manipulation of Qubits/ Quantum Gates

• Let us define a state of a Qubit which is not completely 0 or 1 but a combination of both
|𝑞 =
1
2
|0 +
𝑖
2
|1
• In vector form it can be represented as |𝑞 =
1
2
𝑖
2
• It can also be represented as [0.70710678+0.i 0+0.70710678i] which indicates the portion
of 0 is same as 1
• Therefore, when we measure the Qubit |𝑞 , the probability of getting 0 and 1 is same and it is
0.5
https://en.wikipedia.org/wiki/Qubit
Qubits ....contd.

 The amplitude of the state vector contains the information on probability.
But, once measured, the state is known with certainty (0 or 1)
 For example, |q = α|0 +β|1 is the state before measurement and once
measured, it collapses to 0 or 1 and will not change its state but will
behave like a classical bit
 Therefore, during quantum computation, we must allow the qubits to
explore more complex states and measurements are only done when we
need to extract an output
 Probability of measuring a state |q in the state |x is given by
p(|x )=|⟨x|q |2 (take the inner product of |x and the state we are
measuring (in this case |q ), then square the magnitude
Qubits Measurement

Multiple Qubits
 Like a bit has two possible states (0 and 1), two qubits have four possible states: (00, 01, 10,
11)
|𝜓 = 𝛼|00 + 𝛽|01 + 𝛾|10 + 𝛿|11
● Probability of measuring |00 state is
𝑝(|00 ) = 00 𝜓 2
= 1 0 0 0
𝛼
𝛽
𝛾
𝛿
2
= 𝛼 2
● 𝑆𝑢𝑚 𝑜𝑓 𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑜𝑓 𝑚𝑒𝑎𝑠𝑢𝑟𝑖𝑛𝑔 𝑎𝑙𝑙 𝑠𝑡𝑎𝑡𝑒𝑠 𝑖𝑠
𝛼 2 + 𝛽 2 + 𝛾 2 + 𝛿 2 = 1
 A two-qubit state |ψ is an entangled state if and only if there not exist two one-qubit states
|a = α|0 +β|1 and |b = γ|0 +𝛿|1

Quantum GATES
 In classical computers, we have gates like
AND, OR, NAND, XOR,....
 Quantum logic gates are the building blocks
of quantum circuits, like classical logic gates
are for conventional digital circuits
Gates, Symbols & Unitary Matrices
Typical Quantum Circuit (software)

Quantum GATES .....contd.
 Hadamard Gate – widely used to set a Qubit in superposition
H =
1
2
1 1
1 −1
• It maps the basis states |0 to
|0 + |1
2
and |1 to
|0 − |1
2
• The output states are also represented as |+ and |−
respectively

 Controlled Not (CNOT) – work with 2 Qubits (Control & Target)
• Applying CNOT on |11 ,
1 0 0 0
0 1 0 0
0 0 0 1
0 0 1 0
x
0
0
0
1
=
0
0
1
0
which is |01

 X Gate (also called NOT gate)
 Switches the amplitudes of the states |0 and |1 . Rotates the state
around X axis by π radians
 Y & Z Gates
 X, Y and Z gates are also called Pauli Gates

 Toffoli Gate (also called CCNOT gate)
 This gate has 3-bit inputs and outputs. If the first two bits are set, it flips
the third bit

Until Now
 Qubit can take any state – 0, 1, a combination of both
(superposition)
 Qubits(n) can be entangled to get 2𝑛
states simultaneously
 Minimum knowledge on mathematics is sufficient to start
 States of Qubits can be manipulated to do the required
computation. Algorithms are written with Gates represented by
Unitary matrices
 States of Qubits can be measured, and the outcome will be like
classical bits (one of 2𝑛
possibilities)

Quantum Computation
Simplified
Kathiresan S
Part 4 - Deutsch-Jozsa Algorithm

Deutsch-Jozsa Algorithm
 The significance of the DJ algorithm is it was the first example
(historically) on the benefit of a quantum algorithm over classical
computation
 It is an algorithm designed for execution on Quantum computers
and has the potential to be more efficient than classical algorithm by
taking advantage of the quantum superposition and entanglement
principles

Deutsch-Jozsa Algorithm
 We are given a hidden Boolean function f , which takes as input a string of bits, and returns either 0
or 1, that is:
f:{0,1}n → {0,1}
 The property of the given Boolean function is that it is guaranteed to either be constant or balanced.
A constant function returns all 0s or all 1s for any input, while a balanced function returns 0s
for half of all inputs and 1s for the other half.
Example : f(0,1,1,0) → 1,1,1,1 & f(0,1,0,1) → 0, 0,0,0 - constant
f(0,1,0,1) → 1,0,1,0 (0 for two inputs and 1 for other two inputs) - balanced
 Minimum of 2 and a maximum of 2𝑛−1 + 1 queries/ invocations are required to get a probability of
success = 1 in classical approach
( eg. Results may be 0 and 1, or 0,0,0,..,1,1,1,..)
 1 Query is sufficient to get the right answer in quantum approach

0
Deutsch-Jozsa Algorithm ...contd.
Oracle
Inputs/ data
Ancila/ target
|𝜓0 = |0 ⊗𝑛 |1
|𝜓1 =
1
2𝑛
𝑥∈ 0,1 𝑛
|𝑥
|0 − |1
2
𝐻|0 =
1
2
|0 + |1 , 𝐻|1 =
1
2
|0 − |1
𝐻⊗2
|0 ⊗2
=
1
2
|0 + |1 |0 + |1
=
1
2
|00 + |01 + |10 + |11
=
1
2
𝑥 ∈ 0,1 2
|𝑥

0 Oracle
𝐴𝑝𝑝𝑙𝑦𝑖𝑛𝑔 𝑄𝑢𝑎𝑛𝑡𝑢𝑚 𝑂𝑟𝑎𝑐𝑙𝑒, 𝑥 𝑦 𝑡𝑜 𝑥 𝑦 ⊕ 𝑓 𝑥
|𝜓1 =
1
2𝑛
𝑥∈ 0,1 𝑛
|𝑥
|0 − |1
2
|𝜓2 =
1
2𝑛
𝑥∈ 0,1 𝑛
(−1)𝑓(𝑥)
|𝑥
|0 − |1
2
|𝜓2 =
1
2𝑛
𝑥∈ 0,1 𝑛
|𝑥
|𝑓 𝑥 − |1 ⊕ 𝑓(𝑥)
2
𝐼𝑓 𝑓 𝑥 = 0, |𝜓2 =
1
2𝑛
𝑥∈ 0,1 𝑛
|𝑥
|0 − |1
2
|𝑥 |0 ⊕ 𝑓 𝑥 − |1 ⊕ 𝑓(𝑥) = |𝑥 |𝑓 𝑥 − |1 ⊕ 𝑓(𝑥)
𝐼𝑓 𝑓 𝑥 = 1, |𝜓2 =
1
2𝑛
𝑥∈ 0,1 𝑛
|𝑥
|1 − |0
2

0
Oracle
|𝜓2 =
1
2𝑛
𝑥∈ 0,1 𝑛
(−1)𝑓(𝑥)
|𝑥
|0 − |1
2
|𝜓3 =
1
2𝑛
𝑧∈ 0,1 𝑛 𝑥∈ 0,1 𝑛
(−1)𝑓 𝑥 +𝑥.𝑧
|𝑧
|0 − |1
2
𝐹𝑜𝑟 𝑥 ∈ 0,1 , 𝐻|𝑥 =
1
2
|0 +
1
2
−1 𝑥
|1
𝐻|𝑥 =
1
2 𝑧∈ 0,1
−1 𝑥𝑧
|𝑧
𝐹𝑜𝑟 2 𝑄𝑢𝑏𝑖𝑡𝑠, 𝐻⊗2|𝑥
=
1
22
𝑧∈{0,1}2
(−1)𝑥1𝑧1+𝑥2𝑧2
|𝑧

0 Oracle
|𝜓2 =
1
2𝑛
𝑥∈ 0,1 𝑛
(−1)𝑓(𝑥)
|𝑥
|0 − |1
2
|𝜓3 =
1
2𝑛
𝑧∈ 0,1 𝑛 𝑥∈ 0,1 𝑛
(−1)𝑓 𝑥 +𝑥.𝑧
|𝑧
|0 − |1
2
𝐹𝑜𝑟 2 𝑄𝑢𝑏𝑖𝑡𝑠, 𝐻⊗2
|𝑥
=
1
22
𝑧∈{0,1}2
(−1)𝑥1𝑧1+𝑥2𝑧2|𝑧
𝐹𝑜𝑟 𝑛 𝑄𝑢𝑏𝑖𝑡𝑠, 𝐻⊗𝑛
|𝑥 =
1
2𝑛
𝑧∈{0,1}𝑛
(−1)𝑥𝑧
|𝑧
𝑻𝒉𝒆 𝒑𝒓𝒐𝒃𝒂𝒃𝒊𝒍𝒊𝒕𝒚 𝒐𝒇 𝒎𝒆𝒂𝒔𝒖𝒓𝒆𝒎𝒆𝒏𝒕 𝒇𝒐𝒓 𝒂𝒍𝒍 𝟎 𝒊𝒔
𝟏
𝟐𝒏
𝒙∈{𝟎,𝟏}𝒏
(−𝟏)𝒇(𝒙)
𝟐
=
𝟏 𝒊𝒇 𝒇 𝒊𝒔 𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕
𝟎 𝒊𝒇 𝒊𝒔 𝒃𝒂𝒍𝒂𝒏𝒄𝒆𝒅

0 Oracle
Balanced function to be used in the sample code
Input Output
001 001
101 100
011 010
111 111

Quantum Computation
Simplified
Kathiresan S
Part 5 – Programming QC & Summary

Quantum Computer (SW)
 Hardware companies and some cloud service companies provide access and libraries/ tools
for working with Quantum computer/ simulator
 Some of the software available now are:
 IBM QISKIT
 Circuit composer
 Python Jupyter notebook
 Google Cirq
 Riggeti Forest
 Microsoft Azure Quantum/ Q#
 Dwave Leap/ Ocean
 Amazon Braket
 https://algassert.com/quirk (simulator only)

Programming QC ...contd.
https://qiskit.org/

Summary
 Oracles are used for representing block box function
 Ancilla bits are extra bits being used to achieve some specific goals in
computation
 States of Qubits can be manipulated to do the required computation
through gates. Algorithms are written with Gates represented by Unitary
matrices
 States of Qubits are measured at the end of computation and the outcome
will be like classical bits (one of 2𝑛 possibilities).
 Choose the hardware/ software which is comfortable for you

References
 Funny Introduction on Qubit
https://www.youtube.com/watch?v=T2DXrs0OpHU
 Toy Quantum circuit simulator
https://www.youtube.com/watch?v=aloFwlBUwsQ
 For Quantum gates
https://en.wikipedia.org/wiki/Quantum_logic_gate
 Full Quantum computing
https://www.youtube.com/watch?v=X2q1PuI2RFI&list=PL1826E60FD05B44E4
 For Linear Algebra
https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/video-lectures/

References …contd.
 Textbook
Quantum Computation and Quantum Information by Michael A Nielsen and Isaax L.
Chuang
 Documents
https://qiskit.org/textbook/preface.html
https://qiskit.org/documentation/getting_started.html
https://docs.microsoft.com/en-us/quantum/

What Next?
YOU
• You can give feedback on the videos and what you are expecting next.
• You can go through the references and other videos/ documents and
start coding…….
I
• Based on your feedback in terms of likes, subscription, replies,….on this
video, I will come back with what is needed for most of you.
• If this video does not develop much interest on quantum computing now,
I will hold on for some time……

Quantum Computation simplified.pptx

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