MULT20015 Lecture 4 2021 FINAL v2
MULT20015 Elements of Quantum Computing
Lecture 4
Subject outline
Lecture topics (by week)
1 – Introduction to quantum computing and maths basics
2 – Single qubit representations and logic operations
3 – Two qubit states and logic gates
4 – Multi-qubit states and quantum arithmetic
5 – Basic quantum algorithms
6 – Period finding, cryptography and quantum factoring
7 – Shor’s algorithm, post-quantum crypto, quantum key distribution
8 – Quantum search algorithms
9 – Grover search applications, optimisation problems
10 – Solving optimisation problems on quantum computers
11 – Applications in quantum machine learning
12 – Real quantum computer devices
Assignment schedule:
#1: Hand out in Week 2
#2: Hand out in Week 8
MULT20015 Elements of Quantum Computing
Lecture 4
Week 2
Lecture 3
3.1 The Bloch Sphere representation for qubits
3.2 Quantum operations on qubits
3.3 Qubit gates in matrix form and the Pauli matrices
Lecture 4
4.1 The Pauli gates X, Y and Z and the QUI
4.2 Qubit operations around non-cartesian axes – H and R gates
4.3 Programming sequences over the qubit logic gate library
4.4 Note on the context and use of angles
Practice class 2
Bloch sphere and single qubit logic operations on the QUI
MULT20015 Elements of Quantum Computing
Lecture 4
Lecture 3 recap
| i = a0 |0i+ a1 |a1i ! cos
✓B
2
|0i+ sin
✓B
2
ei�B |1i| i = a0 |0i+ a1 |1i =
✓
a0
a1
◆
| ⟩𝜓 | ⟩𝜓!U
Qubits on the Bloch sphere:
Operations on
qubit states:
a0 |0i+ a1 |1i ! a1 |0i+ a0 |1i
X
!”#
!$#
!”
!$
|&#⟩ = * |&⟩
= 2 x 2matrix
Ket notation:
e.g. X-gate
Operations: matrix notation
Pauli Matrices
|0i
|1i
|0i � i|1i
p
2
|0i+ |1i
p
2
|0i � |1i
p
2
|0i+ i|1i
p
2
| i
!B
“B
!B =#/2, “B=0
!B=#/2, “B=#/2
!B=#/2, “B=#
!B =#/2, “B=3#/2
!B=0, “B=0
!B =#, “B=0
| 0i = U | i
|0i
|1i
| i
| 0i
“Z” axis
“X” axis
“Y” axis
! = 0 11 0 Y =
0 −’
‘ 0 Z =
1 0
0 −1
hXi
hY i
|0i
|1i
“Z” axis
“X” axis
“Y” axis
|”!⟩
X-gate: rotate around X-axis by !
|”⟩
hXi
hY i
|0i
|1i
“Z” axis
“X” axis
“Y” axis
Z-gate: rotate around Z-axis by !
|”!⟩
|”⟩
hXi
hY i
|0i
|1i
“Z” axis
“X” axis
“Y” axis
Y-gate: rotate around Y-axis by !
|”!⟩
|”⟩
MULT20015 Elements of Quantum Computing
Lecture 4
4.1 The Pauli gates X, Y and Z and the QUI
MULT20015
Lecture 4
MULT20015 Elements of Quantum Computing
Lecture 4
Recap: the “Cartesian” quantum operations: X, Y, Z
| i ! | 0i
We can specify the state moving across the Bloch sphere in many ways, but the “Cartesian”
operations are very simple – a rotation of 𝜋 (180o) about any of X, Y, or Z axes:
The cartesian rotations are usually referred to as the “Pauli” operators X, Y, Z
NB. Perspectives not 100%
accurate! Rotations should
follow right-hand-rule.
hXi
hY i
|0i
|1i
“Z” axis
“X” axis
“Y” axis
|”!⟩
X-gate: rotate around X-axis by !
|”⟩
hXi
hY i
|0i
|1i
“Z” axis
“X” axis
“Y” axis
Z-gate: rotate around Z-axis by !
|”!⟩
|”⟩
hXi
hY i
|0i
|1i
“Z” axis
“X” axis
“Y” axis
Y-gate: rotate around Y-axis by !
|”!⟩
|”⟩
MULT20015 Elements of Quantum Computing
Lecture 4
Recap: the X gate in matrix form
|0i =
1
0
�
|1i =
0
1
�
| i = a0 |0i+ a1 |1i =
a0
a1
�
Recall “matrix”
notation:
| i =
a0
a1
�
a0, a1 2 C
𝑎” ⟩0 + 𝑎# ⟩1 =
𝑎”
𝑎#
⟶ 𝑎# ⟩0 + 𝑎” ⟩1 =
𝑎#
𝑎”
a0 |0i+ a1 |1i ! a1 |0i+ a0 |1i
X
Action of X-gate in “ket” form:
What is the X-gate in “matrix” form?
𝑎!
𝑎”
⟶
𝑎”
𝑎!
X
𝑎!
#
𝑎”
#
𝑎!
𝑎”
| ⟩𝜓! = 𝑈 | ⟩𝜓
= 2 x 2matrix
Operations in matrix representation:
Action of X-gate in matrix form:
i.e.
In matrix notation, in general:
X
𝑎”
𝑎#
= 0 1
1 0
𝑎”
𝑎#
=
𝑎#
𝑎”𝑋 =
0 1
1 0
MULT20015 Elements of Quantum Computing
Lecture 4
Recap: the X gate in matrix form – the Pauli matrices
| ⟩𝜓! = 𝑋 | ⟩𝜓
𝑎!
#
𝑎”
#
𝑎!
𝑎”=
0 1
1 0
𝑋 = 0 1
1 0
This is the so-called
Pauli X matrix…one of
three Pauli matrices
representing X, Y and Z
operations…
Y= 0 −𝑖
𝑖 0
Z= 1 0
0 −1
𝑋 = 0 1
1 0
All cartesian axes – the Pauli matrices for X, Y and Z:
hXi
hY i
|0i
|1i
“Z” axis
“X” axis
“Y” axis
|”!⟩
X-gate: rotate around X-axis by !
|”⟩
hXi
hY i
|0i
|1i
“Z” axis
“X” axis
“Y” axis
Z-gate: rotate around Z-axis by !
|”!⟩
|”⟩
hXi
hY i
|0i
|1i
“Z” axis
“X” axis
“Y” axis
Y-gate: rotate around Y-axis by !
|”!⟩
|”⟩
MULT20015 Elements of Quantum Computing
Lecture 4
X Gate (the X operator): 𝜋 around X-axis
X =
0 1
1 0
�
Circuit symbol:
Matrix representation:
Action on ket states:
QUI example:
Re
Im
✓
i
-i
-1 +1
x
z
y
p
3
2
|0i+
�i
2
|1i complexamplitudes
↵|0i+ �|1i ! ↵|1i+ �|0i↵|0i+ �|1i ! ↵|1i+ �|0i↵|0i+ �|1i ! ↵|1i+ �|0i↵|0i+ �|1i ! ↵|1i+ �|0i
X (a0 |0i+ a1 |1i) =
0 1
1 0
�
a0
a1
�
=
a1
a0
�
i.e. X (a0 |0i+ a1 |1i) = a1 |0i+ a0 |1i
a0 |0i+ a1 |1i ! a1 |0i+ a0 |1i
�i
2
|0i+
p
3
2
|1i
MULT20015 Elements of Quantum Computing
Lecture 4
Y Gate (the Y operator ): 𝜋 around Y-axis
Circuit symbol:
Matrix representation:
Action on ket states:
x
z
y
Y =
0 �i
i 0
�
Y (a0 |0i+ a1 |1i) =
0 �i
i 0
�
a0
a1
�
=
�ia1
ia0
�
i.e. Y (a0 |0i+ a1 |1i) = �ia1 |0i+ ia0 |1i
a0 |0i+ a1 |1i ! �ia1 |0i+ ia0 |1i
Y (a0 |0i+ a1 |1i) =
0 �i
i 0
�
a0
a1
�
=
�ia1
ia0
�
i.e. Y (a0 |0i+ a1 |1i) = �ia1 |0i+ ia0 |1i
a0 |0i+ a1 |1i ! �ia1 |0i+ ia0 |1i
QUI example:
Re
Im
✓
i
-i
-1 +1p
3
2
|0i+
�i
2
|1i complexamplitudes
↵|0i+ �|1i ! ↵|1i+ �|0i↵|0i+ �|1i ! ↵|1i+ �|0i↵|0i+ �|1i ! ↵|1i+ �|0i↵|0i+ �|1i ! ↵|1i+ �|0i
�1
2
|0i+
i
p
3
2
|1i
MULT20015 Elements of Quantum Computing
Lecture 4
Z Gate (the Z operator): 𝜋 around Z-axis
Circuit symbol:
Matrix representation:
Action on ket states:
Z =
1 0
0 �1
� x
z
y
QUI example:
Re
Im
✓
i
-i
-1 +1p
3
2
|0i+
�i
2
|1i complexamplitudes
Z (a0 |0i+ a1 |1i) =
1 0
0 �1
�
a0
a1
�
=
a0
�a1
�
i.e. Z (a0 |0i+ a1 |1i) = a0 |0i � a1 |1i
a0 |0i+ a1 |1i ! a0 |0i � a1 |1i
↵|0i+ �|1i ! ↵|1i+ �|0i↵|0i+ �|1i ! ↵|1i+ �|0i↵|0i+ �|1i ! ↵|1i+ �|0i↵|0i+ �|1i ! ↵|1i+ �|0i
Z (a0 |0i+ a1 |1i) =
1 0
0 �1
�
a0
a1
�
=
a0
�a1
�
i.e. Z (a0 |0i+ a1 |1i) = a0 |0i � a1 |1i
a0 |0i+ a1 |1i ! a0 |0i � a1 |1i
p
3
2
|0i+
i
2
|1i
MULT20015 Elements of Quantum Computing
Lecture 4 The Paulis and their action on computational states
(useful results for later on…)
𝑋 = 0 1
1 0
Y = 0 −𝑖
𝑖 0
Z = 1 0
0 −1
⟩𝑋|0 = 0 1
1 0
1
0
= 0
1
= ⟩|1
|0i =
1
0
�
|1i =
0
1
�
⟩𝑋|1 = 0 1
1 0
0
1
= 1
0
= ⟩|0
⟩𝑌|0 = 0 −𝑖
𝑖 0
1
0
= 𝑖 0
1
= ⟩𝑖|1
⟩𝑌|1 = 0 −𝑖
𝑖 0
0
1
= −𝑖 1
0
= −𝑖 ⟩|0
⟩𝑍|0 = 1 0
0 −1
1
0
= 1
0
= ⟩|0
⟩𝑍|1 = 1 0
0 −1
0
1
= − 0
1
= − ⟩|1
Computational basis states: ket <-> matrix conversion:
⟩𝑍|0 = ⟩|0
⟩𝑍|1 = ⟩−|1
⟩𝑌|0 = 𝑖 ⟩|1
⟩𝑌|1 = ⟩−𝑖|0
⟩𝑋|0 = ⟩|1
⟩𝑋|1 = ⟩|0
x
z
y
MULT20015 Elements of Quantum Computing
Lecture 4
The QUI gate library
So far we have covered the Pauli operations X, Y, Z…but there are other single qubit
gates in the QUI library related to Z – the so-called S and T operations…
The Pauli
gates X, Y, Z
The S and T
gates are
related to Z
i.e. rotations
about the Z-axis
but by different
angles.
MULT20015 Elements of Quantum Computing
Lecture 4
S Gate (the S operator ): Z-axs, 𝜋/2 rotation
Circuit symbol:
Matrix representation:
Action on ket states:
S =
1 0
0 i
� x
z
y
QUI example:
Re
Im
✓
i
-i
-1 +1p
3
2
|0i+
�i
2
|1i complexamplitudes
↵|0i+ �|1i ! ↵|1i+ �|0i↵|0i+ �|1i ! ↵|1i+ �|0i↵|0i+ �|1i ! ↵|1i+ �|0i↵|0i+ �|1i ! ↵|1i+ �|0i
p
3
2
|0i+
1
2
|1i
S(a0 |0i+ a1 |1i) =
✓
1 0
0 i
◆✓
a0
a1
◆
=
✓
a0
ia1
◆
= a0 |0i+ ia1 |1i
a0 |0i+ a1 |1i ! a0 |0i+ ia1 |1i
S (a0 |0i+ a1 |1i) =
1 0
0 i
�
a0
a1
�
=
a0
ia1
�
i.e. S (a0 |0i+ a1 |1i) = a0 |0i+ ia1 |1i
a0 |0i+ a1 |1i ! a0 |0i+ ia1 |1i
MULT20015 Elements of Quantum Computing
Lecture 4
T Gate (the T operator ): Z-axis, 𝜋/4 rotation
Circuit symbol:
Matrix representation:
Action on ket states:
T =
1 0
0 ei⇡/4
� x
z
y
QUI example:
Re
Im
✓
i
-i
-1 +1p
3
2
|0i+
�i
2
|1i complexamplitudes
T (a0 |0i+ a1 |1i) =
1 0
0 ei⇡/4
�
a0
a1
�
=
a0
ei⇡/4a1
�
i.e. T (a0 |0i+ a1 |1i) = a0 |0i+ ei⇡/4a1 |1i
a0 |0i+ a1 |1i ! a0 |0i+ ei⇡/4a1 |1i
↵|0i+ �|1i ! ↵|1i+ �|0i↵|0i+ �|1i ! ↵|1i+ �|0i↵|0i+ �|1i ! ↵|1i+ �|0i↵|0i+ �|1i ! ↵|1i+ �|0i
T (a0 |0i+ a1 |1i) =
1 0
0 ei⇡/4
�
a0
a1
�
=
a0
ei⇡/4a1
�
i.e. T (a0 |0i+ a1 |1i) = a0 |0i+ ei⇡/4a1 |1i
a0 |0i+ a1 |1i ! a0 |0i+ ei⇡/4a1 |1i
p
3
2
|0i+
�iei⇡/4
2
|1i
MULT20015 Elements of Quantum Computing
Lecture 4
4.2 Qubit operations around non-cartesian axes – H and R gates
MULT20015
Lecture 4
MULT20015 Elements of Quantum Computing
Lecture 4
The Hadamard gate H
Rotation about “X+Z” axis,
given by 𝒏 = (1,0,1)/ 2
The Hadamard gate H is one of the most important – it generates superposition states.
Unlike the X, Y and Z gates which rotate about one of the cartesian axes, the H-gate
rotates about the X+Z axis (or unit vector) by an angle 𝜋.
hXi
hY i
|0i
|1i
| i
| 0i
!R = ”
“Z” axis
“X” axis
“Y” axis
|0i
|0i+ |1i
p
2
X+Z
X+Z
|0i � |1i
p
2
|1i
⟩|0 ⟶
⟩0 + ⟩1
2
⟩|1 ⟶
⟩0 − ⟩1
2
MULT20015 Elements of Quantum Computing
Lecture 4
The Hadamard gate H
H-gate on a general superposition state:
|0i
|0i+ |1i
p
2
X+Z X+Z
|0i � |1i
p
2
|1i
⟩|0 ⟶
⟩0 + ⟩1
2
⟩|1 ⟶
⟩0 − ⟩1
2
H-gate on the computational states:
𝑎! ⟩0 + 𝑎” ⟩1 =
𝑎!
𝑎”
⟶ 𝑎!
⟩0 + ⟩1
2
+ 𝑎”
⟩0 − ⟩1
2
=
𝑎! + 𝑎”
2
⟩|0 +
𝑎! − 𝑎”
2
⟩|1 ⟶
1
2
𝑎! + 𝑎”
𝑎! − 𝑎”
𝑎!
#
𝑎”
#
𝑎!
𝑎”
| ⟩𝜓! = 𝐻 | ⟩𝜓
= 2 x 2matrix
H =
1
p
2
1 1
1 �1
�
(Ex. check it)
MULT20015 Elements of Quantum Computing
Lecture 4
H Gate (the H operator ): 𝜋 around X+Z-axis
Circuit symbol:
Matrix representation:
Action on ket states:
x
z
y
QUI example:
Re
Im
✓
i
-i
-1 +1p
3
2
|0i+
�i
2
|1i complexamplitudes
H =
1
p
2
1 1
1 �1
�
|0i !
|0i+ |1i
p
2
|1i !
|0i � |1i
p
2
a0 |0i+ a1 |1i !
a0 + a1p
2
|0i+
a0 � a1p
2
|1i
|0i !
|0i+ |1i
p
2
|1i !
|0i � |1i
p
2
a0 |0i+ a1 |1i !
a0 + a1p
2
|0i+
a0 � a1p
2
|1i
↵|0i+ �|1i ! ↵|1i+ �|0i↵|0i+ �|1i ! ↵|1i+ �|0i↵|0i+ �|1i ! ↵|1i+ �|0i↵|0i+ �|1i ! ↵|1i+ �|0i
p
3� i
2
p
2
|0i+
p
3 + i
2
p
2
|1i
MULT20015 Elements of Quantum Computing
Lecture 4
Rotations about arbitrary axes – R-gate
The R-gate will allow you to perform a rotation around an arbitrary axis 𝒏 by angle 𝜃R :
Exercise 3.3 To construct our example state, start from a blank 1-qubit circuit
and choose the R-gate from the gate library (for now, don’t worry about what it
is or how it works). Click the R-gate into the first time block. Once placed in the
circuit right click on the R-gate in the circuit to bring up the editable gate menu:
Edit Parameters -> set axis to X, rotation angle to !” = p$, and global phase to
zero.
Hit compute and the QUI output will look like the following (bottom right):
X axis, i.e. 𝒏 = (1,0,0)
Rotation angle set to 𝜃$ =
p
%
|0i !
p
3
2
|0i+
�i
2
|1i = |0i+ |1i
| ⟩𝜓 =| ⟩0 | ⟩𝜓! =
3
2
| ⟩0 +
−𝑖
2
| ⟩1
hXi
hY i
|0i
|1i
| i
| 0i
Rotation
axis n
!R
“Z” axis
“X” axis
“Y” axis
You have done this in Prac Class 1
𝑅𝒏 𝜃$ → 𝑅(𝟏,𝟎,𝟎)
p
3
or 𝑅𝑿
p
%
𝑅𝒏(𝜃$)
p/3
0o
90o
180o
270o -90o
-180o
MULT20015 Elements of Quantum Computing
Lecture 4
Arbitrary axis rotation – coding in QUI
Coding an arbitrary rotation gate in QUI – R-gate
The “parameters” menu allows you to specify axis and angle:
Cartesian cords for
axis of rotation n
Angle of rotation
𝜃R about n
Global phase 𝜃g :
generally
set this to zero unless
otherwise directed!
NB. QUI normalises axis for you:
e.g. X axis, i.e. 𝒏 = (1,0,0) -> enter (1,0,0)
hXi
|0i
|1i
| i
| 0i
Rotation
axis n
!R
“Z” axis
“X” axis
“Y” axis
e.g. X+Z axis, i.e. 𝒏 = (1,0,1)/ 2 -> enter (1,0,1)
(entries can be decimals)
MULT20015 Elements of Quantum Computing
Lecture 4
4.3 Programming sequences over the qubit logic gate library
MULT20015
Lecture 4
MULT20015 Elements of Quantum Computing
Lecture 4
The QUI gate library
Now we have covered the entire QUI library of single qubit logic operations…
The QUI gate
library for
single qubit
logic operations
& measurement
2 & 3 qubit
logic gates
-> allows us
to process
“quantum”
information
MULT20015 Elements of Quantum Computing
Lecture 4
Recall: Paulis and their action on computational states
𝑋 = 0 1
1 0
Y = 0 −𝑖
𝑖 0
Z = 1 0
0 −1
⟩𝑋|0 = 0 1
1 0
1
0
= 0
1
= ⟩|1
|0i =
1
0
�
|1i =
0
1
�
⟩𝑋|1 = 0 1
1 0
0
1
= 1
0
= ⟩|0
⟩𝑌|0 = 0 −𝑖
𝑖 0
1
0
= 𝑖 0
1
= ⟩𝑖|1
⟩𝑌|1 = 0 −𝑖
𝑖 0
0
1
= −𝑖 1
0
= −𝑖 ⟩|0
⟩𝑍|0 = 1 0
0 −1
1
0
= 1
0
= ⟩|0
⟩𝑍|1 = 1 0
0 −1
0
1
= − 0
1
= − ⟩|1
Computational basis states: ket <-> matrix conversion:
⟩𝑍|0 = ⟩|0
⟩𝑍|1 = ⟩−|1
⟩𝑌|0 = 𝑖 ⟩|1
⟩𝑌|1 = ⟩−𝑖|0
⟩𝑋|0 = ⟩|1
⟩𝑋|1 = ⟩|0
MULT20015 Elements of Quantum Computing
Lecture 4
Programming sequences of qubit logic operations
Let’s look at a simple quantum program.
You are asked to code the following sequence of operations in QUI:
How does this look mathematically, in say “ket” notation?
You are tempted to write | ⟩𝜓 𝑡 = 3 = 𝑋 𝑌 𝑍 ⟩|0 -> this is not correct!
In QUI (or any quantum circuit), time naturally goes from left to right, but in maths the
operations are in the order they are applied:
Time step (t) Operation | ⟩𝜓(𝒕)
0 – ⟩|0
1 X X ⟩|0 = ⟩|1
2 Y Y ⟩|1 = – i ⟩|0
3 Z Z(- i ⟩|0 ) = – i ⟩|0
| ⟩𝜓 𝑡 = 3 = 𝑍 ⟩|𝜓 𝑡 = 2 = 𝑍 𝑌 ⟩|𝜓 𝑡 = 1 = 𝑍 (𝑌(𝑋 ⟩|0 ))
| ⟩𝜓 𝑡 = 3 = 𝑍 𝑌𝑋 ⟩|0
i.e. exactly the opposite sequence
to what you program in the QUIi.e. in ket form the program is:
MULT20015 Elements of Quantum Computing
Lecture 4
Operations and operators – fun facts
In quantum mechanics, unitary operators acting on quantum states produce new
quantum states. These operators can be described by unitary matrices.
The operator U is a unitary operator (reversible), i.e. we have: U†U = I
Matrix rep: 𝑈- <-> taking the transpose (t) and complex conjugate (*): U † = U t⇤
| 0i = U | i
The new state is given by:
𝑋 = 0 1
1 0
Y = 0 −𝑖
𝑖 0
Z = 1 0
0 −1
e.g. consider the Pauli operations in matrix representation:
Ex.
Show that 𝑈- = 𝑈 for X, Y and Z
and
Show that 𝑈-𝑈 = 𝐼 for X, Y and Z
(𝐼 is the 2×2 identity)
Also, operators generally don’t commute – i.e. for two operators acting sequentially, in general
the order matters:
⟩𝑈=𝑈>|𝜓 ≠ ⟩𝑈>𝑈=|𝜓
MULT20015 Elements of Quantum Computing
Lecture 4
Operations don’t commute – order matters!
Time step
(t)
Operatio
n
| ⟩𝜓(𝒕)
0 – ⟩|0
1 X X ⟩|0 = ⟩|1
2 Y Y ⟩|1 = – i ⟩|0
⟩|𝜓 𝑡 = 2 = 𝑌𝑋 ⟩|0 = − i ⟩|0
Time step
(t)
Operatio
n
| ⟩𝜓(𝒕)
0 – ⟩|0
1 Y Y ⟩|0 = i ⟩|1
2 X X(i ⟩|1 ) = i ⟩|0
⟩|𝜓 𝑡 = 2 = 𝑋𝑌 ⟩|0 = i ⟩|0
Same |a0|2 (same probability), but different phase -> different states
MULT20015 Elements of Quantum Computing
Lecture 4
Follow the evolution on the Bloch sphere…
etc…
MULT20015 Elements of Quantum Computing
Lecture 4
4.4 Note on the context and use of angles
MULT20015
Lecture 4
MULT20015 Elements of Quantum Computing
Lecture 4
Note angles in context – abundant use of 𝜃
Angle specifying position on the Bloch sphere:
Phase angle of complex amplitudes in polar coordinates:
Re a
Im a
✓a
|a|
Angle of rotation of a qubit state on the Bloch sphere about a
specified axis (unit vector), n:
| i = a0 |0i+ a1 |a1i ! cos
✓B
2
|0i+ sin
✓B
2
ei�B |1i
| i = a0 |0i+ a1 |1i =
✓
a0
a1
◆
! | i = |a0|ei✓0 |0i+ |a1|ei✓1 |1i
| 0i = Rn̂(✓R) | i
𝜃”
𝜙”
MULT20015 Elements of Quantum Computing
Lecture 4
Week 2
Lecture 3
3.1 The Bloch Sphere representation for qubits
3.2 Quantum operations on qubits
3.3 Qubit gates in matrix form and the Pauli matrices
Lecture 4
4.1 The Pauli gates X, Y and Z and the QUI
4.2 Qubit operations around non-cartesian axes – H and R gates
4.3 Programming sequences over the qubit logic gate library
4.4 Note on the context and use of angles
Practice class 2
Bloch sphere and single qubit logic operations on the QUI
MULT20015 Elements of Quantum Computing
Lecture 4
Subject outline
Lecture topics (by week)
1 – Introduction to quantum computing and maths basics
2 – Single qubit representations and logic operations
3 – Two qubit states and logic gates
4 – Multi-qubit states and quantum arithmetic
5 – Basic quantum algorithms
6 – Period finding, cryptography and quantum factoring
7 – Shor’s algorithm, post-quantum crypto, quantum key distribution
8 – Quantum search algorithms
9 – Grover search applications, optimisation problems
10 – Solving optimisation problems on quantum computers
11 – Applications in quantum machine learning
12 – Real quantum computer devices
Assignment schedule:
#1: Hand out in Week 2
#2: Hand out in Week 8