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MULT90063 Introduction to Quantum Computing
Week by week
(1) Introduction to quantum computing
(2) Single qubit representation and operations

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(3) Multi-qubit systems
(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
3.1 Two qubit systems and operations 3.2 Entanglement
4.1 Dense coding 4.2 Teleportation
Two qubit operations, entanglement, dense coding, teleportation

MULT90063 Introduction to Quantum Computing
Recap: A zoo of one-qubit gates
X=01 10
Y = 0 i i 0
111 H=p2 1 1
T = 1 0 𝜋/4rotationaboutthez-axis. 0 ei⇡/4
𝜋 rotation about the x-, y- and z- axes.
“Hadamard” gate
𝜋/2 rotation about the z- axis.

MULT90063 Introduction to Quantum Computing
Recap: General qubit rotation
General rotation matrix:

MULT90063 Introduction to Quantum Computing
5.1 Multi-qubit systems and operations
MULT90063 Lecture 5

MULT90063 Introduction to Quantum Computing
Lecture overview
In this lecture:
5.1 Two qubit systems and operations
– Mulitple qubits and binary numbers
– Linear algebra of two-qubit systems
– Two-qubit logic gates
– Universality
5.2 Entanglement
– Separable states
– Entangled states
– Entropy of entanglement
– Entanglement in the QUI
• Reiffel, Chapter 3
• Kaye, 2.6
• Mike and Ike, 1.3.2-1.3.4

MULT90063 Introduction to Quantum Computing
Recap: qubits and binary numbers
Computers represent integers as binary numbers. Similarly, we can think as the state of several qubits as a binary digits.
|1i |1i |1i |0i |0i |0i
For example, the number 5 can be represented in binary as 101, and this can be encoded directly in the state of three individual atoms.
This lecture we will talk about multi-qubit systems (i.e. 2-qubit systems).

MULT90063 Introduction to Quantum Computing
Two qubits: tensor product
Two atoms, each with one electron in a superposition of
the bit states:
| i=a|0i+b|1i Then the joint state of both atoms is:
|i=c|0i+d|1i
Tensor product!

MULT90063 Introduction to Quantum Computing
Ordering of amplitudes in vectors
We want to order the elements of the vector so that they correspond to binary numbers.
Three qubit state

MULT90063 Introduction to Quantum Computing
Tensor product
Two atoms, each with one electron in a superposition of
the bit states:
| i=a|0i+b|1i For these two atoms in the states indicated:
|i=c|0i+d|1i
| i⌦|i = (a|0i+b|1i)⌦(c|0i+d|1i)
= ac |0i ⌦ |0i + ad |0i ⌦ |1i + bc |1i ⌦ |0i + bd |1i ⌦ |1i = ac |00i + ad |01i + bc |10i + bd |11i

MULT90063 Introduction to Quantum Computing
Tensor product of vectors
  26ac37 a ⌦ c =64ad75
00 amplitude 01 amplitude 10 amplitude 11 amplitude
| i=a|0i+b|1i |i=c|0i+d|1i

MULT90063 Introduction to Quantum Computing
Tensor product of operators
Similarly, we can define a Kronecker tensor product of qubit operators:
  26 a m a n b m b n 37 a b ⌦ m n = 64 a p a q b p b q 75
c d p q cm cn dm dn cp cq dp dq

MULT90063 Introduction to Quantum Computing
Single qubit gates on multi-qubit systems
We can apply single-qubit operators to multi-qubit systems:
(X ⌦ I) |0i ⌦ |0i X1 |00i = |10i (I ⌦ X ) |0i ⌦ |0i X2 |00i = |01i
Simplest way to think of it: the subscript represents which qubit the operation is applied to.
To work out the operator we are applying in matrix representation, we use the Kronecker (tensor) product with the identity:
Identity is applied to second qubit where there’s no operation
X⌦I=0 1⌦1 0 10 01
26 0 0 1 0 37 = 64 0 0 0 1 75

MULT90063 Introduction to Quantum Computing
Two qubit logic gate: CNOT
Two qubit gates can be constructed using an interaction between the two systems. Most important is the Controlled-NOT (CNOT) gate.
Control qubit Target qubit
Symbol for “control”
Symbol for binary addition (flip)
How states transform: CNOT truth table
|00i ! |00i |01i ! |01i |10i ! |11i |11i ! |10i
Rule: The target is flipped iff the control qubit is “1”.
a|00i+b|01i+c|10i+d|11i !a|00i+b|01i+d|10i+c|11i
As a matrix: 26 1 0 0 0 37 CNOT =64 0 1 0 0 75
0 0 0 1 0010

MULT90063 Introduction to Quantum Computing
Example: CNOT on superposition
↵|0i+|1i |0i
|i |0i Before the CNOT, the state is:
| i=(↵|0i+|1i)⌦|0i=↵|00i+|10i
After the CNOT, the state is:
| 0i=↵|00i+|11i

MULT90063 Introduction to Quantum Computing
Control Phase Gate
Two qubit gates can be constructed using an interaction between the two systems.
Control qubit Target qubit
|00i ! |00i |01i ! |01i |10i ! |10i |11i ! |11i
a|00i + b|01i + c|10i + d|11i
! a |00i + b |01i + c |10i d |11i
Asamatrix: 26 1 0 0 0 37 CZ=64 0 1 0 0 75
How states transform:
Rule:thephaseofthetarget flippediffthecontrolqubitis“1”.
0010 0 0 0 1
Fun fact: CZ doesn’t matter which one you think of as control/target.

MULT90063 Introduction to Quantum Computing
A swap operation can be implemented using an interaction between the two qubits – the states of the two qubits are swapped (not the physics qubits).
Qubit 1 Qubit 2
|00i ! |00i |01i ! |10i |10i ! |01i |11i ! |11i
a|00i + b|01i + c|10i + d|11i
! a |00i + c |01i + b |10i + d |11i
Rule: the two qubits are swapped.
NB. Unlike CNOT, swap gates do not generate entanglement (but sqrt SWAP does!) .
How states transform:
Asamatrix: 261 0 0 037 SWAP=640 0 1 075

MULT90063 Introduction to Quantum Computing
Toffoli gate
Toffoli gate plus NOT is universal for classical computation. It was used in the proof that classical computation can be made reversible!
Control qubit Control qubit
Target qubit
Rule: the target is flipped iff both the control qubits are in “1” state.
How states transform:
|000i ! |001i ! |010i ! |011i ! |100i ! |101i ! |110i ! |111i !
|000i |001i |010i |011i |100i |101i |111i |110i
a|000i + b|001i + c|010i + d|011i e |100i + f |101i + g |110i + h |111i
! a |000i + b |001i + c |010i + d |011i e |100i + f |101i + h |110i + g |111i

MULT90063 Introduction to Quantum Computing
Universality
In classical computing the NAND gate is universal, i.e. every Boolean function can be implemented as a sequence of NAND (NOT AND) gates:
In quantum computing every quantum circuit can be expressed as a sequence of:
CNOT Single Qubit Rotations

MULT90063 Introduction to Quantum Computing
Universality
In classical computing the NAND gate is universal, i.e. every Boolean function can be implemented as a sequence of NAND (NOT AND) gates:
In quantum computing every quantum circuit can be expressed as a sequence of:
CNOT Hadamard
T Gate (pi/8)

MULT90063 Introduction to Quantum Computing
Construction for any two qubit unitary
Any two qubit gate can be decomposed into just 3 CNOTs and single qubit rotations:

MULT90063 Introduction to Quantum Computing
Challenge problem
How can you decompose Tofolli as CNOTs and single qubit rotations?

MULT90063 Introduction to Quantum Computing
Two-qubit projective measurement
Examples of two-qubit projectors. Eg. For measuring the first qubit in the computational basis:
P0 =|0ih0|⌦I P1 =|1ih1|⌦I
=1 0⌦1 0 =0 0⌦1 0
26 1 0 0 0 37 26 0 0 0 0 37
= 64 0 1 0 0 75 = 64 0 0 0 0 75 0000 0010

MULT90063 Introduction to Quantum Computing
Two-qubit measurement
Measurement on a two-qubit state:
(1) Apply projector into the measured state (2) Renormalize the state
For example, consider the general two-qubit state:
If the first qubit were measured to be “0”, apply
P0 and renormalize to get the collapsed state: |
If the first qubit were measured to be “1”, apply
P1 and renormalize to get the collapsed state: |
0 Pm|i i = pp(m)
| i=a|00i+b|01i+c|10i+d|11i
a |00i + b |01i
p|a|2 + |b|2 c |10i + d |11i
p|c|2 + |d|2 Later (and Lab 3): this generalizes to measurements on multi-qubit states.
normalization

MULT90063 Introduction to Quantum Computing
5.2 Entanglement
MULT90063 Lecture 5

MULT90063 Introduction to Quantum Computing
Separable states
| i=a|0i+b|1i
|i = c|0i + d|1i
A separable state is one which can be written as
|i=| i⌦|i
All separable states (of two qubits) can be written as:
| i=ac|00i+ad|01i+bc|10i+bd|11i

MULT90063 Introduction to Quantum Computing
Examples of separable states
Consider the state:
It is separable because:
|00i + |01i |i= p2
|0i + |1i | i = |0i ⌦ p2
Considerthestate:
It is also separable because:
| i=|00i+|01i+|10i+|11i 2
|0i + |1i |0i + |1i | i = p2 ⌦ p2

MULT90063 Introduction to Quantum Computing
Constructing a Bell state
This is one of four states named after the physicist (who figured out how to experimentally explore reality of entanglement).
Consider the following circuit in the QUI:
Execution:
|00i + |10i |00i + |11i |00i! p ! p
|00i + |11i
Question: Is p2 separable?

MULT90063 Introduction to Quantum Computing
Entanglement
Answer: No! We can never find a, b, c, d, i.e. |00i + |11i
p2 6= (a |0i + b |1i) ⌦ (c |0i + d |1i)
A state which is not separable is called an entangled state.
Entanglement is a uniquely quantum mechanical property, with no direct classical analogue.

MULT90063 Introduction to Quantum Computing
Entanglement Measure
We would like to have a measure of how much entanglement a state has. Some states are more entangled than others:
p0.99 |00i + p0.01 |11i
p2 |00i + p2 |11i
Not entangled, separable
Entangled, but close to a separable state Maximally entangled
One simple measure is to ask: How many Bell states (asymptotically) do would Alice and Bob need to share to construct this state, given they are allowed to do Local Operations and Classical Communication (LOCC), but not interact their qubits directly. For pure states, this is equivalent to the Entropy of Entanglement.

MULT90063 Introduction to Quantum Computing
Entropy of entanglement
Entanglement is a type of correlation between two systems, say A and B.
To see how much correlation there is between A and B: We will measure B, throw away the result, and ask how many bits information of information do we need to determine the state of A?
For example, taking the state (with Alice controlling the first qubit, Bob the second):
|00i + |11i p2
We measure the state of the second (Bob’s) qubit, and forget the result.
50% of the time, the first qubit (Alice’s) collapses to the state |0i 50% of the time, the first qubit (Alice’s) collapses to the state |1i

MULT90063 Introduction to Quantum Computing
Entropy of Entanglement
Bob needs some information to determine Alice’s state. How much? That’s measured by the entropy.
Entanglement entropy is giveXn by:
S = pi log pi
where pi is the probability of measuring ith state of Alice’s qubit.
For this case of a Bell state, p0=50%, p1=50%,
S = 12 log 12 12 log 12 = 12 + 12 = 1
Therefore, a Bell State has 1 bit of entanglement (max possible).
Here, and throughout this subject, logarithms are taken base 2.

MULT90063 Introduction to Quantum Computing
Entropy measure of a separable state
For the separable state: |00i
We measure the state of the second (Bob’s) qubit.
100%ofthetime,thefirstqubit(Alice’s)collapsestothestate |0i
S = 1 ⇥ log 1 = 0 p0.99 |00i + p0.01 |11i S = 0.08
The entropy of entanglement is therefore:
All separable states have an entropy of entanglement of 0.
For the state:
This measure of entanglement generalizes between two subsystems of qubits A and B, and is how the measure of entanglement is calculated in the QUI.

MULT90063 Introduction to Quantum Computing

How much entanglement is present in a general state?
| i=a00|00i+a01|01i+a10|10i+a11|11i
Can be hard to tell. It’s not in anything like product form. For that we will use SVD.
Arrange as a matrix:
T a k i n g S i n g u l a r V a l u e D e c o m p o s i t i o n X( S V D ) :
A = i |uiihvi|
A=a00 a01 a10 a11
Allows us to express the state in thisXconvenient form:
This from is known as
| i= i|uii|vii

MULT90063 Introduction to Quantum Computing

| i = Xi |uii|vii
Several terms might have a singular value of 0. The number of non-zero terms is
called the Schmidt rank.
If a state has a Schmidt rank of 1:
| i=|u0i⌦|v0i Then the state is separable, and not entangled.
If a state has a Schmidt rank greater than 1, then the state is entangled. Schmidt rank is a very coarse measure of entanglement. We would like a finer measure.

MULT90063 Introduction to Quantum Computing
Entanglement Entropy
| i = Xi |uii|vii
A more fine-grained measure of entanglement is the entanglement entropy. Form
a probability distribution:
p i = 2i
From which you can calculate the entanglement entropy:
S = Xpi logpi i
This is a measure of entanglement. The higher the entanglement entropy, the more entanglement.

MULT90063 Introduction to Quantum Computing
Entanglement in the QUI
The time scrubber is the vertical bar which moves left and right to show the quantum state at each time step.
The entropy of entanglement is shown in a red colour scale between min and max values possible. Each segment corresponds to the entropy between the system of qubits above and below for that particular bi-partition.
qubit-1 qubit-2 qubit-3 qubit-4
Entanglement entropy between qubit 1 and qubits {2 & 3 & 4} partition
Entanglement entropy between qubits {1 &2} and qubits {3 & 4} partitions
Entanglement entropy between qubit 4 and qubits {1 & 2 & 3} partition

MULT90063 Introduction to Quantum Computing
Lecture 5 Summary
  2 ac 3 Twoqubitgates
a ⌦ c = 6 64 a d 7 75 26 1 0 0 0 37 b d bc CZ=64010075
Tensor Product
bd 0010 0 0 0 1
A state which is not separable is called an entangled state.
26 1 0 0 0 37 CNOT=640 1 0 075
Universal set of gates
26 1 0 0 0 37 SWAP=640 0 1 075
Single Qubit Rotations

MULT90063 Introduction to Quantum Computing
Week by week
(1) Introduction to quantum computing
(2) Single qubit representation and operations
(3) Multi-qubit systems
(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