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Intro to Quantum Algorithms
SUNY Polytechnic Institute
Chen-Fu Chiang
Fall 2015
Foreword …

There are 2 parts in this lecture
 Part
I: Scientists’ vision & current status
 Part
II: The basics & simple algorithms showing
exponential speedup
Part I : Scientists’ vision and current status

See @
https://www.youtube.com/watch?v=CMdHDHEuOUE
Part II: Content

Quantum computing & the basics
 Quantum bits & superposition
 Quantum circuit & data manipulation

Elementary quantum algorithms
 Deutsch’s algorithm
 Deutsch-Jozsa algorithm
Quantum Computing

A quantum computer is a machine that performs
calculations based on the laws of quantum mechanics,
which is the behavior of particles at the sub-atomic level.
Quantum Computing

A quantum computer is a machine that performs
calculations based on the laws of quantum mechanics,
which is the behavior of particles at the sub-atomic level.

Quantum computing uses the quantum mechanical
properties in order to build computers and algorithms that
have a better performance than current computer
technology.
Quantum Computing

Before we continue, I assume that you have the
knowledge of the following

Circuit models (computer organization)
 Gates (NOT, Controlled-NOT, XOR, etc.)
 Wires

Linear algebra
 Matrix operations (multiplication, eigenvector, eigenvalue)
 Tensor product
 Vector space
Representation of Data - Qubits


A bit of data is represented by a single atom that is in one
of two states denoted by |0> and |1>. A single bit of this
form is known as a qubit
A physical implementation of a qubit could use the two
energy levels of an atom. An excited state representing
|1> and a ground state representing |0>.
Light pulse of
frequency  for
time interval t
Excited
State
Ground
State
Nucleu
s
State |0>
Electro
n
State |1>
Represent Data – Superposition

A quantum state can be described by a vector
state. Thus a qubit maybe written as
 |ψ˃
= α|0> + β|1> where
and |α|2 +|β|2= 1
Represent Data – Superposition

A quantum state can be described by a vector
state. Thus a qubit maybe written as
 |ψ˃
= α|0> + β|1> where
and |α|2 +|β|2= 1
 Hadamard
Transform
Simple Illustration – Deutsch’s Problem

Problem:


Given a black box function
Task: determine whether
is constant or balanced
Simple Illustration – Deutsch’s Problem

Problem:

Given a black box function
Task: determine whether

Balanced

is constant or balanced
Constant
x
f1(x)
f2(x)
x
f3(x) f4(x)
0
0
1
0
1
0
1
1
0
1
1
0
Q: How many queries do you need classically?
Simple Illustration – Deutsch’s Problem

Problem:

Given a black box function
Task: determine whether

Balanced

is constant or balanced
Constant
x
f1(x)
f2(x)
x
f3(x)
f4(x)
0
0
1
0
1
0
1
1
0
1
1
0
Q: How many queries do you need classically?
2
Q: Quantumly ?
A quantum algorithm for Deutsch’s Problem

Simple quantum circuit
A quantum algorithm for Deutsch’s Problem

Simple quantum circuit
A quantum algorithm for Deutsch’s Problem

Simple quantum circuit
A quantum algorithm for Deutsch’s Problem

Simple quantum circuit
A quantum algorithm for Deutsch’s Problem

Simple quantum circuit
A quantum algorithm for Deutsch’s Problem

Simple quantum circuit
recall that
A quantum algorithm for Deutsch’s Problem

Simple quantum circuit
A quantum algorithm for Deutsch’s Problem

Simple quantum circuit
A Deutsch-Jozsa Problem

Problem:


Given a black box function
Promise:
is either constant or balanced



Balanced : for exactly half values of x, we have
Constant :
is independent of x
Determine whether if
is constant or balanced
A Deutsch-Jozsa Problem

Problem:


Given a black box function
Promise:
is either constant or balanced



Balanced : for exactly half values of x, we have
Constant :
is independent of x
Determine whether if
is constant or balanced
Q: How many queries are needed?
A Deutsch-Jozsa Problem

Problem:


Given a black box function
Promise:
is either constant or balanced



Balanced : for exactly half values of x, we have
Constant :
is independent of x
Determine whether if
is constant or balanced
Q: How many queries are needed?
Classically,
queries with certainty (at least)
Quantumly ?
A Deutsch-Jozsa Problem

Simple quantum Circuit
A Deutsch-Jozsa Problem

Simple quantum Circuit
A Deutsch-Jozsa Problem

Simple quantum Circuit
Look at Hadamard Again
Look at Hadamard Again
Look at Hadamard Again
A Deutsch-Jozsa Problem – finishing up

Recall the state before the last Hadamard gates

The operation of n Hadamard could have the effect

Hence, we obtain the state
(let us ignore the last qubit)
A Deutsch-Jozsa Problem – finishing up

The final state before measurement is

If
is constant, the amplitude of
is?
A Deutsch-Jozsa Problem – finishing up

The final state before measurement is

If

is constant, the amplitude of
is?
Since the amplitude is
, we will only see
after the measurement. So, we are sure the function is a
constant function.
A Deutsch-Jozsa Problem – finishing up

The final state before measurement is

If
is balanced, the amplitude of
is?
A Deutsch-Jozsa Problem – finishing up

The final state before measurement is

If

is balanced, the amplitude of
is?
Since the amplitude is 0, that means if we measure and
obtain a non
state, then the function must be
balanced.
Opportunity – For Undergraduates

Quantum summer school @ IQC Canada

USEQIP