full description on quantum computing.ppt

Quantum
Computing
Joseph Stelmach

Overview
 Introduction and History
 Data Representation
 Operations on Data
 Shor’s Algorithm
 Conclusion and Open Questions

Introduction
What is a quantum computer?
 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.

Introduction
 “I think I can safely say that nobody
understands quantum mechanics” - Feynman
 1982 - Feynman proposed the idea of creating
machines based on the laws of quantum
mechanics instead of the laws of classical
physics.
 1985 - David Deutsch developed the quantum turing
machine, showing that quantum circuits are universal.
 1994 - Peter Shor came up with a quantum
algorithm to factor very large numbers in polynomial
time.
1997 - Lov Grover develops a quantum search
algorithm with O(√N) complexity

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>.
Excited
State
Ground
State
Nucleus
Light pulse of
frequency  for
time interval t
Electron
State |0> State |1>

Representation of Data - Superposition
A single qubit can be forced into a superposition of the two states
denoted by the addition of the state vectors:
|> =  |0> +  |1>
Where  and  are complex numbers and | | + |  | = 1
1 2
1 2 1 2
2 2
A qubit in superposition is in both of the
states |1> and |0 at the same time

Representation of Data - Superposition
Light pulse of
frequency  for time
interval t/2
State |0> State |0> + |1>
Consider a 3 bit qubit register. An equally weighted
superposition of all possible states would be denoted by:
|> = |000> + |001> + . . . + |111>
1
√8
1
√8
1
√8

Data Retrieval
 In general, an n qubit register can represent the numbers 0
through 2^n-1 simultaneously.
Sound too good to be true?…It is!
 If we attempt to retrieve the values represented within a
superposition, the superposition randomly collapses to
represent just one of the original values.
In our equation: |> =  |0> +  |1> ,  represents the
probability of the superposition collapsing to |0>. The ’s
are called probability amplitudes. In a balanced
superposition,  = 1/√2 where n is the number of qubits.
1 2 1
n

Overview
 Introduction and History
 Data Representation
 Operations on Data
 Shor’s Algorithm

Due to the nature of quantum physics, the destruction of
information in a gate will cause heat to be evolved which can
destroy the superposition of qubits.
Operations on Qubits - Reversible Logic
A B C
0 0 0
0 1 0
1 0 0
1 1 1
Input Output
A
B
C
In these 3 cases,
information is
being destroyed
Ex.
The AND Gate
This type of gate cannot be used. We must use
Quantum Gates.

Quantum Gates
 Quantum Gates are similar to classical gates, but do not have
a degenerate output. i.e. their original input state can be derived
from their output state, uniquely. They must be reversible.
This means that a deterministic computation can be performed
on a quantum computer only if it is reversible. Luckily, it has
been shown that any deterministic computation can be made
reversible.(Charles Bennet, 1973)

Quantum Gates - Hadamard
Simplest gate involves one qubit and is called a Hadamard
Gate (also known as a square-root of NOT gate.) Used to put
qubits into superposition.
H
State
|0>
State |
0> + |1>
H
State
|1>
Note: Two Hadamard gates used in
succession can be used as a NOT gate

Example Operation - Multiplication By 2
Carry Bit
Carry
Bit
Ones
Bit
Carry
Bit
Ones
Bit
0 0 0 0
0 1 1 0
Input Output
Ones Bit
 We can build a reversible logic circuit to calculate multiplication
by 2 using CN gates arranged in the following manner:
0
H

Quantum Gates - Controlled Controlled NOT (CCN)
A - Target
B - Control 1
C - Control 2
A B C A’ B’ C’
0 0 0 0 0 0
0 0 1 0 0 1
0 1 0 0 1 0
0 1 1 1 1 1
1 0 0 1 0 0
1 0 1 1 0 1
1 1 0 1 1 0
1 1 1 0 1 1
Input Output
A’
B’
C’
A gate which operates on three qubits is called a
Controlled Controlled NOT (CCN) Gate. Iff the bits on
both of the control lines is 1,then the target bit is inverted.

A Universal Quantum Computer
 The CCN gate has been shown to be a universal reversible
logic gate as it can be used as a NAND gate.
A - Target
B - Control 1
C - Control 2
A B C A’ B’ C’
0 0 0 0 0 0
0 0 1 0 0 1
0 1 0 0 1 0
0 1 1 1 1 1
1 0 0 1 0 0
1 0 1 1 0 1
1 1 0 1 1 0
1 1 1 0 1 1
Input Output
A’
B’
C’
When our target input is 1, our target
output is a result of a NAND of B and C.

Shor’s Algorithm
Shor’s algorithm shows (in principle,) that a quantum
computer is capable of factoring very large numbers in
polynomial time.
The algorithm is dependant on
Modular Arithmetic
Quantum Parallelism
Quantum Fourier Transform

Shor’s Algorithm - Periodicity
 Choose N = 15 and x = 7 and we get the following:
7 mod 15 = 1
7 mod 15 = 7
7 mod 15 = 4
7 mod 15 = 13
7 mod 15 = 1
0
1
2
3
4
 An important result from Number Theory:
F(a) = x mod N is a periodic function
a
.
.
.

Shor’s Algorithm - Preparing Data
4. Load the input register with an equally weighted
superposition of all integers from 0 to q-1. 0 to 255
5. Load the output register with all zeros.
The total state of the system at this point will be:
1
√256
∑ |a, 000>
a=0
255
Input
Register
Output
Register
Note: the comma here
denotes that the
registers are entangled

Shor’s Algorithm - Modular Arithmetic
6. Apply the transformation x mod N to each number in
the input register, storing the result of each computation
in the output register.
a
Input Register 7 Mod 15 Output Register
|0> 7 Mod 15 1
|1> 7 Mod 15 7
|2> 7 Mod 15 4
|3> 7 Mod 15 13
|4> 7 Mod 15 1
|5> 7 Mod 15 7
|6> 7 Mod 15 4
|7> 7 Mod 15 13
a
0
1
7
6
5
4
3
2
Note that we are using decimal
numbers here only for simplicity.
.
.

Shor’s Algorithm - Superposition Collapse
7. Now take a measurement on the output register. This will
collapse the superposition to represent just one of the results
of the transformation, let’s call this value c.
Our output register will collapse to represent one of
the following:
|1>, |4>, |7>, or |13
For sake of example, lets choose |1>

Shor’s Algorithm - QFT
We now apply the Quantum Fourier transform on the
partially collapsed input register. The fourier transform
has the effect of taking a state |a> and transforming it into a
state given by:
1
√q
∑ |c> *
e
c=0
q-1
2iac / q

Shor’s Algorithm - QFT
1
√256
∑ |c> *
e
c=0
255
2iac / 256
1
√64
∑ |a> , |1>
a  A
Note: A is the set of all values that 7 mod 15 yielded 1.
In our case A = {0, 4, 8, …, 252}
So the final state of the input register after the QFT is:
a
1
√64
∑ , |1>
a  A
1
√256
∑ |c> *
e
c=0
255
2iac / 256

Shor’s Algorithm - The Factors :)
10. Now that we have the period, the factors of N can be
determined by taking the greatest common divisor of N
with respect to x ^ (P/2) + 1 and x ^ (P/2) - 1. The idea
here is that this computation will be done on a classical
computer.
We compute:
Gcd(7 + 1, 15) = 5
Gcd(7 - 1, 15) = 3
We have successfully factored 15!
4/2
4/2

Overview
 Introduction and History
 Data Representation
 Operations on Data
 Shor’s Algorithm

Conclusion
 In 2001, a 7 qubit machine was built and programmed to run
Shor’s algorithm to successfully factor 15.
 What algorithms will be discovered next?
Can quantum computers solve NP Complete problems in
polynomial time?

full description on quantum computing.ppt

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