Quantum computing uses quantum mechanics principles to perform calculations. A qubit can represent a 1, 0, or superposition of both simultaneously. Operations are performed by reversible logic gates like CNOT. Shor's algorithm shows quantum computers can factor large numbers faster by using quantum parallelism and Fourier transforms to find the period of a function, revealing the factors. While progress is being made, challenges remain in building larger quantum computers and developing new algorithms to solve other hard problems.
Quantum computing - A Compilation of ConceptsGokul Alex
Excerpts of the Talk Delivered at the 'Bio-Inspired Computing' Workshop conducted by Department of Computational Biology and Bioinformatics, University of Kerala.
Shor's algorithm is for quantum computer. Using this algorithm any arbitrarily large number can be factored in polynomial time. which is not possible in classical computer
Quantum computing - A Compilation of ConceptsGokul Alex
Excerpts of the Talk Delivered at the 'Bio-Inspired Computing' Workshop conducted by Department of Computational Biology and Bioinformatics, University of Kerala.
Shor's algorithm is for quantum computer. Using this algorithm any arbitrarily large number can be factored in polynomial time. which is not possible in classical computer
Quantum computers are incredibly powerful machines that take a new approach to processing information. Built on the principles of quantum mechanics, they exploit complex and fascinating laws of nature that are always there, but usually remain hidden from view. By harnessing such natural behavior, quantum computing can run new types of algorithms to process information more holistically. They may one day lead to revolutionary breakthroughs in materials and drug discovery, the optimization of complex manmade systems, and artificial intelligence. We expect them to open doors that we once thought would remain locked indefinitely. Acquaint yourself with the strange and exciting world of quantum computing.
Cryptanalysis with a Quantum Computer - An Exposition on Shor's Factoring Alg...Daniel Hutama
Integer factorization is a problem that has been studied by mathematicians for centuries, but has yet to see an efficient classical solution. The apparent intractability of the factorization problem has become the cornerstone of several cryptosystems, such as the widely used RSA encryption scheme for securing financial transactions and communications.
In this presentation, we show an in-depth study of quantum circuit designs for a quantum computer running Shor's algorithm. In particular, we present a classical-based reversible quantum circuit design of Vedral et. al., and a Fourier space circuit designed by Draper and Beauregard. Included in the appendix are detailed descriptions of Shor's full algorithm and a fully worked (classically simulated) example for factoring a 5-bit semiprime number.
Readers should have a basic knowledge of quantum computing concepts, such as qubits, quantum logic gates, entanglement, and their mathematical descriptions.
This is my second version of the quantum notes collected as part of my study.
This organizes content from various open source for study and reference only.
a ppt on based on quantum computing and in very short manner and all the basic areas are covered
and Logical gates are also included
and observation and conclusion also
this will lead you to get a brief knowledge about quantum computers and its explanation
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
Descripcion about IBM quantum experience. In this presentation I introduce the IBM Tools for quantum programming. Also it serves as a general introduction to Quantum Computing
Experimental realisation of Shor's quantum factoring algorithm using qubit r...XequeMateShannon
Quantum computational algorithms exploit quantum mechanics to solve problems exponentially faster than the best classical algorithms. Shor's quantum algorithm for fast number factoring is a key example and the prime motivator in the international effort to realise a quantum computer. However, due to the substantial resource requirement, to date, there have been only four small-scale demonstrations. Here we address this resource demand and demonstrate a scalable version of Shor's algorithm in which the n qubit control register is replaced by a single qubit that is recycled n times: the total number of qubits is one third of that required in the standard protocol. Encoding the work register in higher-dimensional states, we implement a two-photon compiled algorithm to factor N=21. The algorithmic output is distinguishable from noise, in contrast to previous demonstrations. These results point to larger-scale implementations of Shor's algorithm by harnessing scalable resource reductions applicable to all physical architectures.
The basics of quantum computing, associated mathematics, DJ algorithms and coding details are covered.
These slides are used in my videos https://youtu.be/6o2jh25lrmI, https://youtu.be/Wj73E4pObRk, https://youtu.be/OkFkSXfGawQ and https://youtu.be/OkFkSXfGawQ
Quantum computers are incredibly powerful machines that take a new approach to processing information. Built on the principles of quantum mechanics, they exploit complex and fascinating laws of nature that are always there, but usually remain hidden from view. By harnessing such natural behavior, quantum computing can run new types of algorithms to process information more holistically. They may one day lead to revolutionary breakthroughs in materials and drug discovery, the optimization of complex manmade systems, and artificial intelligence. We expect them to open doors that we once thought would remain locked indefinitely. Acquaint yourself with the strange and exciting world of quantum computing.
Cryptanalysis with a Quantum Computer - An Exposition on Shor's Factoring Alg...Daniel Hutama
Integer factorization is a problem that has been studied by mathematicians for centuries, but has yet to see an efficient classical solution. The apparent intractability of the factorization problem has become the cornerstone of several cryptosystems, such as the widely used RSA encryption scheme for securing financial transactions and communications.
In this presentation, we show an in-depth study of quantum circuit designs for a quantum computer running Shor's algorithm. In particular, we present a classical-based reversible quantum circuit design of Vedral et. al., and a Fourier space circuit designed by Draper and Beauregard. Included in the appendix are detailed descriptions of Shor's full algorithm and a fully worked (classically simulated) example for factoring a 5-bit semiprime number.
Readers should have a basic knowledge of quantum computing concepts, such as qubits, quantum logic gates, entanglement, and their mathematical descriptions.
This is my second version of the quantum notes collected as part of my study.
This organizes content from various open source for study and reference only.
a ppt on based on quantum computing and in very short manner and all the basic areas are covered
and Logical gates are also included
and observation and conclusion also
this will lead you to get a brief knowledge about quantum computers and its explanation
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
Descripcion about IBM quantum experience. In this presentation I introduce the IBM Tools for quantum programming. Also it serves as a general introduction to Quantum Computing
Experimental realisation of Shor's quantum factoring algorithm using qubit r...XequeMateShannon
Quantum computational algorithms exploit quantum mechanics to solve problems exponentially faster than the best classical algorithms. Shor's quantum algorithm for fast number factoring is a key example and the prime motivator in the international effort to realise a quantum computer. However, due to the substantial resource requirement, to date, there have been only four small-scale demonstrations. Here we address this resource demand and demonstrate a scalable version of Shor's algorithm in which the n qubit control register is replaced by a single qubit that is recycled n times: the total number of qubits is one third of that required in the standard protocol. Encoding the work register in higher-dimensional states, we implement a two-photon compiled algorithm to factor N=21. The algorithmic output is distinguishable from noise, in contrast to previous demonstrations. These results point to larger-scale implementations of Shor's algorithm by harnessing scalable resource reductions applicable to all physical architectures.
The basics of quantum computing, associated mathematics, DJ algorithms and coding details are covered.
These slides are used in my videos https://youtu.be/6o2jh25lrmI, https://youtu.be/Wj73E4pObRk, https://youtu.be/OkFkSXfGawQ and https://youtu.be/OkFkSXfGawQ
About
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
• Remote control: Parallel or serial interface.
• Compatible with MAFI CCR system.
• Compatible with IDM8000 CCR.
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
• Easy in configuration using DIP switches.
Technical Specifications
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
Key Features
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
• Remote control: Parallel or serial interface
• Compatible with MAFI CCR system
• Copatiable with IDM8000 CCR
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
Application
• Remote control: Parallel or serial interface.
• Compatible with MAFI CCR system.
• Compatible with IDM8000 CCR.
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
• Easy in configuration using DIP switches.
Water scarcity is the lack of fresh water resources to meet the standard water demand. There are two type of water scarcity. One is physical. The other is economic water scarcity.
Hybrid optimization of pumped hydro system and solar- Engr. Abdul-Azeez.pdffxintegritypublishin
Advancements in technology unveil a myriad of electrical and electronic breakthroughs geared towards efficiently harnessing limited resources to meet human energy demands. The optimization of hybrid solar PV panels and pumped hydro energy supply systems plays a pivotal role in utilizing natural resources effectively. This initiative not only benefits humanity but also fosters environmental sustainability. The study investigated the design optimization of these hybrid systems, focusing on understanding solar radiation patterns, identifying geographical influences on solar radiation, formulating a mathematical model for system optimization, and determining the optimal configuration of PV panels and pumped hydro storage. Through a comparative analysis approach and eight weeks of data collection, the study addressed key research questions related to solar radiation patterns and optimal system design. The findings highlighted regions with heightened solar radiation levels, showcasing substantial potential for power generation and emphasizing the system's efficiency. Optimizing system design significantly boosted power generation, promoted renewable energy utilization, and enhanced energy storage capacity. The study underscored the benefits of optimizing hybrid solar PV panels and pumped hydro energy supply systems for sustainable energy usage. Optimizing the design of solar PV panels and pumped hydro energy supply systems as examined across diverse climatic conditions in a developing country, not only enhances power generation but also improves the integration of renewable energy sources and boosts energy storage capacities, particularly beneficial for less economically prosperous regions. Additionally, the study provides valuable insights for advancing energy research in economically viable areas. Recommendations included conducting site-specific assessments, utilizing advanced modeling tools, implementing regular maintenance protocols, and enhancing communication among system components.
Student information management system project report ii.pdfKamal Acharya
Our project explains about the student management. This project mainly explains the various actions related to student details. This project shows some ease in adding, editing and deleting the student details. It also provides a less time consuming process for viewing, adding, editing and deleting the marks of the students.
Hierarchical Digital Twin of a Naval Power SystemKerry Sado
A hierarchical digital twin of a Naval DC power system has been developed and experimentally verified. Similar to other state-of-the-art digital twins, this technology creates a digital replica of the physical system executed in real-time or faster, which can modify hardware controls. However, its advantage stems from distributing computational efforts by utilizing a hierarchical structure composed of lower-level digital twin blocks and a higher-level system digital twin. Each digital twin block is associated with a physical subsystem of the hardware and communicates with a singular system digital twin, which creates a system-level response. By extracting information from each level of the hierarchy, power system controls of the hardware were reconfigured autonomously. This hierarchical digital twin development offers several advantages over other digital twins, particularly in the field of naval power systems. The hierarchical structure allows for greater computational efficiency and scalability while the ability to autonomously reconfigure hardware controls offers increased flexibility and responsiveness. The hierarchical decomposition and models utilized were well aligned with the physical twin, as indicated by the maximum deviations between the developed digital twin hierarchy and the hardware.
Final project report on grocery store management system..pdfKamal Acharya
In today’s fast-changing business environment, it’s extremely important to be able to respond to client needs in the most effective and timely manner. If your customers wish to see your business online and have instant access to your products or services.
Online Grocery Store is an e-commerce website, which retails various grocery products. This project allows viewing various products available enables registered users to purchase desired products instantly using Paytm, UPI payment processor (Instant Pay) and also can place order by using Cash on Delivery (Pay Later) option. This project provides an easy access to Administrators and Managers to view orders placed using Pay Later and Instant Pay options.
In order to develop an e-commerce website, a number of Technologies must be studied and understood. These include multi-tiered architecture, server and client-side scripting techniques, implementation technologies, programming language (such as PHP, HTML, CSS, JavaScript) and MySQL relational databases. This is a project with the objective to develop a basic website where a consumer is provided with a shopping cart website and also to know about the technologies used to develop such a website.
This document will discuss each of the underlying technologies to create and implement an e- commerce website.
2. Overview
Introduction and History
Data Representation
Operations on Data
Shor’s Algorithm
Conclusion and Open Questions
3. 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.
4. 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
5. Overview
Introduction and History
Data Representation
Operations on Data
Shor’s Algorithm
Conclusion and Open Questions
6. 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>
7. 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
8. 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
9. 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
10. Relationships among data - Entanglement
Entanglement is the ability of quantum systems to exhibit
correlations between states within a superposition.
Imagine two qubits, each in the state |0> + |1> (a superposition
of the 0 and 1.) We can entangle the two qubits such that the
measurement of one qubit is always correlated to the
measurement of the other qubit.
11. Overview
Introduction and History
Data Representation
Operations on Data
Shor’s Algorithm
Conclusion and Open Questions
12. 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.
13. 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)
14. 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
15. Quantum Gates - Controlled NOT
A gate which operates on two qubits is called a Controlled-
NOT (CN) Gate. If the bit on the control line is 1, invert
the bit on the target line.
A - Target
B - Control
A B A’ B’
0 0 0 0
0 1 1 1
1 0 1 0
1 1 0 1
Input Output
Note: The CN gate has a similar
behavior to the XOR gate with some
extra information to make it reversible.
A’
B’
16. 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
17. 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.
18. 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.
19. Overview
Introduction and History
Data Representation
Operations on Data
Shor’s Algorithm
Conclusion and Open Questions
20. 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
21. 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
.
.
.
22. Shor’s Algorithm - In Depth Analysis
To Factor an odd integer N (Let’s choose 15) :
1. Choose an integer q such that N < q < 2N let’s pick 256
2. Choose a random integer x such that GCD(x, N) = 1 let’s pick 7
3. Create two quantum registers (these registers must also be
entangled so that the collapse of the input register corresponds to
the collapse of the output register)
• Input register: must contain enough qubits to represent
numbers as large as q-1. up to 255, so we need 8 qubits
• Output register: must contain enough qubits to represent
numbers as large as N-1. up to 14, so we need 4 qubits
2 2
23. 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
24. 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.
.
.
25. 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>
26. Shor’s Algorithm - Entanglement
8. Since the two registers are entangled, measuring the output
register will have the effect of partially collapsing the input
register into an equal superposition of each state between 0
and q-1 that yielded c (the value of the collapsed output
register.)
Now things really get interesting !
Since the output register collapsed to |1>, the input register
will partially collapse to:
|0> + |4> + |8> + |12>, . . .
The probabilities in this case are since our register is
now in an equal superposition of 64 values (0, 4, 8, . . . 252)
1
√64
1
√64
1
√64
1
√64
1
√64
27. 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
2iac / q
28. Shor’s Algorithm - QFT
1
√256
∑ |c> * e
c=0
255
2iac / 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
2iac / 256
29. Shor’s Algorithm - QFT
The QFT will essentially peak the probability amplitudes at
integer multiples of q/4 in our case 256/4, or 64.
|0>, |64>, |128>, |192>, …
So we no longer have an equal superposition of states, the
probability amplitudes of the above states are now higher
than the other states in our register. We measure the register,
and it will collapse with high probability to one of these
multiples of 64, let’s call this value p.
With our knowledge of q, and p, there are methods of
calculating the period (one method is the continuous fraction
expansion of the ratio between q and p.)
30. 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
31. Shor’s Algorithm - Problems
The QFT comes up short and reveals the wrong period. This
probability is actually dependant on your choice of q. The
larger the q, the higher the probability of finding the correct
probability.
The period of the series ends up being odd
If either of these cases occur, we go back to
the beginning and pick a new x.
32. Overview
Introduction and History
Data Representation
Operations on Data
Shor’s Algorithm
Conclusion and Open Questions
33. 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?