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
Slide 1: Title Slide
Extrachromosomal Inheritance
Slide 2: Introduction to Extrachromosomal Inheritance
Definition: Extrachromosomal inheritance refers to the transmission of genetic material that is not found within the nucleus.
Key Components: Involves genes located in mitochondria, chloroplasts, and plasmids.
Slide 3: Mitochondrial Inheritance
Mitochondria: Organelles responsible for energy production.
Mitochondrial DNA (mtDNA): Circular DNA molecule found in mitochondria.
Inheritance Pattern: Maternally inherited, meaning it is passed from mothers to all their offspring.
Diseases: Examples include Leber’s hereditary optic neuropathy (LHON) and mitochondrial myopathy.
Slide 4: Chloroplast Inheritance
Chloroplasts: Organelles responsible for photosynthesis in plants.
Chloroplast DNA (cpDNA): Circular DNA molecule found in chloroplasts.
Inheritance Pattern: Often maternally inherited in most plants, but can vary in some species.
Examples: Variegation in plants, where leaf color patterns are determined by chloroplast DNA.
Slide 5: Plasmid Inheritance
Plasmids: Small, circular DNA molecules found in bacteria and some eukaryotes.
Features: Can carry antibiotic resistance genes and can be transferred between cells through processes like conjugation.
Significance: Important in biotechnology for gene cloning and genetic engineering.
Slide 6: Mechanisms of Extrachromosomal Inheritance
Non-Mendelian Patterns: Do not follow Mendel’s laws of inheritance.
Cytoplasmic Segregation: During cell division, organelles like mitochondria and chloroplasts are randomly distributed to daughter cells.
Heteroplasmy: Presence of more than one type of organellar genome within a cell, leading to variation in expression.
Slide 7: Examples of Extrachromosomal Inheritance
Four O’clock Plant (Mirabilis jalapa): Shows variegated leaves due to different cpDNA in leaf cells.
Petite Mutants in Yeast: Result from mutations in mitochondrial DNA affecting respiration.
Slide 8: Importance of Extrachromosomal Inheritance
Evolution: Provides insight into the evolution of eukaryotic cells.
Medicine: Understanding mitochondrial inheritance helps in diagnosing and treating mitochondrial diseases.
Agriculture: Chloroplast inheritance can be used in plant breeding and genetic modification.
Slide 9: Recent Research and Advances
Gene Editing: Techniques like CRISPR-Cas9 are being used to edit mitochondrial and chloroplast DNA.
Therapies: Development of mitochondrial replacement therapy (MRT) for preventing mitochondrial diseases.
Slide 10: Conclusion
Summary: Extrachromosomal inheritance involves the transmission of genetic material outside the nucleus and plays a crucial role in genetics, medicine, and biotechnology.
Future Directions: Continued research and technological advancements hold promise for new treatments and applications.
Slide 11: Questions and Discussion
Invite Audience: Open the floor for any questions or further discussion on the topic.
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
Slide 1: Title Slide
Extrachromosomal Inheritance
Slide 2: Introduction to Extrachromosomal Inheritance
Definition: Extrachromosomal inheritance refers to the transmission of genetic material that is not found within the nucleus.
Key Components: Involves genes located in mitochondria, chloroplasts, and plasmids.
Slide 3: Mitochondrial Inheritance
Mitochondria: Organelles responsible for energy production.
Mitochondrial DNA (mtDNA): Circular DNA molecule found in mitochondria.
Inheritance Pattern: Maternally inherited, meaning it is passed from mothers to all their offspring.
Diseases: Examples include Leber’s hereditary optic neuropathy (LHON) and mitochondrial myopathy.
Slide 4: Chloroplast Inheritance
Chloroplasts: Organelles responsible for photosynthesis in plants.
Chloroplast DNA (cpDNA): Circular DNA molecule found in chloroplasts.
Inheritance Pattern: Often maternally inherited in most plants, but can vary in some species.
Examples: Variegation in plants, where leaf color patterns are determined by chloroplast DNA.
Slide 5: Plasmid Inheritance
Plasmids: Small, circular DNA molecules found in bacteria and some eukaryotes.
Features: Can carry antibiotic resistance genes and can be transferred between cells through processes like conjugation.
Significance: Important in biotechnology for gene cloning and genetic engineering.
Slide 6: Mechanisms of Extrachromosomal Inheritance
Non-Mendelian Patterns: Do not follow Mendel’s laws of inheritance.
Cytoplasmic Segregation: During cell division, organelles like mitochondria and chloroplasts are randomly distributed to daughter cells.
Heteroplasmy: Presence of more than one type of organellar genome within a cell, leading to variation in expression.
Slide 7: Examples of Extrachromosomal Inheritance
Four O’clock Plant (Mirabilis jalapa): Shows variegated leaves due to different cpDNA in leaf cells.
Petite Mutants in Yeast: Result from mutations in mitochondrial DNA affecting respiration.
Slide 8: Importance of Extrachromosomal Inheritance
Evolution: Provides insight into the evolution of eukaryotic cells.
Medicine: Understanding mitochondrial inheritance helps in diagnosing and treating mitochondrial diseases.
Agriculture: Chloroplast inheritance can be used in plant breeding and genetic modification.
Slide 9: Recent Research and Advances
Gene Editing: Techniques like CRISPR-Cas9 are being used to edit mitochondrial and chloroplast DNA.
Therapies: Development of mitochondrial replacement therapy (MRT) for preventing mitochondrial diseases.
Slide 10: Conclusion
Summary: Extrachromosomal inheritance involves the transmission of genetic material outside the nucleus and plays a crucial role in genetics, medicine, and biotechnology.
Future Directions: Continued research and technological advancements hold promise for new treatments and applications.
Slide 11: Questions and Discussion
Invite Audience: Open the floor for any questions or further discussion on the topic.
Introduction:
RNA interference (RNAi) or Post-Transcriptional Gene Silencing (PTGS) is an important biological process for modulating eukaryotic gene expression.
It is highly conserved process of posttranscriptional gene silencing by which double stranded RNA (dsRNA) causes sequence-specific degradation of mRNA sequences.
dsRNA-induced gene silencing (RNAi) is reported in a wide range of eukaryotes ranging from worms, insects, mammals and plants.
This process mediates resistance to both endogenous parasitic and exogenous pathogenic nucleic acids, and regulates the expression of protein-coding genes.
What are small ncRNAs?
micro RNA (miRNA)
short interfering RNA (siRNA)
Properties of small non-coding RNA:
Involved in silencing mRNA transcripts.
Called “small” because they are usually only about 21-24 nucleotides long.
Synthesized by first cutting up longer precursor sequences (like the 61nt one that Lee discovered).
Silence an mRNA by base pairing with some sequence on the mRNA.
Discovery of siRNA?
The first small RNA:
In 1993 Rosalind Lee (Victor Ambros lab) was studying a non- coding gene in C. elegans, lin-4, that was involved in silencing of another gene, lin-14, at the appropriate time in the
development of the worm C. elegans.
Two small transcripts of lin-4 (22nt and 61nt) were found to be complementary to a sequence in the 3' UTR of lin-14.
Because lin-4 encoded no protein, she deduced that it must be these transcripts that are causing the silencing by RNA-RNA interactions.
Types of RNAi ( non coding RNA)
MiRNA
Length (23-25 nt)
Trans acting
Binds with target MRNA in mismatch
Translation inhibition
Si RNA
Length 21 nt.
Cis acting
Bind with target Mrna in perfect complementary sequence
Piwi-RNA
Length ; 25 to 36 nt.
Expressed in Germ Cells
Regulates trnasposomes activity
MECHANISM OF RNAI:
First the double-stranded RNA teams up with a protein complex named Dicer, which cuts the long RNA into short pieces.
Then another protein complex called RISC (RNA-induced silencing complex) discards one of the two RNA strands.
The RISC-docked, single-stranded RNA then pairs with the homologous mRNA and destroys it.
THE RISC COMPLEX:
RISC is large(>500kD) RNA multi- protein Binding complex which triggers MRNA degradation in response to MRNA
Unwinding of double stranded Si RNA by ATP independent Helicase
Active component of RISC is Ago proteins( ENDONUCLEASE) which cleave target MRNA.
DICER: endonuclease (RNase Family III)
Argonaute: Central Component of the RNA-Induced Silencing Complex (RISC)
One strand of the dsRNA produced by Dicer is retained in the RISC complex in association with Argonaute
ARGONAUTE PROTEIN :
1.PAZ(PIWI/Argonaute/ Zwille)- Recognition of target MRNA
2.PIWI (p-element induced wimpy Testis)- breaks Phosphodiester bond of mRNA.)RNAse H activity.
MiRNA:
The Double-stranded RNAs are naturally produced in eukaryotic cells during development, and they have a key role in regulating gene expression .
What is greenhouse gasses and how many gasses are there to affect the Earth.moosaasad1975
What are greenhouse gasses how they affect the earth and its environment what is the future of the environment and earth how the weather and the climate effects.
Deep Behavioral Phenotyping in Systems Neuroscience for Functional Atlasing a...Ana Luísa Pinho
Functional Magnetic Resonance Imaging (fMRI) provides means to characterize brain activations in response to behavior. However, cognitive neuroscience has been limited to group-level effects referring to the performance of specific tasks. To obtain the functional profile of elementary cognitive mechanisms, the combination of brain responses to many tasks is required. Yet, to date, both structural atlases and parcellation-based activations do not fully account for cognitive function and still present several limitations. Further, they do not adapt overall to individual characteristics. In this talk, I will give an account of deep-behavioral phenotyping strategies, namely data-driven methods in large task-fMRI datasets, to optimize functional brain-data collection and improve inference of effects-of-interest related to mental processes. Key to this approach is the employment of fast multi-functional paradigms rich on features that can be well parametrized and, consequently, facilitate the creation of psycho-physiological constructs to be modelled with imaging data. Particular emphasis will be given to music stimuli when studying high-order cognitive mechanisms, due to their ecological nature and quality to enable complex behavior compounded by discrete entities. I will also discuss how deep-behavioral phenotyping and individualized models applied to neuroimaging data can better account for the subject-specific organization of domain-general cognitive systems in the human brain. Finally, the accumulation of functional brain signatures brings the possibility to clarify relationships among tasks and create a univocal link between brain systems and mental functions through: (1) the development of ontologies proposing an organization of cognitive processes; and (2) brain-network taxonomies describing functional specialization. To this end, tools to improve commensurability in cognitive science are necessary, such as public repositories, ontology-based platforms and automated meta-analysis tools. I will thus discuss some brain-atlasing resources currently under development, and their applicability in cognitive as well as clinical neuroscience.
(May 29th, 2024) Advancements in Intravital Microscopy- Insights for Preclini...Scintica Instrumentation
Intravital microscopy (IVM) is a powerful tool utilized to study cellular behavior over time and space in vivo. Much of our understanding of cell biology has been accomplished using various in vitro and ex vivo methods; however, these studies do not necessarily reflect the natural dynamics of biological processes. Unlike traditional cell culture or fixed tissue imaging, IVM allows for the ultra-fast high-resolution imaging of cellular processes over time and space and were studied in its natural environment. Real-time visualization of biological processes in the context of an intact organism helps maintain physiological relevance and provide insights into the progression of disease, response to treatments or developmental processes.
In this webinar we give an overview of advanced applications of the IVM system in preclinical research. IVIM technology is a provider of all-in-one intravital microscopy systems and solutions optimized for in vivo imaging of live animal models at sub-micron resolution. The system’s unique features and user-friendly software enables researchers to probe fast dynamic biological processes such as immune cell tracking, cell-cell interaction as well as vascularization and tumor metastasis with exceptional detail. This webinar will also give an overview of IVM being utilized in drug development, offering a view into the intricate interaction between drugs/nanoparticles and tissues in vivo and allows for the evaluation of therapeutic intervention in a variety of tissues and organs. This interdisciplinary collaboration continues to drive the advancements of novel therapeutic strategies.
Seminar of U.V. Spectroscopy by SAMIR PANDASAMIR PANDA
Spectroscopy is a branch of science dealing the study of interaction of electromagnetic radiation with matter.
Ultraviolet-visible spectroscopy refers to absorption spectroscopy or reflect spectroscopy in the UV-VIS spectral region.
Ultraviolet-visible spectroscopy is an analytical method that can measure the amount of light received by the analyte.
Richard's aventures in two entangled wonderlandsRichard Gill
Since the loophole-free Bell experiments of 2020 and the Nobel prizes in physics of 2022, critics of Bell's work have retreated to the fortress of super-determinism. Now, super-determinism is a derogatory word - it just means "determinism". Palmer, Hance and Hossenfelder argue that quantum mechanics and determinism are not incompatible, using a sophisticated mathematical construction based on a subtle thinning of allowed states and measurements in quantum mechanics, such that what is left appears to make Bell's argument fail, without altering the empirical predictions of quantum mechanics. I think however that it is a smoke screen, and the slogan "lost in math" comes to my mind. I will discuss some other recent disproofs of Bell's theorem using the language of causality based on causal graphs. Causal thinking is also central to law and justice. I will mention surprising connections to my work on serial killer nurse cases, in particular the Dutch case of Lucia de Berk and the current UK case of Lucy Letby.
Professional air quality monitoring systems provide immediate, on-site data for analysis, compliance, and decision-making.
Monitor common gases, weather parameters, particulates.
A brief information about the SCOP protein database used in bioinformatics.
The Structural Classification of Proteins (SCOP) database is a comprehensive and authoritative resource for the structural and evolutionary relationships of proteins. It provides a detailed and curated classification of protein structures, grouping them into families, superfamilies, and folds based on their structural and sequence similarities.
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?