User:Quantum Information Retrieval/sandbox

INTRODUCTION – QUANTUM MECHANICS
Information Retrieval (IR) is the process of searching and retrieve information that is relevant to the need of a user. But during the last two decades the way in which people consume information has changed a lot and due to this change there has been advancement in web search engines.

There are many investigations by IR researchers to improve IR systems from system view point as well as user view point. There are many areas developed in IR such as neural IR, cognitive IR, dynamic IR.

Quantum Information retrieval is also one of the areas of IR, where it uses framework of quantum theory which is mathematical in nature, it is used to develop user models so that the alignment is better with human cognitive information processing.

There are many benefits of quantum IR. In this case it provides a new method for calculating the probability of events and also representing the events. Quantum theory represents events as subspaces of an abstract, complex vector space (Hilbert space). This type of representation helps a lot in abstraction as well as contextualization of information objects like documents and queries.

For example, if a set of basis vectors correspond to a set of documents, another set of basis vectors in the same Hilbert space can represent the same set of documents in a different context. Hence a query will be represented by this different basis which will depend upon the retrieval context.

The application of Quantum theory in IR is divided in 2 aspects:

·      Representation and Ranking

·      User interaction

The traditional IR system and Quantum IR system is different from each other as quantum uses tools to retrieve information that is it uses mathematical framework of QT which includes complex Hilbert space model to representing learning.


 * Information Retrieval as Quantum Mechanics:

Using this framework Van Rijsbergen has presented a new theory for IR foundations. He showed that a document can be represented in a Hilbert space as a vector and the relevance of the document by the Hermitian operator.

There are usual quantum mechanics notion for example, uncertainty, superposition and observable, these notions have their IR theoretic analogies. The problems in Information retrieval can be addressed by standard theorems which are pseudo feedback, relevance feedback and ostensive retrieval.

HISTORY AND DEVOLOPMENT
The history of quantum theory began at 20th century when classical physical revolutionized into quantum physics. Quantum mechanics was formulated by Schrödinger using wave mechanics and Heisenberg using matrix mechanics.

The development of the quantum theory has undergone several aspects which are:


 * Development from communication:

The first appearance of quantum theory was proposed by using a formulation of optical communications by using quantum mechanics.


 * Development from atomic physics and relativity:

During the 1970s there was development of techniques for manipulating single atom states such as atom trap and scanning tunneling microscope which made possible to isolate atoms and arranging them in single array.


 * Development from cryptography:

Bennett and Brassard developed a communication channel on which it is impossible to eavesdrop without being detected, a way of communicating secretly at long distances using the BB84 quantum cryptographic protocol.


 * Development from computer science and mathematics:

When the first computers are made it grew at a faster rate and the transistors were becoming less in size with time so as to accommodate more power surface area then the quantum effects started to show, which led to quantum computing using quantum mechanics for algorithm design.


 * Development from information theory:

During this time Shannon developed two fundamental theorems of information theory: Noiseless channel coding theorem and noisy channel coding theorem. He also showed that error correcting codes could be used to protect information being sent.

QUANTUM INFORMATION SCIENCE (QIS)
Quantum Information Science (QIS) is an interdisciplinary field that merges principles of quantum mechanics with information theory to develop new technologies for computing, communication, and cryptography. It leverages quantum phenomena like superposition, entanglement, and quantum tunneling to process and transmit information in ways that classical systems cannot achieve. Here’s an overview of its key components and applications:

Core Concepts

1. Qubits

- The basic unit of quantum information. Unlike classical bits that can be either 0 or 1, qubits can exist in a superposition of both states simultaneously, described by a quantum state

|Ψ> = α|0〉+ β|1〉

here α and β are complex numbers.

2. Superposition

- Enables a qubit to represent multiple states at once, allowing quantum computers to perform many calculations in parallel.

3. Entanglement

- A phenomenon where qubits become interdependent such that the state of one qubit instantly influences the state of another, no matter the distance separating them. This property is crucial for quantum communication and computing.

4. Quantum Gates

- The building blocks of quantum circuits. They manipulate qubits through unitary operations, analogous to logic gates in classical computing but capable of creating and manipulating superposition and entanglements.

5. Quantum Measurement

- The process of observing a quantum state, which causes the superposition to collapse to a definite state. Measurement is probabilistic, with outcomes determined by the quantum state's probability amplitudes.

Key Technologies

1.   Quantum Computing

- Utilizes quantum bits and quantum gates to perform computations. Quantum computers have the potential to solve certain problems exponentially faster than classical computers. Notable models include:

    - Quantum Circuit Model: Uses a sequence of quantum gates to perform calculations.

 - Adiabatic Quantum Computing: Solves optimization problems by evolving the ground state of a quantum system.

- Topological Quantum Computing: Utilizes anyons and topological states of matter for fault-tolerant computation.

2.   Quantum Communication

- Involves the transmission of quantum states over distances. Key applications include:

    - Quantum Key Distribution (QKD): Allows secure communication by generating encryption keys through the exchange of entangled particles, ensuring that any eavesdropping attempts are detectable.

    - Quantum Teleportation: Transfers quantum states between distant locations using entanglement, preserving the integrity of the information.

3. Quantum Cryptography

- Exploits quantum principles to create secure cryptographic systems. Quantum algorithms, such as Shor’s algorithm for factoring large integers, pose a threat to classical cryptographic schemes, prompting the development of quantum-resistant cryptography.

Post-quantum cryptography, also known as quantum-resistant cryptography, is a burgeoning field dedicated to creating cryptographic methods resilient to potential attacks from quantum computers. The advent of sufficiently powerful quantum computers poses a significant threat to the security of many current cryptographic schemes. This threat arises from the ability of quantum computers, as demonstrated by Shor's seminal work, to efficiently solve certain mathematical problems upon which many cryptographic protocols rely, such as factoring large numbers and computing discrete logarithms in abelian groups.

While public key cryptography is particularly vulnerable to quantum attacks due to its reliance on these hard-mathematical problems, symmetric cryptography initially appears less threatened. Grover's algorithm, a quantum algorithm for searching unstructured databases, provides only a quadratic speedup compared to classical exhaustive search, suggesting that doubling the key length of symmetric ciphers could counteract its impact, maintaining security against quantum adversaries.

However, recent research has shown that Grover's algorithm may not be the only concern for symmetric cryptography. Works by Kuwakado and Morii demonstrated that constructions like Even-Mansour could be broken in polynomial time in the quantum chosen-plaintext attack (CPA) setting. The Even-Mansour construction involves a public permutation and two secret keys used for encryption, and exploiting its structure, researchers showed that Simon's quantum algorithm could be applied to compute the period in polynomial time, compromising security.

Numerous other studies have followed, revealing vulnerabilities in symmetric cryptosystems based on quantum period finding and other quantum algorithms. These findings suggest that the security of some symmetric primitives, depending on the quantum adversary model, may fall significantly below the threshold provided by Grover's algorithm alone.

In response, efforts in post-quantum cryptography focus on understanding and enhancing quantum algorithms, as well as designing new attacks, to inform the development of quantum-safe cryptographic primitives. Only through this iterative process can we effectively fortify cryptographic systems against the threats posed by quantum adversaries.


 *  Prominent Algorithms 

1. Shor's Algorithm

- Efficiently factors large integers, which could break widely-used cryptographic schemes like RSA. It demonstrates the potential of quantum computers to solve problems intractable for classical computers.

2. Grover's Algorithm

- Provides a quadratic speedup for unstructured search problems, enabling faster database searches and optimization tasks.


 *  Quantum Error Correction 

- Quantum systems are highly susceptible to errors due to decoherence and noise. Quantum error correction codes (QECC) are essential for maintaining coherence and enabling reliable quantum computation. These include:

- Shor Code

- Steane Code

- Surface Codes


 *  Applications and Impact 

1. Material Science and Chemistry

- Quantum simulations can model molecular structures and reactions with high precision, aiding in the development of new materials and drugs.

2. Optimization Problems

- Quantum algorithms can efficiently solve complex optimization problems in logistics, finance, and machine learning.

3. Cryptography and Cybersecurity

- Quantum technologies promise new methods for secure communication and challenge existing cryptographic systems, driving the development of quantum-resistant encryption methods

4. Artificial Intelligence and Machine Learning

- Quantum computing has the potential to accelerate machine learning algorithms, offering faster data processing and enhanced learning capabilities.


 *  CLASSICAL INFORMATION AND QUANTUM INFORMATION 

Classical information and quantum information are two fundamentally different paradigms in information theory and computing, each with distinct properties, storage, and processing methods. Here's a comparison of the two:

Classical Information

Basic Unit: Bit

- Definition: The basic unit of classical information is the bit, which can have one of two values: 0 or 1.

- Representation: Bits are typically represented by physical states like voltage levels in a circuit, magnetic orientation in a storage medium, or any two-state system.

Operations and Logic Gates

- Classical Gates: Logical operations on bits are performed using gates such as AND, OR, NOT, XOR. These gates follow deterministic rules.

- Processing: Classical computers process information sequentially or in parallel using these gates to perform computations.

Storage and Transmission

- Reliability: Bits can be stored and transmitted reliably using classical error-correction techniques.

- Redundancy: Information is often duplicated or encoded with error-correcting codes to ensure integrity.

Information Theory

- Shannon Entropy: Measures the amount of uncertainty or information content. It is defined for a set of possible outcomes with probabilities.

- Classical Channel Capacity: The maximum rate at which information can be reliably transmitted over a communication channel, as defined by Shannon's theorem.

Quantum Information

Basic Unit: Qubit

- Definition: The basic unit of quantum information is the qubit, which can exist in a superposition of states 0 and 1 simultaneously.

- Representation: A qubit is represented by a quantum state

|Ψ〉 = α |0〉 + β |1〉

where α and β are complex numbers such that |α|2+ |β|2 = 1.

Properties of Qubits

- Superposition: A qubit can represent both 0 and 1 at the same time, enabling quantum computers to process a vast amount of information simultaneously.

- Entanglement: Qubits can be entangled, creating correlations between them that are stronger than any classical correlations.

- Quantum Interference: Quantum algorithms can exploit interference between different paths of computation to amplify correct results and cancel out incorrect ones.

Operations and Quantum Gates

- Quantum Gates: Operations on qubits are performed using quantum gates such as the Hadamard (H), Pauli-X, Pauli-Y, Pauli-Z, and CNOT gates. These gates are unitary, meaning they preserve the total probability.

- Reversibility: Quantum operations are inherently reversible due to the unitary nature of quantum gates.

Measurement

- Collapsing States: Measuring a qubit collapses its state to either 0 or 1, probabilistically based on the state’s amplitudes. This collapse destroys the superposition.

- Non-Deterministic: Measurement outcomes are inherently probabilistic, unlike classical measurements.

Quantum Information Theory

- Von Neumann Entropy: Analogous to Shannon entropy for classical systems, it measures the quantum information content and uncertainty of a quantum state.

- Quantum Channel Capacity: Defines the maximum rate at which quantum information can be reliably transmitted over a quantum channel, considering quantum effects like entanglement

 Key Differences 

1. Information Representation

- Classical: Information is binary and deterministic.

  - Quantum: Information is stored in qubits that can be in superpositions and entangled states.

2. Computation and Algorithms

  - Classical: Algorithms are based on deterministic or probabilistic Turing machines. Examples include sorting algorithms, search algorithms, and arithmetic computations.

  - Quantum: Quantum algorithms exploit superposition and entanglement. Examples include Shor’s algorithm for factoring, Grover’s algorithm for search, and quantum simulation algorithms.

3. Error Correction

  - Classical: Classical error correction uses redundancy and checksums to detect and correct errors.

- Quantum: Quantum error correction is more complex, involving the use of entanglement and redundancy to protect quantum information without directly measuring it. Examples include the Shor code and surface codes.

4. Communication

  - Classical: Information is transmitted through classical channels, such as electrical signals, optical fibers, or radio waves.

- Quantum: Quantum communication uses quantum channels, employing photons and quantum entanglement. Techniques include Quantum Key Distribution (QKD) and quantum teleportation.

5. Security

- Classical: Security is based on computational hardness assumptions (e.g., RSA, which relies on the difficulty of factoring large integers).

- Quantum: Quantum cryptography promises theoretically unbreakable security based on the laws of quantum mechanics (e.g., QKD ensures that any eavesdropping attempt can be detected)

Applications

- Classical Information Systems: General-purpose computing, classical cryptography, data storage, internet communications.

- Quantum Information Systems: Quantum computing (solving specific problems faster than classical computers), quantum cryptography (secure communication), quantum simulations (modeling physical systems at the quantum level).

QUANTUM INFORMATION : THEORY & PROPONENTS
In a quantum mechanical experiment, quantum information is the type of data that is transferred by quantum systems from the preparation apparatus to the measuring apparatus.

Thus, a quantum information "transmitter" is essentially a device that prepares quantum particles, while a quantum information "receiver" is just a measuring device. This is something different from the viewpoint of Classical information theory. According to that theory, information is not concerned with its physical medium, else one would also need to make distinctions between "electro dynamical information," "printed information," "magnetic information," and a host of other terms. The key to the success of classical information theory is to focus on the general principles that underpin all information flow rather than the actual carrier.

It is evident from a brief analysis that the successful theory resulting from the abstraction from the physical carrier of information is related to the flexibility with which information may be converted between all those carriers.

Quantum information is new kind theory. We can see for a successful translation, what it needs. Suppose we are going to convert quantum information into classical information. For this conversion, as input quantum system is going to be used and we will get output of classical information. But for the converse one, i.e. from classical information to quantum information, there need preparation of quantum systems. The preparing device's parameters are controlled by the classical input, and it is acceptable for the preparation process to depend on classical information.

From the above discussed two mechanisms, we can constitute two type of elements. First one is like Classical information to Quantum to again Classical information. This type of operation is generally common.

But for the converse part, there is a concept called ‘Classical Teleportation’ has arrived. This type is about Quantum information to Classical and again Quantum information. From the first paragraph, we can say that we need a measuring device say A and it operates with some quantum mechanism. After that, the measurement data are supplied into a preparation device P, which generates the combined device's final result. Setting everything up so that the combined device's outputs and the quantum inputs can't be distinguished.

It is necessary to define "indistinguishable" exactly. This obviously cannot represent that "the same" system is output on the other end. This is also not required in the classical instance. In quantum mechanics’ language, this can only mean that a statistical test cannot detect a difference.

We will always obtain the same probability distribution of results as if the inputs were directly observed, regardless of how well the input systems are prepared or what observable we measure on the teleportation device's outputs. Also take note of the fact that this criterion only considers states as the distribution parameters of ensembles of similarly prepared systems, not the states of individual systems.

Here, we are concerned with transmission issues rather than issues with meaning or content. This is precisely how traditional information theory works. There, too, it is frequently difficult to avoid confusion with the other, more commonly used definition of "information," which is the type that is offered at an information desk. Information theory can tell all about what is required to ensure the technical quality of the final visuals, regardless of whether a TV channel is utilized for "misinformation." Therefore, the quantitative measures of "information" are all related to the capacity for transmission and storage, to the possibilities for error correction and compression, and others. Similarly, the meaning of a "quantum message" is not understandable by quantum information theory, and it is presumably useless anyhow. However, quantum information theory provides exact ideas about the resources required for reliable transmission of this kind of data.

In the context present-day physics, quantum information transmission is by no means an alien concept. However, the information metaphor is helpful because it generates new questions, leads to quantitative concepts where there was only a qualitative understanding previously, and offers new applications. It's also possible that this will help us to understand the long-standing mysteries surrounding the basic principles of quantum mechanics.

Fundamentally, quantum theory offers a mathematical representation of a reality that differs significantly from our macroscopic experience. We'll talk about how some aspects of this mathematical description have been used in LIS.


 * Indeterminism/probability: Quantum theory provides only probabilistic results rather than precise deterministic predictions; additionally, quantum probabilities differ from classical probabilities;


 * Complementarity/duality: Quantum systems can exhibit two distinct and incompatible natures simultaneously; the most well-known example of this is the wave/particle duality;


 * Measurement: Any measurement can impact and modify a quantum system; superposition: Quantum systems can exist in multiple states simultaneously;


 * Entanglement: The properties of two or more quantum systems become correlated;


 * Non-locality: a component of a quantum system can be instantly impacted by modifications to a distant part of the system.


 * Interference: Interaction that cannot occur in a classical system between two quantum systems or between a system and itself


 * Contextuality: A quantum system's nature varies based on its context;
 * Indeterminacy/uncertainty: There is a limit to what can be understood about a quantum system.

Over time, many scientists created and built upon quantum theory, which profoundly changed our understanding of physics and the nature of reality. The following are a few of the most important contributors:


 * Max Planck:

Max Planck proposed that energy exists in discrete packets called quanta, which he published in 1900. This implies that only quantized amounts of energy can be transported. This is a fundamental principle of Planck's quantum theory that holds true for all types of radiation. Scientists started applying Planck's quantum theory to systems other than radiation in the 1920s and 1930s, including heat and electricity. The Planck Constant, 6.62 x 10-34 J·s·kelvin (where J·s·Kelvin is the unit of energy equal to 1/1 second and kilo-electron volts is the unit of energy equal to 1/1 electron volt), is one of the most significant features that were measured using quantum theory.


 * Albert Einstein:

The main figure behind the quantum theory of light was Einstein. In 1905, he postulated and clarified that light is made up of energy packets called photons. When he suggested that electromagnetic radiation is granular and made up of quanta with an energy of hf, he provided the physical interpretation for Planck's mathematical results. He also clarified how intense electric pulses, such as those produced by lightning, cause metals to release electrons.


 * Niels Bohr:

Based on the idea that certain physical quantities can only have discrete values according to quantum theory, Niels Bohr put up a theory for the hydrogen atom in 1913. Only in their designated orbits can electrons move around a nucleus; if they deviate from this path, the energy difference is released as radiation


 * Werner Heisenberg:

One of the key ideas in quantum physics is the uncertainty principle, sometimes referred to as Heisenberg's indeterminacy principle. It asserts that the precision with which some pairings of physical attributes, like position and momentum, may be determined at the same time is finite. Stated otherwise, the greater the precision of one measurement, the lower the accuracy of the other attribute.

Formally speaking, the uncertainty principle is any one of several mathematical inequalities that state that the product of the accuracy of some related pairs of observations on a quantum system, like position, x, and momentum, p, has a fundamental limit. Complementary variables or canonically conjugate variables are the terms used to describe such paired variables


 * Erwin Schrödinger:

A quantum-mechanical system's wave function is determined by the linear partial differential equation known as the Schrödinger equation. An important turning point in the evolution of quantum mechanics was reached with its discovery. It bears Erwin Schrödinger's name, who proposed the equation in 1925 and published it in 1926, providing the foundation for the research that led to his 1933 Nobel Prize in Physics. The Schrödinger equation can be thought of as the quantum equivalent of Newton's second law in classical physics. Newton's second law predicts mathematically the course a given physical system will take over time given a set of specified initial conditions. The quantum-mechanical description of a single physical system can be obtained from the Schrödinger equation, which also provides the evolution of a wave function over time. Schrödinger derived the equation using Louis de Broglie's postulate, which states that every matter has a corresponding matter wave. The atom's bound states were predicted by the equation and agreed with experimental findings


 * Paul Dirac:

Paul Dirac developed a completely relativistic quantum theory in 1928. He deduced from the equation's solutions that a positively charged particle similar to the electron was the source. The existence of this particle—the positron—was subsequently verified by experiments


 * Max Born:

According to Niels Bohr's theory of the atom, when electrons move between orbits around a nucleus, they absorb and release radiation at defined wavelengths. The hypothesis needed to be refined to account for more complex atoms and molecules, but it gave a decent description of the spectrum produced by the hydrogen atom. Max Born made contributions to the advancement of quantum mechanics after Werner Heisenberg's pioneering work about 1925. Additionally, he demonstrated that Schrödinger's wave equation might be understood to provide statistical predictions of variables rather than precise ones.


 * Wolfgang Pauli:

The Pauli exclusion principle in quantum mechanics asserts that in a system that complies with the laws of quantum mechanics, two or more identical particles with half-integer spins, or fermions, cannot concurrently occupy the same quantum state. Wolfgang Pauli, an Austrian scientist, first proposed this idea in 1925 for electrons. His 1940 spin-statistics theorem later expanded it to include all fermions.

All fermions or particles with half-integer spin behave according to the Pauli exclusion principle; bosons, or particles with integer spin, follow different rules. Examples of fermions are elementary particles, including quarks, electrons and neutrinos. In addition, certain atoms (such as helium-3) and baryons (subatomic particles composed of three quarks), such as protons and neutrons, are fermions and thus subject to the Pauli exclusion principle. The total surface area of ​​an atom can distinguish it from a boson or fermion. For example, helium-3 has a spin of 1/2 and is a fermion, while helium-4 has a spin of 0 and is a boson.


 * Richard Feynman:

Science's understanding of the nature of waves and particles has changed as a result of Feynman's reconstruction of quantum electrodynamics, the theory governing the interaction of light and matter. This study, which brought together all the various phenomena involved in light, radio, electricity, and magnetism in an experimentally flawless package, earned him a share of the 1965 Nobel Prize in Physics. Although Julian S. Schwinger of the United States and Tomonaga Shin'ichirō of Japan, the other two Nobel Prize winners, had separately developed similar theories, Feynman's turned out to be the most inventive and comprehensive. In the latter part of the 20th century, Feynman diagrams, which are visual depictions of particle interactions, were among the problem-solving instruments he created and were widely used in many branches of theoretical physics


 * John von Neumann:

John von Neumann stated that the collapse of the wave function can occur at any point in the causal chain from the measurement device to the "subjective perception" of the human observer in his 1932 book The Mathematical Foundations of Quantum Mechanics. Edmond Bauer and Fritz London argued for the latter barrier (awareness) in 1939. Eugene Wigner rephrased the "Schrödinger's cat" thought experiment as "Wigner's friend" in the 1960s and postulated that, regardless of realist interpretation, observer consciousness is the demarcation line that triggers the collapse of the wave function. His work on the Hilbert space formalism and operator theory is fundamental to the mathematical structure of quantum mechanics.


 * David Bohm:

One of the most influential theoretical physicists of the 20th century, David Joseph Bohm was an American, Brazilian, and British scientist who made unconventional contributions to quantum theory, neuropsychology, and the philosophy of mind. De Broglie-Bohm theory, a causal and deterministic explanation of quantum theory, is one of his several contributions to physics


 * Hugh Everett:

A philosophical viewpoint regarding the relationship between physical reality and the mathematics utilized in quantum physics is known as the many-worlds interpretation, or MWI. It states that there is no wave function collapse and that the universal wave function is objectively genuine. This suggests that in some "world" or reality, every possible result of quantum measurements is physically realized. In MWI, the evolution of reality as a whole is strictly deterministic and local, in contrast to certain alternative views. The concept of many-worlds is also known as the Everett interpretation or the relative state formulation, following physicist Hugh Everett, who initially put it forth in 1957. In the 1970s, Bryce DeWitt popularized the formulation and gave it the moniker "many-worlds"

FRAMEWORK OF QUANTUM INFORMATION RETRIEVAL
To discuss about the framework of Quantum Information Retrieval, first we need to know about the Quantum Probability Formalism. Quantum Probability Formalism uses ‘Hilbert Spaces’, ‘unit vectors’, ‘vector subspaces’ with standard theory of probability. Therefore, we should have a clear idea about these three.


 * Vector space and Sub-spaces

A non-empty set V is said to form a vector space over a field F if,

(i) there is a binary composition + on V, called 'addition', satisfying the conditions

V1. α + β ∈ V for all α, β ∈ V,

V2. α + β = β + α for all α, β ∈ V,

V3. α + (β + γ) = (α + β) + γ for all α, β, γ ∈ V

V4. there exists an element θ in V such that α + θ = α for all α ∈ V,

V5. for each α in V there exists an element – α in V such that,

α + (- α) = θ;

(ii) there is an external composition of F with V, called 'multi- plication by the elements of F' satisfying the conditions

V6. cα ∈ V for all c ∈ F and all α ∈ V,

V7. c(dα) = (cd) α for all c, d ∈ F and all α ∈ V,

V8. c (α + β) = cα+ cβ for all c ∈ F and all α, β ∈ V,

V9. (c + d) α= cα+ dα for all c, d ∈ F and all α ∈ V,

V10. 1α= α for all α ∈ V, 1 being the identity element in F.

A non-empty subset W of a vector space V over a field is a subspace of V if and only if

i.  a ∈ W, β ∈ W ⇒ α + β ∈ W, and

ii. a ∈ W, c ∈ F ⇒ cα ∈ W.


 * Proof.

Let the conditions hold in W.

Let α, β ∈ W. Since F is a field, -1 ∈ F where 1 is the identity element in F.

By (ii) we have, -1β ∈ W, i.e., -β ∈ W.

Then by (i) we have α + (- β) ∈ W, i.e., α – β ∈ W

Thus α, β ∈ W ⇒ α – β ∈ W

This proves that W is a subgroup of the additive group V. Since V is a commutative group, W is also a commutative subgroup of V.

Therefore, the conditions V1-V5 for a vector space are satisfied in W. V6 is satisfied in W by (ii). The conditions V7-V10 are satisfied in W, since they are hereditary properties. Thus W is by itself a vector space over F and so W is a subspace of V.

The necessity of the conditions (i) and (ii) follows from the definition of a vector space


 * Unit Vector

A vector α in a Euclidean Space V is said to be a unit vector if ||α|| = 1 If α be a non-zero vector in V, then (1/||α||) * α is a unit vector


 * Hilbert Spaces

In mathematics, the techniques of linear algebra and calculus can be extended from (finite-dimensional) Euclidean vector spaces to potentially infinite-dimensional spaces thanks to Hilbert spaces, which bear David Hilbert's name. In mathematics and science, Hilbert spaces naturally occur and are often encountered as function spaces. Formally, a vector space with an inner product that generates a distance function for which the space is a complete metric space is called a Hilbert space

A Hilbert space is an arbitrary large number of dimensions that generalizes the well-known concept of three-dimensional Euclidean space. Since it contains the ideas of magnitude and direction, it is known as a vector space. Points in such a space indicate all potential states that a system may be in.

The characteristics that set a Hilbert space apart from other abstract spaces make it the perfect mathematical setting for a quantum-like formalism. Given that the same formalism applies to both, it makes sense to think that IR could contain analogs for quantum phenomena. The connection between the two is probability, which is a key topic in both quantum theory and in both IR

Quantum theory talks about how the behavior of the matter takes place. It talks about atomic scales and also the sub-atomic scales of the matter which may be in a state space.

The range of physical sizes that includes items smaller than an atom is known as the subatomic scale. It is the scale at which the components of an atom, such as its protons and neutrons-containing nucleus and the electrons in their orbitals, become visible

In physics, a quantum state space is an abstract space in which different "positions" represent, not literal locations, but rather quantum states of some physical system. This state space can be called as a Hilbert space, and here we denote it with “H’. A Unit vector’ in ‘state space’ denotes the state of the system and we can call it as ‘state vector’. In this case, the probability that a particle is at a given position determines the state vector. A system state may be completely known or may be unknown. If it is known, we can use a state vector, but if it is unknown then we need more than one state vector to represent a set of state vectors or can say possible state vectors, each of which has probability to be occurred


 * Single-part Systems


 * 1) States and Probabilities : Consider a state space, H, and a one-particle state vector, denoted by ϕ, and a probabilistic event is presented as a subspace S of the state space H. Then a state ϕ passes on a kind of definite “probability distribution” over events. The likelihood of an event is a square of length of the projection of ϕ onto the regular space for the certain event subspace S, it means by computing the value || S1 ϕ ||2. Where S1 stands for the projection onto the subspace S. This value is identified as the probability of the event S with respect to the distribution specified by ϕ.
 * 2) Uncertain States and Weighted Sets : In this case, we will use ‘ensembles’ for our purpose. It denotes the possible state vectors’ set. A weighted set is a function of V that associates a weight V to each of its elements, the state vectors ϕ. We assume that the weight V of ϕ is the probability of the system that is, for ϕ. If the weight of the state ϕ is non-zero, then ϕ belongs to V.

For notations, as usual, all operations on function spaces are standard. Thus, we add two weighted sets: V1 + V2, or scale by a factor αV, with α a constant. We denote ϕ → w, the weighted set that associates w to ϕ and 0 to the other vectors.

As states are mutually exclusive since a system can’t be in two different states at the same time, we need that a weighted set representing a system has weights that sum up to 1.

If the system state is uncertain & the weighted set V containing of possible state vectors, we can define that the probability of an event S as,

Pr (S|V) = ∑ V (ϕ) Pr (S| ϕ), for all ϕ € V

= ∑ V (ϕ) ||S1 ϕ||2  …………… (i)

= tr (ρv S1)

Where, ρ = ∑ V (ϕ) ϕ ϕT, for all ϕ € V

Here, ‘tr’ is the trace operator, ϕT is the transpose of ϕ and ρv is the density operator.

Equation (i) reduces to || S1 ϕ ||2, if V consists of only one vector ϕ, i.e. when there is no uncertainty about the system state.


 * Multi-part Systems

Multi-part systems are those which are made of single part systems. In this case, there are multiple particles. Each of the particle is denoting a single part system. Combining all of them we get the result.

In this case we need to have concept of tensor product and we will use that in case of Hilbert spaces, i.e. the state spaces for the systems. Tensor product is denoted by ⊗.

Suppose there are two Hilbert spaces namely, H1 & H2. The dimension of these space be ‘n’, & ‘m’. Dimension denotes the number of unit vector in each spaces. Then the tensor space will be H1 ⊗ H2 and the dimension of that space will be ‘n*m’. It is also a Hilbert space with ‘n*m’ number of unit vectors.

We will now use the property of vector space. Let ϕ1 be a unit vector in H1 and ϕ2 be a unit vector in H2, then ϕ1 ⊗ ϕ2 will be a vector in H1 ⊗ H2.

If X and Y be two subspaces in H1 & H2 respectively, then X ⊗ Y will be a subspace in H1 ⊗ H2. It will also happen for the projection mapping too. The projection of ϕ1 ⊗ ϕ2 onto X ⊗ Y is the tensor product of the two projected vectors. In this case, if one of the projections is null, then the result is the null vector in H1 ⊗ H2. The norm of a vector ϕ1 ⊗ ϕ2 is the product of the norms of the component vectors.

From Equation (i), we can find the probability of any event.

For example, the probability of the composite event X ⊗ Y is,

Pr (X ⊗ Y | ϕ1 ⊗ ϕ2) = Pr (X | ϕ1) × Pr (Y | ϕ2)

Where, ϕ1 is a state vector in the first state space and ϕ2 is a state vector in the second state space. This can be used for more than two spaces and any weighted set.

Mathematically we can see that, with ‘n’ number of spaces, the weighted set

⊗ Vi = V1 ⊗. . .⊗Vn, where each Vi is a weighted space.



(Source: )

……………… (ii)

where the weight of an element ⊗ ϕi ∈ ⊗ Vi is the product of the weights of its component elements.

From Equations (i) and (ii), the probability will be,

(Source: )

…………….. (iii)


 *  Implementation into the Quantum IR (QIR) 

Now we can use the discussed theory in the framework for Information Retrieval.

Information Need Spaces

Here we will assume that there exist a Hilbert space containing all the queries of users, i.e. there need for information. We can call this Hilbert space as Information Need Space. Now if a user wants a specific information, i.e. he needs knowledge about something, that need can be denoted through ϕ, which means pure information need.

Now, the collection of documents can denote a subspace in Information need space. A system can rank its documents as per relevancy order, if it knows the information need of users’. From the theory discussed earlier, we can easily conclude that if the state vector denoting the query contained in the document sub-space, the query or need of information by the user can be served. Again the query partially answered if there is a non-null projection on to the document sub-space.

Let us consider a subspace be Sd containing the documents for users’ need for information and if Sd1 be the projector, then from the equation (iii), we can find the probability of document’s relevancy.


 * Aspect Spaces and Multi Part Systems

Sometimes there are several aspects of users’ query, that there should be multiple needs to answer a single need. For example, we can say that, what were the two batsmen of India at the moment of winning cricket world cup? It can be easily seen that there are actually more than one aspects. One is to know about the batsmen and another is winning cricket world cup. For this purpose, the IR system will search two different sub-space containing results about two different queries. Here we can say each sub-space as IN aspect space. IN means information need and the information need that is the queries are called pure IN aspects. The asked batsmen are MS Dhoni and Yuvraj Singh. Therefore, the pure IN aspects will be the vectors that represent ‘MS Dhoni’ and ‘Yuvraj Singh’.

Now for a multi part system, if we divide the whole information need of any user into several parts and let us say that each part is answered by different document sub-space then, for the total query, we can use the tensor product which we have already discussed earlier. The total information need space will be denoted as the tensor product ⊗ Hi =  H (say)

Now in an Information need space, H, a document will be represented by S, where S will be equal to ⊗ Sd and Sd is the sub-space containing the documents in topical spaces of each Hi. Here we are using the theory which we have already discussed earlier.

Therefore, the actual information need is basically denoting the weighted set

V = ⊗ Vi, where each Vi denotes here the corresponding users’ information need aspect space. The probability of document’s relevance has already discussed in formulating the equation (iii).

QUANTUM MACHINE LEARNING FOR INFORMATION

 * What Is Machine Learning?

Machine Learning is a discipline of artificial intelligence that provides machines with the capability to learn by themselves from data and past experiences while recognizing patterns for predicting output with less human involvement.

Machine learning methods allow computers to operate by themselves with no explicit programming. ML applications are fed with new data, and they can learn, grow, develop, and adapt on their own. Machine Learning derives insightful information from large quantities of data by leveraging algorithms to recognize patterns and learn in an iterative manner. ML algorithms leverage computation methods to learn directly from data instead of relying on any pre-set equation that may serve as a model.

The performance of ML algorithms improves adaptively with an increase in the number of available samples during the 'learning' processes. For instance, deep learning is a sub-domain of machine learning that trains computers to imitate natural human attributes like learning from examples. It offers better performance parameters than conventional ML algorithms.

While machine learning is not a new concept – the idea dating back to World War II with the implementation of the Enigma Machine, the ability to apply complicated mathematical calculations by itself to growing volumes and varieties of available data is a relatively recent development. .


 * Quantum Machine Learning

Quantum computing uses quantum mechanics to perform calculations. While a classical computer works on bits, which could exist at either 0 or 1, a quantum computer works on qubits, which can exist in multiple states simultaneously. Therefore, this permits the quantum computer to handle special categories of calculations at a very high speed in contrast to classical computers, which include tasks associated with optimization and machine learning or cryptography.

However, this is a tremendous technical challenge in the building of quantum computers because qubits are really sensitive to the noise of the environment, and error correction techniques at their best are complex. Even with these challenges, the potential applications of quantum computing in areas such as drug discovery, materials science, and artificial intelligence are great and therefore of immense interest.

Quantum computing has the potential to revolutionise many fields by solving problems that are currently intractable using classical computing. There is a huge scope quantum computing in assorted domains, and it has the potential to impact nearly every field of science and technology. However, the technology is still in its early stages of development, and there are significant challenges to be addressed in terms of hardware limitations, error correction, and algorithm development, as well as scalability and reliability. Nonetheless, with continued research and development, quantum computing could lead to significant breakthroughs and advancements in many areas of science and technology.

In 2020, Google claimed to have achieved quantum supremacy with its Sycamore quantum computer, which took just 200 seconds to complete a calculation that would have taken the world's fastest super computer 10,000 years to complete.

Other major players in quantum computing include IBM, Microsoft, Intel, and Honeywell, all of whom are working on their own quantum computing technologies and making them available to researchers and developers via cloud-based services.


 * Key advantages of quantum computing

Speed: Quantum computers can solve certain problems much faster than classical computers, especially those related to data optimization, machine learning, and cryptography.

Parallelism: Quantum computing allows for massive parallelism, which means that many calculations can be performed at the same time.

Quantum superposition: Quantum computing can make use of quantum superposition, which allows quantum bits (qubits) to exist in multiple states simultaneously. This allows for more complex calculations and faster problem-solving.

Quantum entanglement: Quantum computing can also make use of quantum entanglement, which allows qubits to be connected in such a way that the state of one qubit affects the state of the others. This can use to perform certain types of calculations much faster than classical computing.

Improved accuracy: Quantum computing can offer improved accuracy over classical computing in certain calculations, such as in the simulation of chemical reactions and the modelling of financial markets.

Security: Quantum computing can potentially offer improved accuracy over classical computing in areas such as cryptography, as certain algorithms that are difficult to break using classical computing can be easily broken by quantum computing.

Innovative applications: Quantum computing is a new and rapidly evolving field, with the potential for a wide range of innovative applications in areas such as drug discovery, materials science, and artificial intelligence.


 * How does quantum computing work?

Quantum computing is a highly specialized field that requires expertise in quantum mechanics, computer science, and electrical engineering. Here is a general overview of how quantum computing works:

Qubit (Quantum bit):

A qubit (or quantum bit) is the quantum mechanical analogue of a classical bit. In classical computing the information is encoded in bits, where each bit can have the value zero or one. In quantum computing the information is encoded in qubits. While classical bits can only have a value of either 0 or 1, qubits can exist in both states at the same time.

Quantum Gates:

Quantum gates are operations performed on qubits that enable the manipulation of the state of the qubits. They have direct analogs in classical logic gates but are distinct due to some important features of quantum mechanics. Quantum gates are operations performed on qubits that enable the manipulation of the state of the qubits. As opposed to classical gates, quantum gates can operate on qubits in superposition.

Quantum Circuits:

Quantum circuits are very similar to classical circuits, constructed from a sequence of gates that operate on qubits. However, while classical circuits operate only on single bits at a time, quantum circuits can operate on many qubits simultaneously due to the property of entanglement.

Quantum Algorithms:

Quantum algorithms are algorithms designed to run in quantum computers. They are typically devised to harness the unique properties of qubits and quantum gates to achieve calculations far more quickly than classical algorithms

Quantum Hardware:

Quantum hardware is the physical realization of a quantum computer. Currently, there are several different varieties of quantum hardware, including superconducting qubits, ion trap qubits, and topological qubits.


 * Quantum computing algorithms

Quantum computing algorithms leverage the principles of quantum mechanics to perform computational tasks in ways that traditional computers cannot.

1.   Quantum Support Vector Machines (QSVM):

QSVM is an algorithm used for classification tasks, which is an extension of classical support vector machines (SVMs). It aims to find the optimal hyperplane that separates data points into different classes. Unlike classical SVMs, QSVM utilizes quantum algorithms to process and classify data. This algorithm has the potential to handle large datasets more efficiently than classical SVMs by exploiting quantum parallelism and interference.

2.   Quantum Neural Networks (QNN):

QNNs are quantum counterparts of classical neural networks. They leverage the principles of quantum mechanics, such as superposition and entanglement, to perform computations. QNNs consist of quantum gates and quantum circuits that process quantum information, enabling them to potentially solve certain computational tasks more efficiently than classical neural networks. QNNs are still in their early stages of development but hold promise for tasks like pattern recognition, optimization, and machine learning.

3.   Quantum Annealing:

Quantum annealing is a quantum computing technique used to solve combinatorial optimization problems. It's based on the principles of adiabatic quantum computing, where a system starts in a simple Hamiltonian (the energy function of the system) and gradually evolves into a more complex Hamiltonian that represents the optimization problem to be solved. Quantum annealing aims to find the lowest energy state of a system, which corresponds to the optimal solution of the optimization problem. This technique has been applied to various problems such as the traveling salesman problem, protein folding, and portfolio optimization.


 * Some of the potential applications of quantum computing are listed below

Cryptography: Quantum computers have the potential to break many of the cryptographic algorithms that are used to secure sensitive data. However, they can also be used to develop new, more secure encryption methods.

Optimization: Many real-world optimization problems, such as supply chain management and logistics, are extremely difficult to solve with classical computers. Quantum computers can provide faster and more efficient solutions to these problems.

Machine learning:  Quantum machine learning algorithms can be used to analyse and classify large amounts of data more efficiently than classical algorithms.

Chemistry: Quantum computers can simulate the behaviour of molecules and chemical reactions more accurately than classical computers, which can lead to the development of new materials and drugs.

Financial Modelling: Quantum computing can be used to perform financial modelling and risk analysis more efficiently, enabling faster and more accurate predictions of financial outcomes.

Weather forecasting: Quantum computing can provide more accurate and precise weather forecasts by simulating complex weather patterns and climate models.

Particle physics: Quantum computing can be used to simulate particle interactions, and accelerate the development of new theories and technologies in particle physics.


 * Machine learning integration with quantum computing

Quantum machine learning (QML) is a field that explores the intersection between quantum computing and machine learning. It is focused on developing algorithms and techniques that can leverage the unique properties of quantum computing to improve the efficiency and accuracy of machine learning tasks.

Quantum computers use qubits (quantum bits) to perform operations that can solve certain problems exponentially faster than classical computers. This speed can be particularly advantageous for large scale data analysis tasks, such as those encountered in machine learning.

One of the main goals of quantum machine learning is to develop quantum algorithms that outperform classical machine learning algorithms for tasks such as classification, clustering, and regression. Proposed quantum machine learning algorithms include the quantum support vector machine (QSVM), quantum principal component analysis (QPCA), and quantum k-means.

One example of a quantum machine learning algorithm is the quantum approximate optimisation algorithm (QAOA), which is used to solve optimisation problems. QAOA is a hybrid algorithm that combines classical optimisation with quantum operations to find the optimal solution to a problem.

Another example of a quantum machine learning technique is a quantum-inspired classical algorithm. These algorithms are designed to mimic the behaviour of quantum systems using classical computers, with the potential for improved performance in certain tasks.


 * Platforms and libraries for quantum machine learning

QML is an interdisciplinary research area at the intersection of quantum computing and machine learning. In recent years, several libraries and platforms have emerged to facilitate the development of QML algorithms and applications. Here are some popular ones.

·      TensorFlow Quantum (TFQ)

https://www.tensorflow.org/quantum

TFQ is a library developed by Google that enables the creation of quantum machine learning models in TensorFlow. It provides a high-level interface for constructing quantum circuits and integrating them into classical machine learning models.

·      PennyLane

https://pennylane.ai/

PennyLane is an open source software library for building and training quantum machine learning models. It provides a unified interface to different quantum hardware and simulators, allowing researchers to develop and test their algorithms on a range of platforms.

·      Qiskit Machine Learning

https://qiskit.org/ecosystem/machine-learning/

Qiskit is an open source framework for programming quantum computers, and Qiskit Machine Learning is an extension that adds quantum machine learning algorithms to the toolkit. It provides a range of machine learning tools, including classical machine learning models that can be trained on quantum data.

·      Pyquil

https://pyquil-docs.rigetti.com/en/stable/

Pyquil is a library for quantum programming in Python, developed by Rigetti Computing. It provides a simple interface for constructing and simulating quantum circuits and allows for the creation of hybrid quantum-classical models for machine learning. Forest is a suite of software tools for developing and running quantum applications, also developed by Rigetti Computing. It includes Pyquil and other tools for quantum programming, as well as a cloud based platform for running quantum simulations and experiments.

·      IBM Q Experience

https://quantum-computing.ibm.com/

IBM Q Experience is a cloud based platform for programming and running quantum circuits on IBM's quantum computers. It includes a range of tools for building and testing quantum algorithms, including quantum machine learning algorithms. These are just some of the platforms and libraries available for quantum machine learning. As the field continues to grow, we can expect to see more tools and platforms emerge to support this exciting field of research.

QUANTUM ALGORITHMS AND SEARCHING DATABASE STRATEGY

 * Quantum algorithms principles

Quantum algorithms leverage the principles of quantum mechanics, such as superposition, entanglement, and interference, to solve problems more efficiently than classical algorithms. Here are some of the most well-known quantum algorithms:

1.   Shor's Algorithm

Purpose: Factoring large integers and computing discrete logarithms.

Significance: Exponentially faster than the best-known classical algorithms for these problems.

Applications: Breaking widely-used cryptographic schemes like RSA, which rely on the difficulty of factoring large numbers.

Principle: Utilizes quantum Fourier transform to find the period of a function related to the number being factored, leading to the determination of its factors.

Shor's algorithm provides an efficient method for integer factorization, which classical computers struggle with for large numbers. Here’s a mathematical explanation of the algorithm:

Mathematical Background

Problem Definition

Given an integer N, the goal is to find its non-trivial factors p and q such that

N = p ×q

Steps of Shor's Algorithm

1. Choose a Random Integer a

Select a random integer a such that 1 < a < N

2. Compute the Greatest Common Divisor (GCD)

Compute gcd (a, N). If gcd (a, N) ≠ 1, then a shares a factor with N and you are done (this factor is gcd (a, N). Otherwise, proceed to the next step.

3. Period Finding with Quantum Computing

The goal here is to find the period r of the function f(x) = ax mod N. The period r is the smallest positive integer such that:

  ar ≡ 1 (mod N)

4. Quantum Fourier Transform (QFT)

Use a quantum computer to perform the Quantum Fourier Transform on a superposition of states to find the period r.

Quantum Steps

Superposition

Initialize two quantum registers. The first register is put into a superposition of all possible states.

where Q is a power of 2, typically Q = 2n for some integer n (greater than log N)

Function Evaluation

Apply the function f on the second register:

f(x)⟩ = |a x mod N⟩

Quantum Fourier Transform

Apply the Quantum Fourier Transform (QFT) on the first register to find the period r.

Classical Post-Processing

5. Continued Fraction Expansion

Measure the first register to obtain a value y. This value will be of the form  where s is an integer and r is the period. Use continued fraction expansion to approximate  and determine r.

6. Factorization

Once r is found, use it to compute potential factors of N:

gcd (ar/2 – 1, N)

  gcd (ar/2 + 1, N)

If either gcd is a non-trivial factor, you have successfully factorized N.

Example: Factoring N = 15

Let's go through a simplified version of the algorithm to factor N = 15:

1. Choose a

Suppose a = 7

2. Compute GCD

gcd (7, 15) = 1

3. Quantum Period Finding

Find the period r of the function f(x) = 7x mod 15. The possible values of x modulo 15 are:

71 mod 15 = 7

72 mod 15 = 4

73 mod 15 = 13

74 mod 15 = 1

So, the period r is 4 because 74 ≡ 1 (mod 15)

4. Classical Post-Processing

Compute:

7r/2 – 1 = 72 – 1 = 49 – 1 = 48

7r/2 + 1 = 72 + 1 = 49 + 1 = 50

Now, find the GCD with N = 15:

gcd (48, 15) = 3

gcd (50, 15) = 5

Therefore, the non-trivial factors of 15 are 3 and 5

2. Grover’s Search Algorithm

Grover’s Search Algorithm is a quantum algorithm designed to search an unsorted database or solve unstructured search problems with quadratic speedup compared to classical algorithms. Here’s a detailed overview:

Purpose

Unstructured Search: Finding a specific item in an unsorted database.

Significance: Grover's algorithm provides a quadratic speedup, requiring ON queries to search a database of N entries, compared to O(N) queries required by classical algorithms.

Key Concepts

I. Quantum Superposition:

- The algorithm starts by placing the qubits in a superposition of all possible states, representing all potential solutions simultaneously.

'II. Oracle Function:'

- An oracle is a quantum subroutine used to mark the correct solution. When the oracle is applied, it flips the phase of the correct solution (target state) while leaving other states unchanged.

'III. Amplitude Amplification:'

- Grover’s algorithm amplifies the probability amplitude of the correct solution, increasing the likelihood of measuring the target state.

Algorithm Steps

i. Initialization:

- Begin with a register of n qubits, each initialized to |0〉.

- Apply the Hadamard gate to each qubit to create an equal superposition of all 2n possible states:


 * Ψ〉 = 1/√N ∑ (x=0)^(N-1) |x >

where N = 2n

'ii. Oracle Operation:'

- Apply the oracle O that inverts the phase of the target state |x0>:

'iii. Grover Diffusion Operator:'

- Apply the Grover diffusion operator (also known as the inversion about the mean) to amplify the amplitude of the target state:

D = 2| Ψ 〉< Ψ | - I

This step increases the probability amplitude of the correct solution while decreasing the amplitudes of incorrect solutions.

'iv. Iterate:'

- Repeat the oracle and diffusion operations O times. The optimal number of iterations is approximately

v. Measurement:

- Measure the qubits. The result will be the index of the target state with high probability.


 * Quantum Circuit Representation

i. Hadamard Gates: Initialize the qubits to create a superposition of all possible states.

'ii. Oracle:' A black-box function that flips the phase of the target state.

'iii. Diffusion Operator:' Sequence of gates implementing the inversion about the mean.

'iv. Repetition:' Apply the oracle and diffusion steps iteratively.

v. Measurement: Measure the final state to obtain the target index.

Example

Suppose you have a database of 8 elements, and you want to find a specific item. Classically, you might need to check each item, potentially up to 8 checks. Grover's algorithm will find the item with high probability in about  ≈ 3 iterations.

Applications

Search Problems: Finding a specific item in large datasets.

Optimization: Solving NP-complete problems by searching for the optimal solution.

Cryptography: Breaking symmetric cryptographic systems (e.g., finding the key in a brute-force search).

Advantages and Limitations

Advantages:


 * 1) Provides a significant speedup for search problems.
 * 2) Conceptually simple and widely applicable to various search and optimization tasks.

Limitations:


 * 1) Requires a well-defined oracle function for marking the correct solution.
 * 2) Only offers quadratic speedup, which is substantial but not exponential.

Purpose: Searching an unsorted database or solving unstructured search problems.

Significance: Quadratically faster than any classical algorithm for the same task.

Applications: Searching large databases, solving NP-complete problems, optimization problems.

Principle: Uses amplitude amplification to increase the probability of finding the correct result, effectively reducing the number of searches needed.

2.   Quantum Fourier Transform (QFT)

Purpose: Transforming a quantum state into its frequency domain, similar to the classical discrete Fourier transform but more efficient.

Significance: Integral to many quantum algorithms, including Shor's algorithm.

Applications: Signal processing, solving linear systems of equations, periodicity finding.

Principle: Maps quantum states to superpositions that represent the frequency components of the input state.

3.   Quantum Phase Estimation

Purpose: Estimating the phase (or eigenvalue) corresponding to an eigenvector of a unitary operator.

Significance: A fundamental subroutine used in many other quantum algorithms.

Applications: Eigenvalue problems, solving linear equations, factoring, simulation of quantum systems.

Principle: Combines QFT and controlled unitary operations to measure the phase of an eigenstate.

4.   Variational Quantum Algorithms (VQAs)

Purpose: Solving optimization problems and quantum simulations.

Significance: Suitable for near-term quantum devices (NISQ era).

Applications: Finding ground states of molecules (quantum chemistry), solving combinatorial optimization problems.

Principle: Uses a hybrid quantum-classical approach where a quantum computer evaluates a parameterized quantum circuit, and a classical computer optimizes the parameters.

5.   Quantum Annealing

Purpose: Finding the global minimum of a function, solving optimization problems.

Significance: Provides a different approach to quantum computation, potentially faster for certain problems.

Applications: Optimization in logistics, finance, machine learning.

Principle: Encodes the problem into the ground state of a quantum system and gradually evolves the system to find the solution.

6.   Quantum Walk Algorithms

Purpose: Quantum analogs of classical random walk algorithms.

Significance: Provides quadratic speedup for various search and graph traversal problems.

Applications: Network analysis, element distinctness, searching marked elements.

Principle: Uses superposition and interference to explore multiple paths simultaneously, enhancing search efficiency.

7.   Hamiltonian Simulation

Purpose: Simulating the time evolution of quantum systems governed by a Hamiltonian.

Significance: Central to quantum chemistry and materials science.

Applications: Studying chemical reactions, material properties, high-energy physics.

Principle: Approximates the evolution operator e-iHt using techniques like Trotter-Suzuki decomposition and Taylor series expansion.

8.   HHL Algorithm (Harrow-Hassidim-Lloyd)

Purpose: Solving linear systems of equations.

Significance: Exponential speedup under certain conditions compared to classical algorithms.

Applications: Machine learning, numerical simulations, data fitting.

Principle: Uses quantum phase estimation and amplitude amplification to find the solution vector.

9.   Quantum Approximate Optimization Algorithm (QAOA)

Purpose: Approximate solutions for combinatorial optimization problems.

Significance: Tailored for near-term quantum devices, balancing classical and quantum resources.

Applications: Max-Cut problem, scheduling, portfolio optimization.

Principle: Combines classical optimization with quantum state evolution to find approximate solutions efficiently


 * Quantum Complexity Theory

Quantum Complexity Theory is a branch of computational complexity theory that studies the complexity of problems in the context of quantum computing. It aims to classify problems based on the resources required by quantum computers to solve them, such as time (quantum gate operations), space (qubits), and other quantum-specific resources like entanglement. Here's an overview of the key concepts and classes in quantum complexity theory:

Key Concepts

1. Quantum Turing Machine (QTM):

The quantum analog of a classical Turing machine.

Provides a formal model for quantum computation, defining what it means for an algorithm to run on a quantum computer.

2. Quantum Circuit Model:

A more practical model used to design and analyze quantum algorithms.

Represents computations as sequences of quantum gates acting on qubits.

3. Complexity Measures:

  Time Complexity: Number of quantum gates required to solve a problem.

Space Complexity: Number of qubits used during computation.

  Query Complexity: Number of queries to an oracle (black-box function) required by a quantum algorithm.


 * Quantum Complexity Classes

1.   BQP (Bounded-Error Quantum Polynomial Time):

Class of decision problems solvable by a quantum computer in polynomial time, with an error probability of less than 1/3 for all instances.

Quantum analog of the classical complexity class BPP (Bounded-Error Probabilistic Polynomial Time).

  Example: Factoring integers (Shor's algorithm).

2. QMA (Quantum Merlin-Arthur):

Quantum analog of NP (nondeterministic polynomial time).

Problems where a "yes" answer can be verified by a quantum computer in polynomial time, given a quantum state (proof) as a witness.

  Example: Quantum versions of certain optimization problems.

3. QCMA (Quantum Classical Merlin-Arthur):

Similar to QMA, but the witness is a classical string instead of a quantum state.

  Example: Some variants of satisfiability problems.

4. QIP (Quantum Interactive Polynomial time):

Quantum analog of IP (Interactive Polynomial time).

Involves interactive proof systems where a quantum verifier interacts with a quantum prover.

  Example: Certain graph non-isomorphism problems.

5. QMA-Complete:

Subclass of QMA problems that are as hard as any other problem in QMA.

  Example: The Local Hamiltonian problem.

6. BQP vs. P and NP:

It is unknown whether BQP is contained within P or NP.

BQP includes problems that are believed to be outside P (e.g., integer factorization) but not all NP problems (e.g., NP-complete problems are not known to be in BQP)


 * Quantum Lower Bounds:

Proofs showing that certain problems cannot be solved faster than a given lower bound by any quantum algorithm.

Example: Grover’s algorithm is optimal for unstructured search, requiring ON queries


 * Quantum Supremacy:

Demonstrates that quantum computers can solve specific problems faster than classical computers.

Achieved in practice by Google in 2019, showing a problem that a quantum computer solved exponentially faster than the best-known classical algorithms


 *  Implementation 

To develop and run quantum algorithms, you need a quantum computing development environment. Here's a comprehensive guide to setting up such an environment, focusing on popular tools and frameworks:

Choose a Quantum Computing Framework

Several frameworks are available for developing quantum algorithms. Some of the most popular ones include:


 * 1) Qiskit: Developed by IBM, it’s a comprehensive framework for quantum computing.
 * 2) Cirq: Developed by Google, it’s designed for creating, simulating, and running quantum circuits.
 * 3) PyQuil: Developed by Rigetti Computing, it’s used for writing and running quantum programs.
 * 4) Quipper: A functional programming language for quantum computing.
 * 5) Microsoft Quantum Development Kit (QDK): Includes Q#, a language for expressing quantum algorithms.

For this guide, we’ll focus on setting up Qiskit, which is widely used and well-supported.


 * Setting Up Qiskit

Prerequisites

1. Python: Ensure you have Python installed. Qiskit is compatible with Python 3.7 or later.

Download and install Python from [python.org] ( https://www.python.org/ )

2. Pip: Python's package installer, which usually comes with Python.

Installation Steps

1. Create a Virtual Environment (Optional but recommended):

- Open a terminal or command prompt.

- Navigate to your project directory.

- Create a virtual environment:

                       python -m venv qiskit_env

- Activate the virtual environment:

- On Windows:

                     qiskit_env\Scripts\activate

- On macOS/Linux:

                       source qiskit_env/bin/activate

2. Install Qiskit:

- Install Qiskit using pip:

   pip install qiskit

3. Install Additional Tools (Optional):

- For visualization and running on IBM Quantum Experience:

                                      pip install qiskit[visualization]

- For Jupyter Notebooks:

                               pip install notebook

Verify Installation                

To verify the installation, you can run a simple Qiskit program:

python

                        from qiskit import QuantumCircuit, Aer, transpile, assemble

                        from qiskit.visualization import plot_histogram

Create a simple quantum circuit

qc = QuantumCircuit (2)

qc.h(0)  # Apply Hadamard gate to qubit 0

qc.cx(0, 1)  # Apply CNOT gate with control qubit 0 and target qubit 1

Simulate the circuit

simulator = Aer.get_backend('qasm_simulator')

compiled_circuit = transpile(qc, simulator)

qobj = assemble(compiled_circuit)

result = simulator.run(qobj).result

Get and plot the result

counts = result.get_counts

print(counts)

plot_histogram(counts).show

Run this script to ensure everything is working correctly.


 * Development Environment

è Using Jupyter Notebooks

Jupyter Notebooks are a popular choice for developing and testing quantum algorithms interactively:

i. Start Jupyter Notebook:

- If you have installed Jupyter, start it by running:

                         jupyter notebook

- This will open a web interface in your default browser.

'ii. Create a New Notebook:'

- In the Jupyter interface, create a new Python 3 notebook.

- You can now write and execute Qiskit code interactively.

Using an Integrated Development Environment (IDE)

You can also use traditional IDEs like Visual Studio Code, PyCharm, or any other Python-compatible IDE:

i.                  Visual Studio Code:

- Install Visual Studio Code from [code.visualstudio.com] ( https://code.visualstudio.com/ ).

- Install the Python extension from the VS Code marketplace.

- Open your project folder and start coding.

'ii. PyCharm:'

- Install PyCharm from [jetbrains.com/pycharm] ( https://www.jetbrains.com/pycharm/ ).

- Create a new project and configure the Python interpreter to use your virtual environment.

- Start writing and running your Qiskit scripts.


 * Running on Real Quantum Hardware

Qiskit allows you to run your quantum circuits on real quantum computers provided by IBM Quantum Experience:

i. Create an IBM Quantum Account:

- Sign up for an IBM Quantum Experience account at [quantum-computing.ibm.com] ( https://quantum-computing.ibm.com/ ).

'''ii. Obtain an API Token:'''

- Once logged in, go to your account settings to retrieve your API token.

'''iii. Set Up IBMQ Provider:'''

- Initialize your account in your script or notebook:

python

from qiskit import IBMQ

                                 IBMQ.save_account('YOUR_API_TOKEN')

'''iii. Load Your Account:'''

- In subsequent scripts or sessions, load your account:

python

from qiskit import IBMQ

    IBMQ.load_account

    provider = IBMQ.get_provider(hub='ibm-q')

    backend = provider.get_backend('ibmq_qasm_simulator')

'''iv. Run a Circuit on a Real Backend:'''

- Choose a backend and execute your circuit:

python

    job = backend.run(qobj)

    result = job.result

    counts = result.get_counts

    plot_histogram(counts).show 


 * Grover's Search Algorithm Implementation in Qiskit

To implement a quantum algorithm, we'll use Qiskit, one of the most popular frameworks for quantum computing. We'll focus on implementing Grover's Search Algorithm as an example. This algorithm is designed to search an unsorted database or solve any search problem where the correct solution can be verified by a binary oracle.

1. Setup the Environment

Ensure you have Qiskit installed and set up. You can install it using pip if you haven't already:

pip install qiskit

2. Import Required Libraries

We'll start by importing the necessary Qiskit modules:

python

from qiskit import QuantumCircuit, Aer, transpile, assemble, IBMQ

from qiskit.visualization import plot_histogram

import numpy as np

3. Define the Oracle

The oracle is a quantum gate that flips the phase of the target state. For simplicity, we'll search for the state |101〉

python

def grover_oracle(qc, target):

   n = len(target)

   for i in range(n):

       if target[i] == '0':

           qc.x(i)

   qc.h(n - 1)

   qc.mcx(list(range(n - 1)), n - 1)

   qc.h(n - 1)

   for i in range(n):

       if target[i] == '0':

           qc.x(i)

4. Define the Diffusion Operator

The diffusion operator (or Grover diffusion operator) inverts the amplitudes about the mean:

python

def diffusion_operator(qc, n):

   qc.h(range(n))

   qc.x(range(n))

   qc.h(n - 1)

   qc.mcx(list(range(n - 1)), n - 1)

   qc.h(n - 1)

   qc.x(range(n))

   qc.h(range(n))

5. Initialize and Build the Quantum Circuit

Now, let's build the quantum circuit and apply Grover's algorithm:

python


 * 1) Number of qubits

n = 3

num_iterations = int(np.pi/4 * np.sqrt(2**n))


 * 1) Create a quantum circuit with n qubits

qc = QuantumCircuit(n, n)


 * 1) Apply Hadamard gates to all qubits

qc.h(range(n))


 * 1) Apply Grover's Algorithm

for _ in range(num_iterations):

   grover_oracle(qc, '101')

   diffusion_operator(qc, n)


 * 1) Measure all qubits

qc.measure(range(n), range(n))


 * 1) Simulate the quantum circuit

simulator = Aer.get_backend('qasm_simulator')

compiled_circuit = transpile(qc, simulator)

qobj = assemble(compiled_circuit)

result = simulator.run(qobj).result


 * 1) Get the counts of the results

counts = result.get_counts

print(counts)

plot_histogram(counts).show

6. Running the Algorithm

You can run the code above in a Python script or Jupyter Notebook. The steps are:

i. Initialization: We create a quantum circuit with 3 qubits and apply Hadamard gates to create a superposition of all possible states.

'ii. Oracle and Diffusion Operator': We define the oracle for the target state |101> and the diffusion operator. We then apply these operators iteratively.

'iii. Measurement:' Finally, we measure the qubits to read out the result.


 * Running on Real Quantum Hardware

To run the algorithm on real quantum hardware, you need to have an IBM Quantum Experience account and set up your credentials.

1. Initialize IBMQ Account:

python

IBMQ.save_account('YOUR_API_TOKEN')

IBMQ.load_account

provider = IBMQ.get_provider(hub='ibm-q')

backend = provider.get_backend('ibmq_qasm_simulator')

2. Run the Circuit on Real Backend:

python

job = backend.run(qobj)

result = job.result

counts = result.get_counts

print(counts)

plot_histogram(counts).show

Replace `'YOUR_API_TOKEN'` with your actual IBM Quantum Experience API token

Optimize the implementation of Grover's Search Algorithm

To optimize the implementation of Grover's Search Algorithm, we should focus on the following aspects:

1. Code Readability: Make the code modular and clear.

2. Reusability: Encapsulate repeated operations in functions.

3. Efficiency: Use efficient methods for circuit creation and execution.

4. Parallelization: If possible, leverage parallel computation, especially for simulations.

Here’s the optimized and well-structured code for Grover's Search Algorithm using Qiskit:

Optimized Grover's Search Algorithm Implementation


 * Setup the Environment

Ensure you have Qiskit installed. If not, you can install it using pip:

pip install qiskit


 * Import Required Libraries

We start by importing the necessary Qiskit modules:

python

import numpy as np

from qiskit import QuantumCircuit, Aer, transpile, assemble, IBMQ, execute

from qiskit.visualization import plot_histogram


 * Define Utility Functions

We'll create utility functions for the oracle and the diffusion operator. This modular approach improves readability and reusability.

python

def grover_oracle(qc, target):

   """Oracle that flips the phase of the target state."""

   n = len(target)

   for i in range(n):

       if target[i] == '0':

           qc.x(i)

   qc.h(n - 1)

   qc.mcx(list(range(n - 1)), n - 1)

   qc.h(n - 1)

   for i in range(n):

       if target[i] == '0':

           qc.x(i)

   qc.barrier

def diffusion_operator(qc, n):

   """Diffusion operator (inversion about the mean)."""

   qc.h(range(n))

   qc.x(range(n))

   qc.h(n - 1)

   qc.mcx(list(range(n - 1)), n - 1)

   qc.h(n - 1)

   qc.x(range(n))

   qc.h(range(n))

   qc.barrier


 * Build the Quantum Circuit

We create and initialize the quantum circuit and apply Grover’s algorithm:

python

def grover_circuit(n, target):

   """Creates a Grover's search algorithm quantum circuit."""

   num_iterations = int(np.pi / 4 * np.sqrt(2**n))

   qc = QuantumCircuit(n, n)

   # Apply Hadamard gates to all qubits to create superposition

   qc.h(range(n))

   qc.barrier

   # Apply Grover's iterations

   for _ in range(num_iterations):

       grover_oracle(qc, target)

       diffusion_operator(qc, n)

   # Measure all qubits

   qc.measure(range(n), range(n))

   return qc

# Parameters

n = 3  # Number of qubits

target = '101'  # Target state

# Build the Grover circuit

qc = grover_circuit(n, target)


 * Simulate the Quantum Circuit

Use Qiskit’s Aer simulator to run the quantum circuit and display the results:

python

# Simulate the quantum circuit

simulator = Aer.get_backend('qasm_simulator')

compiled_circuit = transpile(qc, simulator)

qobj = assemble(compiled_circuit)

result = simulator.run(qobj).result


 * 1) Get the counts of the results

counts = result.get_counts

print("Simulation results:", counts)

plot_histogram(counts).show


 * Running on Real Quantum Hardware

To run the algorithm on real quantum hardware, you need an IBM Quantum Experience account and your API token. The setup steps are as follows:

python

# Initialize IBMQ Account (replace 'YOUR_API_TOKEN' with your actual API token)

IBMQ.save_account('YOUR_API_TOKEN')

IBMQ.load_account

# Choose a real backend

provider = IBMQ.get_provider(hub='ibm-q')

backend = provider.get_backend('ibmq_qasm_simulator')

# Execute the circuit on the real quantum backend

job = execute(qc, backend=backend)

result = job.result

# Get the counts of the results

counts = result.get_counts

print("Hardware results:", counts)

plot_histogram(counts).show


 * Explanation of the Optimizations

1. Modular Functions: Encapsulating the oracle and diffusion operator in functions makes the code cleaner and more maintainable.

2. Barriers: Barriers are used to separate different stages of the algorithm for better visualization and debugging.

3. Efficient Iterations: The number of iterations is calculated once and used consistently.

4. Simulation and Hardware Execution: Clear separation between simulation and hardware execution code.

This optimized code for Grover’s Search Algorithm improves readability, maintainability, and reusability. It’s ready for both simulation and execution on real quantum hardware. This approach can be extended to other quantum algorithms by following similar principles of modular design and efficient computation

APPLICATION OF QUANTUM COMPUTING IN INFORMATION RETRIEVAL
Quantum computing has the potential to fundamentally transform information retrieval (IR) by using the unique properties of quantum mechanics to increase the speed, efficiency and accuracy of the search process. This project explores the application of quantum computing in IR, focusing on the potential to improve search algorithms, increase the relevance of search results, handle complex queries, and scale efficiently with big data. These applications are briefly discussed below -

1. Enhanced Search Algorithm

One of the most promising applications of quantum computing in IR is the development of faster and more efficient search algorithms.

Grover's algorithm is a quantum algorithm that provides a quadratic speedup for searching unstructured databases. Classical search algorithms require O(N) time to search through N entries, but Grover's algorithm reduces this to O(√N), making it significantly faster for large datasets

This speedup can be particularly beneficial in IR systems where quick retrieval of relevant information is important. It can also significantly reduce the time required to retrieve information from large data sets and help make searches more efficient.

2. Improved Relevance of Search Results

Quantum Machine Learning (QML) can also enhance the relevance of search results by processing and analyzing vast amounts of data to identify patterns and correlations that classical algorithms might miss. Quantum algorithms can handle complex relationships within data, leading to more accurate and contextually relevant search results.

For example, quantum-inspired algorithms have been shown to improve machine learning tasks (the techniques used in this case are quantum support vector mechanism and quantum neural network) that are critical for IR, such as clustering and classification, which directly affect the relevance of retrieved documents

3. Handling Complex Queries

Quantum computing allows for more sophisticated query processing by efficiently handling multi-dimensional data and intricate relationships within data sets. This capability is particularly valuable for IR tasks involving natural language processing (NLP), where understanding context and semantics is critical.

QNLP involves applying quantum computing to natural language processing tasks, enhancing the ability to understand and process complex queries. QNLP can improve context-aware search capabilities, enhancing the retrieval of relevant documents based on complex queries.

4. Scalability and Big Data

The ability of quantum computers to process large amounts of data compared to classical computers makes them ideal for big data applications in IR. It helps in speed and efficiency as well as resource management.

As data volumes continue to grow, the scalability of quantum computing could provide significant advantages in indexing, searching, and retrieving information from extensive databases. This scalability is essential for managing the ever-increasing amounts of digital information generated across various domains

Finally, it is said that, quantum computing increases the speed, efficiency and relevance of information searching processes and simplifies the process of accurate information retrieval. By implementing quantum algorithms, using quantum machine learning and applying QNLP techniques, IR systems can achieve significant progress.

RECENT TREND AND RESEARCH IN QUANTUM INFORMATION RETRIEVAL
Quantum Information Retrieval (QIR) is an interdisciplinary field at the intersection of quantum computing and information retrieval, which uses quantum principles to improve traditional Information Retrieval (IR) methods. Recent trends and research areas in quantum information retrieval include developing quantum algorithms, improving hardware capabilities, and integrating classical IR methods with quantum systems

Below is a brief overview of the key trends and areas of research in QIR.

1.    Quantum Algorithms for IR

One of the primary areas of QIR research focus is the development and optimization of quantum algorithms for information retrieval tasks. As well as improving Quantum Natural Language Processing (QNLP) and developing Quantum Machine Learning (QML) algorithms such as Quantum Support Vector Machines (QSVMs) and Quantum Neural Networks (QNNs).

Grover's algorithm, a fundamental quantum algorithm, offers a quadratic speedup for unstructured search problems (Grover, 1996). By efficiently searching through unstructured databases, Grover's algorithm has significant implications for document retrieval and database searching in IR systems.

2.    Quantum Hardware and Platform

Quantum processors are important for converting theoretical quantum algorithms into practical solutions that can improve the performance of information retrieval.

Advancements in quantum hardware, such as IBM's Qiskit and Google's Sycamore processor, provide researchers with experimental platforms to implement and test quantum algorithms for IR applications

Addressing the limitations of current quantum hardware, such as qubit decoherence and gate errors, remains a significant area of research. Technologies such as error correction and noise reduction are now essential to implement reliable QIR

3. Theoretical Foundation of QIR

Researchers are exploring the theoretical foundations of quantum information retrieval, including quantum query complexity and quantum information theory principles, such as using quantum entropy measures for information retrieval and relevance ranking in search results.

This theoretical research refers to the development of optimized quantum algorithms for IR tasks, such as semantic search and context retrieval.

4. Scalability and Big Data

Scalability of quantum computing to handle large-scale data is a prominent research area of QIR. Quantum-enhanced data structures improve data storage and retrieval efficiency.

Researchers are developing quantum algorithms that can process, index, and retrieve information from massive datasets, addressing the scalability issues faced by classical IR systems.

5. Hybrid Quantum-Classical Approach

Hybrid approaches combine quantum and classical computing strengths. Hybrid quantum-classical computing approaches are gaining traction in QIR research. These approaches leverage the strengths of both quantum and classical computing paradigms to address complex information retrieval challenges

Middleware development plays an important role in facilitating seamless integration between quantum-enhanced components and classical IR systems. This includes developing protocols and interfaces that allow classical IR systems to offload certain computations to quantum processors.

Finally, it is said that, Quantum Information Retrieval (QIR) is a rapidly evolving field with significant potential to change the way information is retrieved and processed. As quantum computing technology continues to advance, the practical applications of QIR are expected to expand, providing new capabilities and efficiencies for information retrieval systems.

CONCLUSION
QIR is a growing multi-disciplinary area and has been attracting an increasing attention of researchers in IR. Especially, the recent several years have witnessed a large number of models and applications of QIR which have shown good results and a great potential. However, the field lacks a comprehensive review of the literature, and the individual works are largely segmented. This is why a survey paper is urgently needed. It is important and timely to review the literature systematically to provide a clear picture of the landscape and a road-map for the future. Although this is not the first work to accumulate findings in QIR. Around ten years ago, a position paper [Song et al. 2010], organized QIR into three themes: frameworks, spaces, and interference. However, it was more on a conceptual level and the field of QIR was in its infancy. After a decade of development since then, the landscape of QIR research has significantly changed. A large number of more comprehensive and larger scale QIR approaches have been developed covering different aspects of IR, and have achieved remarkable experimental results.