Entropy (information theory)
Information entropy is the average rate at which information is produced by a stochastic source of data. The measure of information entropy associated with each possible data value is the negative logarithm of the probability mass function for the value: S = − ∑ i P i log P i S=-\sum _P_\log; when the data source produces a low-probability value, the event carries more "information" than when the source data produces a high-probability value. The amount of information conveyed by each event defined in this way becomes a random variable whose expected value is the information entropy. Entropy refers to disorder or uncertainty, the definition of entropy used in information theory is directly analogous to the definition used in statistical thermodynamics; the concept of information entropy was introduced by Claude Shannon in his 1948 paper "A Mathematical Theory of Communication". The basic model of a data communication system is composed of three elements, a source of data, a communication channel, a receiver, – as expressed by Shannon – the "fundamental problem of communication" is for the receiver to be able to identify what data was generated by the source, based on the signal it receives through the channel.
The entropy provides an absolute limit on the shortest possible average length of a lossless compression encoding of the data produced by a source, if the entropy of the source is less than the channel capacity of the communication channel, the data generated by the source can be reliably communicated to the receiver. Information entropy is measured in bits or sometimes in "natural units" or decimal digits; the unit of the measurement depends on the base of the logarithm, used to define the entropy. The logarithm of the probability distribution is useful as a measure of entropy because it is additive for independent sources. For instance, the entropy of a fair coin toss is 1 bit, the entropy of m tosses is m bits. In a straightforward representation, log2 bits are needed to represent a variable that can take one of n values if n is a power of 2. If these values are probable, the entropy is equal to this number. If one of the values is more probable to occur than the others, an observation that this value occurs is less informative than if some less common outcome had occurred.
Conversely, rarer events provide more information. Since observation of less probable events occurs more the net effect is that the entropy received from non-uniformly distributed data is always less than or equal to log2. Entropy is zero; the entropy quantifies these considerations when a probability distribution of the source data is known. The meaning of the events observed. Entropy only takes into account the probability of observing a specific event, so the information it encapsulates is information about the underlying probability distribution, not the meaning of the events themselves; the basic idea of information theory is that the more one knows about a topic, the less new information one is apt to get about it. If an event is probable, it is no surprise when it happens and provides little new information. Inversely, if the event was improbable, it is much more informative; the information content is an increasing function of the reciprocal of the probability of the event. If more events may happen, entropy measures the average information content you can expect to get if one of the events happens.
This implies that casting a die has more entropy than tossing a coin because each outcome of the die has smaller probability than each outcome of the coin. Entropy is a measure of unpredictability of the state, or equivalently, of its average information content. To get an intuitive understanding of these terms, consider the example of a political poll; such polls happen because the outcome of the poll is not known. In other words, the outcome of the poll is unpredictable, performing the poll and learning the results gives some new information. Now, consider the case that the same poll is performed a second time shortly after the first poll. Since the result of the first poll is known, the outcome of the second poll can be predicted well and the results should not contain much new information. Consider the example of a coin toss. Assuming the probability of heads is the same as the probability of tails the entropy of the coin toss is as high as it could be. There is no way to predict the outcome of the coin toss ahead of time: if one has to choose, the best one can do is predict that the coin will come up heads, this prediction will be correct with probability 1/2.
Such a coin toss has one bit of entropy since there are two possible outcomes that occur with equal probability, learning the actual outcome contains one bit of information. In contrast, a coin toss using a coin that has two heads and no tails has zero entropy since the coin will always come up heads, the outcome can be predicted pe
The Einstein–Podolsky–Rosen paradox is a thought experiment proposed by physicists Albert Einstein, Boris Podolsky and Nathan Rosen that they interpreted as indicating that the explanation of physical reality provided by Quantum Mechanics was incomplete. In a 1935 paper titled Can Quantum-Mechanical Description of Physical Reality be Considered Complete?, they attempted to mathematically show that the wave function does not contain complete information about physical reality, hence the Copenhagen interpretation is unsatisfactory. The work was done at the Institute for Advanced Study in 1934, which Einstein had joined the prior year after he had fled Nazi Germany; the essence of the paradox is that particles can interact in such a way that it is possible to measure both their position and their momentum more than Heisenberg's uncertainty principle allows, unless measuring one particle instantaneously affects the other to prevent this accuracy, which would involve information being transmitted faster than light as forbidden by the theory of relativity.
This consequence had not been noticed and seemed unreasonable at the time. The article that first brought forth these matters, "Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?" was published in 1935. The paper prompted a response by Bohr, which he published in the same journal, in the same year, using the same title. There followed a debate between Einstein about the fundamental nature of reality. Einstein had been skeptical of the Heisenberg uncertainty principle and the role of chance in quantum theory, but the crux of this debate was not about chance, but something deeper: Is there one objective physical reality, which every observer sees from his own vantage? Or does the observer co-create physical reality by the questions he poses with experiments? Einstein struggled to the end of his life for a theory that could better comply with his idea of causality, protesting against the view that there exists no objective physical reality other than that, revealed through measurement interpreted in terms of quantum mechanical formalism.
However, since Einstein's death, experiments analogous to the one described in the EPR paper have been carried out, starting in 1976 by French scientists Lamehi-Rachti and Mittig at the Saclay Nuclear Research Centre. These experiments appear vindicating Bohr. Per EPR, the paradox demonstrated that quantum theory was incomplete, needed to be extended with hidden variables. One modern resolution is as follows: for two "entangled" particles created at once, measurable properties have well-defined meaning only for the ensemble system. Properties of constituent subsystems, considered individually, remain undefined. Therefore, if analogous measurements are performed on the two entangled subsystems, there will always be a correlation between the outcomes, a well-defined global outcome for the ensemble. However, the outcomes for each subsystem, considered separately, at each repetition of the experiment, will not be well defined or predictable; this correlation does not imply that measurements performed on one particle influence measurements on the other.
This modern resolution eliminates the need for hidden variables, action at a distance, or other schemes introduced over time, in order to explain the phenomenon. According to quantum mechanics, under some conditions, a pair of quantum systems may be described by a single wave function, which encodes the probabilities of the outcomes of experiments that may be performed on the two systems, whether jointly or individually. At the time the EPR article discussed below was written, it was known from experiments that the outcome of an experiment sometimes cannot be uniquely predicted. An example of such indeterminacy can be seen when a beam of light is incident on a half-silvered mirror. One half of the beam will reflect, the other will pass. If the intensity of the beam is reduced until only one photon is in transit at any time, whether that photon will reflect or transmit cannot be predicted quantum mechanically; the routine explanation of this effect was, at that time, provided by Heisenberg's uncertainty principle.
Physical quantities come in pairs called conjugate quantities. Examples of such conjugate pairs are, and; when one quantity was measured, became determined, the conjugated quantity became indeterminate. Heisenberg explained this uncertainty as due to the quantization of the disturbance from measurement; the EPR paper, written in 1935, was intended to illustrate. It considered two entangled particles, referred to as A and B, pointed out that measuring a quantity of a particle A will cause the conjugated quantity of particle B to become undetermined if there was no contact, no classical disturbance; the basic idea was that the quantum states of two particles in a system cannot always be decomposed from the joint state of the two, as is the case for the Bell state, | Φ + ⟩ = 1 2. Heisenberg's principle was an attempt to provide a classical explanation of a quantum e
Quantum mechanics, including quantum field theory, is a fundamental theory in physics which describes nature at the smallest scales of energy levels of atoms and subatomic particles. Classical physics, the physics existing before quantum mechanics, describes nature at ordinary scale. Most theories in classical physics can be derived from quantum mechanics as an approximation valid at large scale. Quantum mechanics differs from classical physics in that energy, angular momentum and other quantities of a bound system are restricted to discrete values. Quantum mechanics arose from theories to explain observations which could not be reconciled with classical physics, such as Max Planck's solution in 1900 to the black-body radiation problem, from the correspondence between energy and frequency in Albert Einstein's 1905 paper which explained the photoelectric effect. Early quantum theory was profoundly re-conceived in the mid-1920s by Erwin Schrödinger, Werner Heisenberg, Max Born and others; the modern theory is formulated in various specially developed mathematical formalisms.
In one of them, a mathematical function, the wave function, provides information about the probability amplitude of position and other physical properties of a particle. Important applications of quantum theory include quantum chemistry, quantum optics, quantum computing, superconducting magnets, light-emitting diodes, the laser, the transistor and semiconductors such as the microprocessor and research imaging such as magnetic resonance imaging and electron microscopy. Explanations for many biological and physical phenomena are rooted in the nature of the chemical bond, most notably the macro-molecule DNA. Scientific inquiry into the wave nature of light began in the 17th and 18th centuries, when scientists such as Robert Hooke, Christiaan Huygens and Leonhard Euler proposed a wave theory of light based on experimental observations. In 1803, Thomas Young, an English polymath, performed the famous double-slit experiment that he described in a paper titled On the nature of light and colours.
This experiment played a major role in the general acceptance of the wave theory of light. In 1838, Michael Faraday discovered cathode rays; these studies were followed by the 1859 statement of the black-body radiation problem by Gustav Kirchhoff, the 1877 suggestion by Ludwig Boltzmann that the energy states of a physical system can be discrete, the 1900 quantum hypothesis of Max Planck. Planck's hypothesis that energy is radiated and absorbed in discrete "quanta" matched the observed patterns of black-body radiation. In 1896, Wilhelm Wien empirically determined a distribution law of black-body radiation, known as Wien's law in his honor. Ludwig Boltzmann independently arrived at this result by considerations of Maxwell's equations. However, it underestimated the radiance at low frequencies. Planck corrected this model using Boltzmann's statistical interpretation of thermodynamics and proposed what is now called Planck's law, which led to the development of quantum mechanics. Following Max Planck's solution in 1900 to the black-body radiation problem, Albert Einstein offered a quantum-based theory to explain the photoelectric effect.
Around 1900–1910, the atomic theory and the corpuscular theory of light first came to be accepted as scientific fact. Among the first to study quantum phenomena in nature were Arthur Compton, C. V. Raman, Pieter Zeeman, each of whom has a quantum effect named after him. Robert Andrews Millikan studied the photoelectric effect experimentally, Albert Einstein developed a theory for it. At the same time, Ernest Rutherford experimentally discovered the nuclear model of the atom, for which Niels Bohr developed his theory of the atomic structure, confirmed by the experiments of Henry Moseley. In 1913, Peter Debye extended Niels Bohr's theory of atomic structure, introducing elliptical orbits, a concept introduced by Arnold Sommerfeld; this phase is known as old quantum theory. According to Planck, each energy element is proportional to its frequency: E = h ν, where h is Planck's constant. Planck cautiously insisted that this was an aspect of the processes of absorption and emission of radiation and had nothing to do with the physical reality of the radiation itself.
In fact, he considered his quantum hypothesis a mathematical trick to get the right answer rather than a sizable discovery. However, in 1905 Albert Einstein interpreted Planck's quantum hypothesis realistically and used it to explain the photoelectric effect, in which shining light on certain materials can eject electrons from the material, he won the 1921 Nobel Prize in Physics for this work. Einstein further developed this idea to show that an electromagnetic wave such as light could be described as a particle, with a discrete quantum of energy, dependent on its frequency; the foundations of quantum mechanics were established during the first half of the 20th century by Max Planck, Niels Bohr, Werner Heisenberg, Louis de Broglie, Arthur Compton, Albert Einstein, Erwin Schrödinger, Max Born, John von Neumann, Paul Dirac, Enrico Fermi, Wolfgang Pauli, Max von Laue, Freeman Dyson, David Hilbert, Wi
In mathematics and computer science, an algorithm is an unambiguous specification of how to solve a class of problems. Algorithms can perform calculation, data processing, automated reasoning, other tasks; as an effective method, an algorithm can be expressed within a finite amount of space and time and in a well-defined formal language for calculating a function. Starting from an initial state and initial input, the instructions describe a computation that, when executed, proceeds through a finite number of well-defined successive states producing "output" and terminating at a final ending state; the transition from one state to the next is not deterministic. The concept of algorithm has existed for centuries. Greek mathematicians used algorithms in the sieve of Eratosthenes for finding prime numbers, the Euclidean algorithm for finding the greatest common divisor of two numbers; the word algorithm itself is derived from the 9th century mathematician Muḥammad ibn Mūsā al-Khwārizmī, Latinized Algoritmi.
A partial formalization of what would become the modern concept of algorithm began with attempts to solve the Entscheidungsproblem posed by David Hilbert in 1928. Formalizations were framed as attempts to define "effective calculability" or "effective method"; those formalizations included the Gödel–Herbrand–Kleene recursive functions of 1930, 1934 and 1935, Alonzo Church's lambda calculus of 1936, Emil Post's Formulation 1 of 1936, Alan Turing's Turing machines of 1936–37 and 1939. The word'algorithm' has its roots in Latinizing the name of Muhammad ibn Musa al-Khwarizmi in a first step to algorismus. Al-Khwārizmī was a Persian mathematician, astronomer and scholar in the House of Wisdom in Baghdad, whose name means'the native of Khwarazm', a region, part of Greater Iran and is now in Uzbekistan. About 825, al-Khwarizmi wrote an Arabic language treatise on the Hindu–Arabic numeral system, translated into Latin during the 12th century under the title Algoritmi de numero Indorum; this title means "Algoritmi on the numbers of the Indians", where "Algoritmi" was the translator's Latinization of Al-Khwarizmi's name.
Al-Khwarizmi was the most read mathematician in Europe in the late Middle Ages through another of his books, the Algebra. In late medieval Latin, English'algorism', the corruption of his name meant the "decimal number system". In the 15th century, under the influence of the Greek word ἀριθμός'number', the Latin word was altered to algorithmus, the corresponding English term'algorithm' is first attested in the 17th century. In English, it was first used in about 1230 and by Chaucer in 1391. English adopted the French term, but it wasn't until the late 19th century that "algorithm" took on the meaning that it has in modern English. Another early use of the word is from 1240, in a manual titled Carmen de Algorismo composed by Alexandre de Villedieu, it begins thus: Haec algorismus ars praesens dicitur, in qua / Talibus Indorum fruimur bis quinque figuris. Which translates as: Algorism is the art by which at present we use those Indian figures, which number two times five; the poem is a few hundred lines long and summarizes the art of calculating with the new style of Indian dice, or Talibus Indorum, or Hindu numerals.
An informal definition could be "a set of rules that defines a sequence of operations". Which would include all computer programs, including programs that do not perform numeric calculations. A program is only an algorithm if it stops eventually. A prototypical example of an algorithm is the Euclidean algorithm to determine the maximum common divisor of two integers. Boolos, Jeffrey & 1974, 1999 offer an informal meaning of the word in the following quotation: No human being can write fast enough, or long enough, or small enough† to list all members of an enumerably infinite set by writing out their names, one after another, in some notation, but humans can do something useful, in the case of certain enumerably infinite sets: They can give explicit instructions for determining the nth member of the set, for arbitrary finite n. Such instructions are to be given quite explicitly, in a form in which they could be followed by a computing machine, or by a human, capable of carrying out only elementary operations on symbols.
An "enumerably infinite set" is one whose elements can be put into one-to-one correspondence with the integers. Thus and Jeffrey are saying that an algorithm implies instructions for a process that "creates" output integers from an arbitrary "input" integer or integers that, in theory, can be arbitrarily large, thus an algorithm can be an algebraic equation such as y = m + n – two arbitrary "input variables" m and n that produce an output y. But various authors' attempts to define the notion indicate that the word implies much more than this, something on the order of: Precise instructions for a fast, efficient, "good" process that specifies the "moves" of "the computer" to find and process arbitrary input integers/symbols m and n, symbols + and =... and "effectively" produce, in a "reasonable" time, output-integer y at a specified place and in a specified format
Trapped ion quantum computer
A trapped ion quantum computer is one proposed approach to a large-scale quantum computer. Ions, or charged atomic particles, can be confined and suspended in free space using electromagnetic fields. Qubits are stored in stable electronic states of each ion, quantum information can be transferred through the collective quantized motion of the ions in a shared trap. Lasers are applied to induce coupling between the qubit states or coupling between the internal qubit states and the external motional states; the fundamental operations of a quantum computer have been demonstrated experimentally with the highest accuracy in trapped ion systems. Promising schemes in development to scale the system to arbitrarily large numbers of qubits include transporting ions to spatially distinct locations in an array of ion traps, building large entangled states via photonically connected networks of remotely entangled ion chains, combinations of these two ideas; this makes the trapped ion quantum computer system one of the most promising architectures for a scalable, universal quantum computer.
As of April 2018, the largest number of particles to be controllably entangled is 20 trapped ions. The electrodynamic ion trap used in trapped ion quantum computing research was invented in the 1950s by Wolfgang Paul. Charged particles cannot be trapped in 3D by just electrostatic forces because of Earnshaw's theorem. Instead, an electric field oscillating at radio frequency is applied, forming a potential with the shape of a saddle spinning at the RF frequency. If the RF field has the right parameters, the charged particle becomes trapped at the saddle point by a restoring force, with the motion described by a set of Mathieu equations; this saddle point is the point of minimized energy magnitude, | E |, for the ions in the potential field. The Paul trap is described as a harmonic potential well that traps ions in two dimensions and does not trap ions in the z ^ direction; when multiple ions are at the saddle point and the system is at equilibrium, the ions are only free to move in z ^. Therefore, the ions will repel each other and create a vertical configuration in z ^, the simplest case being a linear strand of only a few ions.
Coulomb interactions of increasing complexity will create a more intricate ion configuration if many ions are initialized in the same trap. Furthermore, the additional vibrations of the added ions complicate the quantum system, which makes initialization and computation more difficult. Once trapped, the ions should be cooled such that k B T << ℏ ω z. This can be achieved by a combination of Resolved sideband cooling. At this low temperature, vibrational energy in the ion trap is quantized into phonons by the energy eigenstates of the ion strand, which are called the center of mass vibrational modes. A single phonon's energy is given by the relation ℏ ω z; these quantum states occur when the trapped ions vibrate together and are isolated from the external environment. If the ions are not properly isolated, noise can result from ions interacting with external electromagnetic fields, which creates random movement and destroys the quantized energy states; the first implementation scheme for a controlled-NOT quantum gate was proposed by Ignacio Cirac and Peter Zoller in 1995 for the trapped ion system.
The same year, a key step in the controlled-NOT gate was experimentally realized at NIST Ion Storage Group, research in quantum computing began to take off worldwide. Many traditional ion trapping research groups have made the transition to quantum computing research, more many other new research groups have joined the effort. An enormous amount of progress in this field has been made in the past decade and trapped ions remain a leading candidate for quantum computation; the full requirements for a functional quantum computer are not known, but there are many accepted requirements. DiVincenzo outlined several of these criterion for quantum computing. Any two-level quantum system can form a qubit, there are two ways to form a qubit using the electronic states of an ion: Two ground state hyperfine levels A ground state level and an excited level Hyperfine qubits are long-lived and phase/frequency stable. Optical qubits are relatively long-lived, compared to the logic gate operation time; the use of each type of qubit poses its own distinct challenges in the laboratory.
Ionic qubit states can be prepared in a specific qubit state using a process called optical pumping. In this process, a laser couples the ion to some excited states
Claude Elwood Shannon was an American mathematician, electrical engineer, cryptographer known as "the father of information theory". Shannon is noted for having founded information theory with a landmark paper, A Mathematical Theory of Communication, that he published in 1948, he is well known for founding digital circuit design theory in 1937, when—as a 21-year-old master's degree student at the Massachusetts Institute of Technology —he wrote his thesis demonstrating that electrical applications of Boolean algebra could construct any logical numerical relationship. Shannon contributed to the field of cryptanalysis for national defense during World War II, including his fundamental work on codebreaking and secure telecommunications. Shannon was born in Petoskey and grew up in Gaylord, Michigan, his father, Claude, Sr. a descendant of early settlers of New Jersey, was a self-made businessman, for a while, a Judge of Probate. Shannon's mother, Mabel Wolf Shannon, was a language teacher, served as the principal of Gaylord High School.
Most of the first 16 years of Shannon's life were spent in Gaylord, where he attended public school, graduating from Gaylord High School in 1932. Shannon showed an inclination towards electrical things, his best subjects were science and mathematics. At home he constructed such devices as models of planes, a radio-controlled model boat and a barbed-wire telegraph system to a friend's house a half-mile away. While growing up, he worked as a messenger for the Western Union company, his childhood hero was Thomas Edison, who he learned was a distant cousin. Both Shannon and Edison were descendants of John Ogden, a colonial leader and an ancestor of many distinguished people. Shannon was an atheist. In 1932, Shannon entered the University of Michigan, where he was introduced to the work of George Boole, he graduated in 1936 with two bachelor's degrees: one in electrical engineering and the other in mathematics. In 1936, Shannon began his graduate studies in electrical engineering at MIT, where he worked on Vannevar Bush's differential analyzer, an early analog computer.
While studying the complicated ad hoc circuits of this analyzer, Shannon designed switching circuits based on Boole's concepts. In 1937, he wrote A Symbolic Analysis of Relay and Switching Circuits. A paper from this thesis was published in 1938. In this work, Shannon proved that his switching circuits could be used to simplify the arrangement of the electromechanical relays that were used in telephone call routing switches. Next, he expanded this concept, proving that these circuits could solve all problems that Boolean algebra could solve. In the last chapter, he presented diagrams of several circuits, including a 4-bit full adder. Using this property of electrical switches to implement logic is the fundamental concept that underlies all electronic digital computers. Shannon's work became the foundation of digital circuit design, as it became known in the electrical engineering community during and after World War II; the theoretical rigor of Shannon's work superseded the ad hoc methods. Howard Gardner called Shannon's thesis "possibly the most important, the most noted, master's thesis of the century."Shannon received his Ph.
D. degree from MIT in 1940. Vannevar Bush had suggested that Shannon should work on his dissertation at the Cold Spring Harbor Laboratory, in order to develop a mathematical formulation for Mendelian genetics; this research resulted in Shannon's PhD thesis, called An Algebra for Theoretical Genetics. In 1940, Shannon became a National Research Fellow at the Institute for Advanced Study in Princeton, New Jersey. In Princeton, Shannon had the opportunity to discuss his ideas with influential scientists and mathematicians such as Hermann Weyl and John von Neumann, he had occasional encounters with Albert Einstein and Kurt Gödel. Shannon worked across disciplines, this ability may have contributed to his development of mathematical information theory. Shannon joined Bell Labs to work on fire-control systems and cryptography during World War II, under a contract with section D-2 of the National Defense Research Committee. Shannon is credited with the invention of signal-flow graphs, in 1942, he discovered the topological gain formula while investigating the functional operation of an analog computer.
For two months early in 1943, Shannon came into contact with the leading British mathematician Alan Turing. Turing had been posted to Washington to share with the U. S. Navy's cryptanalytic service the methods used by the British Government Code and Cypher School at Bletchley Park to break the ciphers used by the Kriegsmarine U-boats in the north Atlantic Ocean, he was interested in the encipherment of speech and to this end spent time at Bell Labs. Shannon and Turing met at teatime in the cafeteria. Turing showed Shannon his 1936 paper that defined what is now known as the "Universal Turing machine". In 1945, as the war was coming to an end, the NDRC was issuing a summary of technical reports as a last step prior to its eventual closing down. Inside the volume on fire control, a special essay titled Data Smoothing and Prediction in Fire-Control Systems, coauthored by Shannon, Ralph Beebe Blackman, Hendrik Wade Bode, formally treated the problem of smoothing the data in fire-control by analogy with "the problem of separating a signal from interfering noise in communications systems."
In other words, it modeled the problem in terms of data and signal processing and thus heralded the coming of the Information Age. Shannon's w
Computer science is the study of processes that interact with data and that can be represented as data in the form of programs. It enables the use of algorithms to manipulate and communicate digital information. A computer scientist studies the theory of computation and the practice of designing software systems, its fields can be divided into practical disciplines. Computational complexity theory is abstract, while computer graphics emphasizes real-world applications. Programming language theory considers approaches to the description of computational processes, while computer programming itself involves the use of programming languages and complex systems. Human–computer interaction considers the challenges in making computers useful and accessible; the earliest foundations of what would become computer science predate the invention of the modern digital computer. Machines for calculating fixed numerical tasks such as the abacus have existed since antiquity, aiding in computations such as multiplication and division.
Algorithms for performing computations have existed since antiquity before the development of sophisticated computing equipment. Wilhelm Schickard designed and constructed the first working mechanical calculator in 1623. In 1673, Gottfried Leibniz demonstrated a digital mechanical calculator, called the Stepped Reckoner, he may be considered the first computer scientist and information theorist, among other reasons, documenting the binary number system. In 1820, Thomas de Colmar launched the mechanical calculator industry when he released his simplified arithmometer, the first calculating machine strong enough and reliable enough to be used daily in an office environment. Charles Babbage started the design of the first automatic mechanical calculator, his Difference Engine, in 1822, which gave him the idea of the first programmable mechanical calculator, his Analytical Engine, he started developing this machine in 1834, "in less than two years, he had sketched out many of the salient features of the modern computer".
"A crucial step was the adoption of a punched card system derived from the Jacquard loom" making it infinitely programmable. In 1843, during the translation of a French article on the Analytical Engine, Ada Lovelace wrote, in one of the many notes she included, an algorithm to compute the Bernoulli numbers, considered to be the first computer program. Around 1885, Herman Hollerith invented the tabulator, which used punched cards to process statistical information. In 1937, one hundred years after Babbage's impossible dream, Howard Aiken convinced IBM, making all kinds of punched card equipment and was in the calculator business to develop his giant programmable calculator, the ASCC/Harvard Mark I, based on Babbage's Analytical Engine, which itself used cards and a central computing unit; when the machine was finished, some hailed it as "Babbage's dream come true". During the 1940s, as new and more powerful computing machines were developed, the term computer came to refer to the machines rather than their human predecessors.
As it became clear that computers could be used for more than just mathematical calculations, the field of computer science broadened to study computation in general. In 1945, IBM founded the Watson Scientific Computing Laboratory at Columbia University in New York City; the renovated fraternity house on Manhattan's West Side was IBM's first laboratory devoted to pure science. The lab is the forerunner of IBM's Research Division, which today operates research facilities around the world; the close relationship between IBM and the university was instrumental in the emergence of a new scientific discipline, with Columbia offering one of the first academic-credit courses in computer science in 1946. Computer science began to be established as a distinct academic discipline in the 1950s and early 1960s; the world's first computer science degree program, the Cambridge Diploma in Computer Science, began at the University of Cambridge Computer Laboratory in 1953. The first computer science degree program in the United States was formed at Purdue University in 1962.
Since practical computers became available, many applications of computing have become distinct areas of study in their own rights. Although many believed it was impossible that computers themselves could be a scientific field of study, in the late fifties it became accepted among the greater academic population, it is the now well-known IBM brand that formed part of the computer science revolution during this time. IBM released the IBM 704 and the IBM 709 computers, which were used during the exploration period of such devices. "Still, working with the IBM was frustrating if you had misplaced as much as one letter in one instruction, the program would crash, you would have to start the whole process over again". During the late 1950s, the computer science discipline was much in its developmental stages, such issues were commonplace. Time has seen significant improvements in the effectiveness of computing technology. Modern society has seen a significant shift in the users of computer technology, from usage only by experts and professionals, to a near-ubiquitous user base.
Computers were quite costly, some degree of humanitarian aid was needed for efficient use—in part from professional computer operators. As computer adoption became more widespread and affordable, less human assistance was needed for common usage. Despite its short history as a formal academic discipline, computer science has made a number of fundamental contributions to science and society—in fact, along with electronics, it is