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163-454: A quantum computer is a computer that exploits quantum mechanical phenomena. On small scales, physical matter exhibits properties of both particles and waves , and quantum computing leverages this behavior using specialized hardware. Classical physics cannot explain the operation of these quantum devices, and a scalable quantum computer could perform some calculations exponentially faster than any modern "classical" computer. Theoretically

326-521: A binary system meant that Zuse's machines were easier to build and potentially more reliable, given the technologies available at that time. The Z3 was not itself a universal computer but could be extended to be Turing complete . Zuse's next computer, the Z4 , became the world's first commercial computer; after initial delay due to the Second World War, it was completed in 1950 and delivered to

489-526: A black box with a quantum state in superposition , sometimes referred to as quantum parallelism . Peter Shor built on these results with his 1994 algorithm for breaking the widely used RSA and Diffie–Hellman encryption protocols, which drew significant attention to the field of quantum computing. In 1996, Grover's algorithm established a quantum speedup for the widely applicable unstructured search problem. The same year, Seth Lloyd proved that quantum computers could simulate quantum systems without

652-632: A central processing unit (CPU) in the form of a microprocessor , together with some type of computer memory , typically semiconductor memory chips. The processing element carries out arithmetic and logical operations, and a sequencing and control unit can change the order of operations in response to stored information . Peripheral devices include input devices ( keyboards , mice , joysticks , etc.), output devices ( monitors , printers , etc.), and input/output devices that perform both functions (e.g. touchscreens ). Peripheral devices allow information to be retrieved from an external source, and they enable

815-419: A keyboard , and computed and printed the results, demonstrating the feasibility of an electromechanical analytical engine. During the first half of the 20th century, many scientific computing needs were met by increasingly sophisticated analog computers, which used a direct mechanical or electrical model of the problem as a basis for computation . However, these were not programmable and generally lacked

978-524: A mass-production basis, which limited them to a number of specialized applications. At the University of Manchester , a team under the leadership of Tom Kilburn designed and built a machine using the newly developed transistors instead of valves. Their first transistorized computer and the first in the world, was operational by 1953 , and a second version was completed there in April 1955. However,

1141-429: A monolithic integrated circuit (IC) chip. Kilby's IC had external wire connections, which made it difficult to mass-produce. Noyce also came up with his own idea of an integrated circuit half a year later than Kilby. Noyce's invention was the first true monolithic IC chip. His chip solved many practical problems that Kilby's had not. Produced at Fairchild Semiconductor, it was made of silicon , whereas Kilby's chip

1304-407: A randomized algorithm , quantum mechanical notions like superposition and interference are largely irrelevant for program analysis . Quantum programs , in contrast, rely on precise control of coherent quantum systems. Physicists describe these systems mathematically using linear algebra . Complex numbers model probability amplitudes , vectors model quantum states , and matrices model

1467-467: A superposition , which means that the qubit can have non-zero probability amplitude in its both states simultaneously (popularly expressed as "it can be in both states simultaneously"). A qubit requires two complex numbers to describe its two probability amplitudes, and these two complex numbers can together be viewed as a 2-dimensional complex vector, which is called a quantum state vector , or superposition state vector. Alternatively and equivalently,

1630-564: A 1984 paper, Charles Bennett and Gilles Brassard applied quantum theory to cryptography protocols and demonstrated that quantum key distribution could enhance information security . Quantum algorithms then emerged for solving oracle problems , such as Deutsch's algorithm in 1985, the Bernstein–Vazirani algorithm in 1993, and Simon's algorithm in 1994. These algorithms did not solve practical problems, but demonstrated mathematically that one could gain more information by querying

1793-652: A 1998 retrospective, it was the first working machine to contain all of the elements essential to a modern electronic computer. As soon as the Baby had demonstrated the feasibility of its design, a project began at the university to develop it into a practically useful computer, the Manchester Mark 1 . The Mark 1 in turn quickly became the prototype for the Ferranti Mark 1 , the world's first commercially available general-purpose computer. Built by Ferranti , it

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1956-767: A 54-qubit machine, performing a computation that is impossible for any classical computer. However, the validity of this claim is still being actively researched. In December 2023, physicists, for the first time, reported the entanglement of individual molecules, which may have significant applications in quantum computing. Computer engineers typically describe a modern computer 's operation in terms of classical electrodynamics . Within these "classical" computers, some components (such as semiconductors and random number generators ) may rely on quantum behavior, but these components are not isolated from their environment, so any quantum information quickly decoheres . While programmers may depend on probability theory when designing

2119-512: A Chip (SoCs) are complete computers on a microchip (or chip) the size of a coin. They may or may not have integrated RAM and flash memory . If not integrated, the RAM is usually placed directly above (known as Package on package ) or below (on the opposite side of the circuit board ) the SoC, and the flash memory is usually placed right next to the SoC. This is done to improve data transfer speeds, as

2282-490: A classical bit, which can be in one of two states (a binary ), a qubit can exist in a superposition of its two "basis" states, which loosely means that it is in both states simultaneously. When measuring a qubit, the result is a probabilistic output of a classical bit. If a quantum computer manipulates the qubit in a particular way, wave interference effects can amplify the desired measurement results. The design of quantum algorithms involves creating procedures that allow

2445-410: A classical computer in any reasonable amount of time. This concept of extra ability has been called " quantum supremacy ". While such claims have drawn significant attention to the discipline, near-term practical use cases remain limited. For many years, the fields of quantum mechanics and computer science formed distinct academic communities. Modern quantum theory developed in the 1920s to explain

2608-454: A coherent superposition, represented by a point on the surface of the Bloch sphere as described above. Coherence is essential for a qubit to be in a superposition state. With interactions, quantum noise and decoherence , it is possible to put the qubit in a mixed state , a statistical combination or "incoherent mixture" of different pure states. Mixed states can be represented by points inside

2771-463: A computation, because the measurement at the end of the computation gives only one value. To be useful, a quantum algorithm must also incorporate some other conceptual ingredient. There are a number of models of computation for quantum computing, distinguished by the basic elements in which the computation is decomposed. A quantum gate array decomposes computation into a sequence of few-qubit quantum gates . A quantum computation can be described as

2934-744: A control for some specified operation. In particular, the controlled NOT gate (or CNOT or CX) acts on 2 qubits, and performs the NOT operation on the second qubit only when the first qubit is | 1 ⟩ {\displaystyle |1\rangle } , and otherwise leaves it unchanged. With respect to the unentangled product basis { | 00 ⟩ {\displaystyle \{|00\rangle } , | 01 ⟩ {\displaystyle |01\rangle } , | 10 ⟩ {\displaystyle |10\rangle } , | 11 ⟩ } {\displaystyle |11\rangle \}} , it maps

3097-429: A large-scale quantum computer could break some widely used encryption schemes and aid physicists in performing physical simulations ; however, the current state of the art is largely experimental and impractical, with several obstacles to useful applications. The basic unit of information in quantum computing, the qubit (or "quantum bit"), serves the same function as the bit in classical computing. However, unlike

3260-722: A mathematical consequence of this definition, CNOT ⁡ | 00 ⟩ = | 00 ⟩ {\textstyle \operatorname {CNOT} |00\rangle =|00\rangle } , CNOT ⁡ | 01 ⟩ = | 01 ⟩ {\textstyle \operatorname {CNOT} |01\rangle =|01\rangle } , CNOT ⁡ | 10 ⟩ = | 11 ⟩ {\textstyle \operatorname {CNOT} |10\rangle =|11\rangle } , and CNOT ⁡ | 11 ⟩ = | 10 ⟩ {\textstyle \operatorname {CNOT} |11\rangle =|10\rangle } . In other words,

3423-403: A medieval European counting house , a checkered cloth would be placed on a table, and markers moved around on it according to certain rules, as an aid to calculating sums of money. The Antikythera mechanism is believed to be the earliest known mechanical analog computer , according to Derek J. de Solla Price . It was designed to calculate astronomical positions. It was discovered in 1901 in

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3586-638: A more complicated Hamiltonian whose ground state represents the solution to the problem in question. The adiabatic theorem states that if the evolution is slow enough the system will stay in its ground state at all times through the process. Adiabatic optimization may be helpful for solving computational biology problems. Computer A computer is a machine that can be programmed to automatically carry out sequences of arithmetic or logical operations ( computation ). Modern digital electronic computers can perform generic sets of operations known as programs . These programs enable computers to perform

3749-525: A much more general design, an analytical engine , was possible. The input of programs and data was to be provided to the machine via punched cards , a method being used at the time to direct mechanical looms such as the Jacquard loom . For output, the machine would have a printer, a curve plotter and a bell. The machine would also be able to punch numbers onto cards to be read in later. The engine would incorporate an arithmetic logic unit , control flow in

3912-471: A network of quantum logic gates and measurements. However, any measurement can be deferred to the end of quantum computation, though this deferment may come at a computational cost, so most quantum circuits depict a network consisting only of quantum logic gates and no measurements. Quantum parallelism is the heuristic that quantum computers can be thought of as evaluating a function for multiple input values simultaneously. This can be achieved by preparing

4075-561: A network of quantum logic gates and measurements. However, any measurement can be deferred to the end of quantum computation, though this deferment may come at a computational cost, so most quantum circuits depict a network consisting only of quantum logic gates and no measurements. Any quantum computation (which is, in the above formalism, any unitary matrix of size 2 n × 2 n {\displaystyle 2^{n}\times 2^{n}} over n {\displaystyle n} qubits) can be represented as

4238-524: A network of quantum logic gates from a fairly small family of gates. A choice of gate family that enables this construction is known as a universal gate set , since a computer that can run such circuits is a universal quantum computer . One common such set includes all single-qubit gates as well as the CNOT gate from above. This means any quantum computation can be performed by executing a sequence of single-qubit gates together with CNOT gates. Though this gate set

4401-529: A number of successes at breaking encrypted German military communications. The German encryption machine, Enigma , was first attacked with the help of the electro-mechanical bombes which were often run by women. To crack the more sophisticated German Lorenz SZ 40/42 machine, used for high-level Army communications, Max Newman and his colleagues commissioned Flowers to build the Colossus . He spent eleven months from early February 1943 designing and building

4564-491: A polynomial time algorithm for solving the dihedral hidden subgroup problem , which would break many lattice based cryptosystems, is a well-studied open problem. It has been proven that applying Grover's algorithm to break a symmetric (secret key) algorithm by brute force requires time equal to roughly 2 invocations of the underlying cryptographic algorithm, compared with roughly 2 in the classical case, meaning that symmetric key lengths are effectively halved: AES-256 would have

4727-475: A quantum algorithm for integer factorization, could potentially break widely used public-key cryptography schemes like RSA, which rely on the difficulty of factoring large numbers. Post-quantum cryptography, which involves the development of cryptographic algorithms that are resistant to attacks by both classical and quantum computers, is an active area of research aimed at addressing this concern. Ongoing research in quantum cryptography and post-quantum cryptography

4890-658: A quantum computer to perform calculations efficiently and quickly. Quantum computers are not yet practical for real work. Physically engineering high-quality qubits has proven challenging. If a physical qubit is not sufficiently isolated from its environment, it suffers from quantum decoherence , introducing noise into calculations. National governments have invested heavily in experimental research that aims to develop scalable qubits with longer coherence times and lower error rates. Example implementations include superconductors (which isolate an electrical current by eliminating electrical resistance ) and ion traps (which confine

5053-426: A quantum system in a superposition of input states and applying a unitary transformation that encodes the function to be evaluated. The resulting state encodes the function's output values for all input values in the superposition, allowing for the computation of multiple outputs simultaneously. This property is key to the speedup of many quantum algorithms. However, "parallelism" in this sense is insufficient to speed up

Quantum computing - Misplaced Pages Continue

5216-950: A qubit α | 0 ⟩ + β | 1 ⟩ {\displaystyle \alpha |0\rangle +\beta |1\rangle } , the state collapses to | 0 ⟩ {\displaystyle |0\rangle } with probability | α | 2 {\displaystyle |\alpha |^{2}} , or to | 1 ⟩ {\displaystyle |1\rangle } with probability | β | 2 {\displaystyle |\beta |^{2}} . Any valid qubit state has coefficients α {\displaystyle \alpha } and β {\displaystyle \beta } such that | α | 2 + | β | 2 = 1 {\displaystyle |\alpha |^{2}+|\beta |^{2}=1} . As an example, measuring

5379-451: A qubit would destroy its coherence and irrevocably disturb the superposition state. It is possible to fully encode one bit in one qubit. However, a qubit can hold more information, e.g., up to two bits using superdense coding . A bit is always completely in either one of its two states, and a set of n bits (e.g. a processor register or some bit array) can only hold a single of its 2 possible states at any time. A quantum state can be in

5542-913: A sender and receiver exchange quantum states, they can guarantee that an adversary does not intercept the message, as any unauthorized eavesdropper would disturb the delicate quantum system and introduce a detectable change. With appropriate cryptographic protocols , the sender and receiver can thus establish shared private information resistant to eavesdropping. Modern fiber-optic cables can transmit quantum information over relatively short distances. Ongoing experimental research aims to develop more reliable hardware (such as quantum repeaters), hoping to scale this technology to long-distance quantum networks with end-to-end entanglement. Theoretically, this could enable novel technological applications, such as distributed quantum computing and enhanced quantum sensing . Progress in finding quantum algorithms typically focuses on this quantum circuit model, though exceptions like

5705-562: A sequence of sets of values. The whole machine was to be controlled by a read-only program, which was complete with provisions for conditional branching . He also introduced the idea of floating-point arithmetic . In 1920, to celebrate the 100th anniversary of the invention of the arithmometer , Torres presented in Paris the Electromechanical Arithmometer, which allowed a user to input arithmetic problems through

5868-498: A single atomic particle using electromagnetic fields ). In principle, a classical computer can solve the same computational problems as a quantum computer, given enough time. Quantum advantage comes in the form of time complexity rather than computability , and quantum complexity theory shows that some quantum algorithms are exponentially more efficient than the best-known classical algorithms. A large-scale quantum computer could in theory solve computational problems unsolvable by

6031-460: A successful demonstration of its use in computing tables in 1906. In his work Essays on Automatics published in 1914, Leonardo Torres Quevedo wrote a brief history of Babbage's efforts at constructing a mechanical Difference Engine and Analytical Engine. The paper contains a design of a machine capable to calculate formulas like a x ( y − z ) 2 {\displaystyle a^{x}(y-z)^{2}} , for

6194-638: A super-polynomial speedup, which is believed to be unlikely. Some quantum algorithms, like Grover's algorithm and amplitude amplification , give polynomial speedups over corresponding classical algorithms. Though these algorithms give comparably modest quadratic speedup, they are widely applicable and thus give speedups for a wide range of problems. Since chemistry and nanotechnology rely on understanding quantum systems, and such systems are impossible to simulate in an efficient manner classically, quantum simulation may be an important application of quantum computing. Quantum simulation could also be used to simulate

6357-460: A superposition state vector in 2 dimensional Hilbert space. A pure qubit state is a coherent superposition of the basis states. This means that a single qubit ( ψ {\displaystyle \psi } ) can be described by a linear combination of | 0 ⟩ {\displaystyle |0\rangle } and | 1 ⟩ {\displaystyle |1\rangle } : where α and β are

6520-460: A technique called quantum gate teleportation . An adiabatic quantum computer , based on quantum annealing , decomposes computation into a slow continuous transformation of an initial Hamiltonian into a final Hamiltonian, whose ground states contain the solution. Neuromorphic quantum computing (abbreviated as ‘n.quantum computing’) is an unconventional computing type of computing that uses neuromorphic computing to perform quantum operations. It

6683-402: A universal Turing machine. Early computing machines had fixed programs. Changing its function required the re-wiring and re-structuring of the machine. With the proposal of the stored-program computer this changed. A stored-program computer includes by design an instruction set and can store in memory a set of instructions (a program ) that details the computation . The theoretical basis for

Quantum computing - Misplaced Pages Continue

6846-475: A vector labeled ψ {\displaystyle \psi } . Because a qubit is a two-state system, any qubit state takes the form α | 0 ⟩ + β | 1 ⟩ {\displaystyle \alpha |0\rangle +\beta |1\rangle } , where | 0 ⟩ {\displaystyle |0\rangle } and | 1 ⟩ {\displaystyle |1\rangle } are

7009-456: A vital step towards the development of quantum computing . In 2013, a modification of similar systems (using charged rather than neutral donors) has dramatically extended this time, to 3 hours at very low temperatures and 39 minutes at room temperature. Room temperature preparation of a qubit based on electron spins instead of nuclear spin was also demonstrated by a team of scientists from Switzerland and Australia. An increased coherence of qubits

7172-577: A wide range of tasks. The term computer system may refer to a nominally complete computer that includes the hardware , operating system , software , and peripheral equipment needed and used for full operation; or to a group of computers that are linked and function together, such as a computer network or computer cluster . A broad range of industrial and consumer products use computers as control systems , including simple special-purpose devices like microwave ovens and remote controls , and factory devices like industrial robots . Computers are at

7335-577: Is | β | 2 {\displaystyle |\beta |^{2}} . Because the absolute squares of the amplitudes equate to probabilities, it follows that α {\displaystyle \alpha } and β {\displaystyle \beta } must be constrained according to the second axiom of probability theory by the equation The probability amplitudes, α {\displaystyle \alpha } and β {\displaystyle \beta } , encode more than just

7498-414: Is 2-dimensional, and this makes it challenging for a classical computer to simulate a quantum one: representing a 100-qubit system requires storing 2 classical values. The state of this one-qubit quantum memory can be manipulated by applying quantum logic gates , analogous to how classical memory can be manipulated with classical logic gates . One important gate for both classical and quantum computation

7661-415: Is a qubit register . Quantum computers perform calculations by manipulating qubits within a register. The term qudit denotes the unit of quantum information that can be realized in suitable d -level quantum systems. A qubit register that can be measured to N states is identical to an N -level qudit. A rarely used synonym for qudit is quNit , since both d and N are frequently used to denote

7824-455: Is an actively researched topic under the field of post-quantum cryptography . Some public-key algorithms are based on problems other than the integer factorization and discrete logarithm problems to which Shor's algorithm applies, like the McEliece cryptosystem based on a problem in coding theory . Lattice-based cryptosystems are also not known to be broken by quantum computers, and finding

7987-419: Is an incomplete list of physical implementations of qubits, and the choices of basis are by convention only. In 2008 a team of scientists from the U.K. and U.S. reported the first relatively long (1.75 seconds) and coherent transfer of a superposition state in an electron spin "processing" qubit to a nuclear spin "memory" qubit. This event can be considered the first relatively consistent quantum data storage,

8150-427: Is analogous to the unit of classical information trit of ternary computers . Besides the advantage associated with the enlarged computational space, the third qutrit level can be exploited to implement efficient compilation of multi-qubit gates. Any two-level quantum-mechanical system can be used as a qubit. Multilevel systems can be used as well, if they possess two states that can be effectively decoupled from

8313-408: Is believed to be computationally infeasible with an ordinary computer for large integers if they are the product of few prime numbers (e.g., products of two 300-digit primes). By comparison, a quantum computer could solve this problem exponentially faster using Shor's algorithm to find its factors. This ability would allow a quantum computer to break many of the cryptographic systems in use today, in

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8476-573: Is crucial for ensuring the security of communication and data in the face of evolving quantum computing capabilities. Advances in these fields, such as the development of new QKD protocols, the improvement of QRNGs, and the standardization of post-quantum cryptographic algorithms, will play a key role in maintaining the integrity and confidentiality of information in the quantum era. Quantum cryptography enables new ways to transmit data securely; for example, quantum key distribution uses entangled quantum states to establish secure cryptographic keys . When

8639-672: Is in the quantum query model , which is a restricted model where lower bounds are much easier to prove and doesn't necessarily translate to speedups for practical problems. Other problems, including the simulation of quantum physical processes from chemistry and solid-state physics, the approximation of certain Jones polynomials , and the quantum algorithm for linear systems of equations have quantum algorithms appearing to give super-polynomial speedups and are BQP -complete. Because these problems are BQP-complete, an equally fast classical algorithm for them would imply that no quantum algorithm gives

8802-465: Is infinite, it can be replaced with a finite gate set by appealing to the Solovay-Kitaev theorem . Implementation of Boolean functions using the few-qubit quantum gates is presented here. A measurement-based quantum computer decomposes computation into a sequence of Bell state measurements and single-qubit quantum gates applied to a highly entangled initial state (a cluster state ), using

8965-463: Is known as a superposition of | 0 ⟩ {\displaystyle |0\rangle } and | 1 ⟩ {\displaystyle |1\rangle } . A two-dimensional vector mathematically represents a qubit state. Physicists typically use Dirac notation for quantum mechanical linear algebra , writing | ψ ⟩ {\displaystyle |\psi \rangle } ' ket psi ' for

9128-496: Is no way to tell if the first qubit has value "0" or "1" and likewise for the second qubit. Imagine that these two entangled qubits are separated, with one each given to Alice and Bob. Alice makes a measurement of her qubit, obtaining—with equal probabilities—either | 0 ⟩ {\displaystyle |0\rangle } or | 1 ⟩ {\displaystyle |1\rangle } , i.e., she can now tell if her qubit has value "0" or "1". Because of

9291-471: Is removed by the normalization constraint | α | + | β | = 1 . This means, with a suitable change of coordinates, one can eliminate one of the degrees of freedom. One possible choice is that of Hopf coordinates : Additionally, for a single qubit the global phase of the state e i δ {\displaystyle e^{i\delta }} has no physically observable consequences, so we can arbitrarily choose α to be real (or β in

9454-1134: Is simply to select a qubit and apply that gate to the target qubit while leaving the remainder of the memory unaffected. Another way is to apply the gate to its target only if another part of the memory is in a desired state. These two choices can be illustrated using another example. The possible states of a two-qubit quantum memory are | 00 ⟩ := ( 1 0 0 0 ) ; | 01 ⟩ := ( 0 1 0 0 ) ; | 10 ⟩ := ( 0 0 1 0 ) ; | 11 ⟩ := ( 0 0 0 1 ) . {\displaystyle |00\rangle :={\begin{pmatrix}1\\0\\0\\0\end{pmatrix}};\quad |01\rangle :={\begin{pmatrix}0\\1\\0\\0\end{pmatrix}};\quad |10\rangle :={\begin{pmatrix}0\\0\\1\\0\end{pmatrix}};\quad |11\rangle :={\begin{pmatrix}0\\0\\0\\1\end{pmatrix}}.} The controlled NOT (CNOT) gate can then be represented using

9617-436: Is that multiple qubits can exhibit quantum entanglement ; the qubit itself is an exhibition of quantum entanglement. In this case, quantum entanglement is a local or nonlocal property of two or more qubits that allows a set of qubits to express higher correlation than is possible in classical systems. The simplest system to display quantum entanglement is the system of two qubits. Consider, for example, two entangled qubits in

9780-463: Is the stored program , where all the instructions for computing are stored in memory. Von Neumann acknowledged that the central concept of the modern computer was due to this paper. Turing machines are to this day a central object of study in theory of computation . Except for the limitations imposed by their finite memory stores, modern computers are said to be Turing-complete , which is to say, they have algorithm execution capability equivalent to

9943-482: Is the NOT gate, which can be represented by a matrix X := ( 0 1 1 0 ) . {\displaystyle X:={\begin{pmatrix}0&1\\1&0\end{pmatrix}}.} Mathematically, the application of such a logic gate to a quantum state vector is modelled with matrix multiplication . Thus The mathematics of single qubit gates can be extended to operate on multi-qubit quantum memories in two important ways. One way

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10106-570: The | Φ + ⟩ {\displaystyle |\Phi ^{+}\rangle } Bell state : In this state, called an equal superposition , there are equal probabilities of measuring either product state | 00 ⟩ {\displaystyle |00\rangle } or | 11 ⟩ {\displaystyle |11\rangle } , as | 1 / 2 | 2 = 1 / 2 {\displaystyle |1/{\sqrt {2}}|^{2}=1/2} . In other words, there

10269-619: The Antikythera wreck off the Greek island of Antikythera , between Kythera and Crete , and has been dated to approximately c.  100 BCE . Devices of comparable complexity to the Antikythera mechanism would not reappear until the fourteenth century. Many mechanical aids to calculation and measurement were constructed for astronomical and navigation use. The planisphere was a star chart invented by Abū Rayhān al-Bīrūnī in

10432-507: The E6B circular slide rule used for time and distance calculations on light aircraft. In the 1770s, Pierre Jaquet-Droz , a Swiss watchmaker , built a mechanical doll ( automaton ) that could write holding a quill pen. By switching the number and order of its internal wheels different letters, and hence different messages, could be produced. In effect, it could be mechanically "programmed" to read instructions. Along with two other complex machines,

10595-641: The ETH Zurich . The computer was manufactured by Zuse's own company, Zuse KG , which was founded in 1941 as the first company with the sole purpose of developing computers in Berlin. The Z4 served as the inspiration for the construction of the ERMETH , the first Swiss computer and one of the first in Europe. Purely electronic circuit elements soon replaced their mechanical and electromechanical equivalents, at

10758-415: The University of Innsbruck succeeded in developing a universal qudit quantum processor with trapped ions. In the same year, researchers at Tsinghua University 's Center for Quantum Information implemented the dual-type qubit scheme in trapped ion quantum computers using the same ion species. Also in 2022, researchers at the University of California, Berkeley developed a technique to dynamically control

10921-419: The dimension of the state space . As an example, the vector ⁠ 1 / √2 ⁠ |00⟩ + ⁠ 1 / √2 ⁠ |01⟩ represents a two-qubit state, a tensor product of the qubit |0⟩ with the qubit ⁠ 1 / √2 ⁠ |0⟩ + ⁠ 1 / √2 ⁠ |1⟩ . This vector inhabits a four-dimensional vector space spanned by

11084-414: The hidden subgroup problem for abelian finite groups. These algorithms depend on the primitive of the quantum Fourier transform . No mathematical proof has been found that shows that an equally fast classical algorithm cannot be discovered, but evidence suggests that this is unlikely. Certain oracle problems like Simon's problem and the Bernstein–Vazirani problem do give provable speedups, though this

11247-497: The microcomputer revolution in the 1970s. The speed, power, and versatility of computers have been increasing dramatically ever since then, with transistor counts increasing at a rapid pace ( Moore's law noted that counts doubled every two years), leading to the Digital Revolution during the late 20th and early 21st centuries. Conventionally, a modern computer consists of at least one processing element , typically

11410-533: The probability amplitudes , and are both complex numbers . When we measure this qubit in the standard basis, according to the Born rule , the probability of outcome | 0 ⟩ {\displaystyle |0\rangle } with value "0" is | α | 2 {\displaystyle |\alpha |^{2}} and the probability of outcome | 1 ⟩ {\displaystyle |1\rangle } with value "1"

11573-427: The quantum adiabatic algorithm exist. Quantum algorithms can be roughly categorized by the type of speedup achieved over corresponding classical algorithms. Quantum algorithms that offer more than a polynomial speedup over the best-known classical algorithm include Shor's algorithm for factoring and the related quantum algorithms for computing discrete logarithms , solving Pell's equation , and more generally solving

11736-565: The wave–particle duality observed at atomic scales, and digital computers emerged in the following decades to replace human computers for tedious calculations. Both disciplines had practical applications during World War II ; computers played a major role in wartime cryptography , and quantum physics was essential for nuclear physics used in the Manhattan Project . As physicists applied quantum mechanical models to computational problems and swapped digital bits for qubits ,

11899-418: The "South Pole", in the locations where | 0 ⟩ {\displaystyle |0\rangle } and | 1 ⟩ {\displaystyle |1\rangle } are respectively. This particular choice of the polar axis is arbitrary, however. The rest of the surface of the Bloch sphere is inaccessible to a classical bit, but a pure qubit state can be represented by any point on

12062-504: The "second generation" of computers. Compared to vacuum tubes, transistors have many advantages: they are smaller, and require less power than vacuum tubes, so give off less heat. Junction transistors were much more reliable than vacuum tubes and had longer, indefinite, service life. Transistorized computers could contain tens of thousands of binary logic circuits in a relatively compact space. However, early junction transistors were relatively bulky devices that were difficult to manufacture on

12225-525: The 1920s, Vannevar Bush and others developed mechanical differential analyzers. In the 1890s, the Spanish engineer Leonardo Torres Quevedo began to develop a series of advanced analog machines that could solve real and complex roots of polynomials , which were published in 1901 by the Paris Academy of Sciences . Charles Babbage , an English mechanical engineer and polymath , originated

12388-573: The Bloch sphere (or in the Bloch ball). A mixed qubit state has three degrees of freedom: the angles φ {\displaystyle \varphi } and θ {\displaystyle \theta } , as well as the length r {\displaystyle r} of the vector that represents the mixed state. Quantum error correction can be used to maintain the purity of qubits. There are various kinds of physical operations that can be performed on qubits. An important distinguishing feature between qubits and classical bits

12551-606: The Bloch sphere is a two-dimensional space , which represents the observable state space of the pure qubit states. This state space has two local degrees of freedom, which can be represented by the two angles φ {\displaystyle \varphi } and θ {\displaystyle \theta } . A pure state is fully specified by a single ket, | ψ ⟩ = α | 0 ⟩ + β | 1 ⟩ , {\displaystyle |\psi \rangle =\alpha |0\rangle +\beta |1\rangle ,\,}

12714-410: The CNOT applies a NOT gate ( X {\textstyle X} from before) to the second qubit if and only if the first qubit is in the state | 1 ⟩ {\textstyle |1\rangle } . If the first qubit is | 0 ⟩ {\textstyle |0\rangle } , nothing is done to either qubit. In summary, quantum computation can be described as

12877-619: The Cambridge EDSAC of 1949, became operational in April 1951 and ran the world's first routine office computer job . The concept of a field-effect transistor was proposed by Julius Edgar Lilienfeld in 1925. John Bardeen and Walter Brattain , while working under William Shockley at Bell Labs , built the first working transistor , the point-contact transistor , in 1947, which was followed by Shockley's bipolar junction transistor in 1948. From 1955 onwards, transistors replaced vacuum tubes in computer designs, giving rise to

13040-591: The EDVAC in 1945. The Manchester Baby was the world's first stored-program computer . It was built at the University of Manchester in England by Frederic C. Williams , Tom Kilburn and Geoff Tootill , and ran its first program on 21 June 1948. It was designed as a testbed for the Williams tube , the first random-access digital storage device. Although the computer was described as "small and primitive" by

13203-455: The ENIAC were six women, often known collectively as the "ENIAC girls". It combined the high speed of electronics with the ability to be programmed for many complex problems. It could add or subtract 5000 times a second, a thousand times faster than any other machine. It also had modules to multiply, divide, and square root. High speed memory was limited to 20 words (about 80 bytes). Built under

13366-531: The MOS transistor, was invented at Bell Labs between 1955 and 1960 and was the first truly compact transistor that could be miniaturized and mass-produced for a wide range of uses. With its high scalability , and much lower power consumption and higher density than bipolar junction transistors, the MOSFET made it possible to build high-density integrated circuits . In addition to data processing, it also enabled

13529-455: The Scottish scientist Sir William Thomson in 1872 was of great utility to navigation in shallow waters. It used a system of pulleys and wires to automatically calculate predicted tide levels for a set period at a particular location. The differential analyser , a mechanical analog computer designed to solve differential equations by integration , used wheel-and-disc mechanisms to perform

13692-493: The U.S. Although the ENIAC was similar to the Colossus, it was much faster, more flexible, and it was Turing-complete. Like the Colossus, a "program" on the ENIAC was defined by the states of its patch cables and switches, a far cry from the stored program electronic machines that came later. Once a program was written, it had to be mechanically set into the machine with manual resetting of plugs and switches. The programmers of

13855-586: The US, John Vincent Atanasoff and Clifford E. Berry of Iowa State University developed and tested the Atanasoff–Berry Computer (ABC) in 1942, the first "automatic electronic digital computer". This design was also all-electronic and used about 300 vacuum tubes, with capacitors fixed in a mechanically rotating drum for memory. During World War II, the British code-breakers at Bletchley Park achieved

14018-437: The acknowledgments of his 1995 paper, Schumacher states that the term qubit was created in jest during a conversation with William Wootters . A binary digit , characterized as 0 or 1, is used to represent information in classical computers. When averaged over both of its states (0,1), a binary digit can represent up to one bit of Shannon information , where a bit is the basic unit of information . However, in this article,

14181-898: The advent of the integrated circuit (IC). The idea of the integrated circuit was first conceived by a radar scientist working for the Royal Radar Establishment of the Ministry of Defence , Geoffrey W.A. Dummer . Dummer presented the first public description of an integrated circuit at the Symposium on Progress in Quality Electronic Components in Washington, D.C. , on 7 May 1952. The first working ICs were invented by Jack Kilby at Texas Instruments and Robert Noyce at Fairchild Semiconductor . Kilby recorded his initial ideas concerning

14344-416: The algorithm iterates is that of all possible answers. An example and possible application of this is a password cracker that attempts to guess a password. Breaking symmetric ciphers with this algorithm is of interest to government agencies. Quantum annealing relies on the adiabatic theorem to undertake calculations. A system is placed in the ground state for a simple Hamiltonian, which slowly evolves to

14507-647: The basic concept which underlies all electronic digital computers. By 1938, the United States Navy had developed an electromechanical analog computer small enough to use aboard a submarine . This was the Torpedo Data Computer , which used trigonometry to solve the problem of firing a torpedo at a moving target. During World War II similar devices were developed in other countries as well. Early digital computers were electromechanical ; electric switches drove mechanical relays to perform

14670-729: The basis states as follows: A common application of the CNOT gate is to maximally entangle two qubits into the | Φ + ⟩ {\displaystyle |\Phi ^{+}\rangle } Bell state . To construct | Φ + ⟩ {\displaystyle |\Phi ^{+}\rangle } , the inputs A (control) and B (target) to the CNOT gate are: 1 2 ( | 0 ⟩ + | 1 ⟩ ) A {\displaystyle {\frac {1}{\sqrt {2}}}(|0\rangle +|1\rangle )_{A}} and | 0 ⟩ B {\displaystyle |0\rangle _{B}} After applying CNOT,

14833-434: The basis vectors |00⟩ , |01⟩ , |10⟩ , and |11⟩ . The Bell state ⁠ 1 / √2 ⁠ |00⟩ + ⁠ 1 / √2 ⁠ |11⟩ is impossible to decompose into the tensor product of two individual qubits—the two qubits are entangled because their probability amplitudes are correlated . In general, the vector space for an n -qubit system

14996-606: The behavior of atoms and particles at unusual conditions such as the reactions inside a collider . In June 2023, IBM computer scientists reported that a quantum computer produced better results for a physics problem than a conventional supercomputer. About 2% of the annual global energy output is used for nitrogen fixation to produce ammonia for the Haber process in the agricultural fertilizer industry (even though naturally occurring organisms also produce ammonia). Quantum simulations might be used to understand this process and increase

15159-530: The best Arithmetician that euer [ sic ] breathed, and he reduceth thy dayes into a short number." This usage of the term referred to a human computer , a person who carried out calculations or computations . The word continued to have the same meaning until the middle of the 20th century. During the latter part of this period, women were often hired as computers because they could be paid less than their male counterparts. By 1943, most human computers were women. The Online Etymology Dictionary gives

15322-597: The bit is the basic concept of classical information theory, the qubit is the fundamental unit of quantum information . The same term qubit is used to refer to an abstract mathematical model and to any physical system that is represented by that model. A classical bit, by definition, exists in either of two physical states, which can be denoted 0 and 1. A qubit is also described by a state, and two states often written | 0 ⟩ {\displaystyle |0\rangle } and | 1 ⟩ {\displaystyle |1\rangle } serve as

15485-570: The calculation. These devices had a low operating speed and were eventually superseded by much faster all-electric computers, originally using vacuum tubes . The Z2 , created by German engineer Konrad Zuse in 1939 in Berlin , was one of the earliest examples of an electromechanical relay computer. In 1941, Zuse followed his earlier machine up with the Z3 , the world's first working electromechanical programmable , fully automatic digital computer. The Z3

15648-400: The case that α is zero), leaving just two degrees of freedom: where e i φ {\displaystyle e^{i\varphi }} is the physically significant relative phase . The possible quantum states for a single qubit can be visualised using a Bloch sphere (see picture). Represented on such a 2-sphere , a classical bit could only be at the "North Pole" or

15811-1403: The computational basis, are said to span the two-dimensional linear vector (Hilbert) space of the qubit. Qubit basis states can also be combined to form product basis states. A set of qubits taken together is called a quantum register . For example, two qubits could be represented in a four-dimensional linear vector space spanned by the following product basis states: | 00 ⟩ = [ 1 0 0 0 ] {\displaystyle |00\rangle ={\biggl [}{\begin{smallmatrix}1\\0\\0\\0\end{smallmatrix}}{\biggr ]}} , | 01 ⟩ = [ 0 1 0 0 ] {\displaystyle |01\rangle ={\biggl [}{\begin{smallmatrix}0\\1\\0\\0\end{smallmatrix}}{\biggr ]}} , | 10 ⟩ = [ 0 0 1 0 ] {\displaystyle |10\rangle ={\biggl [}{\begin{smallmatrix}0\\0\\1\\0\end{smallmatrix}}{\biggr ]}} , and | 11 ⟩ = [ 0 0 0 1 ] {\displaystyle |11\rangle ={\biggl [}{\begin{smallmatrix}0\\0\\0\\1\end{smallmatrix}}{\biggr ]}} . In general, n qubits are represented by

15974-565: The concept of a programmable computer. Considered the " father of the computer ", he conceptualized and invented the first mechanical computer in the early 19th century. After working on his difference engine he announced his invention in 1822, in a paper to the Royal Astronomical Society , titled "Note on the application of machinery to the computation of astronomical and mathematical tables". He also designed to aid in navigational calculations, in 1833 he realized that

16137-448: The conventional Dirac —or "bra–ket" —notation; the | 0 ⟩ {\displaystyle |0\rangle } and | 1 ⟩ {\displaystyle |1\rangle } are pronounced "ket 0" and "ket 1", respectively. These two orthonormal basis states, { | 0 ⟩ , | 1 ⟩ } {\displaystyle \{|0\rangle ,|1\rangle \}} , together called

16300-704: The core of general-purpose devices such as personal computers and mobile devices such as smartphones . Computers power the Internet , which links billions of computers and users. Early computers were meant to be used only for calculations. Simple manual instruments like the abacus have aided people in doing calculations since ancient times. Early in the Industrial Revolution , some mechanical devices were built to automate long, tedious tasks, such as guiding patterns for looms . More sophisticated electrical machines did specialized analog calculations in

16463-444: The cross-Kerr interactions between fixed-frequency qutrits, achieving high two-qutrit gate fidelities. This was followed by a demonstration of extensible control of superconducting qudits up to d = 4 {\displaystyle d=4} in 2024 based on programmable two-photon interactions. Similar to the qubit, the qutrit is the unit of quantum information that can be realized in suitable 3-level quantum systems. This

16626-499: The data signals do not have to travel long distances. Since ENIAC in 1945, computers have advanced enormously, with modern SoCs (such as the Snapdragon 865) being the size of a coin while also being hundreds of thousands of times more powerful than ENIAC, integrating billions of transistors, and consuming only a few watts of power. The first mobile computers were heavy and ran from mains power. The 50 lb (23 kg) IBM 5100

16789-716: The database, quadratically fewer than the Ω ( n ) {\displaystyle \Omega (n)} queries required for classical algorithms. In this case, the advantage is not only provable but also optimal: it has been shown that Grover's algorithm gives the maximal possible probability of finding the desired element for any number of oracle lookups. Many examples of provable quantum speedups for query problems are based on Grover's algorithm, including Brassard, Høyer, and Tapp's algorithm for finding collisions in two-to-one functions, and Farhi, Goldstone, and Gutmann's algorithm for evaluating NAND trees. Problems that can be efficiently addressed with Grover's algorithm have

16952-515: The decision of the British Government to cease funding. Babbage's failure to complete the analytical engine can be chiefly attributed to political and financial difficulties as well as his desire to develop an increasingly sophisticated computer and to move ahead faster than anyone else could follow. Nevertheless, his son, Henry Babbage , completed a simplified version of the analytical engine's computing unit (the mill ) in 1888. He gave

17115-580: The dimension of a quantum system. Qudits are similar to the integer types in classical computing, and may be mapped to (or realized by) arrays of qubits. Qudits where the d -level system is not an exponent of 2 cannot be mapped to arrays of qubits. It is for example possible to have 5-level qudits. In 2017, scientists at the National Institute of Scientific Research constructed a pair of qudits with 10 different states each, giving more computational power than 6 qubits. In 2022, researchers at

17278-460: The direction of John Mauchly and J. Presper Eckert at the University of Pennsylvania, ENIAC's development and construction lasted from 1943 to full operation at the end of 1945. The machine was huge, weighing 30 tons, using 200 kilowatts of electric power and contained over 18,000 vacuum tubes, 1,500 relays, and hundreds of thousands of resistors, capacitors, and inductors. The principle of

17441-483: The doll is at the Musée d'Art et d'Histoire of Neuchâtel , Switzerland , and still operates. In 1831–1835, mathematician and engineer Giovanni Plana devised a Perpetual Calendar machine , which through a system of pulleys and cylinders could predict the perpetual calendar for every year from 0 CE (that is, 1 BCE) to 4000 CE, keeping track of leap years and varying day length. The tide-predicting machine invented by

17604-481: The early 11th century. The astrolabe was invented in the Hellenistic world in either the 1st or 2nd centuries BCE and is often attributed to Hipparchus . A combination of the planisphere and dioptra , the astrolabe was effectively an analog computer capable of working out several different kinds of problems in spherical astronomy . An astrolabe incorporating a mechanical calendar computer and gear -wheels

17767-494: The early 2000s. These smartphones and tablets run on a variety of operating systems and recently became the dominant computing device on the market. These are powered by System on a Chip (SoCs), which are complete computers on a microchip the size of a coin. Computers can be classified in a number of different ways, including: Qubits The coining of the term qubit is attributed to Benjamin Schumacher . In

17930-399: The early 20th century. The first digital electronic calculating machines were developed during World War II , both electromechanical and using thermionic valves . The first semiconductor transistors in the late 1940s were followed by the silicon -based MOSFET (MOS transistor) and monolithic integrated circuit chip technologies in the late 1950s, leading to the microprocessor and

18093-493: The energy efficiency of production. It is expected that an early use of quantum computing will be modeling that improves the efficiency of the Haber–Bosch process by the mid-2020s although some have predicted it will take longer. A notable application of quantum computation is for attacks on cryptographic systems that are currently in use. Integer factorization , which underpins the security of public key cryptographic systems,

18256-477: The exact definition of the term "microprocessor", it is largely undisputed that the first single-chip microprocessor was the Intel 4004 , designed and realized by Federico Faggin with his silicon-gate MOS IC technology, along with Ted Hoff , Masatoshi Shima and Stanley Mazor at Intel . In the early 1970s, MOS IC technology enabled the integration of more than 10,000 transistors on a single chip. System on

18419-488: The exponential overhead present in classical simulations, validating Feynman's 1982 conjecture. Over the years, experimentalists have constructed small-scale quantum computers using trapped ions and superconductors. In 1998, a two-qubit quantum computer demonstrated the feasibility of the technology, and subsequent experiments have increased the number of qubits and reduced error rates. In 2019, Google AI and NASA announced that they had achieved quantum supremacy with

18582-638: The fields of cryptography and cybersecurity. Quantum cryptography, which relies on the principles of quantum mechanics, offers the possibility of secure communication channels that are resistant to eavesdropping. Quantum key distribution (QKD) protocols, such as BB84, enable the secure exchange of cryptographic keys between parties, ensuring the confidentiality and integrity of communication. Moreover, quantum random number generators (QRNGs) can produce high-quality random numbers, which are essential for secure encryption. However, quantum computing also poses challenges to traditional cryptographic systems. Shor's algorithm,

18745-489: The fields of quantum mechanics and computer science began to converge. In 1980, Paul Benioff introduced the quantum Turing machine , which uses quantum theory to describe a simplified computer. When digital computers became faster, physicists faced an exponential increase in overhead when simulating quantum dynamics , prompting Yuri Manin and Richard Feynman to independently suggest that hardware based on quantum phenomena might be more efficient for computer simulation. In

18908-508: The first Colossus. After a functional test in December 1943, Colossus was shipped to Bletchley Park, where it was delivered on 18 January 1944 and attacked its first message on 5 February. Colossus was the world's first electronic digital programmable computer. It used a large number of valves (vacuum tubes). It had paper-tape input and was capable of being configured to perform a variety of boolean logical operations on its data, but it

19071-725: The first attested use of computer in the 1640s, meaning 'one who calculates'; this is an "agent noun from compute (v.)". The Online Etymology Dictionary states that the use of the term to mean " 'calculating machine' (of any type) is from 1897." The Online Etymology Dictionary indicates that the "modern use" of the term, to mean 'programmable digital electronic computer' dates from "1945 under this name; [in a] theoretical [sense] from 1937, as Turing machine ". The name has remained, although modern computers are capable of many higher-level functions. Devices have been used to aid computation for thousands of years, mostly using one-to-one correspondence with fingers . The earliest counting device

19234-400: The following matrix: CNOT := ( 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 ) . {\displaystyle \operatorname {CNOT} :={\begin{pmatrix}1&0&0&0\\0&1&0&0\\0&0&0&1\\0&0&1&0\end{pmatrix}}.} As

19397-412: The following properties: For problems with all these properties, the running time of Grover's algorithm on a quantum computer scales as the square root of the number of inputs (or elements in the database), as opposed to the linear scaling of classical algorithms. A general class of problems to which Grover's algorithm can be applied is a Boolean satisfiability problem , where the database through which

19560-409: The form of conditional branching and loops , and integrated memory , making it the first design for a general-purpose computer that could be described in modern terms as Turing-complete . The machine was about a century ahead of its time. All the parts for his machine had to be made by hand – this was a major problem for a device with thousands of parts. Eventually, the project was dissolved with

19723-466: The integrated circuit in July 1958, successfully demonstrating the first working integrated example on 12 September 1958. In his patent application of 6 February 1959, Kilby described his new device as "a body of semiconductor material ... wherein all the components of the electronic circuit are completely integrated". However, Kilby's invention was a hybrid integrated circuit (hybrid IC), rather than

19886-411: The integration. In 1876, Sir William Thomson had already discussed the possible construction of such calculators, but he had been stymied by the limited output torque of the ball-and-disk integrators . In a differential analyzer, the output of one integrator drove the input of the next integrator, or a graphing output. The torque amplifier was the advance that allowed these machines to work. Starting in

20049-590: The machine did make use of valves to generate its 125 kHz clock waveforms and in the circuitry to read and write on its magnetic drum memory , so it was not the first completely transistorized computer. That distinction goes to the Harwell CADET of 1955, built by the electronics division of the Atomic Energy Research Establishment at Harwell . The metal–oxide–silicon field-effect transistor (MOSFET), also known as

20212-402: The measurement of a qubit—usually taken to have the value "0" and "1", like a bit. However, whereas the state of a bit can only be binary (either 0 or 1), the general state of a qubit according to quantum mechanics can arbitrarily be a coherent superposition of all computable states simultaneously. Moreover, whereas a measurement of a classical bit would not disturb its state, a measurement of

20375-452: The modern computer was proposed by Alan Turing in his seminal 1936 paper, On Computable Numbers . Turing proposed a simple device that he called "Universal Computing machine" and that is now known as a universal Turing machine . He proved that such a machine is capable of computing anything that is computable by executing instructions (program) stored on tape, allowing the machine to be programmable. The fundamental concept of Turing's design

20538-403: The more famous Sir William Thomson. The art of mechanical analog computing reached its zenith with the differential analyzer , built by H. L. Hazen and Vannevar Bush at MIT starting in 1927. This built on the mechanical integrators of James Thomson and the torque amplifiers invented by H. W. Nieman. A dozen of these devices were built before their obsolescence became obvious. By the 1950s,

20701-556: The near future, but noise in quantum gates limits their reliability. Scientists at Harvard University successfully created "quantum circuits" that correct errors more efficiently than alternative methods, which may potentially remove a major obstacle to practical quantum computers. The Harvard research team was supported by MIT , QuEra Computing , Caltech , and Princeton University and funded by DARPA 's Optimization with Noisy Intermediate-Scale Quantum devices (ONISQ) program. Quantum computing has significant potential applications in

20864-571: The operations that can be performed on these states. Programming a quantum computer is then a matter of composing operations in such a way that the resulting program computes a useful result in theory and is implementable in practice. As physicist Charlie Bennett describes the relationship between quantum and classical computers, A classical computer is a quantum computer ... so we shouldn't be asking about "where do quantum speedups come from?" We should say, "well, all computers are quantum. ... Where do classical slowdowns come from?" Just as

21027-453: The output is the | Φ + ⟩ {\displaystyle |\Phi ^{+}\rangle } Bell State: 1 2 ( | 00 ⟩ + | 11 ⟩ ) {\displaystyle {\frac {1}{\sqrt {2}}}(|00\rangle +|11\rangle )} . The | Φ + ⟩ {\displaystyle |\Phi ^{+}\rangle } Bell state forms part of

21190-592: The physical problem at hand and then leverage their respective physics properties of the system to seek the “minimum”. Neuromorphic quantum computing and quantum computing share similar physical properties during computation. A topological quantum computer decomposes computation into the braiding of anyons in a 2D lattice. A quantum Turing machine is the quantum analog of a Turing machine . All of these models of computation—quantum circuits, one-way quantum computation , adiabatic quantum computation, and topological quantum computation—have been shown to be equivalent to

21353-486: The practical use of MOS transistors as memory cell storage elements, leading to the development of MOS semiconductor memory , which replaced earlier magnetic-core memory in computers. The MOSFET led to the microcomputer revolution , and became the driving force behind the computer revolution . The MOSFET is the most widely used transistor in computers, and is the fundamental building block of digital electronics . The next great advance in computing power came with

21516-814: The probabilities of the outcomes of a measurement; the relative phase between α {\displaystyle \alpha } and β {\displaystyle \beta } is for example responsible for quantum interference , as seen in the double-slit experiment . It might, at first sight, seem that there should be four degrees of freedom in | ψ ⟩ = α | 0 ⟩ + β | 1 ⟩ {\displaystyle |\psi \rangle =\alpha |0\rangle +\beta |1\rangle \,} , as α {\displaystyle \alpha } and β {\displaystyle \beta } are complex numbers with two degrees of freedom each. However, one degree of freedom

21679-568: The quantum Turing machine; given a perfect implementation of one such quantum computer, it can simulate all the others with no more than polynomial overhead. This equivalence need not hold for practical quantum computers, since the overhead of simulation may be too large to be practical. The threshold theorem shows how increasing the number of qubits can mitigate errors, yet fully fault-tolerant quantum computing remains "a rather distant dream". According to some researchers, noisy intermediate-scale quantum ( NISQ ) machines may have specialized uses in

21842-406: The quantum counterparts of the classical states 0 and 1. However, the quantum states | 0 ⟩ {\displaystyle |0\rangle } and | 1 ⟩ {\displaystyle |1\rangle } belong to a vector space , meaning that they can be multiplied by constants and added together, and the result is again a valid quantum state. Such a combination

22005-443: The qubit 1 / 2 | 0 ⟩ + 1 / 2 | 1 ⟩ {\displaystyle 1/{\sqrt {2}}|0\rangle +1/{\sqrt {2}}|1\rangle } would produce either | 0 ⟩ {\displaystyle |0\rangle } or | 1 ⟩ {\displaystyle |1\rangle } with equal probability. Each additional qubit doubles

22168-484: The qubit is in superposition. Such a quantum state vector acts similarly to a (classical) probability vector , with one key difference: unlike probabilities, probability amplitudes are not necessarily positive numbers. Negative amplitudes allow for destructive wave interference. When a qubit is measured in the standard basis , the result is a classical bit. The Born rule describes the norm-squared correspondence between amplitudes and probabilities—when measuring

22331-797: The qubits' entanglement, Bob must now get exactly the same measurement as Alice. For example, if she measures a | 0 ⟩ {\displaystyle |0\rangle } , Bob must measure the same, as | 00 ⟩ {\displaystyle |00\rangle } is the only state where Alice's qubit is a | 0 ⟩ {\displaystyle |0\rangle } . In short, for these two entangled qubits, whatever Alice measures, so would Bob, with perfect correlation, in any basis, however far apart they may be and even though both can not tell if their qubit has value "0" or "1"—a most surprising circumstance that cannot be explained by classical physics. Controlled gates act on 2 or more qubits, where one or more qubits act as

22494-434: The rest (e.g., the ground state and first excited state of a nonlinear oscillator). There are various proposals. Several physical implementations that approximate two-level systems to various degrees have been successfully realized. Similarly to a classical bit where the state of a transistor in a processor, the magnetization of a surface in a hard disk and the presence of current in a cable can all be used to represent bits in

22657-548: The results of operations to be saved and retrieved. It was not until the mid-20th century that the word acquired its modern definition; according to the Oxford English Dictionary , the first known use of the word computer was in a different sense, in a 1613 book called The Yong Mans Gleanings by the English writer Richard Brathwait : "I haue [ sic ] read the truest computer of Times, and

22820-672: The same computer, an eventual quantum computer is likely to use various combinations of qubits in its design. All physical implementations are affected by noise. The so-called T 1 lifetime and T 2 dephasing time are a time to characterize the physical implementation and represent their sensitivity to noise. A higher time does not necessarily mean that one or the other qubit is better suited for quantum computing because gate times and fidelities need to be considered, too. Different applications like quantum sensing , quantum computing and quantum communication use different implementations of qubits to suit their application. The following

22983-509: The same security against an attack using Grover's algorithm that AES-128 has against classical brute-force search (see Key size ). The most well-known example of a problem that allows for a polynomial quantum speedup is unstructured search , which involves finding a marked item out of a list of n {\displaystyle n} items in a database. This can be solved by Grover's algorithm using O ( n ) {\displaystyle O({\sqrt {n}})} queries to

23146-591: The same time that digital calculation replaced analog. The engineer Tommy Flowers , working at the Post Office Research Station in London in the 1930s, began to explore the possible use of electronics for the telephone exchange . Experimental equipment that he built in 1934 went into operation five years later, converting a portion of the telephone exchange network into an electronic data processing system, using thousands of vacuum tubes . In

23309-735: The sense that there would be a polynomial time (in the number of digits of the integer) algorithm for solving the problem. In particular, most of the popular public key ciphers are based on the difficulty of factoring integers or the discrete logarithm problem, both of which can be solved by Shor's algorithm. In particular, the RSA , Diffie–Hellman , and elliptic curve Diffie–Hellman algorithms could be broken. These are used to protect secure Web pages, encrypted email, and many other types of data. Breaking these would have significant ramifications for electronic privacy and security. Identifying cryptographic systems that may be secure against quantum algorithms

23472-478: The setup of the superdense coding , quantum teleportation , and entangled quantum cryptography algorithms. Quantum entanglement also allows multiple states (such as the Bell state mentioned above) to be acted on simultaneously, unlike classical bits that can only have one value at a time. Entanglement is a necessary ingredient of any quantum computation that cannot be done efficiently on a classical computer. Many of

23635-437: The standard basis states , and α {\displaystyle \alpha } and β {\displaystyle \beta } are the probability amplitudes , which are in general complex numbers . If either α {\displaystyle \alpha } or β {\displaystyle \beta } is zero, the qubit is effectively a classical bit; when both are nonzero,

23798-490: The stored-program computer was laid out by Alan Turing in his 1936 paper. In 1945, Turing joined the National Physical Laboratory and began work on developing an electronic stored-program digital computer. His 1945 report "Proposed Electronic Calculator" was the first specification for such a device. John von Neumann at the University of Pennsylvania also circulated his First Draft of a Report on

23961-443: The success of digital electronic computers had spelled the end for most analog computing machines, but analog computers remained in use during the 1950s in some specialized applications such as education ( slide rule ) and aircraft ( control systems ). Claude Shannon 's 1937 master's thesis laid the foundations of digital computing, with his insight of applying Boolean algebra to the analysis and synthesis of switching circuits being

24124-466: The successes of quantum computation and communication, such as quantum teleportation and superdense coding , make use of entanglement, suggesting that entanglement is a resource that is unique to quantum computation. A major hurdle facing quantum computing, as of 2018, in its quest to surpass classical digital computing, is noise in quantum gates that limits the size of quantum circuits that can be executed reliably. A number of qubits taken together

24287-458: The surface. For example, the pure qubit state ( | 0 ⟩ + | 1 ⟩ ) / 2 {\displaystyle (|0\rangle +|1\rangle )/{\sqrt {2}}} would lie on the equator of the sphere at the positive X-axis. In the classical limit , a qubit, which can have quantum states anywhere on the Bloch sphere, reduces to the classical bit, which can be found only at either poles. The surface of

24450-901: The value stored in a qubit can be described as a single point in a 2-dimensional complex coordinate space . Similarly, a set of n qubits, which is also called a register , requires 2 complex numbers to describe its superposition state vector. In quantum mechanics, the general quantum state of a qubit can be represented by a linear superposition of its two orthonormal basis states (or basis vectors ). These vectors are usually denoted as | 0 ⟩ = [ 1 0 ] {\displaystyle |0\rangle ={\bigl [}{\begin{smallmatrix}1\\0\end{smallmatrix}}{\bigr ]}} and | 1 ⟩ = [ 0 1 ] {\displaystyle |1\rangle ={\bigl [}{\begin{smallmatrix}0\\1\end{smallmatrix}}{\bigr ]}} . They are written in

24613-412: The versatility and accuracy of modern digital computers. The first modern analog computer was a tide-predicting machine , invented by Sir William Thomson (later to become Lord Kelvin) in 1872. The differential analyser , a mechanical analog computer designed to solve differential equations by integration using wheel-and-disc mechanisms, was conceptualized in 1876 by James Thomson , the elder brother of

24776-441: The word bit is synonymous with a binary digit. In classical computer technologies, a processed bit is implemented by one of two levels of low direct current voltage , and whilst switching from one of these two levels to the other, a so-called "forbidden zone" between two logic levels must be passed as fast as possible, as electrical voltage cannot change from one level to another instantly. There are two possible outcomes for

24939-406: Was a 16-transistor chip built by Fred Heiman and Steven Hofstein at RCA in 1962. General Microelectronics later introduced the first commercial MOS IC in 1964, developed by Robert Norman. Following the development of the self-aligned gate (silicon-gate) MOS transistor by Robert Kerwin, Donald Klein and John Sarace at Bell Labs in 1967, the first silicon-gate MOS IC with self-aligned gates

25102-625: Was an early example. Later portables such as the Osborne 1 and Compaq Portable were considerably lighter but still needed to be plugged in. The first laptops, such as the Grid Compass , removed this requirement by incorporating batteries – and with the continued miniaturization of computing resources and advancements in portable battery life, portable computers grew in popularity in the 2000s. The same developments allowed manufacturers to integrate computing resources into cellular mobile phones by

25265-537: Was built with 2000 relays , implementing a 22 bit word length that operated at a clock frequency of about 5–10 Hz . Program code was supplied on punched film while data could be stored in 64 words of memory or supplied from the keyboard. It was quite similar to modern machines in some respects, pioneering numerous advances such as floating-point numbers . Rather than the harder-to-implement decimal system (used in Charles Babbage 's earlier design), using

25428-511: Was delivered to the University of Manchester in February 1951. At least seven of these later machines were delivered between 1953 and 1957, one of them to Shell labs in Amsterdam . In October 1947 the directors of British catering company J. Lyons & Company decided to take an active role in promoting the commercial development of computers. Lyons's LEO I computer, modelled closely on

25591-443: Was developed by Federico Faggin at Fairchild Semiconductor in 1968. The MOSFET has since become the most critical device component in modern ICs. The development of the MOS integrated circuit led to the invention of the microprocessor , and heralded an explosion in the commercial and personal use of computers. While the subject of exactly which device was the first microprocessor is contentious, partly due to lack of agreement on

25754-825: Was developed in the late 16th century and found application in gunnery, surveying and navigation. The planimeter was a manual instrument to calculate the area of a closed figure by tracing over it with a mechanical linkage. The slide rule was invented around 1620–1630, by the English clergyman William Oughtred , shortly after the publication of the concept of the logarithm . It is a hand-operated analog computer for doing multiplication and division. As slide rule development progressed, added scales provided reciprocals, squares and square roots, cubes and cube roots, as well as transcendental functions such as logarithms and exponentials, circular and hyperbolic trigonometry and other functions . Slide rules with special scales are still used for quick performance of routine calculations, such as

25917-449: Was invented by Abi Bakr of Isfahan , Persia in 1235. Abū Rayhān al-Bīrūnī invented the first mechanical geared lunisolar calendar astrolabe, an early fixed- wired knowledge processing machine with a gear train and gear-wheels, c.  1000 AD . The sector , a calculating instrument used for solving problems in proportion, trigonometry , multiplication and division, and for various functions, such as squares and cube roots,

26080-477: Was made of germanium . Noyce's monolithic IC was fabricated using the planar process , developed by his colleague Jean Hoerni in early 1959. In turn, the planar process was based on Carl Frosch and Lincoln Derick work on semiconductor surface passivation by silicon dioxide. Modern monolithic ICs are predominantly MOS ( metal–oxide–semiconductor ) integrated circuits, built from MOSFETs (MOS transistors). The earliest experimental MOS IC to be fabricated

26243-643: Was most likely a form of tally stick . Later record keeping aids throughout the Fertile Crescent included calculi (clay spheres, cones, etc.) which represented counts of items, likely livestock or grains, sealed in hollow unbaked clay containers. The use of counting rods is one example. The abacus was initially used for arithmetic tasks. The Roman abacus was developed from devices used in Babylonia as early as 2400 BCE. Since then, many other forms of reckoning boards or tables have been invented. In

26406-434: Was not Turing-complete. Nine Mk II Colossi were built (The Mk I was converted to a Mk II making ten machines in total). Colossus Mark I contained 1,500 thermionic valves (tubes), but Mark II with 2,400 valves, was both five times faster and simpler to operate than Mark I, greatly speeding the decoding process. The ENIAC (Electronic Numerical Integrator and Computer) was the first electronic programmable computer built in

26569-434: Was suggested that quantum algorithms, which are algorithms that run on a realistic model of quantum computation, can be computed equally efficiently with neuromorphic quantum computing. Both, traditional quantum computing and neuromorphic quantum computing are physics-based unconventional computing approaches to computations and do not follow the von Neumann architecture . They both construct a system (a circuit) that represents

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