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For physically implementing a quantum computer, many different candidates are being pursued, among them distinguished by the physical system used to realize the qubits :. A large number of candidates demonstrates that the topic, in spite of rapid progress, is still in its infancy. There is also a vast amount of flexibility. In , Paul Benioff describes the first quantum mechanical model of a computer. In this work, Benioff showed that a computer could operate under the laws of quantum mechanics by describing a Schrodinger equation description of Turing machines , laying a foundation for further work in quantum computing.

The paper [60] was submitted in June and published in April of Russian mathematician Yuri Manin then motivates the development of quantum computers. In , Paul Benioff further develops his original model of a quantum mechanical Turing machine. In , David Deutsch describes the first universal quantum computer. Just as a Universal Turing machine can simulate any other Turing machine efficiently Church-Turing thesis , so the universal quantum computer is able to simulate any other quantum computer with at most a polynomial slowdown.

In , Bikas K. In , David Deutsch and Richard Jozsa propose a computational problem that can be solved efficiently with the determinist Deutsch—Jozsa algorithm on a quantum computer, but for which no deterministic classical algorithm is possible. This was perhaps the earliest result in the computational complexity of quantum computers, proving that they were capable of performing some well-defined computational task more efficiently than any classical computer.

In , an international group of six scientists, including Charles Bennett, showed that perfect quantum teleportation is possible [67] in principle, but only if the original is destroyed. Shor's algorithm can theoretically break many of the Public-key cryptography systems in use today, [68] sparking a tremendous interest in quantum computers. In , the DiVincenzo's criteria are published, which are a list of conditions that are necessary for constructing a quantum computer, proposed by the theoretical physicist David P.

In , researchers demonstrated Shor's algorithm to factor 15 using a 7- qubit NMR computer.

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In , researchers at the University of Michigan built a semiconductor chip ion trap. Such devices from standard lithography may point the way to scalable quantum computing. In , researchers at Yale University created the first solid-state quantum processor. The 2-qubit superconducting chip had artificial atom qubits made of a billion aluminum atoms that acted like a single atom that could occupy two states.

A team at the University of Bristol also created a silicon chip based on quantum optics , able to run Shor's algorithm. In February , Digital Combinational Circuits like an adder, subtractor etc. In April , a team of scientists from Australia and Japan made a breakthrough in quantum teleportation , successfully transferring a complex set of quantum data with full transmission integrity, without affecting the qubits' superpositions. In , D-Wave Systems announced the first commercial quantum annealer , the D-Wave One, claiming a qubit processor.

Investors liked this more than academics, who said D-Wave had not demonstrated that they really had a quantum computer. Criticism softened after a D-Wave paper in Nature that proved that the chips have some quantum properties. During the same year, researchers at the University of Bristol created an all-bulk optics system that ran a version of Shor's algorithm to successfully factor In September , researchers proved quantum computers can be made with a Von Neumann architecture separation of RAM.

In November , researchers factorized using 4 qubits.

Quantum Computing: From Linear Algebra to Physical Realizations

In February , IBM scientists said that they had made several breakthroughs in quantum computing with superconducting integrated circuits. In April , a multinational team of researchers from the University of Southern California, the Delft University of Technology , the Iowa State University of Science and Technology , and the University of California, Santa Barbara constructed a 2-qubit quantum computer on a doped diamond crystal that can easily be scaled up and is functional at room temperature.

Two logical qubit directions of electron spin and nitrogen kernels spin were used, with microwave pulses.

In September , Australian researchers at the University of New South Wales said the world's first quantum computer was just 5 to 10 years away, after announcing a global breakthrough enabling the manufacture of its memory building blocks. A research team led by Australian engineers created the first working qubit based on a single atom in silicon, invoking the same technological platform that forms the building blocks of modern-day computers. Wineland and Serge Haroche for their basic work on understanding the quantum world, which may help make quantum computing possible.

In November , the first quantum teleportation from one macroscopic object to another was reported by scientists at the University of Science and Technology of China. In February , a new technique, boson sampling , was reported by two groups using photons in an optical lattice that is not a universal quantum computer, but may be good enough for practical problems. The Universities Space Research Association USRA will invite researchers to share time on it with the goal of studying quantum computing for machine learning.

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In , researchers at University of New South Wales used silicon as a protectant shell around qubits, making them more accurate, increasing the length of time they will hold information, and possibly making quantum computers easier to build. In April , IBM scientists claimed two critical advances towards the realization of a practical quantum computer, claiming the ability to detect and measure both kinds of quantum errors simultaneously, as well as a new, square quantum bit circuit design that could scale to larger dimensions.

In October , QuTech successfully conducted the Loophole-free Bell inequality violation test using electron spins separated by 1. In October , researchers at the University of New South Wales built a quantum logic gate in silicon for the first time. The device was purchased in via a partnership with Google and Universities Space Research Association. The presence and use of quantum effects in the D-Wave quantum processing unit is more widely accepted. Watson Research Center. In August , scientists at the University of Maryland successfully built the first reprogrammable quantum computer.

In October , the University of Basel described a variant of the electron-hole based quantum computer, which instead of manipulating electron spins, uses electron holes in a semiconductor at low mK temperatures, which are much less vulnerable to decoherence.

This has been dubbed the "positronic" quantum computer, as the quasi-particle behaves as if it has a positive electrical charge. The company also released a new API for the IBM Quantum Experience that enables developers and programmers to begin building interfaces between its existing 5-qubit cloud-based quantum computer and classical computers, without needing a deep background in quantum physics. In May , IBM announced [] that it had successfully built and tested its most powerful universal quantum computing processors.

The first is a qubit processor that will allow for more complex experimentation than the previously available 5-qubit processor. The second is IBM's first prototype commercial processor with 17 qubits, and leverages significant materials, device, and architecture improvements to make it the most powerful quantum processor created to date by IBM. In July , a group of U. Solving a different equation would require building a new system, whereas a computer can solve many different equations.

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In September , IBM Research scientists used a 7-qubit device to model beryllium hydride molecule, the largest molecule to date by a quantum computer. In October , IBM Research scientists successfully "broke the qubit simulation barrier" and simulated and qubit short-depth circuits , using the Lawrence Livermore National Laboratory's Vulcan supercomputer , and the University of Illinois' Cyclops Tensor Framework originally developed at the University of California.

In November , the University of Sydney research team successfully made a microwave circulator , an important quantum computer part, that was times smaller than a conventional circulator, by using topological insulators to slow down the speed of light in a material.

Quantum computing: From linear algebra to physical realizations

In December , Microsoft released a preview version of a "Quantum Development Kit", [] which includes a programming language, Q that can be used to write programs that are run on an emulated quantum computer. In , D-Wave was reported to be selling a 2,qubit quantum computer. In late and early , IBM, [] Intel, [] and Google [] each reported testing quantum processors containing 50, 49, and 72 qubits, respectively, all realized using superconducting circuits.

By number of qubits, these circuits are approaching the range in which simulating their quantum dynamics is expected to become prohibitive on classical computers, although it has been argued that further improvements in error rates are needed to put classical simulation out of reach. In February , scientists reported, for the first time, the discovery of a new form of light , which may involve polaritons , that could be useful in the development of quantum computers.

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In February , QuTech reported successfully testing a silicon-based two-spin-qubits quantum processor. In June , Intel began testing a silicon-based spin-qubit processor, manufactured in the company's D1D Fab in Oregon.

Connecting qubits to bits: the computational basis states

In July , a team led by the University of Sydney achieved the world's first multi-qubit demonstration of a quantum chemistry calculation performed on a system of trapped ions, one of the leading hardware platforms in the race to develop a universal quantum computer. In December , IonQ reported that its machine could be built as large as qubits.

In March , a group of Russian scientists used the open-access IBM quantum computer to demonstrate a protocol for the complex conjugation of the probability amplitudes needed for time reversal of a physical process , [] in this case, for an electron scattered on a two-level impurity, a two- qubit experiment. The class of problems that can be efficiently solved by quantum computers is called BQP , for "bounded error, quantum, polynomial time".

Quantum computers only run probabilistic algorithms , so BQP on quantum computers is the counterpart of BPP "bounded error, probabilistic, polynomial time" on classical computers. It is defined as the set of problems solvable with a polynomial-time algorithm, whose probability of error is bounded away from one half. If that solution runs in polynomial time, then that problem is in BQP.

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Both integer factorization and discrete log are in BQP. Both are suspected to not be NP-complete.

The Mathematics of Quantum Computers - Infinite Series

There is a common misconception that quantum computers can solve NP-complete problems in polynomial time. That is not known to be true, and is generally suspected to be false. The capacity of a quantum computer to accelerate classical algorithms has rigid limits—upper bounds of quantum computation's complexity. The overwhelming part of classical calculations cannot be accelerated on a quantum computer. Bohmian Mechanics is a non-local hidden variable interpretation of quantum mechanics. Neither search method will allow quantum computers to solve NP-Complete problems in polynomial time. Although quantum computers may be faster than classical computers for some problem types, those described above cannot solve any problem that classical computers cannot already solve.

A Turing machine can simulate these quantum computers, so such a quantum computer could never solve an undecidable problem like the halting problem. The existence of "standard" quantum computers does not disprove the Church—Turing thesis. Currently, defining computation in such theories is an open problem due to the problem of time , i.