What is a recent development in quantum computation?
Integrated quantum photonics
As the field has progressed new quantum algorithms have been developed which provide short and long term routes towards the demonstration of the superiority of quantum computers over their classical counterparts. Cluster state computation is now generally accepted as the approach that will be used to develop a fully fledged quantum computer. Whilst development of quantum computer will require the synthesis of many different aspects of integrated optics, boson sampling seeks to demonstrate the power of quantum computation via technologies readily available and is therefore a very promising near term algorithm to doing so. In fact shortly after its proposal there were several small scale experimental demonstrations of the boson sampling algorithm
Centre for Quantum Computation
Initially the CQC was based at the Clarendon Laboratory, but it has now grown to span several departments at the University of Oxford:The Centre has its origins in the early 80's when the computer industry began to worry about the limits of computing. In 1981, Oxford physicist David Deutsch attended a party in Texas given by the famous American physicist John Wheeler who had invited a number of scientists interested in the foundations of computing. It was at this party that Deutsch gained the crucial insight that would lead to an entirely new branch of physics. At the time, computer scientists were turning to Newtonian physics to try to resolve certain fundamental puzzles in the field. But during a conversation at Wheeler’s party, Deutsch realised that this was the wrong approach. Physics is fundamentally governed by quantum theory, and Deutsch could see immediately that using quantum theory instead of Newtonian physics would give a different result. As a consequence of this insight, Deutsch published the paper in 1985 that is now generally regarded as a classic in the field. The paper describes how a computer might run using quantum mechanics and why such a computer is fundamentally different from ordinary computers.
Quantum algorithm
In quantum computing, a quantum algorithm is an algorithm which runs on a realistic model of quantum computation, the most commonly used model being the quantum circuit model of computation. A classical (or non-quantum) algorithm is a finite sequence of instructions, or a step-by-step procedure for solving a problem, where each step or instruction can be performed on a classical computer. Similarly, a quantum algorithm is a step-by-step procedure, where each of the steps can be performed on a quantum computer. Although all classical algorithms can also be performed on a quantum computer, the term quantum algorithm is usually used for those algorithms which seem inherently quantum, or use some essential feature of quantum computation such as quantum superposition or quantum entanglement.
Quantum Turing machine
A quantum Turing machine (QTM), also a universal quantum computer, is an abstract machine used to model the effect of a quantum computer. It provides a very simple model which captures all of the power of quantum computation. Any quantum algorithm can be expressed formally as a particular quantum Turing machine. Such Turing machines were first proposed in a 1985 article written by Oxford University physicist David Deutsch suggesting quantum gates could function in a similar fashion to traditional digital computing binary logic gates.
Quantum computing
In May 2013, Google announced that it was launching the Quantum Artificial Intelligence Lab, hosted by NASA Ames Research Center, with a 512-qubit D-Wave quantum computer. The USRA (Universities Space Research Association) will invite researchers to share time on it with the goal of studying quantum computing for machine learning. Google added that they had "already developed some quantum machine learning algorithms" and had "learned some useful principles", such as that "best results" come from "mixing quantum and classical computing".
Natural computing
Quantum cryptography is not based on the complexity of the computation, but on the special properties of quantum information, such as the fact that quantum information cannot be measured reliably and any attempt at measuring it results in an unavoidable and irreversible disturbance. 
A successful open air experiment in quantum cryptography was reported in 2007, where data was transmitted securely over a distance of 144 km. 
Quantum teleportation is another promising application, in which a quantum state (not matter or energy) is transferred to an arbitrary distant location. Implementations of practical quantum computers are based on various substrates such as ion-traps, 
superconductors, nuclear magnetic resonance, etc. 
As of 2006, the largest quantum computing experiment used liquid state nuclear magnetic resonance quantum information processors, and could operate on up to 12 qubits.
Continuous-variable quantum information
One example of a scientific problem that is naturally expressed in continuous terms is path integration. The general technique of path integration has numerous applications including quantum mechanics, quantum chemistry, statistical mechanics, and computational finance. Because randomness is present throughout quantum theory, one typically requires that a quantum computational procedure yield the correct answer, not with certainty, but with high probability. For example, one might aim for a procedure that computes the correct answer with probability at least 3/4. One also specifies a degree of uncertainty, typically by setting the maximum acceptable error. Thus, the goal of a quantum computation could be to compute the numerical result of a path-integration problem to within an error of at most ε with probability 3/4 or more. In this context, it is known that quantum algorithms can outperform their classical counterparts, and the computational complexity of path integration, as measured by the number of times one would expect to have to query a quantum computer to get a good answer, grows as the inverse of ε.
Quantum shape dynamics
Quantum Shape Dynamics is an emerging research endeavour that aims to provide a quantum understanding of quantum mechanics of shapes, quantum fields on shape dynamics background, and quantization of shape dynamics.
Cavitation
A natural continuation of these studies was recently presented in "The Hydrodynamics of Cavitating Flows" – an encyclopedic work encompassing all the best advances in this domain for the last three decades, and blending the classical methods of mathematical research with the modern capabilities of computer technologies. These include elaboration of nonlinear numerical methods of solving 3D cavitation problems, refinement of the known plane linear theories, development of asymptotic theories of axisymmetric and nearly axisymmetric flows, etc. As compared to the classical approaches, the new trend is characterized by expansion of the theory into the 3D flows. It also reflects a certain correlation with current works of an applied character on the hydrodynamics of supercavitating bodies.
Quantum probability
Some recent advances are based on quantum filtering and feedback control theory as applications of quantum stochastic calculus.
Lov Grover
Lov Kumar Grover (born 1961) is an Indian-American computer scientist. He is the originator of the Grover database search algorithm used in quantum computing. Grover's 1996 algorithm won renown as the second major algorithm proposed for quantum computing (after Shor's 1994 algorithm), and in 2017 was finally implemented in a scalable physical quantum system. Grover's algorithm has been the subject of numerous popular science articles. Grover has been ranked as the 9th most prominent computer scientist from India.
QxBranch
In April 2017, QxBranch, in partnership with the Commonwealth Bank of Australia, released a quantum computing simulator aiming to enable software and algorithm development to assess the feasibility and performance of applications ahead of the development of silicon-based quantum computers. The simulator was modeled on the hardware being developed by the University of New South Wales and made accessible as part of the bank’s internal cloud-based systems, allowing developers to design and evaluate software and algorithms concurrently with the hardware's on-going development.
Quantum sensor
The Defense Advanced Research Projects Agency has recently launched a research program in optical quantum sensors that seeks to exploit ideas from quantum metrology and quantum imaging, such as quantum lithography and the NOON state, in order to achieve these goals with optical sensor systems such as lidar.
Quantum computing
In 2014, a group of researchers from ETH Zürich, USC, Google ,and Microsoft reported a definition of quantum speedup, and were not able to measure quantum speedup with the D-Wave Two device, but did not explicitly rule it out.
Quantum algorithm for linear systems of equations
Wiebe et al. provide a new quantum algorithm to determine the quality of a least-squares fit in which a continuous function is used to approximate a set of discrete points by extending the quantum algorithm for linear systems of equations. As the amount of discrete points increases, the time required to produce a least-squares fit using even a quantum computer running a quantum state tomography algorithm becomes very large. Wiebe et al. find that in many cases, their algorithm can efficiently find a concise approximation of the data points, eliminating the need for the higher-complexity tomography algorithm.
Counterfactual quantum computation
More recently the idea of counterfactual quantum communication has been proposed and demonstrated.
Lieb-Robinson bounds
One of the practical applications of Lieb-Robinson bounds is quantum computing. Current proposals to construct quantum computers built out of atomic-like units mostly rely on the existence of this finite speed of propagation to protect against too rapid dispersal of information.
Intelligence Advanced Research Projects Activity
Several programs at IARPA focus on quantum computing and neuroscience. IARPA is a major funder of quantum computing research due to its applications in quantum cryptography. As of 2009, IARPA was said to provide a large portion of quantum computing funding resources in the United States. Quantum computing research funded by IARPA was named Science Magazine's Breakthrough of the Year in 2010, and physicist David Wineland was a winner of the 2012 Nobel Prize in Physics for quantum computing research funded by IARPA. IARPA is also involved in neuromorphic computation efforts as part of the U.S. BRAIN Initiative and the National Nanotechnology Initiative's Grand Challenge for Future Computing. IARPA's MICrONS project seeks to reverse engineer one cubic millimeter of brain tissue and use insights from its study to improve machine learning and artificial intelligence.
Quantum computing
In December 2015 NASA publicly displayed the world's first fully operational $15-million quantum computer made by the Canadian company D-Wave at the Quantum Artificial Intelligence Laboratory at its Ames Research Center in California's Moffett Field. The device was purchased in 2013 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. In some tests, it can be shown that the D-Wave quantum annealing processor outperforms Selby’s algorithm. Only two of this computer have been made so far.
Quantum computing
In September 2012, 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.
Natural computing
A quantum computer processes data stored as quantum bits (qubits), and uses quantum mechanical phenomena such as superposition and entanglement to perform computations.
A qubit can hold a "0", a "1", or a quantum superposition of these. 
A quantum computer operates on qubits with quantum logic gates. 
Through Shor's polynomial algorithm for factoring integers, and Grover's algorithm for quantum database search that has a quadratic time advantage, quantum computers were shown to potentially possess a significant benefit relative to electronic computers.
Quantum computing
Adiabatic quantum computation relies on the adiabatic theorem to undertake calculations. A system is placed in the ground state for a simple Hamiltonian, which is slowly evolved to 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.
Quantum computing
In November 2017, the University of Sydney research team in Australia successfully made a microwave circulator, an important quantum computer part, 1000 times smaller than a conventional circulator by using topological insulators to slow down the speed of light in a material.
Quantum machine learning
Quantum machine learning is an emerging interdisciplinary research area at the intersection of quantum physics and machine learning. The most common use of the term refers to machine learning algorithms for the analysis of classical data executed on a quantum computer. While machine learning algorithms are used to compute immense quantities of data, quantum machine learning increases such capabilities intelligently, by creating opportunities to conduct analysis on quantum states and systems.This includes hybrid methods that involve both classical and quantum processing, where computationally difficult subroutines are outsourced to a quantum device. These routines can be more complex in nature and executed faster with the assistance of quantum devices. Furthermore, quantum algorithms can be used to analyze quantum states instead of classical data. Beyond quantum computing, the term "quantum machine learning" is often associated with machine learning methods applied to data generated from quantum experiments, such as learning quantum phase transitions or creating new quantum experiments. Quantum machine learning also extends to a branch of research that explores methodological and structural similarities between certain physical systems and learning systems, in particular neural networks. For example, some mathematical and numerical techniques from quantum physics are applicable to classical deep learning and vice versa. Finally, researchers investigate more abstract notions of learning theory with respect to quantum information, sometimes referred to as "quantum learning theory".
Quantum computing
In March 2017, IBM announced an industry-first initiative to build commercially available universal quantum computing systems called IBM Q. The company also released a new API (Application Program Interface) for the IBM Quantum Experience that enables developers and programmers to begin building interfaces between its existing five quantum bit (qubit) cloud-based quantum computer and classical computers, without needing a deep background in quantum physics.
Quantum algorithm for linear systems of equations
Quantum computers are devices that harness quantum mechanics to perform computations in ways that classical computers cannot. For certain problems, quantum algorithms supply exponential speedups over their classical counterparts, the most famous example being Shor's factoring algorithm. Few such exponential speedups are known, and those that are (such as the use of quantum computers to simulate other quantum systems) have so far found limited use outside the domain of quantum mechanics. This algorithm provides an exponentially faster method of estimating features of the solution of a set of linear equations, which is a problem ubiquitous in science and engineering, both on its own and as a subroutine in more complex problems.
Quantum computing
The state of a three-qubit quantum computer is similarly described by an eight-dimensional vector formula_11 (or a one dimensional vector with
each vector node holding the amplitude and the state as the bit string of qubits). Here, however, the coefficients formula_12 are complex numbers, and it is the sum of the "squares" of the coefficients' absolute values, formula_13, that must equal 1. For each formula_14, the absolute value squared formula_15 gives the probability of the system being found in the formula_14-th state after a measurement. However, because a complex number encodes not just a magnitude but also a direction in the complex plane, the phase difference between any two coefficients (states) represents a meaningful parameter. This is a fundamental difference between quantum computing and probabilistic classical computing.
Quantum computing
While a classical 3-bit state and a quantum 3-qubit state are each eight-dimensional vectors, they are manipulated quite differently for classical or quantum computation. For computing in either case, the system must be initialized, for example into the all-zeros string, formula_28, corresponding to the vector (1,0,0,0,0,0,0,0). In classical randomized computation, the system evolves according to the application of stochastic matrices, which preserve that the probabilities add up to one (i.e., preserve the L1 norm). In quantum computation, on the other hand, allowed operations are unitary matrices, which are effectively rotations (they preserve that the sum of the squares add up to one, the Euclidean or L2 norm). (Exactly what unitaries can be applied depend on the physics of the quantum device.) Consequently, since rotations can be undone by rotating backward, quantum computations are reversible. (Technically, quantum operations can be probabilistic combinations of unitaries, so quantum computation really does generalize classical computation. See quantum circuit for a more precise formulation.)
Analog computer
With the development of very-large-scale integration (VLSI) technology, Yannis Tsividis' group at Columbia University has been revisiting analog/hybrid computers design in standard CMOS process. Two VLSI chips have been developed, an 80th-order analog computer (250 nm) by Glenn Cowan in 2005 and a 4th-order hybrid computer (65 nm) developed by Ning Guo in 2015, both targeting at energy-efficient ODE/PDE applications. Glenn's chip contains 16 macros, in which there are 25 analog computing blocks, namely integrators, multipliers, fanouts, few nonlinear blocks. Ning's chip contains one macro block, in which there are 26 computing blocks including integrators, multipliers, fanouts, ADCs, SRAMs and DACs. Arbitrary nonlinear function generation is made possible by the ADC+SRAM+DAC chain, where the SRAM block stores the nonlinear function data. The experiments from the related publications revealed that VLSI analog/hybrid computers demonstrated about 1–2 orders magnitude of advantage in both solution time and energy while achieving accuracy within 5%, which points to the promise of using analog/hybrid computing techniques in the area of energy-efficient approximate computing. In 2016, a team of researchers developed a compiler to solve differential equations using analog circuits.
Michel Devoret
His work in association with well-known experimentalists in the field such as Rob Schoelkopf, Steven Girvin and Irfan Siddiqi has brought about valuable insights in quantum computing and in developing a new paradigm of circuit QED using superconducting electrical circuits.
His recent work, in association with a colleague, A. Marblestone, has shown an exponential quantum enhancement in communication channels as a result of entanglement (see Quantum pseudo-telepathy).
Sridhar Tayur
Tayur created the Quantum Group at Carnegie Mellon in 2017 to study optimization problems―motivated by operations research and operations management applications―by bringing ideas and methods from algebraic geometry, geometry of numbers and algebraic topology to quantum computing. The group has conducted numerical testing on D-Wave of their new hybrid quantum-classical algorithm to compute Graver basis, which is known to be useful in many applications of non-linear integer optimization and game theory. Based on a novel algebraic geometry framing, the Quantum Group is also developing a “Translator API” that will make quantum hardware (based on Ising Spin Glass architecture, like D-Wave) more easily accessible to the optimization community. The group is also the first to introduce Morse theory (from algebraic topology) to understand adiabatic quantum computation better.
Quantum technology
The field of quantum technology was first outlined in a 1997 book by Gerard J. Milburn, which was then followed by a 2003 article by Jonathan P. Dowling and Gerard J. Milburn, as well as a 2003 article by David Deutsch. The field of quantum technology has benefited immensely from the influx of new ideas from the field of quantum information processing, particularly quantum computing. Disparate areas of quantum physics, such as quantum optics, atom optics, quantum electronics, and quantum nanomechanical devices, have been unified under the search for a quantum computer and given a common language, that of quantum information theory.
Quantum computing
Besides factorization and discrete logarithms, quantum algorithms offering a more than polynomial speedup over the best known classical algorithm have been found for several problems, including the simulation of quantum physical processes from chemistry and solid state physics, the approximation of Jones polynomials, and solving Pell's equation. No mathematical proof has been found that shows that an equally fast classical algorithm cannot be discovered, although this is considered unlikely. For some problems, quantum computers offer a polynomial speedup. The most well-known example of this is "quantum database search", which can be solved by Grover's algorithm using quadratically fewer queries to the database than that are required by 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. Several other examples of provable quantum speedups for query problems have subsequently been discovered, such as for finding collisions in two-to-one functions and evaluating NAND trees.
Continuous-variable quantum information
Continuous-variable quantum information is the area of quantum information science that makes use of physical observables, like the strength of an electromagnetic field, whose numerical values belong to continuous intervals. One primary application is quantum computing. In a sense, continuous-variable quantum computation is "analog", while quantum computation using qubits is "digital." In more technical terms, the former makes use of Hilbert spaces that are infinite-dimensional, while the Hilbert spaces for systems comprising collections of qubits are finite-dimensional. One motivation for studying continuous-variable quantum computation is to understand what resources are necessary to make quantum computers more powerful than classical ones.
Quantum programming
A quantum programming environment developed by Cambridge Quantum Computing, that will target simulators and quantum hardware. It is planned for use with the hardware produced by the NQIT hub, as well as Oxford Quantum Circuits. Full release is expected in September 2018.
Quantum imaging
Quantum imaging, is a new sub-field of quantum optics that exploits quantum correlations such as quantum entanglement of the electromagnetic field in order to image objects with a resolution or other imaging criteria that is beyond what is possible in classical optics. Examples of quantum imaging are quantum ghost imaging, quantum lithography, and quantum sensing. Quantum imaging may someday be useful for storing patterns of data in quantum computers and transmitting large amounts of highly secure encrypted information. Quantum mechanics has shown that light has inherent “uncertainties” in its features, manifested as moment-to-moment fluctuations in its properties. Controlling these fluctuations—which represent a sort of “noise”—can improve detection of faint objects, produce better amplified images, and allow workers to more accurately position laser beams.
Neural cryptography
A quantum computer is a device that uses quantum mechanisms for computation. In this device the data are stored as qubits (quantum binary digits). That gives a quantum computer in comparison with a conventional computer the opportunity to solve complicated problems in a short time, e.g. discrete logarithm problem or factorization. Algorithms that are not based on any of these number theory problems are being searched because of this property.
Human-based computation game
"Quantum Moves" is a dexterity and spatial problem solving game, where players move slippery particles across quantum space. Players' solutions on various levels are used to program and fine tune a real quantum computer at Aarhus University.
The game was first developed as a graphical interface for quantum simulation and education in 2012. In 2013 it was released to the public in a user-friendly form, and has been continually updated throughout 2014.
Technological and industrial history of 21st-century Canada
Quantum computing in Canada is also gaining a foothold as evidenced by D-Wave Systems, a Burnaby-based quantum computing company founded in 1999 where in May 2013 it was announced that a collaboration between NASA, Google and the Universities Space Research Association (USRA) launched a Quantum Artificial Intelligence Lab using a 512 qubit D-Wave Two that would be used for research into machine learning, among other fields of study. Though the field is still in its infancy, experiments have been carried out in which quantum computational operations were executed on a very small number of qubits. Both practical and theoretical research continues, and many national governments and military funding agencies support quantum computing research to develop quantum computers for both civilian and national security purposes, such as cryptanalysis.
Quantum mechanics
Another goal is the development of quantum computers, which are expected to perform certain computational tasks exponentially faster than classical computers. Instead of using classical bits, quantum computers use qubits, which can be in superpositions of states. Quantum programmers are able to manipulate the superposition of qubits in order to solve problems that classical computing cannot do effectively, such as searching unsorted databases or integer factorization. IBM claims that the advent of quantum computing may progress the fields of medicine, logistics, financial services, artificial intelligence and cloud security.
Artur Ekert
Ekert and colleagues have made a number of contributions to both theoretical aspects of quantum computation and proposals for its experimental realisations. These include proving that almost any quantum logic gate operating on two quantum bits is universal, proposing one of the first realistic implementations of quantum computation, e.g. using the induced dipole-dipole coupling in an optically driven array of quantum dots, introducing more stable geometric quantum logic gates, and proposing "noiseless encoding", which became later known as decoherence free subspaces. His other notable contributions include work on quantum state swapping, optimal quantum state estimation and quantum state transfer. With some of the same collaborators, he has written on connections between the notion of mathematical proofs and the laws of physics. He has also contributed semi-popular writing on the history of science.
Quantum computing
In late 2017 and early 2018 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.
Superconducting computing
In 1983, Bell Labs created niobium/aluminum oxide Josephson junctions that were more reliable and easier to fabricate. In 1985, the Rapid single flux quantum logic scheme, which had improved speed and energy efficiency, was developed by researchers at Moscow State University. These advances led to the United States' Hybrid Technology Multi-Threaded project, started in 1997, which sought to beat conventional semiconductors to the petaflop computing scale. The project was abandoned in 2000, however, and the first conventional petaflop computer was constructed in 2008. After 2000, attention turned to superconducting quantum computing. The 2011 introduction of reciprocal quantum logic by Quentin Herr of Northrop Grumman, as well as energy-efficient rapid single flux quantum by Hypres, were seen as major advances.
Quantum computing
In 1994 Peter Shor, at AT&T's Bell Labs discovered an important quantum algorithm, which allows a quantum computer to factor large integers exponentially much faster than the best known classical algorithm. Shor's algorithm can theoretically break many of the public-key cryptosystems in use today. Its invention sparked a tremendous interest in quantum computers.
Quantum mechanics
By 1930, quantum mechanics had been further unified and formalized by the work of David Hilbert, Paul Dirac and John von Neumann with greater emphasis on measurement, the statistical nature of our knowledge of reality, and philosophical speculation about the 'observer'. It has since permeated many disciplines, including quantum chemistry, quantum electronics, quantum optics, and quantum information science. Its speculative modern developments include string theory and quantum gravity theories. It also provides a useful framework for many features of the modern periodic table of elements, and describes the behaviors of atoms during chemical bonding and the flow of electrons in computer semiconductors, and therefore plays a crucial role in many modern technologies.
Quantum programming
Q|SI> is a platform embedded in .Net language supporting quantum programming in a quantum extension of while-language. This platform includes a compiler of the quantum while-language and a chain of tools for the simulation of quantum computation, optimisation of quantum circuits, termination analysis of quantum programs, and verification of quantum programs.
Theoretical computer science
, quantum computing is still in its infancy but experiments have been carried out in which quantum computational operations were executed on a very small number of qubits. Both practical and theoretical research continues, and many national governments and military funding agencies support quantum computing research to develop quantum computers for both civilian and national security purposes, such as cryptanalysis.
Superconducting quantum computing
Classical computation models rely on physical implementations consistent with the laws of classical mechanics. It is known, however, that the classical description is only accurate for specific cases, while the more general description of nature is given by the quantum mechanics. Quantum computation studies the application of quantum phenomena, that are beyond the scope of classical approximation, for information processing and communication. Various models of quantum computation exist, however the most popular models incorporate the concepts of qubits and quantum gates. A qubit is a generalization of a bit - a system with two possible states, that can be in a quantum superposition of both. A quantum gate is a generalization of a logic gate: it describes the transformation that one or more qubits will experience after the gate is applied on them, given their initial state. The physical implementation of qubits and gates is difficult, for the same reasons that quantum phenomena are hard to observe in everyday life. One approach is to implement the quantum computers in superconductors, where the quantum effects become macroscopic, though at a price of extremely low operation temperatures.
Quantum Artificial Intelligence Lab
The Quantum AI Lab was announced by Google Research in a blog post on May 16, 2013. At the time of launch, the Lab was using the most advanced commercially available quantum computer, D-Wave Two from D-Wave Systems.
Quantum computing
In December 2017, Microsoft released a preview version of a "Quantum Development Kit". It includes a programming language, Q#, which can be used to write programs that are run on an emulated quantum computer.
Quantum cognition
Quantum cognition is an emerging field which applies the mathematical formalism of quantum theory to model cognitive phenomena such as information processing by the human brain, language, decision making, human memory, concepts and conceptual reasoning, human judgment, and perception. The field clearly distinguishes itself from the quantum mind as it is not reliant on the hypothesis that there is something micro-physical quantum mechanical about the brain. Quantum cognition is based on the quantum-like paradigm or generalized quantum paradigm or quantum structure paradigm that information processing by complex systems such as the brain, taking into account contextual dependence of information and probabilistic reasoning, can be mathematically described in the framework of quantum information and quantum probability theory.
Quantum (statistical programming language)
Quantum is a software package and programming language for statistical survey data validation and manipulation and tabulation. Originally developed by Quantime to run on Unix systems, it was incorporated into SPSS Inc.'s "SPSS MR" product line  after its acquisition of Quantime on September 1997.
Jacquiline Romero
Mary Jacquiline Romero is a quantum physicist in the Australian Research Council Centre of Excellence for Engineered Quantum Systems at the University of Queensland, Australia. Her research expertise and interests are in the field of quantum foundations and quantum information. In particular, Romero is an experimental quantum physicist studying the properties of single photons for the development of new quantum alphabets and the nature of quantum causality.
Physics of computation
The study of the physics of computation relates to understanding the fundamental physical limits of computers. This field has led to the investigation of how thermodynamics limits information processing, the understanding of chaos and dynamical systems, and a rapidly growing effort to invent new quantum computers.
Kane quantum computer
The Kane quantum computer is a proposal for a scalable quantum computer proposed by Bruce Kane in 1998, who was then at the University of New South Wales. Often thought of as a hybrid between quantum dot and nuclear magnetic resonance (NMR) quantum computers, the Kane computer is based on an array of individual phosphorus donor atoms embedded in a pure silicon lattice. Both the nuclear spins of the donors and the spins of the donor electrons participate in the computation.
Quantum computing
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. A similar fact takes place for particular computational tasks, like the search problem, for which Grover's algorithm is optimal.
Edward Farhi
Farhi continues to work on quantum computing but keeps a close eye on particle physics and recent developments in cosmology.
Quantum computing
In April 2011, a team of scientists from Australia and Japan made a breakthrough in quantum teleportation. They successfully transferred a complex set of quantum data with full transmission integrity, without affecting the qubits' superpositions.
Quantum mind
There are computers that are specifically designed to compute using quantum mechanical effects. Quantum computing is computing using quantum-mechanical phenomena, such as superposition and entanglement. They are different from binary digital electronic computers based on transistors. Whereas common digital computing requires that the data be encoded into binary digits (bits), each of which is always in one of two definite states (0 or 1), quantum computation uses quantum bits, which can be in superpositions of states. One of the greatest challenges is controlling or removing quantum decoherence. This usually means isolating the system from its environment as interactions with the external world cause the system to decohere. Currently, some quantum computers require their qubits to be cooled to 20 millikelvins in order to prevent significant decoherence. As a result, time consuming tasks may render some quantum algorithms inoperable, as maintaining the state of qubits for a long enough duration will eventually corrupt the superpositions. There aren't any obvious analogies between the functioning of quantum computers and the human brain. Some of the hypothetical models of quantum mind have proposed mechanisms for maintaining quantum coherence in the brain, but they have not been shown to operate.
Quantum computing
In December 2012, the first dedicated quantum computing software company, 1QBit was founded in Vancouver, BC. 1QBit is the first company to focus exclusively on commercializing software applications for commercially available quantum computers, including the D-Wave Two. 1QBit's research demonstrated the ability of superconducting quantum annealing processors to solve real-world problems.
Robert J. Schoelkopf
Robert Schoelkopf focuses his research on the development of superconducting devices for quantum information processing, which might eventually lead to revolutionary advances in computing.
In 2007, a team of scientists led by Schoelkopf and Steven Girvin made a major breakthrough in quantum computing when it engineered a superconducting communication "bus" to store and transfer information between distant quantum bits, or qubits, on a chip. Their work is the first step to making the fundamentals of quantum computing useful. In 2009, their team demonstrated the first electronic quantum processor which could perform a quantum computation.
Topological quantum computer
A topological quantum computer is a theoretical quantum computer that employs two-dimensional quasiparticles called anyons, whose world lines pass around one another to form braids in a three-dimensional spacetime (i.e., one temporal plus two spatial dimensions). These braids form the logic gates that make up the computer. The advantage of a quantum computer based on quantum braids over using trapped quantum particles is that the former is much more stable. Small, cumulative perturbations can cause quantum states to decohere and introduce errors in the computation, but such small perturbations do not change the braids' topological properties. This is like the effort required to cut a string and reattach the ends to form a different braid, as opposed to a ball (representing an ordinary quantum particle in four-dimensional spacetime) bumping into a wall. Alexei Kitaev proposed topological quantum computation in 1997. While the elements of a topological quantum computer originate in a purely mathematical realm, experiments in fractional quantum Hall systems indicate these elements may be created in the real world using semiconductors made of gallium arsenide at a temperature of near absolute zero and subjected to strong magnetic fields.
Seth Lloyd
His research area is the interplay of information with complex systems, especially quantum systems. He has performed seminal work in the fields of quantum computation, quantum communication and quantum biology, including proposing the first technologically feasible design for a quantum computer, demonstrating the viability of quantum analog computation, proving quantum analogs of Shannon's noisy channel theorem, and designing novel methods for quantum error correction and noise reduction.
Trapped ion quantum computer
The first implementation scheme for a controlled-NOT quantum gate was proposed by Ignacio Cirac and Peter Zoller in 1995, specifically for the trapped ion system. The same year, a key step in the controlled-NOT gate was experimentally realized at NIST Ion Storage Group, and research in quantum computing began to take off worldwide. Many traditional ion trapping research groups have made the transition to quantum computing research, while, more recently, 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.
Quantum computing
In July 2017, a group of U.S. researchers announced a quantum simulator with 51 qubits. The announcement was made by Mikhail Lukin of Harvard University at the International Conference on Quantum Technologies in Moscow. A quantum simulator differs from a computer. Lukin’s simulator was designed to solve one equation. Solving a different equation would require building a new system. A computer can solve many different equations.
Quantum computing
In August 2016, scientists at the University of Maryland successfully built the first reprogrammable quantum computer.
Superconducting quantum computing
Superconducting quantum computing is an implementation of a quantum computer in superconducting electronic circuits. Research in superconducting quantum computing is conducted by Google, Microsoft, IBM, Rigetti, and Intel. , up to nine fully controllable qubits are demonstrated in a 1D array, up to sixteen in a 2D architecture.
Linear optical quantum computing
At its root, the KLM scheme induces an effective interaction between photons by making projective measurements with photodetectors, which falls into the category of non-deterministic quantum computation. It is based on a non-linear sign shift between two qubits that uses two ancilla photons and post-selection. It is also based on the demonstrations that the probability of success of the quantum gates can be made close to one by using entangled states prepared non-deterministically and quantum teleportation with single-qubit operations Otherwise, without a high enough success rate of a single quantum gate unit, it may require an exponential amount of computing resources. Meanwhile, the KLM scheme is based on the fact that proper quantum coding can reduce the resources for obtaining accurately encoded qubits efficiently with respect to the accuracy achieved, and can make LOQC fault-tolerant for photon loss, detector inefficiency and phase decoherence. As a result, LOQC can be robustly implemented through the KLM scheme with a low enough resource requirement to suggest practical scalability, making it as promising a technology for QIP as other known implementations.
UK National Quantum Technologies Programme
The "second quantum revolution", or "quantum 2.0" is a term that is often used to describe quantum technologies based on superposition and entanglement. Originally described in a 1997 book by Gerard J. Milburn, which was then followed by a 2003 article by Jonathan P. Dowling and Gerard J. Milburn, as well as a 2003 article by David Deutsch. These technologies use equipment such as highly stabilised laser systems, magneto-optical traps, cryogenic cooled solid state devices, ion traps and vacuum systems to create, manipulate and then use quantum effects for a number of different purposes. These include: quantum information processing, such as quantum computing, quantum simulation, quantum secure communications, quantum sensing and metrology and quantum imaging, and are widely believed to offer capabilities that will out-perform existing and future classical technologies.
Quantum algorithm for linear systems of equations
On February 5, 2013, Barz et al. demonstrated the quantum algorithm for linear systems of equations on a photonic quantum computing architecture. This implementation used two consecutive entangling gates on the same pair of polarization-encoded qubits. Two separately controlled NOT gates were realized where the successful operation of the first was heralded by a measurement of two ancillary photons. Barz et al. found that the fidelity in the obtained output state ranged from 64.7% to 98.1% due to the influence of higher-order emissions from spontaneous parametric down-conversion.
Yoshihisa Yamamoto (scientist)
During the 2010s, his work has continued on exploring quantum dots as a platform for building both quantum repeaters and quantum computers. One highlight was the co-first demonstration (with Ataç İmamoğlu's group) of entanglement between a spin in a quantum dot and a photon emitted by it. Work on exciton-polaritons continued, and since 2012, Yamamoto has pioneered the development of an optical computer, inspired by developments in quantum annealing and adiabatic quantum computing.
Linear optics
One currently active field of research is the use of linear optics versus the use of nonlinear optics in quantum computing. For example, one model of linear optical quantum computing, the KLM model, is universal for quantum computing, and another model, the boson sampling-based model, is believed to be non-universal (for quantum computing) yet still seems to be able to solve some problems exponentially faster than a classical computer.
Quantum machine learning
Since 2016, IBM has launced on online cloud-based platform for quantum software developers, called the IBM Q Experience. This platform consists of several fully operational quantum processors accessible via the IBM Web API. In doing so, the company is encouraging software developers to pursue new algorithms through a development environment with quantum capabilities. New architectures are being explored on an experimental basis, up to 32 qbits, utilizing both trapped-ion and superconductive quantum computing methods.
Institute for Quantum Computing
The Institute for Quantum Computing (IQC) is an affiliate scientific research institute of the University of Waterloo in located in Waterloo, Ontario with a multidisciplinary approach to the field of quantum information processing. IQC was founded in 2002 primarily through a donation made by Mike Lazaridis and his wife Ophelia whose substantial donations have continued over the years. The institute is now located in the Mike & Ophelia Lazaridis Quantum-Nano Centre and the Research Advancement Centre at the University of Waterloo.
Quantum computing
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. It has been speculated that theories of quantum gravity, such as M-theory or loop quantum gravity, may allow even faster computers to be built. Currently, "defining" computation in such theories is an open problem due to the "problem of time", i.e., there currently exists no obvious way to describe what it means for an observer to submit input to a computer and later receive output.
Pittsburgh Quantum Institute
PQI researchers work on various aspects of quantum computing and information. Development in information theory as well as in quantum algorithms is carried out. Qubit platforms, such as superconducting microwave circuits and Majorana Fermions in semiconducting nanowires are designed for quantum computing. Quantum simulation is another approach to quantum computing; the experimental thrust is the design of a 1D solid state quantum simulation platform that can be controlled on the nanoscale, while a theoretical approach consists of the development of powerful numerical methods that would open the door to faster, more accurate simulations of various novel and exotic quantum systems on a classical computer.
Quantum computing
John Preskill has introduced the term "quantum supremacy" to refer to the hypothetical speedup advantage that a quantum computer would have over a classical computer in a certain field. Google announced in 2017 that it expected to achieve quantum supremacy by the end of the year though that did not happen. IBM said in 2018 that the best classical computers will be beaten on some practical task within about five years and views the quantum supremacy test only as a potential future benchmark. Quantum supremacy has not been achieved yet, and skeptics like Gil Kalai doubt that it will ever be. Bill Unruh doubted the practicality of quantum computers in a paper published back in 1994. Paul Davies pointed out that a 400-qubit computer would even come into conflict with the cosmological information bound implied by the holographic principle. Those such as Roger Schlafly have pointed out that the claimed theoretical benefits of quantum computing go beyond the proven theory of quantum mechanics and imply non-standard interpretations, such as the many-worlds interpretation and negative probabilities. Schlafly maintains that the Born rule is just "metaphysical fluff" and that quantum mechanics does not rely on probability any more than other branches of science but simply calculates the expected values of observables. He also points out that arguments about Turing complexity cannot be run backwards. Those who prefer Bayesian interpretations of quantum mechanics have questioned the physical nature of the mathematical abstractions employed.
Florian Neukart
His research continues to attract considerable public international attention and earned him numerous speaking invitations on both conferences and more private venues. He was one of the first researchers to propose and implement quantum neural networks. He is a pioneer in applied quantum computing and among the first ones to solve real-world problems of society and environment by means of quantum computing. He was the first one to have introduced practical quantum computing in academic education worldwide. He currently works as Principal Scientist at Volkswagen Group of America and visiting scientist and lecturer in the field of quantum computing at the University of Leiden. He was named by the IEEE Computer Society as one of the leading experts in predictive analytics. Neukart was awarded by the Science Park Austria for his work in biologically-inspired artificial intelligence software.
Quantum probability
Quantum probability was developed in the 1980s as a noncommutative analog of the Kolmogorovian theory of stochastic processes. One of its aims is to clarify the mathematical foundations of quantum theory and its statistical interpretation.
Kenji Ohmori
National Institutes of Natural Sciences, Japan (NIMS), Institute for Molecular Science (IMS)Kenji Ohmori has succeeded in designing and visualizing spatiotemporal images given by the interference of matter waves of atoms in a molecule with picometer and femtosecond resolution [1,2]. The precision of this processing is the highest to date, higher than that of the current nanotechnology by three orders of magnitudes. This ultrahigh-precision processing has been implemented with the temporal oscillations of laser electric fields engineered with attosecond precision and imprinted on the matter waves of atoms and electrons in a molecule. He has utilized this technique to develop a molecular computer in which a single 0.3-nanometer-size molecule can calculate 1000 times faster than the current fastest supercomputer [3,4]. He has also developed an ultrafast quantum simulator that can simulate non-equilibrium dynamics of quantum many-body systems in one nanosecond, introducing a novel concept where he has combined his ultrafast coherent control with attosecond precision and ultracold atoms cooled down to temperatures close to absolute zero[5].
Rigetti Computing
Rigetti Computing is a Berkeley, California-based developer of quantum integrated circuits used for quantum computers. The company also develops a cloud platform called Forest that enables programmers to write quantum algorithms.
Quantum programming
Quil is an instruction set architecture for quantum computing that first introduced a shared quantum/classical memory model. It was introduced by Robert Smith, Michael Curtis, and William Zeng in "A Practical Quantum Instruction Set Architecture". Many quantum algorithms (including quantum teleportation, quantum error correction, simulation, and optimization algorithms) require a shared memory architecture.
Transmon
In quantum computing, and more specifically in superconducting quantum computing, a transmon is a type of superconducting charge qubit that was designed to have reduced sensitivity to charge noise. The transmon was developed by Robert J. Schoelkopf, Michel Devoret, Steven M. Girvin and their colleagues at Yale University in 2007. Its name is an abbreviation of the term "transmission line shunted plasma oscillation qubit"; one which consists of a Cooper-pair box "where the two superconductors are also capacitatively shunted in order to decrease the sensitivity to charge noise, while maintaining a sufficient anharmonicity for selective qubit control".
The transmon achieves its reduced sensitivity to charge noise by significantly increasing the ratio of the Josephson energy to the charging energy. This is accomplished through the use of a large shunting capacitor. The result is energy level spacings that are approximately independent of offset charge. Planar on-chip transmon qubits have "T" coherence times ~ 30 μs to 40 μs. By replacing the superconducting transmission line cavity with a three-dimensional superconducting cavity, recent work on transmon qubits has shown significantly improved "T" times, as long as 95 μs. These results demonstrate that previous "T" times were not limited by Josephson junction losses. Understanding the fundamental limits on the coherence time in superconducting qubits such as the transmon is an active area of research.
Quantum computing
In April 2015 IBM scientists claimed two critical advances towards the realization of a practical quantum computer. They claimed 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.
Quantum machine learning
The earliest experiments were conducted using the adiabatic D-Wave quantum computer, for instance, to detect cars in digital images using regularized boosting with a nonconvex objective function in a demonstration in 2009. Many experiments followed on the same architecture, and leading tech companies have shown interest in the potential of quantum machine learning for future technological implementations. In 2013, Google Research, NASA, and the Universities Space Research Association launched the Quantum Artificial Intelligence Lab which explores the use of the adiabatic D-Wave quantum computer. A more recent example trained a probabilistic generative models with arbitrary pairwise connectivity, showing that their model is capable of generating handwritten digits as well as reconstructing noisy images of bars and stripes and handwritten digits.
Quantum programming
Quantum lambda calculi are extensions of the classical lambda calculus introduced by Alonzo Church and Stephen Cole Kleene in the 1930s. The purpose of quantum lambda calculi is to extend quantum programming languages with a theory of higher-order functions.
Quantum cryptography
Under the name of 'quantum tagging', the first position-based quantum schemes have been investigated in 2002 by Kent. A US-patent was granted in 2006. The notion of using quantum effects for location verification first appeared in the scientific literature in 2010. After several other quantum protocols for position verification have been suggested in 2010, Buhrman et al. claimed a general impossibility result: using an enormous amount of quantum entanglement (they use a doubly exponential number of EPR pairs, in the number of qubits the honest player operates on), colluding adversaries are always able to make it look to the verifiers as if they were at the claimed position. However, this result does not exclude the possibility of practical schemes in the bounded- or noisy-quantum-storage model (see above). Later Beigi and König improved the amount of EPR pairs needed in the general attack against position-verification protocols to exponential. They also showed that a particular protocol remains secure against adversaries who controls only a linear amount of EPR pairs. It is argued in that due to time-energy coupling the possibility of formal unconditional location verification via quantum effects remains an open problem.
Quantum Fourier transform
In quantum computing, the quantum Fourier transform (for short: QFT) is a linear transformation on quantum bits, and is the quantum analogue of the discrete Fourier transform. The quantum Fourier transform is a part of many quantum algorithms, notably Shor's algorithm for factoring and computing the discrete logarithm, the quantum phase estimation algorithm for estimating the eigenvalues of a unitary operator, and algorithms for the hidden subgroup problem.
Quantum algorithm for linear systems of equations
While there does not yet exist a quantum computer that can truly offer a speedup over a classical computer, implementation of a "proof of concept" remains an important milestone in the development of a new quantum algorithm. Demonstrating the quantum algorithm for linear systems of equations remained a challenge for years after its proposal until 2013 when it was demonstrated by Cai et al., Barz et al. and Pan et al. in parallel.
Quantum operation
In quantum mechanics, a quantum operation (also known as quantum dynamical map or quantum process) is a mathematical formalism used to describe a broad class of transformations that a quantum mechanical system can undergo. This was first discussed as a general stochastic transformation for a density matrix by George Sudarshan. The quantum operation formalism describes not only unitary time evolution or symmetry transformations of isolated systems, but also the effects of measurement and transient interactions with an environment. In the context of quantum computation, a quantum operation is called a quantum channel.
Robert J. Schoelkopf
Schoelkopf's current research focus, together with Michel Devoret and Steven Girvin of the Yale Department of Applied Physics, is to further develop superconducting circuits that might one day lead to a practical quantum computer. Other projects are directed at developing "hybrid" quantum systems based on integrating cold atoms, molecules, or electrons with solid-state circuits.
Philosophy of artificial intelligence
After concluding that human reasoning is non-computable, Penrose went on to controversially speculate that some kind of hypothetical non-computable processes involving the collapse of quantum mechanical states give humans a special advantage over existing computers. Existing quantum computers are only capable of reducing the complexity of Turing computable tasks and are still restricted to tasks within the scope of Turing machines. . By Penrose and Lucas's arguments, existing quantum computers are not sufficient , so Penrose seeks for some other process involving new physics, for instance quantum gravity which might manifest new physics at the scale of the Planck mass via spontaneous quantum collapse of the wave function. These states, he suggested, occur both within neurons and also spanning more than one neuron. However, other scientists point out that there is no plausible organic mechanism in the brain for harnessing any sort of quantum computation, and furthermore that the timescale of quantum decoherence seems too fast to influence neuron firing.
Quantum algorithm for linear systems of equations
In June 2018, Zhao et al. developed an algorithm for performing Bayesian training of deep neural networks in quantum computers with an exponential speedup over classical training due to the use of the quantum algorithm for linear systems of equations, providing also the first general-purpose implementation of the algorithm to be run in cloud-based quantum computers.
Quantum compass
Developed by the United Kingdom Ministry of Defence for future uses in submarines and aircraft, a prototype quantum compass comprises a one metre long container containing clouds of rubidium atoms frozen using lasers. By measuring the movement of these frozen particles over precise periods of time the motion of the device can be calculated or its proximity to areas of high gravity. It is proposed to use three such containers at right angles to each other to measure change in position from a known location (dead reckoning). The device would then provide a tamper proof accurate position in circumstances where satellites are not available for satellite navigation, e.g. a fully submerged submarine.
Quantum neural network
Rinkus proposes that distributed representation, specifically "sparse distributed representation" (SDR), provides a classical implementation of quantum computing. Specifically, the set of SDR codes stored in an SDR coding field will generally intersect with each other to varying degrees. In other work, Rinkus describes a fixed time learning (and inference) algorithm that preserves similarity in the input space into similarity (intersection size) in the SDR code space. Assuming that input similarity correlates with probability, this means that any single active SDR code is also a probability distribution over all stored inputs, with the probability of each input measured by the fraction of its SDR code that is active (i.e., the size of its intersection with the active SDR code). The learning/inference algorithm can also be viewed as a state update operator and because any single active SDR simultaneously represents both the probability of the single input, X, to which it was assigned during learning and the probabilities of all other stored inputs, the same physical process that updates the probability of X also updates all stored probabilities. By 'fixed time', it is meant that the number of computational steps comprising this process (the update algorithm) remains constant as the number of stored codes increases. This theory departs radically from the standard view of quantum computing and quantum physical theory more generally: rather than assuming that the states of the lowest level entities in the system, i.e., single binary neurons, exist in superposition, it assumes only that higher-level, i.e., composite entities, i.e., whole SDR codes (which are sets of binary neurons), exist in superposition.
Computational complexity
Quantum complexity theory has been developed for studying computational complexity of quantum computing. It is used in post-quantum cryptography, which consists of designing cryptographic protocols that will resist to attacks with quantum computers, when such computers will really exist.
Quantum supremacy
Quantum supremacy is the potential ability of quantum computing devices to solve problems that classical computers practically cannot. Quantum advantage is the potential to solve problems faster. In computational complexity-theoretic terms, this generally means providing a superpolynomial speedup over the best known or possible classical algorithm. The term was originally popularized by John Preskill but the concept of a quantum computational advantage, specifically for simulating quantum systems, dates back to Yuri Manin's (1980)
and Richard Feynman's (1981) proposals of quantum computing.
Quantum technology
Quantum technology is a new field of physics and engineering, which transitions some of the properties of quantum mechanics, especially quantum entanglement, quantum superposition and quantum tunnelling, into practical applications such as quantum computing, quantum sensors, quantum cryptography, quantum simulation, quantum metrology and quantum imaging.
Quantum computing
In May 2017, IBM announced that it has successfully built and tested its most powerful universal quantum computing processors. The first is a 16 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.
Anton Zeilinger
He then also applied quantum entanglement to optical quantum computation, where in 2005, he performed the first implementation of one-way quantum computation. This is a protocol based on quantum measurement as proposed by Knill, Laflamme and Milburn. Most recently, it has been shown that one-way quantum computation can be used to implement blind quantum computing. This solves a problem in Cloud computing, namely that, whatever algorithm a client employs on a quantum server is completely unknown, i.e. blind, to the operator of the server.
Continuous-variable quantum information
Occasionally, and somewhat confusingly, the term "continuous quantum computation" is used to refer to a different area of quantum computing: the study of how to use quantum systems having "finite"-dimensional Hilbert spaces to calculate or approximate the answers to mathematical questions involving continuous functions. A major motivation for investigating the quantum computation of continuous functions is that many scientific problems have mathematical formulations in terms of continuous quantities. A second motivation is to explore and understand the ways in which quantum computers can be more capable or powerful than classical ones. The computational complexity of a problem can be quantified in terms of the minimal computational resources necessary to solve it. In quantum computing, resources include the number of qubits available to a computer and the number of queries that can be made to that computer. The classical complexity of many continuous problems is known. Therefore, when the quantum complexity of these problems is obtained, the question as to whether quantum computers are more powerful than classical can be answered. Furthermore, the degree of the improvement can be quantified. In contrast, the complexity of discrete problems is typically unknown. For example, the classical complexity of integer factorization is unknown.