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Study of a model of computation

Quantum computing is a type of computation that harnesses the collective properties of quantum states, such as superposition, interference, and entanglement, to perform calculations. The devices that perform quantum computations are known as quantum computers.[1]:I-5 They are believed to be able to solve certain computational problems, such as integer factorization (which underlies RSA encryption), substantially faster than classical computers. The study of quantum computing is a subfield of quantum information science. Expansion is expected in the next few years[when?] as the field shifts toward real-world use in pharmaceutical, data security and other applications.[2]

Quantum computing began in 1980 when physicist Paul Benioff proposed a quantum mechanical model of the Turing machine.[3]Richard FeynmanandYuri Maninlater suggested that a quantum computer had the potential to simulate things a classical computer could not feasibly do.[4][5] In 1994, Peter Shor developed a quantum algorithm for factoring integers with the potential to decrypt RSA-encrypted communications.[6] Despite ongoing experimental progress since the late 1990s, most researchers believe that "fault-tolerant quantum computing [is] still a rather distant dream."[7] In recent years, investment in quantum computing research has increased in the public and private sectors.[8][9] On 23 October 2019, Google AI, in partnership with the U.S. National Aeronautics and Space Administration (NASA), claimed to have performed a quantum computation that was infeasible on any classical computer,[10][11] but whether this claim was or is still valid is a topic of active research.[12][13]

There are several types of quantum computers (also known as quantum computing systems), including the quantum circuit model, quantum Turing machine, adiabatic quantum computer, one-way quantum computer, and various quantum cellular automata. The most widely used model is the quantum circuit, based on the quantum bit, or "qubit", which is somewhat analogous to the bit in classical computation. A qubit can be in a 1 or 0 quantum state, or in a superposition of the 1 and 0 states. When it is measured, however, it is always 0 or 1; the probability of either outcome depends on the qubit's quantum state immediately prior to measurement.

Efforts towards building a physical quantum computer focus on technologies such as transmons, ion traps and topological quantum computers, which aim to create high-quality qubits.[1]:213 These qubits may be designed differently, depending on the full quantum computer's computing model, whether quantum logic gates, quantum annealing, or adiabatic quantum computation. There are currently a number of significant obstacles to constructing useful quantum computers. It is particularly difficult to maintain qubits' quantum states, as they suffer from quantum decoherence and state fidelity. Quantum computers therefore require error correction.[14][15]

Any computational problem that can be solved by a classical computer can also be solved by a quantum computer.[16] Conversely, any problem that can be solved by a quantum computer can also be solved by a classical computer, at least in principle given enough time. In other words, quantum computers obey the ChurchTuring thesis. This means that while quantum computers provide no additional advantages over classical computers in terms of computability, quantum algorithms for certain problems have significantly lower time complexities than corresponding known classical algorithms. Notably, quantum computers are believed to be able to quickly solve certain problems that no classical computer could solve in any feasible amount of timea feat known as "quantum supremacy." The study of the computational complexity of problems with respect to quantum computers is known as quantum complexity theory.

The prevailing model of quantum computation describes the computation in terms of a network of quantum logic gates.[17] This model can be thought of as an abstract linear-algebraic generalization of a classical circuit. Since this circuit model obeys quantum mechanics, a quantum computer capable of efficiently running these circuits is believed to be physically realizable.

A memory consisting of n {textstyle n} bits of information has 2 n {textstyle 2^{n}} possible states. A vector representing all memory states thus has 2 n {textstyle 2^{n}} entries (one for each state). This vector is viewed as a probability vector and represents the fact that the memory is to be found in a particular state.

In the classical view, one entry would have a value of 1 (i.e. a 100% probability of being in this state) and all other entries would be zero. In quantum mechanics, probability vectors can be generalized to density operators. The quantum state vector formalism is usually introduced first because it is conceptually simpler, and because it can be used instead of the density matrix formalism for pure states, where the whole quantum system is known.

We begin by considering a simple memory consisting of only one bit. This memory may be found in one of two states: the zero state or the one state. We may represent the state of this memory using Dirac notation so that

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 is the NOT gate, which can be represented by a matrix

The mathematics of single qubit gates can be extended to operate on multi-qubit quantum memories in two important ways. One way is simply to select a qubit and apply that gate to the target qubit whilst 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

In summary, a quantum computation can be described as 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 over n {displaystyle n} qubits) can be represented as 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 is infinite, it can be replaced with a finite gate set by appealing to the Solovay-Kitaev theorem.

Progress in finding quantum algorithms typically focuses on this quantum circuit model, though exceptions like the quantum adiabatic algorithm exist. Quantum algorithms can be roughly categorized by the type of speedup achieved over corresponding classical algorithms.[18]

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 the hidden subgroup problem for abelian finite groups.[18] 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, although this is considered unlikely.[19] Certain oracle problems like Simon's problem and the BernsteinVazirani problem do give provable speedups, though this 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 a super-polynomial speedup, which is believed to be unlikely.[20]

Some quantum algorithms, like Grover's algorithm and amplitude amplification, give polynomial speedups over corresponding classical algorithms.[18] Though these algorithms give comparably modest quadratic speedup, they are widely applicable and thus give speedups for a wide range of problems.[21] Many examples of provable quantum speedups for query problems are related to Grover's algorithm, including Brassard, Hyer, and Tapp's algorithm for finding collisions in two-to-one functions,[22] which uses Grover's algorithm, and Farhi, Goldstone, and Gutmann's algorithm for evaluating NAND trees,[23] which is a variant of the search problem.

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, 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).[24] By comparison, a quantum computer could efficiently solve this problem 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 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, DiffieHellman, and elliptic curve DiffieHellman 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 is an actively researched topic under the field of post-quantum cryptography.[25][26] 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.[25][27] Lattice-based cryptosystems are also not known to be broken by quantum computers, and finding a polynomial time algorithm for solving the dihedral hidden subgroup problem, which would break many lattice based cryptosystems, is a well-studied open problem.[28] It has been proven that applying Grover's algorithm to break a symmetric (secret key) algorithm by brute force requires time equal to roughly 2n/2 invocations of the underlying cryptographic algorithm, compared with roughly 2n in the classical case,[29] meaning that symmetric key lengths are effectively halved: AES-256 would have the same security against an attack using Grover's algorithm that AES-128 has against classical brute-force search (see Key size).

Quantum cryptography could potentially fulfill some of the functions of public key cryptography. Quantum-based cryptographic systems could, therefore, be more secure than traditional systems against quantum hacking.[30]

The most well-known example of a problem admitting a polynomial quantum speedup is unstructured search, 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 the database, quadratically fewer than the ( 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.

Problems that can be addressed with Grover's algorithm have the following properties:[citation needed]

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[31] is Boolean satisfiability problem, where the database through which 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. Symmetric ciphers such as Triple DES and AES are particularly vulnerable to this kind of attack.[citation needed] This application of quantum computing is a major interest of government agencies.[32]

Since chemistry and nanotechnology rely on understanding quantum systems, and such systems are impossible to simulate in an efficient manner classically, many believe quantum simulation will be one of the most important applications of quantum computing.[33] Quantum simulation could also be used to simulate the behavior of atoms and particles at unusual conditions such as the reactions inside a collider.[34]Quantum simulations might be used to predict future paths of particles and protons under superposition in the double-slit experiment.[citation needed]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 while naturally occurring organisms also produce ammonia. Quantum simulations might be used to understand this process increasing production.[35]

Quantum annealing or 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.

Since quantum computers can produce outputs that classical computers cannot produce efficiently, and since quantum computation is fundamentally linear algebraic, some express hope in developing quantum algorithms that can speed up machine learning tasks.[36][37]For example, the quantum algorithm for linear systems of equations, or "HHL Algorithm", named after its discoverers Harrow, Hassidim, and Lloyd, is believed to provide speedup over classical counterparts.[38][37] Some research groups have recently explored the use of quantum annealing hardware for training Boltzmann machines and deep neural networks.[39][40]

In the field of computational biology, computing has played a big role in solving many biological problems. One of the well-known examples would be in computational genomics and how computing has drastically reduced the time to sequence a human genome. Given how computational biology is using generic data modeling and storage, its applications to computational biology are expected to arise as well.[41]

Deep generative chemistry models emerge as powerful tools to expedite drug discovery. However, the immense size and complexity of the structural space of all possible drug-like molecules pose significant obstacles, which could be overcome in the future by quantum computers. Quantum computers are naturally good for solving complex quantum many-body problems [42] and thus may be instrumental in applications involving quantum chemistry. Therefore, one can expect that quantum-enhanced generative models[43] including quantum GANs[44] may eventually be developed into ultimate generative chemistry algorithms. Hybrid architectures combining quantum computers with deep classical networks, such as Quantum Variational Autoencoders, can already be trained on commercially available annealers and used to generate novel drug-like molecular structures.[45]

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.[46] 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.[47] Although skeptics like Gil Kalai doubt that quantum supremacy will ever be achieved,[48][49] in October 2019, a Sycamore processor created in conjunction with Google AI Quantum was reported to have achieved quantum supremacy,[50] with calculations more than 3,000,000 times as fast as those of Summit, generally considered the world's fastest computer.[51] In December 2020, a group at USTC implemented a type of Boson sampling on 76 photons with a photonic quantum computer Jiuzhang to demonstrate quantum supremacy.[52][53][54] The authors claim that a classical contemporary supercomputer would require a computational time of 600 million years to generate the number of samples their quantum processor can generate in 20 seconds.[55] Bill Unruh doubted the practicality of quantum computers in a paper published back in 1994.[56] Paul Davies argued that a 400-qubit computer would even come into conflict with the cosmological information bound implied by the holographic principle.[57]

There are a number of technical challenges in building a large-scale quantum computer.[58] Physicist David DiVincenzo has listed these requirements for a practical quantum computer:[59]

Sourcing parts for quantum computers is also very difficult. Many quantum computers, like those constructed by Google and IBM, need Helium-3, a nuclear research byproduct, and special superconducting cables made only by the Japanese company Coax Co.[60]

The control of multi-qubit systems requires the generation and coordination of a large number of electrical signals with tight and deterministic timing resolution. This has led to the development of quantum controllers which enable interfacing with the qubits. Scaling these systems to support a growing number of qubits is an additional challenge.[citation needed]

One of the greatest challenges involved with constructing quantum computers 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. However, other sources of decoherence also exist. Examples include the quantum gates, and the lattice vibrations and background thermonuclear spin of the physical system used to implement the qubits. Decoherence is irreversible, as it is effectively non-unitary, and is usually something that should be highly controlled, if not avoided. Decoherence times for candidate systems in particular, the transverse relaxation time T2 (for NMR and MRI technology, also called the dephasing time), typically range between nanoseconds and seconds at low temperature.[61] Currently, some quantum computers require their qubits to be cooled to 20 millikelvins in order to prevent significant decoherence.[62] A 2020 study argues that ionizing radiation such as cosmic rays can nevertheless cause certain systems to decohere within milliseconds.[63]

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.[64]

These issues are more difficult for optical approaches as the timescales are orders of magnitude shorter and an often-cited approach to overcoming them is optical pulse shaping. Error rates are typically proportional to the ratio of operating time to decoherence time, hence any operation must be completed much more quickly than the decoherence time.

As described in the Quantum threshold theorem, if the error rate is small enough, it is thought to be possible to use quantum error correction to suppress errors and decoherence. This allows the total calculation time to be longer than the decoherence time if the error correction scheme can correct errors faster than decoherence introduces them. An often cited figure for the required error rate in each gate for fault-tolerant computation is 103, assuming the noise is depolarizing.

Meeting this scalability condition is possible for a wide range of systems. However, the use of error correction brings with it the cost of a greatly increased number of required qubits. The number required to factor integers using Shor's algorithm is still polynomial, and thought to be between L and L2, where L is the number of digits in the number to be factored; error correction algorithms would inflate this figure by an additional factor of L. For a 1000-bit number, this implies a need for about 104 bits without error correction.[65] With error correction, the figure would rise to about 107 bits. Computation time is about L2 or about 107 steps and at 1MHz, about 10 seconds.

A very different approach to the stability-decoherence problem is to create a topological quantum computer with anyons, quasi-particles used as threads and relying on braid theory to form stable logic gates.[66][67]

Physicist Mikhail Dyakonov has expressed skepticism of quantum computing as follows:

There are a number of quantum computing models, distinguished by the basic elements in which the computation is decomposed. The four main models of practical importance are:

The quantum Turing machine is theoretically important but the physical implementation of this model is not feasible. All four models of computation have been shown to be equivalent; each can simulate the other with no more than polynomial overhead.

For physically implementing a quantum computer, many different candidates are being pursued, among them (distinguished by the physical system used to realize the qubits):

The large number of candidates demonstrates that quantum computing, despite rapid progress, is still in its infancy.[citation needed]

Any computational problem solvable by a classical computer is also solvable by a quantum computer.[16] Intuitively, this is because it is believed that all physical phenomena, including the operation of classical computers, can be described using quantum mechanics, which underlies the operation of quantum computers.

Conversely, any problem solvable by a quantum computer is also solvable by a classical computer; or more formally, any quantum computer can be simulated by a Turing machine. In other words, quantum computers provide no additional power over classical computers in terms of computability. This means that quantum computers cannot solve undecidable problems like the halting problem and the existence of quantum computers does not disprove the ChurchTuring thesis.[95]

As of yet, quantum computers do not satisfy the strong Church thesis. While hypothetical machines have been realized, a universal quantum computer has yet to be physically constructed. The strong version of Church's thesis requires a physical computer, and therefore there is no quantum computer that yet satisfies the strong Church thesis.

While quantum computers cannot solve any problems that classical computers cannot already solve, it is suspected that they can solve certain problems faster than classical computers. For instance, it is known that quantum computers can efficiently factor integers, while this is not believed to be the case for classical computers.

The class of problems that can be efficiently solved by a quantum computer with bounded error is called BQP, for "bounded error, quantum, polynomial time". More formally, BQP is the class of problems that can be solved by a polynomial-time quantum Turing machine with an error probability of at most 1/3. As a class of probabilistic problems, BQP is the quantum counterpart to BPP ("bounded error, probabilistic, polynomial time"), the class of problems that can be solved by polynomial-time probabilistic Turing machines with bounded error.[96] It is known that BPP {displaystyle subseteq } BQP and is widely suspected that BQP {displaystyle subsetneq } BPP, which intuitively would mean that quantum computers are more powerful than classical computers in terms of time complexity.[97]

The exact relationship of BQP to P, NP, and PSPACE is not known. However, it is known that P {displaystyle subseteq } BQP {displaystyle subseteq } PSPACE; that is, all problems that can be efficiently solved by a deterministic classical computer can also be efficiently solved by a quantum computer, and all problems that can be efficiently solved by a quantum computer can also be solved by a deterministic classical computer with polynomial space resources. It is further suspected that BQP is a strict superset of P, meaning there are problems that are efficiently solvable by quantum computers that are not efficiently solvable by deterministic classical computers. For instance, integer factorization and the discrete logarithm problem are known to be in BQP and are suspected to be outside of P. On the relationship of BQP to NP, little is known beyond the fact that some NP problems that are believed not to be in P are also in BQP (integer factorization and the discrete logarithm problem are both in NP, for example). It is suspected that NP {displaystyle nsubseteq } BQP; that is, it is believed that there are efficiently checkable problems that are not efficiently solvable by a quantum computer. As a direct consequence of this belief, it is also suspected that BQP is disjoint from the class of NP-complete problems (if an NP-complete problem were in BQP, then it would follow from NP-hardness that all problems in NP are in BQP).[98]

The relationship of BQP to the basic classical complexity classes can be summarized as follows:

It is also known that BQP is contained in the complexity class #P (or more precisely in the associated class of decision problems P#P),[98] which is a subclass of PSPACE.

It has been speculated that further advances in physics could lead to even faster computers. For instance, it has been shown that a non-local hidden variable quantum computer based on Bohmian Mechanics could implement a search of an N {displaystyle N} -item database in at most O ( N 3 ) {displaystyle O({sqrt[{3}]{N}})} steps, a slight speedup over Grover's algorithm, which runs in O ( N ) {displaystyle O({sqrt {N}})} steps. Note, however, that neither search method would allow quantum computers to solve NP-complete problems in polynomial time.[99] Theories of quantum gravity, such as M-theory and loop quantum gravity, may allow even faster computers to be built. However, defining computation in these theories is an open problem due to the problem of time; that is, within these physical theories there is currently no obvious way to describe what it means for an observer to submit input to a computer at one point in time and then receive output at a later point in time.[100][101]

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Quantum computing - Wikipedia

Quantum computing is all the rage. It seems like hardly a day goes by without some news outlet describing the extraordinary things this technology promises. Most commentators forget, or just gloss over, the fact that people have been working on quantum computing for decadesand without any practical results to show for it.

We've been told that quantum computers could provide breakthroughs in many disciplines, including materials and drug discovery, the optimization of complex systems, and artificial intelligence." We've been assured that quantum computers will forever alter our economic, industrial, academic, and societal landscape." We've even been told that the encryption that protects the world's most sensitive data may soon be broken" by quantum computers. It has gotten to the point where many researchers in various fields of physics feel obliged to justify whatever work they are doing by claiming that it has some relevance to quantum computing.

Meanwhile, government research agencies, academic departments (many of them funded by government agencies), and corporate laboratories are spending billions of dollars a year developing quantum computers. On Wall Street, Morgan Stanley and other financial giants expect quantum computing to mature soon and are keen to figure out how this technology can help them.

It's become something of a self-perpetuating arms race, with many organizations seemingly staying in the race if only to avoid being left behind. Some of the world's top technical talent, at places like Google, IBM, and Microsoft, are working hard, and with lavish resources in state-of-the-art laboratories, to realize their vision of a quantum-computing future.

In light of all this, it's natural to wonder: When will useful quantum computers be constructed? The most optimistic experts estimate it will take 5 to 10 years. More cautious ones predict 20 to 30 years. (Similar predictions have been voiced, by the way, for the last 20 years.) I belong to a tiny minority that answers, Not in the foreseeable future." Having spent decades conducting research in quantum and condensed-matter physics, I've developed my very pessimistic view. It's based on an understanding of the gargantuan technical challenges that would have to be overcome to ever make quantum computing work.

The idea of quantum computing first appeared nearly 40 years ago, in 1980, when the Russian-born mathematician Yuri Manin, who now works at the Max Planck Institute for Mathematics, in Bonn, first put forward the notion, albeit in a rather vague form. The concept really got on the map, though, the following year, when physicist Richard Feynman, at the California Institute of Technology, independently proposed it.

Realizing that computer simulations of quantum systems become impossible to carry out when the system under scrutiny gets too complicated, Feynman advanced the idea that the computer itself should operate in the quantum mode: Nature isn't classical, dammit, and if you want to make a simulation of nature, you'd better make it quantum mechanical, and by golly it's a wonderful problem, because it doesn't look so easy," he opined. A few years later, University of Oxford physicist David Deutsch formally described a general-purpose quantum computer, a quantum analogue of the universal Turing machine.

The subject did not attract much attention, though, until 1994, when mathematician Peter Shor (then at Bell Laboratories and now at MIT) proposed an algorithm for an ideal quantum computer that would allow very large numbers to be factored much faster than could be done on a conventional computer. This outstanding theoretical result triggered an explosion of interest in quantum computing. Many thousands of research papers, mostly theoretical, have since been published on the subject, and they continue to come out at an increasing rate.

The basic idea of quantum computing is to store and process information in a way that is very different from what is done in conventional computers, which are based on classical physics. Boiling down the many details, it's fair to say that conventional computers operate by manipulating a large number of tiny transistors working essentially as on-off switches, which change state between cycles of the computer's clock.

The state of the classical computer at the start of any given clock cycle can therefore be described by a long sequence of bits corresponding physically to the states of individual transistors. With N transistors, there are 2N possible states for the computer to be in. Computation on such a machine fundamentally consists of switching some of its transistors between their on" and off" states, according to a prescribed program.

Illustration: Christian Gralingen

In quantum computing, the classical two-state circuit element (the transistor) is replaced by a quantum element called a quantum bit, or qubit. Like the conventional bit, it also has two basic states. Although a variety of physical objects could reasonably serve as quantum bits, the simplest thing to use is the electron's internal angular momentum, or spin, which has the peculiar quantum property of having only two possible projections on any coordinate axis: +1/2 or 1/2 (in units of the Planck constant). For whatever the chosen axis, you can denote the two basic quantum states of the electron's spin as and .

Here's where things get weird. With the quantum bit, those two states aren't the only ones possible. That's because the spin state of an electron is described by a quantum-mechanical wave function. And that function involves two complex numbers, and (called quantum amplitudes), which, being complex numbers, have real parts and imaginary parts. Those complex numbers, and , each have a certain magnitude, and according to the rules of quantum mechanics, their squared magnitudes must add up to 1.

That's because those two squared magnitudes correspond to the probabilities for the spin of the electron to be in the basic states and when you measure it. And because those are the only outcomes possible, the two associated probabilities must add up to 1. For example, if the probability of finding the electron in the state is 0.6 (60 percent), then the probability of finding it in the state must be 0.4 (40 percent)nothing else would make sense.

In contrast to a classical bit, which can only be in one of its two basic states, a qubit can be in any of a continuum of possible states, as defined by the values of the quantum amplitudes and . This property is often described by the rather mystical and intimidating statement that a qubit can exist simultaneously in both of its and states.

Yes, quantum mechanics often defies intuition. But this concept shouldn't be couched in such perplexing language. Instead, think of a vector positioned in the x-y plane and canted at 45 degrees to the x-axis. Somebody might say that this vector simultaneously points in both the x- and y-directions. That statement is true in some sense, but it's not really a useful description. Describing a qubit as being simultaneously in both and states is, in my view, similarly unhelpful. And yet, it's become almost de rigueur for journalists to describe it as such.

In a system with two qubits, there are 22 or 4 basic states, which can be written (), (), (), and (). Naturally enough, the two qubits can be described by a quantum-mechanical wave function that involves four complex numbers. In the general case of N qubits, the state of the system is described by 2N complex numbers, which are restricted by the condition that their squared magnitudes must all add up to 1.

While a conventional computer with N bits at any given moment must be in one of its 2N possible states, the state of a quantum computer with N qubits is described by the values of the 2N quantum amplitudes, which are continuous parameters (ones that can take on any value, not just a 0 or a 1). This is the origin of the supposed power of the quantum computer, but it is also the reason for its great fragility and vulnerability.

How is information processed in such a machine? That's done by applying certain kinds of transformationsdubbed quantum gates"that change these parameters in a precise and controlled manner.

Experts estimate that the number of qubits needed for a useful quantum computer, one that could compete with your laptop in solving certain kinds of interesting problems, is between 1,000 and 100,000. So the number of continuous parameters describing the state of such a useful quantum computer at any given moment must be at least 21,000, which is to say about 10300. That's a very big number indeed. How big? It is much, much greater than the number of subatomic particles in the observable universe.

To repeat: A useful quantum computer needs to process a set of continuous parameters that is larger than the number of subatomic particles in the observable universe.

At this point in a description of a possible future technology, a hardheaded engineer loses interest. But let's continue. In any real-world computer, you have to consider the effects of errors. In a conventional computer, those arise when one or more transistors are switched off when they are supposed to be switched on, or vice versa. This unwanted occurrence can be dealt with using relatively simple error-correction methods, which make use of some level of redundancy built into the hardware.

In contrast, it's absolutely unimaginable how to keep errors under control for the 10300 continuous parameters that must be processed by a useful quantum computer. Yet quantum-computing theorists have succeeded in convincing the general public that this is feasible. Indeed, they claim that something called the threshold theorem proves it can be done. They point out that once the error per qubit per quantum gate is below a certain value, indefinitely long quantum computation becomes possible, at a cost of substantially increasing the number of qubits needed. With those extra qubits, they argue, you can handle errors by forming logical qubits using multiple physical qubits.

How many physical qubits would be required for each logical qubit? No one really knows, but estimates typically range from about 1,000 to 100,000. So the upshot is that a useful quantum computer now needs a million or more qubits. And the number of continuous parameters defining the state of this hypothetical quantum-computing machinewhich was already more than astronomical with 1,000 qubitsnow becomes even more ludicrous.

Even without considering these impossibly large numbers, it's sobering that no one has yet figured out how to combine many physical qubits into a smaller number of logical qubits that can compute something useful. And it's not like this hasn't long been a key goal.

In the early 2000s, at the request of the Advanced Research and Development Activity (a funding agency of the U.S. intelligence community that is now part of Intelligence Advanced Research Projects Activity), a team of distinguished experts in quantum information established a road map for quantum computing. It had a goal for 2012 that requires on the order of 50 physical qubits" and exercises multiple logical qubits through the full range of operations required for fault-tolerant [quantum computation] in order to perform a simple instance of a relevant quantum algorithm." It's now the end of 2018, and that ability has still not been demonstrated.

Illustration: Christian Gralingen

The huge amount of scholarly literature that's been generated about quantum-computing is notably light on experimental studies describing actual hardware. The relatively few experiments that have been reported were extremely difficult to conduct, though, and must command respect and admiration.

The goal of such proof-of-principle experiments is to show the possibility of carrying out basic quantum operations and to demonstrate some elements of the quantum algorithms that have been devised. The number of qubits used for them is below 10, usually from 3 to 5. Apparently, going from 5 qubits to 50 (the goal set by the ARDA Experts Panel for the year 2012) presents experimental difficulties that are hard to overcome. Most probably they are related to the simple fact that 25 = 32, while 250 = 1,125,899,906,842,624.

By contrast, the theory of quantum computing does not appear to meet any substantial difficulties in dealing with millions of qubits. In studies of error rates, for example, various noise models are being considered. It has been proved (under certain assumptions) that errors generated by local" noise can be corrected by carefully designed and very ingenious methods, involving, among other tricks, massive parallelism, with many thousands of gates applied simultaneously to different pairs of qubits and many thousands of measurements done simultaneously, too.

A decade and a half ago, ARDA's Experts Panel noted that it has been established, under certain assumptions, that if a threshold precision per gate operation could be achieved, quantum error correction would allow a quantum computer to compute indefinitely." Here, the key words are under certain assumptions." That panel of distinguished experts did not, however, address the question of whether these assumptions could ever be satisfied.

I argue that they can't. In the physical world, continuous quantities (be they voltages or the parameters defining quantum-mechanical wave functions) can be neither measured nor manipulated exactly. That is, no continuously variable quantity can be made to have an exact value, including zero. To a mathematician, this might sound absurd, but this is the unquestionable reality of the world we live in, as any engineer knows.

Sure, discrete quantities, like the number of students in a classroom or the number of transistors in the on" state, can be known exactly. Not so for quantities that vary continuously. And this fact accounts for the great difference between a conventional digital computer and the hypothetical quantum computer.

Indeed, all of the assumptions that theorists make about the preparation of qubits into a given state, the operation of the quantum gates, the reliability of the measurements, and so forth, cannot be fulfilled exactly. They can only be approached with some limited precision. So, the real question is: What precision is required? With what exactitude must, say, the square root of 2 (an irrational number that enters into many of the relevant quantum operations) be experimentally realized? Should it be approximated as 1.41 or as 1.41421356237? Or is even more precision needed? There are no clear answers to these crucial questions.

While various strategies for building quantum computers are now being explored, an approach that many people consider the most promising, initially undertaken by the Canadian company D-Wave Systems and now being pursued by IBM, Google, Microsoft, and others, is based on using quantum systems of interconnected Josephson junctions cooled to very low temperatures (down to about 10 millikelvins).

The ultimate goal is to create a universal quantum computer, one that can beat conventional computers in factoring large numbers using Shor's algorithm, performing database searches by a similarly famous quantum-computing algorithm that Lov Grover developed at Bell Laboratories in 1996, and other specialized applications that are suitable for quantum computers.

On the hardware front, advanced research is under way, with a 49-qubit chip (Intel), a 50-qubit chip (IBM), and a 72-qubit chip (Google) having recently been fabricated and studied. The eventual outcome of this activity is not entirely clear, especially because these companies have not revealed the details of their work.

While I believe that such experimental research is beneficial and may lead to a better understanding of complicated quantum systems, I'm skeptical that these efforts will ever result in a practical quantum computer. Such a computer would have to be able to manipulateon a microscopic level and with enormous precisiona physical system characterized by an unimaginably huge set of parameters, each of which can take on a continuous range of values. Could we ever learn to control the more than 10300 continuously variable parameters defining the quantum state of such a system?

My answer is simple. No, never.

I believe that, appearances to the contrary, the quantum computing fervor is nearing its end. That's because a few decades is the maximum lifetime of any big bubble in technology or science. After a certain period, too many unfulfilled promises have been made, and anyone who has been following the topic starts to get annoyed by further announcements of impending breakthroughs. What's more, by that time all the tenured faculty positions in the field are already occupied. The proponents have grown older and less zealous, while the younger generation seeks something completely new and more likely to succeed.

All these problems, as well as a few others I've not mentioned here, raise serious doubts about the future of quantum computing. There is a tremendous gap between the rudimentary but very hard experiments that have been carried out with a few qubits and the extremely developed quantum-computing theory, which relies on manipulating thousands to millions of qubits to calculate anything useful. That gap is not likely to be closed anytime soon.

To my mind, quantum-computing researchers should still heed an admonition that IBM physicist Rolf Landauer made decades ago when the field heated up for the first time. He urged proponents of quantum computing to include in their publications a disclaimer along these lines: This scheme, like all other schemes for quantum computation, relies on speculative technology, does not in its current form take into account all possible sources of noise, unreliability and manufacturing error, and probably will not work."

Editor's note: A sentence in this article originally stated that concerns over required precision were never even discussed." This sentence was changed on 30 November 2018 after some readers pointed out to the author instances in the literature that had considered these issues. The amended sentence now reads: There are no clear answers to these crucial questions."

Mikhail Dyakonov does research in theoretical physics at Charles Coulomb Laboratory at the University of Montpellier, in France. His name is attached to various physical phenomena, perhaps most famously Dyakonov surface waves.

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The Case Against Quantum Computing - IEEE Spectrum: Technology, Engineering, and Science News

CompanyDate initiatedAreaTechnologyAffiliate University or Research InstituteHeadquartersAccenture[1]June 14, 2017ComputingAEGIQ[2]2019Computing/CommunicationPhotonics and Integrated Photonics, Quantum Dots, CryptographySheffield, UKAlice&Bob2020ComputingSuperconductingParis, FranceAliro Quantum2019Computing/NetworkingQuantum Development Environment, Quantum Network Simulation/EmulationSpun out of Narang Lab, HarvardBoston, MAAlpine Quantum Technologies[3]2018ComputingTrapped IonUniversity of Innsbruck and the Institute for Quantum Optics and Quantum Information (IQOQI)Innsbruck, AustriaAmberFlux2019Computing/CommunicationsQuantum Programming, Classical Simulation, Optimization, Algorithms, Quantum Financial ServicesHyderabad, IndiaAirbus[4]2015ComputingAlgorithmsBlagnac, FranceAT&T[5]2011CommunicationCalifornia Institute of Technology, Fermilab[6]Dallas, TX, USAAliyun (Alibaba Cloud)[7]July 30, 2015Computing/Communication[7][8]SuperconductingChinese Academy of Sciences[8][9][10]Hangzhou, ChinaAtos[11][12]Computing/CommunicationQuantum Programming, Classical Simulation, CryptographyBezons, FranceBaidu[13]2018ComputingAlgorithmsUniversity of Technology Sydney[13]Beijing, ChinaBOLTZ.AI[14]2020ApplicationsQuantum Algorithms|Quantum programming|Quantum ConsultingUniversity of Toronto[13]Toronto, CanadaBooz Allen Hamilton[15]ComputingTysons Corner, VA, USABoxcat Inc.[16]2017ComputingQuantum Algorithms, Quantum Rendering, Quantum Image Processing, Super-Resolution, Quantum Machine LearningUniversity of Toronto, UFRN[17]Toronto, CanadaBT[18]CommunicationLondon, UKCarl Zeiss AG[19]University College LondonOberkochen, GermanyCambridge Quantum Computing[20]2014ComputingQuantum Algorithms Quantum CybersecurityUniversity of CambridgeCambridge, UK London, UKClassiq[21]2019ComputingQuantum Algorithms Quantum SoftwareTechnion Israel Institute of TechnologyTel-Aviv, IsraelCogniFrame Inc[22][23]2019ComputingQuantum Algorithms Quantum for financial servicesUniversity of TorontoToronto, Ontario Toronto, CanadaCube Robot X2004ComputingPhotonic, Trapped Ion, Quantum Algorithms, Quantum Programming, RoboticsUniversity of applied science in Augsburg (FH)Langweid am Lech, Bavaria GermanyD-WaveJanuary 1, 1999ComputingSuperconducting Quantum AnnealerBurnaby, CanadaElyah[24]2018ComputingQuantum Programming,[25] Classical Simulation, Software as a serviceTokyo, JapanEntropica Labs[26]May 2018[27]AlgorithmAlgorithmsCenter for Quantum Technologies, National University of SingaporeSingaporeFujitsu[28]September 28, 2015CommunicationQuantum DotsUniversity of TokyoTokyo, JapanGoogle QuAIL[29]May 16, 2013ComputingSuperconductingUCSBMountain View, CA, USAHP[30][31]Computing[30]/Communication[31]Algorithms, NMRPalo Alto, CA, USAHitachi2012ComputingSilicon CMOS[32]University of Cambridge, University College London, CEA, University of CopenhagenHitachi Cambridge Laboratory[33] and Tokyo, JapanHoneywell[34][35][36]2017ComputingTrapped IonGeorgia Tech,[34]Morris Plains, NJ, USAHorizon Quantum Computing[37]2018ComputingQuantum Algorithms, Quantum Compilation, Quantum Programming, Software as a serviceCQTSingaporeHRL LaboratoriesComputingMalibu, CA, USAHuawei Noah's Ark Lab[38]CommunicationNanjing UniversityShenzhen, ChinaIBM[39]September 10, 1990[40]ComputingSuperconductingMIT[41]Armonk, NY, USAID QuantiqueJuly 1, 2001CommunicationGeneva, Switzerlandimec[42]ComputingSuperconductingBelgiumIntel[43]September 3, 2015ComputingTU DelftSanta Clara, CA, USAInfineon Technologies[44][45]2019ComputingTrapped Ion, Post-Quantum CryptographyUniversity of InnsbruckNeubiberg, GermanyionQ[46][47]2015ComputingTrapped IonUniversity of Maryland, Duke UniversityCollege Park, MD, USAIQM Quantum Computers2019ComputingSuperconductingAalto UniversityEspoo, FinlandKEEQuant2020Quantum cryptographyContinuous Variable QKD, Key Management Systems (KMS)Frth, GermanyKPN[48]CommunicationThe Hague, NetherlandsLockheed MartinComputingQuantum AnnealingUniversity of Southern California, University College LondonBethesda, MD, USAmain incubator2019ComputingQuantum Financial ServicesFrankfurt, GermanyMagiQ1999CommunicationSomerville, MA, USAMicrosoft Research QuArCDecember 19, 2011ComputingAlgorithmsTU Delft, Niels Bohr Institute, University of Sydney, Purdue University, University of Maryland, ETH Zurich, UCSBRedmond, WA, USAMicrosoft Research Station QApril 22, 2005ComputingSuperconductingUCSBSanta Barbara, CA, USAMitsubishi[49]CommunicationTokyo, JapanNEC Corporation[50]April 29, 1999[51]CommunicationQuantum DotsUniversity of TokyoTokyo, JapanNext Generation Quantum[52]2019Computing//NetworkingOptical quantum interconnects for quantum computing clustersCity University of New YorkNew York, NY, USANokia Bell Labs[53][54]ComputingUniversity of OxfordMurray Hill, NJ, USANorthrop GrummanComputingWest Falls Church, VA, USANTT Laboratories[55]Computing/CommunicationPhotonic Quantum Computing, Quantum CommunicationBristol UniversityTokyo, JapanNu Quantum[56][57]2018CommunicationPhotonic Quantum Computing,[57] Quantum Communication[58]University of Cambridge[59]Cambridge, UKPsiQuantum[60]2016ComputingPhotonic Quantum ComputingBristol UniversityPalo Alto, CA, USAQ. BPO Consulting2020Consulting & EngineeringQML Q-ANN OptimizationParis, FRANCEQC Ware2014ComputingQuantum Algorithms

Quantum Computing Software

Quantum Random Number Generator

Orquestra Quantum Operating Environment[80]

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List of companies involved in quantum computing or communication - Wikipedia

Even though banks and hedge funds are still several years out from adding quantum computing to their tech arsenals, that hasn't stopped Wall Street giants from investing time and money into the emerging technology class.

And momentum for QC Ware, a startup looking to cut the time and resources it takes to use quantum computing, is accelerating. The fintech secured a $25 million Series B on September 29 co-led by Koch Disruptive Technologies and Covestro with participation from D.E. Shaw, Citi, and Samsung Ventures.

QC Ware, founded in 2014, builds quantum algorithms for the likes of Goldman Sachs (which led the fintech's Series A), Airbus, and BMW Group. The algorithms, which are effectively code bases that include quantum processing elements, can run on any of the four main public-cloud providers.

Quantum computing allows companies to do complex calculations faster than traditional computers by using a form of physics that runs on quantum bits as opposed to the traditional 1s and 0s that computers use. This is especially helpful in banking for risk analytics or algorithmic trading, where executing calculations milliseconds faster than the competition can give firms a leg up.

"With all of our investors, with every one, there is a strategic dimension to the investment," QC Ware CEO Matt Johnson told Insider. "Almost every one of our investors cares about having a front-row seat as the technology develops."

And while quantum computing has significant potential, the resources required to use it are still too great to be more cost effective than traditional computers.

QC Ware aims to mitigate that by designing algorithms that reduce the resource requirements of quantum computers by using the minimum amount of steps to solve the problem.

For instance, QC Ware's collaboration with Goldman Sachs to design an algorithm to speed derivatives pricing calculations reduced the wait time for the required quantum hardware from 10 years to five, said Yianni Gamvros, QC Ware head of business development.

Since each algorithm is built per use case, QC Ware will use the new funds to double its team to 60, staffing up on quantum engineers to build the specialized algorithms and software engineers to build out a more expansive cloud service.

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Fintech QC Ware Used This Pitch Deck to Raise a $25 Million Series B - Business Insider

University of Maryland President Darryll J. Pines today announced the creation of the Discovery Fund, which will support innovative companies and startups based in College Park and throughout Prince Georges County with up to $1 million a year from the university.

The first round of support is earmarked to help build a network of quantum business focused around UMD, Pines said in an address at the universitys inaugural Quantum Investment Summit. The two-day event was designed to connect investors and innovators in the growing quantum business and technology space, and drew more than 300 in-person and virtual participants from U.S. and international companies and organizations.

The university has long been a powerhouse in quantum physics research as well as a leader in designing and engineering technology based on this revolutionary branch of scienceone expected to result in quantum computers with unprecedented capabilities as well as disruptive advances in material science, digital security, health care and other fields.

UMDs growing commitment to strengthening the industrys foundation further solidifies the universitys status as the heart of the Capital of Quantum, Pines said.

This continual building on the infrastructure needed to catalyze startups and create groundbreaking products is absolutely essential if were to support and accelerate the advancement and commercialization of quantum technologies, he said. The Discovery Fund is the perfect addition to keep the momentum going around the quantum ecosystem we have been building for more than three decades.

The announcement of the new funding comes the same month that a leading quantum computing company, IonQ, went public on the New York Stock exchange with a $2 billion market valuation. The company is based in part on technology developed in UMD labs, and illustrates what the university has to gain: As IonQ works to bring quantum computing to scale, its continued close connection with UMD affords the company access to a pipeline of stellar workforce talent, Pines said today.

Another feature in UMDs expanding ecosystem is the Quantum Startup Foundry (QSF), backed by a $10 million capital investment from UMD, which will function as a business incubator to support nascent firms in the quantum technology field. The university today announced that MITRE, a not-for-profit company that works in the public interest and operates six federally funded research and development centers in areas including aviation, defense, health care, homeland security, and cybersecurity had joined as a founding QSF member.

Julie Lenzer, UMDs chief innovation officer, said offerings like the QSF and the Quantum Investment Summit help make the university central to quantum-based industry as it already is in quantum science and engineering research.

Helping to give rise to a company as successful as IonQ would be a once-in-a-lifetime thing for most schools, if that, Lenzer said. But were continuing to build on this so we can breed more success by connecting innovative quantum research and ideas with investors who want to make a difference in an area thats going to define the future.

Attendees at the investment summit included businesses ranging from giants like Lockheed Martin and IBM to new firms vying to become household names, as well as local and state officials, investors and venture capital firms.

With federal and state agencies and nations worldwide pouring many billions of dollars into quantum researchand hoping to reap the rewards of winning the race to deploy the technologyUMD, the region and the nation must strive to turn deep fundamental understanding of the science into innovation, Pines said.

Make no mistake: This is our generations space race, he said. Who will be the first to unleash the power of quantum? Im hoping its going to be us.

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Discovery Fund to Seed Local Innovation Ecosystem - Maryland Today