Quantum computing

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The Bloch sphere is a representation of a qubit, the fundamental building block of quantum computers.

 

Quantum computing is the study of a currently hypothetical model of computation. Whereas traditional models of computing such as the Turing machine or Lambda calculus rely on "classical" representations of computational memory, a quantum computation could transform the memory into a quantum superposition of possible classical states. A quantum computer is a device that could perform such computation.

Quantum computing began in the early 1980s when physicist Paul Benioff proposed a quantum mechanical model of the Turing machine. Richard Feynman and Yuri Manin later suggested that a quantum computer could perform simulations that are out of reach for classical computers. In 1994, Peter Shor developed a polynomial-time quantum algorithm for factoring integers. This was a major breakthrough in the subject: an important method of asymmetric key exchange known as RSA is based on the belief that factoring integers is computationally difficult. The existence of a polynomial-time quantum algorithm proves that one of the most widely-used cryptographic protocols is vulnerable to an adversary who possesses a quantum computer.

Experimental efforts towards building a quantum computer began after a slew of results known as fault-tolerance threshold theorems. These theorems proved that a quantum computation could be efficiently corrected against the effects of large classes of physically realistic noise models. One early result demonstrated parts of Shor's algorithm in a liquid-state nuclear magnetic resonance experiment. Other notable experiments have been performed in superconducting systems, ion-traps, and photonic systems.

Despite rapid and impressive experimental progress, most researchers believe that "fault-tolerant quantum computing [is] still a rather distant dream". As of September 2019, no scalable quantum computing hardware has been demonstrated. Nevertheless, there is an increasing amount of investment in quantum computing by governments, established companies, and start-ups. Current research focusses on building and using near-term intermediate-scale devices and demonstrating quantum supremacy alongside the long-term goal of building and using a powerful and error-free quantum computer. 

The field of quantum computing is closely related to quantum information science, which includes quantum cryptography and quantum communication.

Basic concept

In most models of classical computation, the computer has access to memory. This is a system that can be found in one of a finite set of possible states, each of which is physically distinct. It is frequently convenient to represent the state of this memory as a string of symbols; most simply, as a string of the symbols 0 and 1. In this scenario, the fundamental unit of memory is called a bit and we can measure the "size" of the memory in terms of the number of bits needed to represent fully the state of the memory.

If the memory obeys the laws of quantum physics, the state of the memory could be found in a quantum superposition of different possible "classical" states. If the classical states are to be represented as a string of bits, the quantum memory could be found in any superposition of the possible bit strings. In the quantum scenario, the fundamental unit of memory is called a qubit.

The defining property of a quantum computer is the ability to turn classical memory states into quantum memory states, and vice-versa. This is not possible with present-day computers because they are carefully designed to ensure that the memory never deviates from clearly defined informational states. To clarify this point, consider that information is normally transmitted through the computer as an electrical signal that could have one of two easily distinguished voltages. If the voltages were to become indistinct (in a classical or quantum sense), the computer would no longer operate correctly.

Of course, in the end we are classical beings and we can only observe classical states. That means the quantum computer must complete its task by returning to us a classical output. To produce these classical outputs, the quantum computer is obliged to measure parts of the memory at various times throughout the computation. The measurement process is inherently probabilistic, meaning that the output of a quantum algorithm is frequently random. The task of a quantum algorithm designer is to ensure that the randomness is tailored to the needs of the problem at hand. For example, if the quantum computer is searching a quantum database for one of several marked items, we can ask the quantum computer to return one of the marked items at random. The quantum computer succeeds in this task as long as it is unlikely to return an unmarked item.

Quantum operations

The prevailing model of quantum computation describes the computation in terms of a network of quantum logic gates. What follows is a brief treatment of the subject based upon Chapter 4 of Nielsen and Chuang.

We may represent the state of a computer memory as a vector whose length is equal to the number of possible states of the memory. So a memory consisting of ${\textstyle n}$bits of information has ${\textstyle 2^{n}}$ possible states, and the vector representing that memory state has ${\textstyle 2^{n}}$ entries. In the classical view, all but one of the entries of this vector would be zero and the remaining entry would be one. The vector should be viewed as a probability vector and represents the fact that the memory is to be found in a particular state with 100% probability (i.e. a probability of one).

In quantum mechanics, probability vectors are generalised to density operators. This is the technically rigorous mathematical foundation for quantum logic gates, but the intermediate quantum state vector formalism is usually introduced first because it is conceptually simpler. Here we focus only on the quantum state vector formalism for simplicity. 

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

$${\displaystyle |0\rangle :={\begin{pmatrix}1\\0\end{pmatrix}};\quad |1\rangle :={\begin{pmatrix}0\\1\end{pmatrix}}}$$

 

A quantum memory may then be found in any quantum superposition ${\textstyle |\psi \rangle }$ of the two classical states ${\textstyle |0\rangle }$ and ${\textstyle |1\rangle }$: 

 

$${\displaystyle |\psi \rangle :=\alpha \,|0\rangle +\beta \,|1\rangle ={\begin{pmatrix}\alpha \\\beta \end{pmatrix}};\quad |\alpha |^{2}+|\beta |^{2}=1.}$$

 

In general, the coefficients ${\textstyle \alpha }$ and ${\textstyle \beta }$ are complex numbers. In this scenario, we say that one qubit of information can be encoded into the quantum memory. The state ${\textstyle |\psi \rangle }$ is not itself a probability vector but can be connected with a probability vector via a measurement operation. If we choose to measure the quantum memory to determine if the state is ${\textstyle |0\rangle }$ or ${\textstyle |1\rangle }$ (this is known as a computational basis measurement), we would observe the zero state with probability ${\textstyle |\alpha |^{2}}$ and the one state with probability ${\textstyle |\beta |^{2}}$. Please see the article on quantum amplitudes for further information.

 

To manipulate the state of this one-qubit quantum memory, we imagine applying quantum logic gates analogous to classical logic gates. One obvious gate is the NOT gate, which can be represented by a matrix

 

$${\displaystyle X:={\begin{pmatrix}0&1\\1&0\end{pmatrix}}.}$$

 

We can formally apply this logic gate to a quantum state vector through matrix multiplication. Thus we find ${\textstyle X|0\rangle =|1\rangle }$ and ${\textstyle X|1\rangle =|0\rangle }$ as expected. But this is not the only interesting single-qubit quantum logic gate. We might, for example, imagine applying one of the other two Pauli matrices.

 

We may imagine extending single qubit gates to operate on multiqubit quantum memories in two important ways. One way to operate a single qubit gate on a multiqubit memory 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. We illustrate this with another example. 

Consider a two-qubit quantum memory. Its possible states are 

$${\displaystyle |00\rangle :={\begin{pmatrix}1\\0\\0\\0\end{pmatrix}};\quad |01\rangle :={\begin{pmatrix}0\\1\\0\\0\end{pmatrix}};\quad |10\rangle :={\begin{pmatrix}0\\0\\1\\0\end{pmatrix}};\quad |11\rangle :={\begin{pmatrix}0\\0\\0\\1\end{pmatrix}}.}$$

 

We may then define the CNOT gate as the following matrix: 

 

$${\displaystyle CNOT:={\begin{pmatrix}1&0&0&0\\0&1&0&0\\0&0&0&1\\0&0&1&0\end{pmatrix}}.}$$

 

It is easy to check that ${\textstyle CNOT|00\rangle =|00\rangle }$, ${\textstyle CNOT|01\rangle =|01\rangle }$, ${\textstyle CNOT|10\rangle =|11\rangle }$, and ${\textstyle CNOT|11\rangle =|10\rangle }$. In other words, the CNOT applies a NOT gate (${\textstyle X}$ from before) to the second qubit if and only if the first qubit is in the state ${\textstyle |1\rangle }$. If the first qubit is ${\textstyle |0\rangle }$, nothing is done to either qubit. 

 

The preceding discussion is of course a very brief introduction to the concept of a quantum logic gate. Please see the article on quantum logic gates for further information. 

To put the story together, we can describe a quantum computation as a network of quantum logic gates and measurements. One can always 'defer' a measurement to the end of a quantum computation, though this can come at a computational cost according to some cost models. Because of this possibility of deferring a measurement, most quantum circuits depict a network consisting only of quantum logic gates and no measurements. For more details on the sequences of operations used for various quantum algorithms, see universal quantum computer, Shor's algorithm, Grover's algorithm, Deutsch–Jozsa algorithm, amplitude amplification, quantum Fourier transform, quantum gate, quantum adiabatic algorithm and quantum error correction. 

One can represent any quantum computation 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. 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.

Potential

Cryptography

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). 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, Diffie–Hellman, and elliptic curve Diffie–Hellman algorithms could be broken. These are used to protect secure Web pages, encrypted email, and many other types of data. Breaking these would have significant ramifications for electronic privacy and security. 

However, other cryptographic algorithms do not appear to be broken by those algorithms. Some public-key algorithms are based on problems other than the integer factorization and discrete logarithm problems to which Shor's algorithm applies, like the McEliece cryptosystem based on a problem in coding theory. Lattice-based cryptosystems are also not known to be broken by quantum computers, and finding a polynomial time algorithm for solving the dihedral hidden subgroup problem, which would break many lattice based cryptosystems, is a well-studied open problem. It has been proven that applying Grover's algorithm to break a symmetric (secret key) algorithm by brute force requires time equal to roughly 2^n/2 invocations of the underlying cryptographic algorithm, compared with roughly 2ⁿ in the classical case, 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. 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.

Quantum search

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. However, quantum computers offer polynomial speedup for some problems. 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. 

Problems that can be addressed with Grover's algorithm have the following properties:

1. There is no searchable structure in the collection of possible answers,
2. The number of possible answers to check is the same as the number of inputs to the algorithm, and
3. There exists a boolean function which evaluates each input and determines whether it is the correct answer

For problems with all these properties, the running time of Grover's algorithm on a quantum computer will scale as the square root of the number of inputs (or elements in the database), as opposed to the linear scaling of classical algorithms. A general class of problems to which Grover's algorithm can be applied is Boolean satisfiability problem. In this instance, the database through which the algorithm is iterating is that of all possible answers. An example (and possible) application of this is a password cracker that attempts to guess the password or secret key for an encrypted file or system. Symmetric ciphers such as Triple DES and AES are particularly vulnerable to this kind of attack. This application of quantum computing is a major interest of government agencies.

Quantum simulation

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. Quantum simulation could also be used to simulate the behavior of atoms and particles at unusual conditions such as the reactions inside a collider.

Quantum annealing and adiabatic optimization

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.

Solving linear equations

The Quantum algorithm for linear systems of equations or "HHL Algorithm", named after its discoverers Harrow, Hassidim, and Lloyd, is expected to provide speedup over classical counterparts.

Quantum supremacy

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. Although skeptics like Gil Kalai doubt that quantum supremacy will ever be achieved, Google has been reported to have done so, with calculations more than 3,000,000 times as fast as those of Summit, generally considered the world's fastest computer. Bill Unruh doubted the practicality of quantum computers in a paper published back in 1994. Paul Davies argued that a 400-qubit computer would even come into conflict with the cosmological information bound implied by the holographic principle.

Obstacles

There are a number of technical challenges in building a large-scale quantum computer, and thus far quantum computers have yet to solve a problem faster than a classical computer. David DiVincenzo, of IBM, listed the following requirements for a practical quantum computer:

• scalable physically to increase the number of qubits;
• qubits that can be initialized to arbitrary values;
• quantum gates that are faster than decoherence time;
• universal gate set;
• qubits that can be read easily.

Quantum decoherence

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. 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. 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.

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 10⁻³, 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 qubits 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. With error correction, the figure would rise to about 10⁷ bits. Computation time is about L² or about 10⁷ steps and at 1 MHz, 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.

Developments

Quantum computing models

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:

• Quantum gate array (computation decomposed into a sequence of few-qubit quantum gates)
• One-way quantum computer (computation decomposed into a sequence of one-qubit measurements applied to a highly entangled initial state or cluster state)
• Adiabatic quantum computer, based on quantum annealing (computation decomposed into a slow continuous transformation of an initial Hamiltonian into a final Hamiltonian, whose ground states contain the solution)
• Topological quantum computer (computation decomposed into the braiding of anyons in a 2D lattice)

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

Physical realizations

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

• Superconducting quantum computing (qubit implemented by the state of small superconducting circuits (Josephson junctions))
• Trapped ion quantum computer (qubit implemented by the internal state of trapped ions)
• Optical lattices (qubit implemented by internal states of neutral atoms trapped in an optical lattice)
• Quantum dot computer, spin-based (e.g. the Loss-DiVincenzo quantum computer) (qubit given by the spin states of trapped electrons)
• Quantum dot computer, spatial-based (qubit given by electron position in double quantum dot)
• Coupled Quantum Wire (qubit implemented by a pair of Quantum Wires coupled by a Quantum Point Contact)
• Nuclear magnetic resonance quantum computer (NMRQC) implemented with the nuclear magnetic resonance of molecules in solution, where qubits are provided by nuclear spins within the dissolved molecule and probed with radio waves
• Solid-state NMR Kane quantum computers (qubit realized by the nuclear spin state of phosphorus donors in silicon)
• Electrons-on-helium quantum computers (qubit is the electron spin)
• Cavity quantum electrodynamics (CQED) (qubit provided by the internal state of trapped atoms coupled to high-finesse cavities)
• Molecular magnet (qubit given by spin states)
• Fullerene-based ESR quantum computer (qubit based on the electronic spin of atoms or molecules encased in fullerenes)
• Linear optical quantum computer (qubits realized by processing states of different modes of light through linear elements e.g. mirrors, beam splitters and phase shifters)
• Diamond-based quantum computer (qubit realized by the electronic or nuclear spin of nitrogen-vacancy centers in diamond)
• Bose-Einstein condensate-based quantum computer
• Transistor-based quantum computer – string quantum computers with entrainment of positive holes using an electrostatic trap
• Rare-earth-metal-ion-doped inorganic crystal based quantum computers (qubit realized by the internal electronic state of dopants in optical fibers)
• Metallic-like carbon nanospheres based quantum computers

A large number of candidates demonstrates that the topic, in spite of rapid progress, is still in its infancy. There is also a vast amount of flexibility.

Timeline

In 1980, Paul Benioff describes the first quantum mechanical model of a computer. In this work, Benioff showed that a computer could operate under the laws of quantum mechanics by describing a Schrodinger equation description of Turing machines, laying a foundation for further work in quantum computing. The paper was submitted in June 1979 and published in April of 1980. Russian mathematician Yuri Manin then motivates the development of quantum computers.

In 1981, at the First Conference on the Physics of Computation held at MIT and co-organized by MIT and IBM, Paul Benioff and Richard Feynman give talks on quantum computing. Benioff built on his earlier 1980 work showing that a computer can operate under the laws of quantum mechanics. The talk was titled “Quantum mechanical Hamiltonian models of discrete processes that erase their own histories: application to Turing machines”. In Feynman’s talk, he observed that it appeared to be impossible to efficiently simulate an evolution of a quantum system on a classical computer, and he proposed a basic model for a quantum computer. Urging the world to build a quantum computer, he said, "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."

In 1982, Paul Benioff further develops his original model of a quantum mechanical Turing machine.

In 1984, IBM scientists Charles Bennett and Gilles Brassard published BB84, the world's first quantum cryptography protocol. 

In 1985, David Deutsch describes the first universal quantum computer. Just as a Universal Turing machine can simulate any other Turing machine efficiently (Church-Turing thesis), so the universal quantum computer is able to simulate any other quantum computer with at most a polynomial slowdown. 

In 1989, Bikas K. Chakrabarti & collaborators proposes the idea that quantum fluctuations could help explore rough energy landscapes by escaping from local minima of glassy systems having tall but thin barriers by tunneling (instead of climbing over using thermal excitations), suggesting the effectiveness of quantum annealing over classical simulated annealing.

In 1992, David Deutsch and Richard Jozsa propose a computational problem that can be solved efficiently with the determinist Deutsch–Jozsa algorithm on a quantum computer, but for which no deterministic classical algorithm is possible. This was perhaps the earliest result in the computational complexity of quantum computers, proving that they were capable of performing some well-defined computational task more efficiently than any classical computer.

In 1993, an international group of six scientists, including Charles Bennett, showed that perfect quantum teleportation is possible in principle, but only if the original is destroyed.

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 cryptography systems in use today, sparking a tremendous interest in quantum computers.

In 1996, the DiVincenzo's criteria are published, which are a list of conditions that are necessary for constructing a quantum computer, proposed by the theoretical physicist David P. DiVincenzo in his 2000 paper "The Physical Implementation of Quantum Computation".

In 2001, researchers demonstrated Shor's algorithm to factor 15 using a 7-qubit NMR computer.

In 2005, researchers at the University of Michigan built a semiconductor chip ion trap. Such devices from standard lithography may point the way to scalable quantum computing.

In 2009, researchers at Yale University created the first solid-state quantum processor. The 2-qubit superconducting chip had artificial atom qubits made of a billion aluminum atoms that acted like a single atom that could occupy two states.

A team at the University of Bristol also created a silicon chip based on quantum optics, able to run Shor's algorithm. Further developments were made in 2010. Springer publishes a journal, Quantum Information Processing, devoted to the subject.

In February 2010, Digital Combinational Circuits like an adder, subtractor etc. are designed with the help of Symmetric Functions organized from different quantum gates.

In April 2011, a team of scientists from Australia and Japan made a breakthrough in quantum teleportation, successfully transferring a complex set of quantum data with full transmission integrity, without affecting the qubits' superpositions.

Photograph of a chip constructed by D-Wave Systems Inc. Mounted and wire-bonded in a sample holder. The D-Wave processor is designed to use 128 superconducting logic elements that exhibit controllable and tunable coupling to perform operations.

 

In 2011, D-Wave Systems announced the first commercial quantum annealer, the D-Wave One, claiming a 128-qubit processor. On 25 May 2011, Lockheed Martin agreed to purchase a D-Wave One system. Lockheed and the University of Southern California (USC) will house the D-Wave One at the newly formed USC Lockheed Martin Quantum Computing Center. D-Wave's engineers designed the chips with an empirical approach, focusing on solving particular problems. Investors liked this more than academics, who said D-Wave had not demonstrated that they really had a quantum computer. Criticism softened after a D-Wave paper in Nature that proved that the chips have some quantum properties. Two published papers have suggested that the D-Wave machine's operation can be explained classically, rather than requiring quantum models. Later work showed that classical models are insufficient when all available data is considered. Experts remain divided on the ultimate classification of the D-Wave systems though their quantum behavior was established concretely with a demonstration of entanglement.

During the same year, researchers at the University of Bristol created an all-bulk optics system that ran a version of Shor's algorithm to successfully factor 21.

In September 2011, researchers proved quantum computers can be made with a Von Neumann architecture (separation of RAM).

In November 2011, researchers factorized 143 using 4 qubits.

In February 2012, IBM scientists said that they had made several breakthroughs in quantum computing with superconducting integrated circuits.

In April 2012, a multinational team of researchers from the University of Southern California, the Delft University of Technology, the Iowa State University of Science and Technology, and the University of California, Santa Barbara constructed a 2-qubit quantum computer on a doped diamond crystal that can easily be scaled up and is functional at room temperature. Two logical qubit directions of electron spin and nitrogen kernels spin were used, with microwave pulses. This computer ran Grover's algorithm, generating the right answer on the first try in 95% of cases.

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.

In October 2012, Nobel Prizes were awarded to David J. Wineland and Serge Haroche for their basic work on understanding the quantum world, which may help make quantum computing possible.

In November 2012, the first quantum teleportation from one macroscopic object to another was reported by scientists at the University of Science and Technology of China.

In December 2012, 1QBit, the first dedicated quantum computing software company, 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.

In February 2013, a new technique, boson sampling, was reported by two groups using photons in an optical lattice that is not a universal quantum computer, but may be good enough for practical problems.

In May 2013, Google announced that it was launching the Quantum Artificial Intelligence Lab, hosted by NASA's Ames Research Center, with a 512-qubit D-Wave quantum computer. The Universities Space Research Association (USRA) 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".

In early 2014, based on documents provided by former NSA contractor Edward Snowden, it was reported that the U.S. National Security Agency (NSA) is running a $79.7 million research program titled "Penetrating Hard Targets", to develop a quantum computer capable of breaking vulnerable encryption.

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.

In 2014, researchers at University of New South Wales used silicon as a protectant shell around qubits, making them more accurate, increasing the length of time they will hold information, and possibly making quantum computers easier to build.

In April 2015, IBM scientists claimed two critical advances towards the realization of a practical quantum computer, claiming the ability to detect and measure both kinds of quantum errors simultaneously, as well as a new, square quantum bit circuit design that could scale to larger dimensions.

In October 2015, QuTech successfully conducted the Loophole-free Bell inequality violation test using electron spins separated by 1.3 kilometres.

In October 2015, researchers at the University of New South Wales built a quantum logic gate in silicon for the first time.

In December 2015, NASA publicly displayed the world's first fully operational quantum computer made by D-Wave Systems at the Quantum Artificial Intelligence Lab at its Ames Research Center. 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 these computers have been made so far. 

In May 2016, IBM Research announced that for the first time ever it is making quantum computing available to members of the public via the cloud, who can access and run experiments on IBM’s quantum processor, calling the service the IBM Quantum Experience. The quantum processor is composed of five superconducting qubits and is housed at IBM's Thomas J. Watson Research Center.

In August 2016, scientists at the University of Maryland successfully built the first reprogrammable quantum computer.

In October 2016, the University of Basel described a variant of the electron-hole based quantum computer, which instead of manipulating electron spins, uses electron holes in a semiconductor at low (mK) temperatures, which are much less vulnerable to decoherence. This has been dubbed the "positronic" quantum computer, as the quasi-particle behaves as if it has a positive electrical charge.

In March 2017, IBM announced an industry-first initiative, called IBM Q, to build commercially available universal quantum computing systems. The company also released a new API for the IBM Quantum Experience that enables developers and programmers to begin building interfaces between its existing 5-qubit cloud-based quantum computer and classical computers, without needing a deep background in quantum physics. 

In May 2017, IBM announced that it had 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.

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, whereas a computer can solve many different equations. 

In September 2017, IBM Research scientists used a 7-qubit device to model beryllium hydride molecule, the largest molecule to date by a quantum computer. The results were published as the cover story in the peer-reviewed journal Nature. 

In October 2017, IBM Research scientists successfully "broke the 49-qubit simulation barrier" and simulated 49- and 56-qubit short-depth circuits, using the Lawrence Livermore National Laboratory's Vulcan supercomputer, and the University of Illinois' Cyclops Tensor Framework (originally developed at the University of California).

In November 2017, the University of Sydney research team successfully made a microwave circulator, an important quantum computer part, that was 1000 times smaller than a conventional circulator, by using topological insulators to slow down the speed of light in a material.

In December 2017, IBM announced its first IBM Q Network clients. The companies, universities, and labs that will explore practical business and science quantum applications, using IBM Q 20-qubit commercial systems, include: JPMorgan Chase, Daimler AG, Samsung, JSR Corporation, Barclays, Hitachi Metals, Honda, Nagase, Keio University, Oak Ridge National Lab, Oxford University and University of Melbourne. 

In December 2017, Microsoft released a preview version of a "Quantum Development Kit", which includes a programming language, Q# that can be used to write programs that are run on an emulated quantum computer. 

In 2017, D-Wave was reported to be selling a 2,000-qubit quantum computer.

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.

In February 2018, scientists reported, for the first time, the discovery of a new form of light, which may involve polaritons, that could be useful in the development of quantum computers.

In February 2018, QuTech reported successfully testing a silicon-based two-spin-qubits quantum processor.

In June 2018, Intel began testing a silicon-based spin-qubit processor, manufactured in the company's D1D Fab in Oregon.

In July 2018, a team led by the University of Sydney achieved the world's first multi-qubit demonstration of a quantum chemistry calculation performed on a system of trapped ions, one of the leading hardware platforms in the race to develop a universal quantum computer.

In December 2018, IonQ reported that its machine could be built as large as 160 qubits.

In January 2019, IBM launched IBM Q System One, its first integrated quantum computing system for commercial use. IBM Q System One is designed by industrial design company Map Project Office and interior design company Universal Design Studio.

In March 2019, a group of Russian scientists used the open-access IBM quantum computer to demonstrate a protocol for the complex conjugation of the probability amplitudes needed for time reversal of a physical process, in this case, for an electron scattered on a two-level impurity, a two-qubit experiment. However, for the three-qubit experiment, the amplitude fell below 50% (failure of time reversal, due to its increased complexity).

In September 2019 Google AI Quantum and NASA published a paper "Quantum supremacy using a programmable superconducting processor" and supplementary material which was later removed from NASA.

Relation to computational complexity theory

The suspected relationship of BQP to other problem spaces.

 

The class of problems that can be efficiently solved by quantum computers is called BQP, for "bounded error, quantum, polynomial time". Quantum computers only run probabilistic algorithms, so BQP on quantum computers is the counterpart of BPP ("bounded error, probabilistic, polynomial time") on classical computers. It is defined as the set of problems solvable with a polynomial-time algorithm, whose probability of error is bounded away from one half. A quantum computer is said to "solve" a problem if, for every instance, its answer will be right with high probability. If that solution runs in polynomial time, then that problem is in BQP.

BQP is contained in the complexity class #P (or more precisely in the associated class of decision problems P^#P), which is a subclass of PSPACE.

BQP is suspected to be disjoint from NP-complete and a strict superset of P, but that is not known. Both integer factorization and discrete log are in BQP. Both of these problems are NP problems suspected to be outside BPP, and hence outside P. Both are suspected to not be NP-complete. There is a common misconception that quantum computers can solve NP-complete problems in polynomial time. That is not known to be true, and is generally suspected to be false.

The capacity of a quantum computer to accelerate classical algorithms has rigid limits—upper bounds of quantum computation's complexity. The overwhelming part of classical calculations cannot be accelerated on a quantum computer. A similar fact prevails for particular computational tasks, like the search problem, for which Grover's algorithm is optimal.

Bohmian Mechanics is a non-local hidden variable interpretation of quantum mechanics. It has been shown that a non-local hidden variable quantum computer could implement a search of an N-item database at most in ${\displaystyle O({\sqrt[{3}]{N}})}$ steps. This is slightly faster than the ${\displaystyle O({\sqrt {N}})}$ steps taken by Grover's algorithm. Neither search method will allow quantum computers to solve NP-Complete problems in polynomial time.

Although quantum computers may be faster than classical computers for some problem types, those described above cannot solve any problem that classical computers cannot already solve. A Turing machine can simulate these quantum computers, so such a quantum computer could never solve an undecidable problem like the halting problem. The existence of "standard" quantum computers does not disprove the Church–Turing thesis. 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.

Quantum network

From Wikipedia, the free encyclopedia

 

Quantum networks form an important element of quantum computing and quantum communication systems. Quantum networks facilitate the transmission of information in the form of quantum bits, also called qubits, between physically separated quantum processors. A quantum processor is a small quantum computer being able to perform quantum logic gates on a certain number of qubits. Quantum networks work in a similar way to classical networks. The main difference, as will be detailed more in later paragraphs, is that quantum networking like quantum computing is better at solving certain problems, such as modeling quantum systems.

Basics

Quantum networks for computation

Networked quantum computing or distributed quantum computing works by linking multiple quantum processors through a quantum network by sending qubits in-between them. Doing this creates a quantum computing cluster and therefore creates more computing potential. Less powerful computers can be linked in this way to create one more powerful processor. This is analogous to connecting several classical computers to form a computer cluster in classical computing. Like classical computing this system is scale-able by adding more and more quantum computers to the network. Currently quantum processors are only separated by short distances.

Quantum networks for communication

In the realm of quantum communication, one wants to send qubits from one quantum processor to another over long distances. This way local quantum networks can be intra connected into a quantum internet. A quantum internet supports many applications, which derive their power from the fact that by creating quantum entangled qubits, information can be transmitted between the remote quantum processors. Most applications of a quantum internet require only very modest quantum processors. For most quantum internet protocols, such as quantum key distribution in quantum cryptography, it is sufficient if these processors are capable of preparing and measuring only a single qubit at a time. This is in contrast to quantum computing where interesting applications can only be realized if the (combined) quantum processors can easily simulate more qubits than a classical computer (around 60). Quantum internet applications require only small quantum processors, often just a single qubit, because quantum entanglement can already be realized between just two qubits. A simulation of an entangled quantum system on a classical computer can not simultaneously provide the same security and speed.

Overview of the elements of a quantum network

The basic structure of a quantum network and more generally a quantum internet is analogous to a classical network. First, we have end nodes on which applications are ultimately run. These end nodes are quantum processors of at least one qubit. Some applications of a quantum internet require quantum processors of several qubits as well as a quantum memory at the end nodes.

Second, to transport qubits from one node to another, we need communication lines. For the purpose of quantum communication, standard telecom fibers can be used. For networked quantum computing, in which quantum processors are linked at short distances, different wavelengths are chosen depending on the exact hardware platform of the quantum processor. 

Third, to make maximum use of communication infrastructure, one requires optical switches capable of delivering qubits to the intended quantum processor. These switches need to preserve quantum coherence, which makes them more challenging to realize than standard optical switches. 

Finally, one requires a quantum repeater to transport qubits over long distances. Repeaters appear in-between end nodes. Since qubits cannot be copied, classical signal amplification is not possible. By necessity, a quantum repeater works in a fundamentally different way than a classical repeater.

Elements of a quantum network

End nodes: quantum processors

End nodes can both receive and emit information. Telecommunication lasers and parametric down-conversion combined with photodetectors can be used for quantum key distribution. In this case, the end nodes can in many cases be very simple devices consisting only of beamsplitters and photodetectors. 

However, for many protocols more sophisticated end nodes are desirable. These systems provide advanced processing capabilities and can also be used as quantum repeaters. Their chief advantage is that they can store and retransmit quantum information without disrupting the underlying quantum state. The quantum state being stored can either be the relative spin of an electron in a magnetic field or the energy state of an electron. They can also perform quantum logic gates. 

One way of realizing such end nodes is by using color centers in diamond, such as the nitrogen-vacancy center. This system forms a small quantum processor featuring several qubits. NV centers can be utilized at room temperatures. Small scale quantum algorithms and quantum error correction has already been demonstrated in this system, as well as the ability to entangle two remote quantum processors, and perform deterministic quantum teleportation.

Another possible platform are quantum processors based on Ion traps, which utilize radio-frequency magnetic fields and lasers. In a multispecies trapped-ion node network, photons entangled with a parent atom are used to entangle different nodes. Also, cavity quantum electrodynamics (Cavity QED) is one possible method of doing this. In Cavity QED, photonic quantum states can be transferred to and from atomic quantum states stored in single atoms contained in optical cavities. This allows for the transfer of quantum states between single atoms using optical fiber in addition to the creation of remote entanglement between distant atoms.

Communication lines: physical layer

Over long distances, the primary method of operating quantum networks is to use optical networks and photon-based qubits. This is due to optical networks having a reduced chance of decoherence. Optical networks have the advantage of being able to re-use existing optical fiber. Alternately, free space networks can be implemented that transmit quantum information through the atmosphere or through a vacuum.

Fiber optic networks

Optical networks using existing telecommunication fiber can be implemented using hardware similar to existing telecommunication equipment. This fiber can be either single-mode or multi-mode, with multi-mode allowing for more precise communication. At the sender, a single photon source can be created by heavily attenuating a standard telecommunication laser such that the mean number of photons per pulse is less than 1. For receiving, an avalanche photodetector can be used. Various methods of phase or polarization control can be used such as interferometers and beam splitters. In the case of entanglement based protocols, entangled photons can be generated through spontaneous parametric down-conversion. In both cases, the telecom fiber can be multiplexed to send non-quantum timing and control signals.

Free space networks

Free space quantum networks operate similar to fiber optic networks but rely on line of sight between the communicating parties instead of using a fiber optic connection. Free space networks can typically support higher transmission rates than fiber optic networks and do not have to account for polarization scrambling caused by optical fiber. However, over long distances, free space communication is subject to an increased chance of environmental disturbance on the photons.

Importantly, free space communication is also possible from a satellite to the ground. A quantum satellite capable of entanglement distribution over a distance of 1,203 km has been demonstrated. The experimental exchange of single photons from a global navigation satellite system at a slant distance of 20,000 km has also been reported. These satellites can play an important role in linking smaller ground-based networks over larger distances.

Repeaters

Long distance communication is hindered by the effects of signal loss and decoherence inherent to most transport mediums such as optical fiber. In classical communication, amplifiers can be used to boost the signal during transmission, but in a quantum network amplifiers cannot be used since qubits cannot be copied – known as the no-cloning theorem. That is, to implement an amplifier, the complete state of the flying qubit would need to be determined, something which is both unwanted and impossible.

Trusted repeaters

An intermediary step which allows the testing of communication infrastructure are trusted repeaters. Importantly, a trusted repeater cannot be used to transmit qubits over long distances. Instead, a trusted repeater can only be used to perform quantum key distribution with the additional assumption that the repeater is trusted. Consider two end nodes A and B, and a trusted repeater R in the middle. A and R now perform quantum key distribution to generate a key ${\displaystyle k_{AR}}$. Similarly, R and B run quantum key distribution to generate a key ${\displaystyle k_{RB}}$. A and B can now obtain a key ${\displaystyle k_{AB}}$ between themselves as follows: A sends ${\displaystyle k_{AB}}$ to R encrypted with the key ${\displaystyle k_{AR}}$. R decrypts to obtain ${\displaystyle k_{AB}}$. R then re-encrypts ${\displaystyle k_{AB}}$ using the key ${\displaystyle k_{RB}}$ and sends it to B. B decrypts to obtain ${\displaystyle k_{AB}}$. A and B now share the key ${\displaystyle k_{AB}}$. The key is secure from an outside eavesdropper, but clearly the repeater R also knows ${\displaystyle k_{AB}}$. This means that any subsequent communication between A and B does not provide end to end security, but is only secure as long as A and B trust the repeater R.

Quantum repeaters

Diagram for quantum teleportation of a photon

 

A true quantum repeater allows the end to end generation of quantum entanglement, and thus - by using quantum teleportation - the end to end transmission of qubits. In quantum key distribution protocols one can test for such entanglement. This means that when making encryption keys, the sender and receiver are secure even if they do not trust the quantum repeater. Any other application of a quantum internet also requires the end to end transmission of qubits, and thus a quantum repeater.

Quantum repeaters allow entanglement and can be established at distant nodes without physically sending an entangled qubit the entire distance.

In this case, the quantum network consists of many short distance links of perhaps tens or hundreds of kilometers. In the simplest case of a single repeater, two pairs of entangled qubits are established: ${\displaystyle |A\rangle }$ and ${\displaystyle |R_{a}\rangle }$ located at the sender and the repeater, and a second pair ${\displaystyle |R_{b}\rangle }$ and ${\displaystyle |B\rangle }$ located at the repeater and the receiver. These initial entangled qubits can be easily created, for example through parametric down conversion, with one qubit physically transmitted to an adjacent node. At this point, the repeater can perform a bell measurement on the qubits ${\displaystyle |R_{a}\rangle }$ and ${\displaystyle |R_{b}\rangle }$ thus teleporting the quantum state of ${\displaystyle |R_{a}\rangle }$ onto ${\displaystyle |B\rangle }$. This has the effect of "swapping" the entanglement such that ${\displaystyle |A\rangle }$ and ${\displaystyle |B\rangle }$ are now entangled at a distance twice that of the initial entangled pairs. It can be seen that a network of such repeaters can be used linearly or in a hierarchical fashion to establish entanglement over great distances.

Hardware platforms suitable as end nodes above can also function as quantum repeaters. However, there are also hardware platforms specific only to the task of acting as a repeater, without the capabilities of performing quantum gates.

Error correction

Error correction can be used in quantum repeaters. Due to technological limitations, however, the applicability is limited to very short distances as quantum error correction schemes capable of protecting qubits over long distances would require an extremely large amount of qubits and hence extremely large quantum computers.

Errors in communication can be broadly classified into two types: Loss errors (due to optical fiber/environment) and operation errors (such as depolarization, dephasing etc.). While redundancy can be used to detect and correct classical errors, redundant qubits cannot be created due to the no-cloning theorem. As a result, other types of error correction must be introduced such as the Shor code or one of a number of more general and efficient codes. All of these codes work by distributing the quantum information across multiple entangled qubits so that operation errors as well as loss errors can be corrected.

In addition to quantum error correction, classical error correction can be employed by quantum networks in special cases such as quantum key distribution. In these cases, the goal of the quantum communication is to securely transmit a string of classical bits. Traditional error correction codes such as Hamming codes can be applied to the bit string before encoding and transmission on the quantum network.

Entanglement purification

Quantum decoherence can occur when one qubit from a maximally entangled bell state is transmitted across a quantum network. Entanglement purification allows for the creation of nearly maximally entangled qubits from a large number of arbitrary weakly entangled qubits, and thus provides additional protection against errors. Entanglement purification (also known as Entanglement distillation) has already been demonstrated in Nitrogen-vacancy centers in diamond.

Applications

A quantum internet supports numerous applications, enabled by quantum entanglement. In general, quantum entanglement is well suited for tasks that require coordination, synchronization or privacy. 

Examples of such applications include quantum key distribution, clock synchronization, protocols for distributed system problems such as leader election or byzantine agreement, extending the baseline of telescopes, as well as position verification, secure identification and two-party cryptography in the noisy-storage model. A quantum internet also enables secure access to a quantum computer in the cloud. Specifically, a quantum internet enables very simple quantum devices to connect to a remote quantum computer in such a way that computations can be performed there without the quantum computer finding out what this computation actually is.

Secure communications

When it comes to communicating in any form the largest issue has always been keeping your communications private. From when couriers were used to send letters between ancient battle commanders to secure radio communications that exist today the main purpose is to ensure that what a sender sends out to the receiver reaches the receiver unmolested. This is an area in which Quantum Networks particularly excel. By applying a quantum operator that the user selects to a system of information the information can then be sent to the receiver without a chance of an eavesdropper being able to accurately be able to record the sent information without either the sender or receiver knowing. This works because if a listener tries to listen in then they will change the information in an unintended way by listening thereby tipping their hand to the people on whom they are attacking. Secondly, without the proper quantum operator to decode the information they will corrupt the sent information without being able to use it themselves.

Jamming protection

Quantum networks can also be used to protect against jamming. A user can use a quantum network by using frequency-hopping spread spectrum. This method is currently used by the United States Army. In this method the user hops from frequency to frequency many times a second so that it is hard for an attacker to keep up and successfully attack the user. Direct-sequence spread spectrum can be used by applying a quantum operator to the system and then freely transmitting the information over the frequencies because an attacker cannot read the information without knowing the key (a quantum operator). These two techniques can be used together to produce a more secure communications system.

Frequency-hopping spread spectrum

Frequency-hopping spread spectrum (FHSS) is a method of protecting information transfer that involves the user switching from one frequency to another frequency hundreds of times a second. For this method to work one computer is set as the main computer and will regulate when the other computers will switch frequencies and how often. By switching frequencies hundreds of times a second a user can be assured that any would be attacker will have an extremely hard time both trying to read the data and trying to jam the frequency.

Direct-sequence spread spectrum

Direct-sequence spread spectrum (DSSS) is a method of protecting information transfer that involves the user applying a predetermined quantum operator to the information that is being sent so that only the receiver and the sender can decipher the information using the operator. This method makes it difficult for a potential listener to eavesdrop because without the operator they will not be able to determine the information. At the same time if a listener does try to decode the sent information by doing so they will change the information which will immediately tell the receiver that someone is listening to them.

Jamming

When using any computer to communicate with another computer the name of the game is security. "Attackers", people who want to receive information that was not intended for them or people who want to stop the proper receiver of the transmission from receiving their information. Quantum networks are particularly useful in this area as there are many different types of jamming techniques that are found in both classical and quantum systems.

Spot jamming

Spot jamming is a process wherein an attacker fully attacks one frequency at a time. For this method to be successful the attacker must send their transmission with more power than the original sender. By doing this the attacker will essentially overpower the original sender's message. The problem with this method is that it takes a tremendous amount of power to overpower a transmission as stated. Another issue with this method is that the original sender can easily switch to another frequency and if the original sender is using frequency-hopping spread spectrum the user will switch frequencies automatically with little hindrance to the original sender.

Sweep jamming

Sweep jamming is similar to spot jamming except it switches rapidly from one frequency to another in rapid succession. In this method the attacker is still attacking by sending a much more powerful message at the same time as the original sender. The advantage of this method over spot jamming is that sweep jamming has a much larger chance of disrupting the sender's frequency and costs the same amount of energy as spot.

Barrage jamming

Barrage jamming is when an attacker attacks many frequencies at one time, but as the range grows the ability to jam decreases. By attacking a few frequencies at a time the attacker increases the change that they might hit one of the sender's frequencies. The main problem with this method is that the attacker's power is greatly lessened because they are attacking many frequencies at once and therefore they decrease their power overall so it is possible that the attacker could hit the sender's frequency and not affect it due to the low power of their jamming frequency.

Current status

Quantum internet

At present, there is no network connecting quantum processors, or quantum repeaters deployed outside a lab.

Quantum key distribution networks

Several test networks have been deployed that are tailored to the task of quantum key distribution either at short distances (but connecting many users), or over larger distances by relying on trusted repeaters. These networks do not yet allow for the end to end transmission of qubits or the end to end creation of entanglement between far away nodes.

Major quantum network projects and QKD protocols implemented

Quantum network	Start	BB84	BBM92	E91	DPS	COW
DARPA Quantum Network	2001	Yes	No	No	No	No
SECOCQ QKD network in Vienna	2003	Yes	Yes	No	No	Yes
Tokyo QKD network	2009	Yes	Yes	No	Yes	No
Hierarchical network in Wuhu, China	2009	Yes	No	No	No	No
Geneva area network (SwissQuantum)	2010	Yes	No	No	No	Yes

DARPA Quantum Network

Starting in the early 2000s, DARPA began sponsorship of a quantum network development project with the aim of implementing secure communication. The DARPA Quantum Network became operational within the BBN Technologies laboratory in late 2003 and was expanded further in 2004 to include nodes at Harvard and Boston Universities. The network consists of multiple physical layers including fiber optics supporting phase-modulated lasers and entangled photons as well free-space links.

SECOQC Vienna QKD network

From 2003 to 2008 the Secure Communication based on Quantum Cryptography (SECOQC) project developed a collaborative network between a number of European institutions. The architecture chosen for the SECOQC project is a trusted repeater architecture which consists of point-to-point quantum links between devices where long distance communication is accomplished through the use of repeaters.

Chinese hierarchical network

In May 2009, a hierarchical quantum network was demonstrated in Wuhu, China. The hierarchical network consists of a backbone network of four nodes connecting a number of subnets. The backbone nodes are connected through an optical switching quantum router. Nodes within each subnet are also connected through an optical switch and are connected to the backbone network through a trusted relay.

Geneva area network (SwissQuantum)

The SwissQuantum network developed and tested between 2009 and 2011 linked facilities at CERN with the University of Geneva and hepia in Geneva. The SwissQuantum program focused on transitioning the technologies developed in the SECOQC and other research quantum networks into a production environment. In particular the integration with existing telecommunication networks, and its reliability and robustness.

Tokyo QKD network

In 2010, a number of organizations from Japan and the European Union setup and tested the Tokyo QKD network. The Tokyo network build upon existing QKD technologies and adopted a SECOQC like network architecture. For the first time, one-time-pad encryption was implemented at high enough data rates to support popular end-user application such as secure voice and video conferencing. Previous large-scale QKD networks typically used classical encryption algorithms such as AES for high-rate data transfer and use the quantum-derived keys for low rate data or for regularly re-keying the classical encryption algorithms.

Beijing-Shanghai Trunk Line

In September 2017, a 2000-km quantum key distribution network between Beijing and Shanghai, China, was officially opened. This trunk line will serve as a backbone connecting quantum networks in Beijing, Shanghai, Jinan in Shandong province and Hefei in Anhui province. During the opening ceremony, two employees from the Bank of Communications completed a transaction from Shanghai to Beijing using the network. The State Grid Corporation of China is also developing a managing application for the link. The line uses 32 trusted nodes as repeaters. A quantum telecommunication network has been also put into service in Wuhan, capital of central China's Hubei Province, which will be connected to the trunk. Other similar city quantum networks along the Yangtze River are planned to follow.

Friendship paradox

From Wikipedia, the free encyclopedia

 

The friendship paradox is the phenomenon first observed by the sociologist Scott L. Feld in 1991 that most people have fewer friends than their friends have, on average. It can be explained as a form of sampling bias in which people with greater numbers of friends have an increased likelihood of being observed among one's own friends. In contradiction to this, most people believe that they have more friends than their friends have.

The same observation can be applied more generally to social networks defined by other relations than friendship: for instance, most people's sexual partners have had (on the average) a greater number of sexual partners than they have.

Mathematical explanation

In spite of its apparently paradoxical nature, the phenomenon is real, and can be explained as a consequence of the general mathematical properties of social networks. The mathematics behind this are directly related to the arithmetic-geometric mean inequality and the Cauchy–Schwarz inequality.

Formally, Feld assumes that a social network is represented by an undirected graph G = (V, E), where the set V of vertices corresponds to the people in the social network, and the set E of edges corresponds to the friendship relation between pairs of people. That is, he assumes that friendship is a symmetric relation: if X is a friend of Y, then Y is a friend of X. He models the average number of friends of a person in the social network as the average of the degrees of the vertices in the graph. That is, if vertex v has d(v) edges touching it (representing a person who has d(v) friends), then the average number μ of friends of a random person in the graph is

${\displaystyle \mu ={\frac {\sum _{v\in V}d(v)}{|V|}}={\frac {2|E|}{|V|}}.}$

The average number of friends that a typical friend has can be modeled by choosing a random person (who has at least one friend), and then calculating how many friends their friends have on average. This amounts to choosing, uniformly at random, an edge of the graph (representing a pair of friends) and an endpoint of that edge (one of the friends), and again calculating the degree of the selected endpoint. The probability of a certain vertex ${\displaystyle v}$ to be chosen is :

${\displaystyle {\frac {d(v)}{|E|}}\times {\frac {1}{2}}}$

The first factor corresponds to how likely it is that the chosen edge contains the vertex, which increases when the vertex has more friends. The halving factor simply comes from the fact that each edge has two vertices. So the expected value of the number of friends of a (randomly chosen) friend is :

${\displaystyle \sum _{v}\left({\frac {d(v)}{|E|}}\times {\frac {1}{2}}\right)\times d(v)={\frac {\sum _{v}d(v)^{2}}{2|E|}}}$

We know from the definition of variance that :

${\displaystyle {\frac {\sum _{v}d(v)^{2}}{|V|}}=\mu ^{2}+\sigma ^{2}}$

where ${\displaystyle \sigma ^{2}}$ is the variance of the degrees in the graph. This allows us to compute the desired expected value :

${\displaystyle {\frac {\sum _{v}d(v)^{2}}{2|E|}}={\frac {|V|}{2|E|}}(\mu ^{2}+\sigma ^{2})={\frac {\mu ^{2}+\sigma ^{2}}{\mu }}=\mu +{\frac {\sigma ^{2}}{\mu }}}$

For a graph that has vertices of varying degrees (as is typical for social networks), both μ and ${\displaystyle {\sigma }^{2}}$ are positive, which implies that the average degree of a friend is strictly greater than the average degree of a random node. 

Another way of understanding how the first term came is as follows. For each friendship (u, v), a node u mentions that v is a friend and v has d(v) friends. There are d(v) such friends who mention this. Hence the square of d(v) term. We add this for all such friendships in the network from both the u's and v's perspective, which gives the numerator. The denominator is the number of total such friendships, which is twice the total edges in the network (one from the u's perspective and the other from the v's). 

After this analysis, Feld goes on to make some more qualitative assumptions about the statistical correlation between the number of friends that two friends have, based on theories of social networks such as assortative mixing, and he analyzes what these assumptions imply about the number of people whose friends have more friends than they do. Based on this analysis, he concludes that in real social networks, most people are likely to have fewer friends than the average of their friends' numbers of friends. However, this conclusion is not a mathematical certainty; there exist undirected graphs (such as the graph formed by removing a single edge from a large complete graph) that are unlikely to arise as social networks but in which most vertices have higher degree than the average of their neighbors' degrees.

Applications

The analysis of the friendship paradox implies that the friends of randomly selected individuals are likely to have higher than average centrality. This observation has been used as a way to forecast and slow the course of epidemics, by using this random selection process to choose individuals to immunize or monitor for infection while avoiding the need for a complex computation of the centrality of all nodes in the network.

A study in 2010 by Christakis and Fowler showed that flu outbreaks can be detected almost 2 weeks before traditional surveillance measures can by using the friendship paradox in monitoring the infection in a social network. They found that using the friendship paradox to analyze the health of central friends is "an ideal way to predict outbreaks, but detailed information doesn't exist for most groups, and to produce it would be time-consuming and costly."

The "generalized friendship paradox" states that the friendship paradox applies to other characteristics as well. For example, one's co-authors are on average likely to be more prominent, with more publications, more citations and more collaborators, or one's followers on Twitter have more followers. The same effect has also been demonstrated for Subjective Well-Being by Bollen et al (2017), who used a large-scale Twitter network and longitudinal data on subjective well-being for each individual in the network to demonstrate that both a Friendship and a "happiness" paradox can occur in online social networks.