Quantum computer
A quantum processor or quantum computer is a processor whose function is based on the laws of quantum mechanics .
In contrast to the classical computer , it does not work on the basis of the laws of classical physics , but on the basis of quantum mechanical states . The processing of these states takes place according to quantum mechanical principles. First of all, the superposition principle (i.e. the quantum mechanical coherence , analogous to the coherence effects, see e.g. holography , in otherwise incoherent optics ) and, secondly, quantum entanglement . Theoretical studies show that when these effects are used, certain problems in computer science, e.g. B. the search in extremely large databases (see Grover algorithm ) and the factoring of large numbers (see Shor algorithm ) can be solved more efficiently than with classical computers. This would make many math problems easier to solve.
The quantum computer has long been a predominantly theoretical concept. There are various proposals how a quantum computer could be realized, and on a small scale some of these concepts have been tested in the laboratory and quantum computers have been realized with few qubits . The record is (early 2020) around 50 to 70 qubits. In addition to the number of qubits, it is also important, for example, to have a low error rate when calculating and reading out and how long the states in the qubits can be maintained without errors.
Since 2018, many governments and research organizations as well as large computer and technology companies around the world have been investing in the development of quantum computers, which many consider to be one of the emerging key technologies of the 21st century. In the context of the 2020 economic crisis caused by the COVID-19 pandemic , the German government decided on June 3, 2020 a € 130 billion economic stimulus program in which quantum technologies are to be promoted with a financial volume of € 2 billion. In particular, it was decided to "immediately award the contract to build at least two quantum computers to suitable consortia."
Possible areas of application
Where classic supercomputers fail due to the complexity of certain tasks, quantum computers could be a solution:
- Optimization tasks (finance, logistics)
- Simulations (find new chemical substances, such as for biotechnology or drugs; or find new materials, for example for novel accumulators)
- Cryptography
- energetic optimizations
technology
Qubits
In a classic computer, all information is represented in bits . Physically, a bit is realized in that an electrical potential is either above a certain level or below it.
In a quantum computer, too, information is usually represented in binary form. To do this, a physical system with two orthogonal base states of a two-dimensional complex space, as it occurs in quantum mechanics, is used. In Dirac notation , one of the basic states is represented by the quantum mechanical state vector , the other by the state vector . In these quantum mechanical two-level systems it can be e.g. E.g. the spin vector of an electron pointing either “up” or “down”. Other implementations use the energy level in atoms or molecules, or the direction of flow of a current in a ring-shaped superconductor . Often only two states are selected from a larger Hilbert space of the physical system, for example the two lowest energy eigenstates of a trapped ion . Such a quantum mechanical two-state system is called a qubit ( quantum bit ).
A property of quantum mechanical state vectors is that they can be a superposition of other states. This is also called superposition. In concrete terms, this means that a qubit does not have to be either or , as is the case for the bits of the classic computer. Rather, the state of a qubit in the above-mentioned two-dimensional complex space generally results
- ,
where, as in coherent optics, any superposition is permitted. The difference between classical and quantum mechanical computing is therefore analogous to that between incoherent or coherent optics (in the first case intensities are added, in the second the field amplitudes directly, such as in holography ).
Where and are complex numbers . For normalization one demands , where is the square of the amount. Without loss of generality, real and non-negative can be chosen. The qubit is usually read out by measuring an observable that is diagonal in the base and not degenerate , e.g. B. . The probability of receiving the value 0 as a result of this measurement on the state is, and that for the result is 1 accordingly . This probabilistic behavior must not be interpreted in such a way that the qubit is in the state with a certain probability and in the state with another probability , while other states are not permitted. Such an exclusive behavior could also be achieved with a classic computer that uses a random number generator to decide whether to continue computing with 0 or 1 when superimposed states occur. Such exclusive behavior occurs in statistical physics , which, in contrast to quantum mechanics, is incoherent. The coherent superposition of the various base states, the relative phase between the various components of the superposition and, in the course of the calculation, the interference between them are of decisive importance for quantum computation .
Quantum register, entanglement
As with the classic computer, several qubits are combined into quantum registers. According to the laws of many-body quantum mechanics, the state of a qubit register is a state from a -dimensional Hilbert space. One possible basis of this vector space is the product basis over the basis . For a register made up of two qubits one would therefore get the basis . The state of the register can also be any superposition of these basic states, i.e. with qubits of the form
- ,
with arbitrary complex numbers and the usual dual basis . Sums or differences of such terms are also allowed, while in classical computers only the basic states themselves occur, i.e. H. only prefactors composed of the digits 0 or 1.
The states of a quantum register cannot always be put together from the states of independent qubits: For example, the state
cannot be decomposed into a product of a state for the first and a state for the second qubit.
Such a state is therefore also called entangled (in English-language literature one speaks of entanglement ). The same goes for the different state
- .
This entanglement is one reason why quantum computers can be more efficient than classical computers; This means that, in principle, they can solve certain problems faster than classic computers: In order to display the status of a classic -bit register, bits of information are required . The state of the quantum register is, however, a vector from a -dimensional vector space, so that complex-valued coefficients are required for its representation . If it is large , the number is much larger than itself.
The superposition principle is often presented in such a way that a quantum computer could simultaneously store all numbers from 0 to in a quantum register made up of qubits . This idea is misleading. Since a measurement made on the register always selects exactly one of the basic states, it can be shown using the so-called Holevo theorem that the maximum accessible information content of a qubit register, like that of a classic -bit register, is exactly bits.
It is still correct that the superposition principle allows a parallelism in the calculations that goes beyond what happens in a classic parallel computer. The main difference to the classic parallel computer is that the parallelism made possible by the superposition principle can only be exploited through interference. For some problems, quantum algorithms can lead to a significantly reduced running time compared to classical methods .
Quantum gate
In classical computers by logic gates (Engl. Gates elementary operations) performed on the bits. Several gates are connected to form a switching network, which can then perform complex operations such as adding two binary numbers. The gates are implemented by physical components such as transistors and the information is passed through these components as an electrical signal.
Calculations on a quantum computer are fundamentally different: A quantum gate is not a technical component, but represents an elementary physical manipulation of one or more qubits. How exactly such a manipulation looks depends on the actual physical nature of the qubit . The spin of an electron can be influenced by irradiated magnetic fields , the excited state of an atom by laser pulses . So even though a quantum gate is not an electronic component, but an action applied to the quantum register over time, quantum algorithms are described with the help of circuit diagrams, cf. see the article List of Quantum Gates .
Formally, a quantum gate is a unitary operation that affects the state of the quantum register:
A quantum gate can therefore be written as a unitary matrix . A gate that reverses (negates) the state of a qubit would correspond to the following matrix in the case of a two-dimensional state space:
More complicated to write are quantum gates (unitary matrices) that modify two or more qubit states, e.g. As the in -defined CNOT gate, with the two-qubit state table , , and . The result can also be symmetrized or antisymmetrized with respect to digit indices and , for example according to the scheme
- ,
whereby entangled states arise.
A quantum circuit consists of several quantum gates that are applied to the quantum register in a fixed time sequence. Examples of this are the quantum Fourier transform or the Shor algorithm . Mathematically, a quantum circuit is also a unitary transformation, the matrix of which is the product of the matrices of the individual quantum gates.
Disposable quantum computers
Another approach to implementing a quantum computer is the so-called one-way quantum computer ( one-way quantum computer , Hans J. Briegel , Robert Raußendorf 2001). This differs from the circuit model in that first a universal (i.e. independent of the problem) entangled quantum state is generated (for example a so-called cluster state), and the actual calculation is carried out through targeted measurements on this state. The results of previous measurements determine which further measurements are carried out.
In contrast to the circuit model, the entangled quantum state is only used as a resource. In the actual calculation, only individual qubits of the state used are measured and classic calculations are carried out. In particular, no multi-qubit operations are carried out (of course, the establishment of the state requires them). Nevertheless, it can be shown that the one-way quantum computer is just as powerful as a quantum computer based on the circuit model.
Adiabatic quantum computers
Another approach for quantum computers is based on a different concept: According to the laws of quantum mechanics, a quantum mechanical system that is in the ground state (state of minimum energy) of a time-independent system remains in the ground state even if the system changes, if the change is slow enough ( so adiabatic happens). The idea of the adiabatic quantum computer is to construct a system that has a ground state, which at that time was still unknown, which corresponds to the solution of a certain problem, and another whose ground state can easily be prepared experimentally. The system, which is easy to prepare, is then transferred to the system whose basic state one is interested in, and its state is then measured. If the transition has been slow enough, that is the solution to the problem.
The company D-Wave Systems announced in 2007 that it had developed a commercially usable quantum computer based on this principle. On May 26, 2011, D-Wave Systems sold the first commercial quantum computer D-Wave One to Lockheed Martin Corporation. However, their results are still controversial. In 2015, D-Wave Systems presented their improved and upwardly scalable version D-Wave-2X to the public. The adiabatic quantum computer, which was specially developed to solve optimization problems, is said to be up to 15 times faster for some problems than conventional classic special computers for the respective problems (this was not the case with the D-Wave One). According to D-Wave, it uses superconducting technology and over 1,000 qubits (called 1000+ qubits, designed for 1,152 qubits) at a working temperature of 15 mK. The company defines a qubit as a superconducting loop on its chip in which the information about the flow direction is encoded. One copy was sold to Google and NASA , which acquired a first-generation D-Wave computer with 512 qubits in 2013. Google uses it to explore the advantages of quantum annealing algorithms, that is, quantum versions of simulated annealing .
Physical realization
The concept described so far is initially abstract and generally applicable. If you want to build a specifically usable quantum computer, you have to consider the natural physical restrictions that are described below.
Relaxation
If a system is left to its own devices, it tends to develop a thermal equilibrium with its surroundings . In the simplest case, this takes place via an energy exchange with the environment, which is accompanied by a change in the state of the qubits. This means that a qubit has jumped from the state after a certain time with a certain probability and vice versa. This process is called relaxation . The relaxation time is the characteristic time in which the system approaches (mostly exponentially ) its stationary state.
Decoherence
With decoherence loss of superposition characteristics is meant a quantum state. Due to the influence of the environment, either the state or the state develops from any superposition state (where ) (with corresponding probabilities, which can be given , for example, by , while mixed terms (e.g. ) do not occur (state reduction; incoherent vs. coherent superposition; thermalization, as in statistical physics)). Then the qubit only behaves like a classic bit. The decoherence time is usually also exponentially distributed and typically shorter than the relaxation time. While relaxation is also a problem for classic computers (magnets on the hard disk could spontaneously reverse polarity), decoherence is a purely quantum mechanical phenomenon.
The reliability of quantum computers can be increased by what is known as quantum error correction .
Computability and Complexity Theory
Since it is formally defined how a quantum computer works, the terms known from theoretical computer science such as predictability or complexity class can also be transferred to a quantum computer. It is found that a quantum computer cannot solve any fundamentally new problems, but that some problems can be solved more quickly.
Predictability
A classic computer can simulate a quantum computer, since the effect of the gates on the quantum register corresponds to a matrix-vector multiplication . The classic computer now simply has to carry out all these multiplications in order to convert the initial into the final state of the register. The consequence of this simulatability is that all problems that can be solved on a quantum computer can also be solved on a classical computer. Conversely, this means that problems like the holding problem cannot be solved on quantum computers either.
It can be shown that the simulation of a quantum computer is in the complexity class PSPACE . It is therefore assumed that there is no simulation algorithm that simulates a quantum computer with a polynomial loss of time .
Conversely, a quantum computer can also simulate a classic computer. To do this, you first have to know that any logic circuit can be formed from NAND gates alone . With the Toffoli gate , with suitable wiring of the three inputs, a quantum gate can be obtained that behaves like a NAND gate on qubits in the basis of the classic bits . The Toffoli gate can also be used to double an input bit. Due to the no-cloning theorem , however, this is only possible for the states . This doubling (also called fan-out ) is necessary because it is possible with a classic circuit to distribute a bit over two lines.
complexity
In the context of complexity theory , algorithmic problems are assigned to so-called complexity classes . The best known and most important representatives are the classes P and NP . P denotes those problems whose solution can be calculated deterministically in a running time polynomial to the input length. The problems for which there are solution algorithms that are non-deterministic polynomial lie in NP. Non-determinism allows different possibilities to be tested at the same time. Since our current computers run deterministically, the non-determinism must be simulated by executing the various options one after the other, which can result in the polynomiality of the solution strategy being lost.
The complexity class BQP is defined for quantum computers (introduced in 1993 by Umesh Vazirani and Ethan Bernstein ). This contains those problems whose runtime depends polynomially on the input length and whose error probability is lower . From the previous section it follows that BQP PSPACE. Furthermore, P BQP applies , since a quantum computer can also simulate classical computers with only a polynomial loss of time.
How BQP relates to the important class NP is still unclear. It is not known whether a quantum computer can efficiently solve an NP-complete problem or not. If one could prove that BQP is a true subset of NP, the P-NP problem would also be solved: Then P NP would hold . On the other hand, from the proof that NP is a real subset of BQP, it would follow that P is a real subset of PSPACE. Both the P-NP problem and the question P PSPACE are important unsolved questions in theoretical computer science.
For a long time it was unclear whether there were problems that quantum computers can demonstrably solve faster and more efficiently than any classical computer, in other words, that are part of BQP, but not of PH , a generalization of NP. In 2018 an example was found by Ran Raz and Avishai Tal that is in BQP ( Scott Aaronson 2009) but not in PH (more precisely, they proved that the problem is oracle-separated for both cases), the forrelation problem. Two random number generators are given. The forrelation problem is to find out from the generated random number sequences whether the two random number generators are independent or whether the sequences are connected in a hidden way, more precisely whether one is the Fourier transform of the other. Raz and Tal proved that quantum computers need far fewer hints (oracles) for the solution than classical computers. A quantum computer even only needs one oracle; in PH, even with an infinite number of oracles, there is no algorithm that solves the problem. The example shows that even for P = NP there are problems that quantum computers can solve, but classical computers cannot.
In the case of other problems, such as the factoring problem of whole numbers, it is assumed that quantum computers are in principle faster (quantum computers solve it in polynomial time with the Shor algorithm ), but it cannot yet be proven because it is unknown whether the problem lies in complexity class P.
Another problem that was expected to be solved efficiently by quantum computers, but not by classical computers, is the recommendation problem, which has even wide practical application. For example, the problem, which is important for online services, is to make predictions about their preferences from the retrieval of services or goods by users, which can be formalized as filling up a matrix that, for example, assigns goods to users. In 2016, Iordanis Kerenidis and Anupam Prakash gave a quantum algorithm that was exponentially faster than any classical algorithm known at the time. In 2018, however, the student Ewin Tang specified a classic algorithm that was just as fast. Tang found the classical algorithm based on the quantum algorithm of Kerenidis and Prakash.
Architecture for quantum computers
All quantum computers demonstrated experimentally so far consisted of a few qubits and were not scalable in terms of decoherence and error rates as well as the architecture used . In this context, architecture is understood to be the concept for the scalable arrangement of a very large number of qubits: How can it be ensured that the error rate per gate is small (below the threshold for error-tolerant computing ), regardless of the number of qubits in the quantum computer and on the spatial distance of the qubits involved in the quantum register.
The problem was summarized by David DiVincenzo in a catalog of five criteria that a scalable, fault-tolerant quantum computer must meet. The DiVincenzo criteria are
- It consists of a scalable system of well-characterized qubits.
- All qubits can be brought into a well-defined initial state (e.g. ).
- A universal set of elementary quantum gates can be implemented.
- Individual qubits (at least one) can be read out ( measured ).
- The relevant decoherence time is much longer than the time it takes to realize an elementary quantum gate , so that with suitable error-correcting code the error rate per gate is below the threshold for error-tolerant quantum computing.
The greatest requirements result from the first and the last point. In this case, scalability means that it must be possible to choose any number of qubits and that the other properties must be fulfilled regardless of the number of qubits. Depending on the code and geometry of the quantum register used, the threshold for error-tolerant computing lies at an error probability of up to (or even smaller values) per gate. So far no universal set of gates has been realized with this accuracy.
The above criteria are often supplemented by two more that relate to the networking within quantum computers:
- A quantum interface between stationary and mobile qubits
- Mobile qubits can be exchanged reliably between different locations.
The search for a scalable architecture for a fault tolerant quantum computer is the subject of current research. The question is how one can achieve that quantum gates can be executed in parallel (simultaneously) on different qubits, even if the interaction between the physical qubits is local, i.e. H. not every qubit is in direct interaction with every other. Depending on the concept used (gate network, one-way quantum computer, adiabatic quantum computer, ...) and the chosen implementation (trapped ions, superconducting circuits, ...) there are various proposals that have so far only been demonstrated for small prototypes. Some of the most specific and advanced proposals include the following:
- Quantum computer in a microstructured ion trap : qubits are realized through the internal state of individual trapped ions. In a microstructured trap, the ions are moved back and forth between storage and interaction regions in a controlled manner. Instead of moving the ions to be coupled to one another into a common interaction region, long-range interactions could also be used between them. In experiments at the University of Innsbruck it was demonstrated that, for example, the electrical dipole interaction between small groups of oscillating ions (which act as antennas) can be used to couple ions that are more than 50 micrometers apart.
- Superconducting qubits in a two-dimensional network of the superconducting strip line resonators ( strip line resonators ).
- Quantum computer based on nitrogen vacancy centers (NV centers) in diamond : nuclear spins of nitrogen atoms in a two-dimensional grid of NV centers function as qubits ; Readout and coupling take place via the electronic spin of the NV center, the coupling being achieved through the magnetic dipole interaction ; Inhomogeneous magnetic fields enable individual addressing and parallel operation on many qubits.
Research history
Quantum computers with a few qubits could already be implemented in the 1990s. Thus, Shor's algorithm in 2001 with a price based on nuclear magnetic resonance system at the IBM Almaden Research Center realized with 7 qubits and increased the number 15 into its prime factors 3 and 5 divide. In 2003, a quantum computer based on particles stored in ion traps was also able to implement the Deutsch-Jozsa algorithm .
In November 2005, Rainer Blatt at the Institute for Experimental Physics at the University of Innsbruck succeeded for the first time in generating a quantum register with 8 entangled qubits . The entanglement of all eight qubits had to be verified by 650,000 measurements and took 10 hours.
In March 2011, the Innsbruck scientists almost doubled the number of qubits again. They held 14 calcium atoms trapped in an ion trap, which they manipulated with laser light according to the principle of a quantum processor.
At Yale University, a team of researchers led by Leo DiCarlo cooled a two-qubit register on a 7 mm long and 2 mm wide quantum processor with a multi-curved channel to a temperature of 13 mK, thus generating a 2-qubit register. Quantum computer. According to a publication by Nature in 2009, the superconducting chip ran through quantum algorithms for the first time.
In 2011, a research group at the National Institute of Standards and Technology (NIST) in Boulder, USA, succeeded in entangling ions using microwaves . The NIST research group has shown that such operations can be carried out not only with a complex, room-filling laser system , but also with miniaturized microwave electronics. To create the entanglement, the physicists integrated the microwave source into the electrodes of a so-called chip trap, a microscopic chip-like structure for storing and manipulating the ions in a vacuum cell. With their experiment, the researchers have shown that entangling the ions with microwaves works in 76% of all cases. The laser-based quantum logic gates that have been used in research for several years are currently even better than the gates based on microwaves with a rate of 99.3%. The new method has the advantage that it only takes up about a tenth of the space of a laser experiment.
On January 2, 2014, the Washington Post reported , citing documents from whistleblower Edward Snowden , that the US National Security Agency (NSA) was working on the development of a "cryptologically useful" quantum computer. Although the current status (2019) of the technology does not yet represent a security threat, post-quantum cryptography is being worked on.
IBM has been providing online access to a superconductor-based quantum processor since 2015. Initially 5 qubits were available, since November 2017 there have been 20. The website includes an editor that can be used to write programs for the quantum computer, as well as an SDK and interactive instructions. As of November 2017, over 35 scientific publications have been published using the IBM Q Experience computer . IBM also offers access to the 50 qubit machine in their laboratory via the cloud. The quantum state of this system is held for 90 microseconds, which was a record in late 2017. In terms of technology for efficient simulation of quantum computers on classic high-performance computers, IBM announced in 2017 that it had reached the 49 qubit limit.
In addition to IBM (as of 2018), many large computer companies develop so-called quantum computers or their technology, such as Google , Microsoft , Intel and startups such as Rigetti in San Francisco. In 2018, Google presented its new Quantum Processor Bristlecone with 72 qubits (scaled from 9 qubits previously) and a low error rate for logical operations and readout. It is also based on superconductors and is mainly used to research the technology and possibly in the near future to prove quantum supremacy ( John Preskill 2012), i.e. a problem in which the quantum computer is superior to a classic supercomputer and its solution is the next big step on the The way to the quantum computer applies (as of August 2018). Google, like other computer companies, estimates that a demonstration of Quantum Supremacy requires at least 49 qubits, a circuit depth greater than 40, and an error rate below half a percent. The number of qubits alone is not decisive, but also, for example, the error rate and the depth of the circuit, i.e. the number of gates (logical operations) that can be implemented in the qubits before the coherence is destroyed due to an excessively high error rate. Before Bristlecone, Google achieved an error rate of around 1 percent for reading and 0.1 percent for logical operations for gates of a single qubit and 0.6 percent for two-qubit gates. According to Google, a commercially usable quantum computer is around 1 million qubits.
Microsoft focuses (as of 2018) on theoretical work on error correction using topological quantum computers (a concept that Alexei Yuryevich Kitayev introduced in 1997) under the direction of mathematician Michael Freedman and developed a simulator with which quantum computers can be simulated on classical computers, and Software for quantum computers. They have their own quantum computer laboratory (Station Q) in Santa Barbara.
Google Researchers demonstrated for the first time in a in the trade magazine on October 23, 2019 Nature published article called quantum superiority (Engl. Quantum Supremacy ). Google's quantum processor Sycamore took about 200 seconds for a complex calculation, for which the most modern supercomputer Summit would need about 10,000 years. The Sycamore chip has 53 qubits. Competitor IBM doubts the Google results and thus the quantum superiority. Google's invoice contains an error. According to IBM, the task can be solved by classic systems in 2.5 days.
literature
- D. Bruß: Quanteninformation Fischer Taschenbuch Verlag, Frankfurt am Main 2015, ISBN 978-3-596-30422-6 .
- M. Homeister: Understanding Quantum Computing: Basics, Applications, Perspectives Springer / Vieweg, Wiesbaden 2018, fifth edition, ISBN 978-3-6582-2883-5 .
- B. Lenze: Mathematik und Quantum Computing Logos Verlag, Berlin 2020, second edition, ISBN 978-3-8325-4716-5 .
- RJ Lipton, KW Regan: Quantum Algorithms via Linear Algebra: A Primer MIT Press, Cambridge MA 2014, ISBN 978-0-262-02839-4 .
- CJ Meier: A Brief History of the Quantum Computer. Heinz Heise Verlag, Hannover 2015, ISBN 978-3-944099-06-4 .
- A. Montanaro: Quantum Algorithms. npj Quantum Information, Volume 2, 2016, p. 15023, Arxiv
- MA Nielsen, IL Chuang: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge MA 2010, ISBN 978-1-1070-0217-3 .
- W. Scherer: Mathematics of quantum informatics. Springer-Verlag, Berlin-Heidelberg 2016, ISBN 978-3-662-49079-2 .
- J. Stolze, D. Suter: Quantum Computing: A Short Course from Theory to Experiment. Wiley-VCH, Weinheim 2004, ISBN 3-527-40787-1 .
- Einstein's unexpected legacy. Quantum Information Technology ( Memento from January 5, 2018 in the Internet Archive ), brochure of the Federal Ministry of Education and Research, 2005.
- Neil Gershenfeld , and Isaac L. Chuang. " Quantum computing with molecules. " Scientific American 278.6 (1998): 66-71.
Web links
- Quantum computing. In: Edward N. Zalta (Ed.): Stanford Encyclopedia of Philosophy .
- Quantum Technology Roadmap , EU, 2018
References and comments
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- ↑ Edwin Pednault et al. a., Breaking the 49-Qubit Barrier in the Simulation of Quantum Circuits , Arxiv 2017.
- ↑ Edwin Pednault, Quantum Computing: Breaking Through the 49 Qubit Simulation Barrier , IBM, October 17, 2017.
- ↑ A Preview of Bristlecone, Google's New Quantum Processor , AI Googleblog, March 5, 2018.
- ↑ Philip Ball, Race for quantum supremacy hits theoretical quagmire , Nature News, November 13, 2017.
- ↑ Frederic Lardinois, Google's new Bristlecone processor brings it one step closer to quantum supremacy , Techcrunch, March 5, 2018.
- ^ Frederic Lardinois, Microsoft places its bets on quantum computing , Techcrunch, September 26, 2017.
- ↑ Quantum supremacy using a programmable superconducting processor nature.com, accessed October 23, 2019.
- ↑ Now also official: Google's quantum computer proves "Quantum Supremacy" heise.de, accessed on October 23, 2019.
- ↑ Google reports quantum superiority tagesschau.de. Retrieved October 24, 2019.