# Quantum computing

Quantum computing is the study of a non-classical 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.[1] A quantum computer is a device that could perform such a computation.[2]

Quantum computing began in the early 1980s, when physicist Paul Benioff proposed a quantum mechanical model of the Turing machine.[3] Richard Feynman and Yuri Manin later suggested that a quantum computer could perform simulations that are out of reach for regular computers.[4][5] In 1994, Peter Shor developed a polynomial-time quantum algorithm for factoring integers.[6] 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. Although it is theoretically accepted that quantum computers could equally be used to counteract against adversarial actions of this kind; with the laws of quantum mechanics applied in this cryptographic capacity (through use of quantum computers) paving the way to quantum cryptography.

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.[7] 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".[8] On 23 October 2019, Google AI, in partnership with the U.S. National Aeronautics and Space Administration (NASA), officially claimed that its Sycamore quantum processor completed in 200 seconds, a task the equivalent of which would take a state-of-the-art supercomputer approximately 10,000 years to complete. In response, one prominent researcher declared that a quantum computing revolution equivalent to the modern digital computer will require "immense engineering, and probably further insights as well." [9]

There is an increasing amount of investment in quantum computing by governments, established companies, and start-ups.[10] Current research focuses on building and using near-term intermediate-scale devices[8] and demonstrating quantum supremacy.[11] 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

A classical computer contains memory made up of fundamental units called bits, where each bit represents either a one or a zero. The memory of a quantum computer, on the other hand, is made up of bits which can represent a one, a zero, or some combination. The combination is known as a quantum superposition, and bits with these quantum properties are known as qubits.

The defining property of a quantum computer is the ability to turn bits into qubits, and vice versa. This is in contrast to classical computers in that they are designed to only perform computations with memory that never deviates from clearly defined values. A quantum computer can perform operations on qubits which include possibilities not available to classical computers. The final result returned by a quantum computer is not deterministic. The measurement process is inherently probabilistic, meaning that the output is frequently random, so special algorithms are required to compensate.

## 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.[12]

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 generalized 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).[13] 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.[14][15] 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.[14][16] 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.[17] It has been proven that applying Grover's algorithm to break a symmetric (secret key) algorithm by brute force requires time equal to roughly 2n/2 invocations of the underlying cryptographic algorithm, compared with roughly 2n in the classical case,[18] meaning that symmetric key lengths are effectively halved: AES-256 would have the same security against an attack using Grover's algorithm that AES-128 has against classical brute-force search (see Key size).

Quantum cryptography could potentially fulfill some of the functions of public key cryptography. Quantum-based cryptographic systems could, therefore, be more secure than traditional systems against quantum hacking.[19]

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,[20] 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.[21] 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[22] 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.[23]

### 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.[24] Quantum simulation could also be used to simulate the behavior of atoms and particles at unusual conditions such as the reactions inside a collider.[25]

### 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.[26]

### 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.[27] 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.[28] Although skeptics like Gil Kalai doubt that quantum supremacy will ever be achieved,[29][30] 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.[31] Bill Unruh doubted the practicality of quantum computers in a paper published back in 1994.[32] Paul Davies argued that a 400-qubit computer would even come into conflict with the cosmological information bound implied by the holographic principle.[33] In October 2019, a Sycamore processor created in conjunction with Google AI Quantum was reported to have achieved quantum supremacy.[34]

## Obstacles

There are a number of technical challenges in building a large-scale quantum computer,[35] 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:[36]

• 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.

Sourcing parts for quantum computers is very difficult: Quantum computers need Helium-3, a nuclear research byproduct, and special cables that are only made by a single company in Japan. [37]

### 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.[38] Currently, some quantum computers require their qubits to be cooled to 20 millikelvins in order to prevent significant decoherence.[39]

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.[40]

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.[41] 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.[42][43]

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

So the number of continuous parameters describing the state of such a useful quantum computer at any given moment must be... about 10300... Could we ever learn to control the more than 10300 continuously variable parameters defining the quantum state of such a system? My answer is simple. No, never.[44]

## 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:

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):

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.

## Relation to computational complexity theory

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.[64] 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),[65] 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.[65]

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.[66] A similar fact prevails for particular computational tasks, like the search problem, for which Grover's algorithm is optimal.[67]

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.[68]

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.[69] 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.[70][71]

## References

1. Overbye, Dennis (October 21, 2019). "Quantum Computing Is Coming, Bit by Qubit - With transmons and entanglement, scientists strive to put subatomic weirdness to work on the human scale". The New York Times. Retrieved October 21, 2019.
2. The National Academies of Sciences, Engineering, and Medicine (2019). Grumbling, Emily; Horowitz, Mark (eds.). Quantum Computing : Progress and Prospects (2018). Washington, DC: National Academies Press. p. I-5. doi:10.17226/25196. ISBN 978-0-309-47969-1. OCLC 1081001288.CS1 maint: multiple names: authors list (link)
3. Benioff, Paul (1980). "The computer as a physical system: A microscopic quantum mechanical Hamiltonian model of computers as represented by Turing machines". Journal of Statistical Physics. 22 (5): 563–591. Bibcode:1980JSP....22..563B. doi:10.1007/bf01011339.
4. Feynman, Richard (June 1982). "Simulating Physics with Computers" (PDF). International Journal of Theoretical Physics. 21 (6/7): 467–488. doi:10.1007/BF02650179. Retrieved 28 February 2019.
5. Manin, Yu. I. (1980). Vychislimoe i nevychislimoe [Computable and Noncomputable] (in Russian). Sov.Radio. pp. 13–15. Archived from the original on 2013-05-10. Retrieved 2013-03-04.
6. Mermin, David (March 28, 2006). "Breaking RSA Encryption with a Quantum Computer: Shor's Factoring Algorithm" (PDF). Cornell University, Physics 481-681 Lecture Notes.
7. Vandersypen, Lieven M. K.; Steffen, Matthias; Breyta, Gregory; Yannoni, Costantino S.; Sherwood, Mark H.; Chuang, Isaac L. (December 2001). "Experimental realization of Shor's quantum factoring algorithm using nuclear magnetic resonance". Nature. 414 (6866): 883–887. arXiv:quant-ph/0112176. doi:10.1038/414883a. ISSN 0028-0836.
8. John Preskill (2018). "Quantum Computing in the NISQ era and beyond". Quantum. 2: 79. arXiv:1801.00862. doi:10.22331/q-2018-08-06-79.
9. Aaronson, Scott (2019-10-30). "Opinion | Why Google's Quantum Supremacy Milestone Matters". The New York Times. ISSN 0362-4331. Retrieved 2019-10-30.
10. "Quantum Computing Report: Players". Retrieved 2019-04-17.
11. John Preskill (2012). "Quantum computing and the entanglement frontier". arXiv:1203.5813 [quant-ph].
12. Nielsen, Michael A.; Chuang, Isaac L. (2010). Quantum Computation and Quantum Information: 10th Anniversary Edition. Cambridge: Cambridge University Press. doi:10.1017/cbo9780511976667. ISBN 9780511976667.
13. Lenstra, Arjen K. (2000). "Integer Factoring" (PDF). Designs, Codes and Cryptography. 19 (2/3): 101–128. doi:10.1023/A:1008397921377. Archived from the original (PDF) on 2015-04-10.
14. Daniel J. Bernstein, Introduction to Post-Quantum Cryptography. Introduction to Daniel J. Bernstein, Johannes Buchmann, Erik Dahmen (editors). Post-quantum cryptography. Springer, Berlin, 2009. ISBN 978-3-540-88701-0
15. See also pqcrypto.org, a bibliography maintained by Daniel J. Bernstein and Tanja Lange on cryptography not known to be broken by quantum computing.
16. Robert J. McEliece. "A public-key cryptosystem based on algebraic coding theory." Jet Propulsion Laboratory DSN Progress Report 42–44, 114–116.
17. Kobayashi, H.; Gall, F.L. (2006). "Dihedral Hidden Subgroup Problem: A Survey". Information and Media Technologies. 1 (1): 178–185.
18. Bennett C.H., Bernstein E., Brassard G., Vazirani U., "The strengths and weaknesses of quantum computation". SIAM Journal on Computing 26(5): 1510–1523 (1997).
19. "What are quantum computers and how do they work? WIRED explains". Wired UK. 2018-02-16.
20. Quantum Algorithm Zoo Archived 2018-04-29 at the Wayback Machine – Stephen Jordan's Homepage
21. Jon Schiller, Phd (2009-06-19). Quantum Computers. ISBN 9781439243497.
22. Ambainis, Andris (2005). "Quantum search algorithms". arXiv:quant-ph/0504012.
23. Rich, Steven; Gellman, Barton (2014-02-01). "NSA seeks to build quantum computer that could crack most types of encryption". Washington Post.
24. Norton, Quinn (2007-02-15). "The Father of Quantum Computing". Wired.
25. Ambainis, Andris (Spring 2014). "What Can We Do with a Quantum Computer?". Institute for Advanced Study.
26. Ambainis, Andris; Hassidim, Avinatan; Lloyd, Seth (2008). "Quantum algorithm for solving linear systems of equations". Physical Review Letters. 103 (15): 150502. arXiv:0811.3171. doi:10.1103/PhysRevLett.103.150502. PMID 19905613.
27. Boixo, Sergio; Isakov, Sergei V.; Smelyanskiy, Vadim N.; Babbush, Ryan; Ding, Nan; Jiang, Zhang; Bremner, Michael J.; Martinis, John M.; Neven, Hartmut (2018). "Characterizing Quantum Supremacy in Near-Term Devices". Nature Physics. 14 (6): 595–600. arXiv:1608.00263. doi:10.1038/s41567-018-0124-x.
28. Savage, Neil. "Quantum Computers Compete for "Supremacy"".
29. "Quantum Supremacy and Complexity". 23 April 2016.
30. Kalai, Gil. "The Quantum Computer Puzzle" (PDF). AMS.
32. Unruh, Bill (1995). "Maintaining coherence in Quantum Computers". Physical Review A. 51 (2): 992–997. arXiv:hep-th/9406058. Bibcode:1995PhRvA..51..992U. doi:10.1103/PhysRevA.51.992. PMID 9911677.
33. Davies, Paul. "The implications of a holographic universe for quantum information science and the nature of physical law" (PDF). Macquarie University.
34. Arute, Frank; Arya, Kunal; Babbush, Ryan; Bacon, Dave; Bardin, Joseph C.; Barends, Rami; Biswas, Rupak; Boixo, Sergio; Brandao, Fernando G. S. L.; Buell, David A.; Burkett, Brian; Chen, Yu; Chen, Zijun; Chiaro, Ben; Collins, Roberto; Courtney, William; Dunsworsth, Andrew; Farhi, Edward; Foxen, Brooks; Fowler, Austin; Gidney, Craig; Giustina, Marissa; Graff, Rob; Guerin, Keith; Habegger, Steve; Harrigan, Matthew P.; Hartmann, Michael J.; Ho, Alan; Hoffman, Markus; Huang, Trent; Humble, Travis S.; Isakov, Sergei V.; Jeffery, Evan; Jiang, Zhang; Kafri, Dvir; Kechedzhi, Kostyantyn; Kelly, Julian; Klimov, Paul V.; Knysh, Sergey; Korotov, Alexander; Kostritsa, Fedor; Landhuis, David; Lindmark, Mike; Lucero, Erik; Lyakh, Dmitry; Mandrà, Salvatore; McClean, Jarrod R.; McEwen, Matthew; Megrant, Anthony; Mi, Xiao; Michielsen, Kristel; Mohseni, Masoud; Mutus, Josh; Naaman, Ofer; Neeley, Matthew; Neill, Charles; Niu, Murphy Yuezhen; Ostby, Eric; Petukhov, Andre; Platt, John C.; Quintana, Chris; Rieffel, Eleanor G.; Roushan, Pedram; Rubin, Nicholas C.; Sank, Daniel; Satzinger, Kevin J.; Smelyanskiy, Vadim; Sung, Kevin J.; Trevithick, Matthew D.; Vainsencher, Amit; Villalonga, Benjamin; White, Theodore; Yao, Z. Jamie; Yeh, Ping; Zalcman, Adam; Neven, Hartmut; Martinis, John M. (23 October 2019). "Quantum supremacy using a programmable superconducting processor". Nature. 574 (7779). Retrieved 25 October 2019.
35. Dyakonov, Mikhail (2018-11-15). "The Case Against Quantum Computing". IEEE Spectrum.
36. DiVincenzo, David P. (2000-04-13). "The Physical Implementation of Quantum Computation". Fortschritte der Physik. 48 (9–11): 771–783. arXiv:quant-ph/0002077. Bibcode:2000ForPh..48..771D. doi:10.1002/1521-3978(200009)48:9/11<771::AID-PROP771>3.0.CO;2-E.
37. https://futurism.com/sourcing-parts-quantum-computers-difficult/amp
38. DiVincenzo, David P. (1995). "Quantum Computation". Science. 270 (5234): 255–261. Bibcode:1995Sci...270..255D. CiteSeerX 10.1.1.242.2165. doi:10.1126/science.270.5234.255. (subscription required)
39. Jones, Nicola (19 June 2013). "Computing: The quantum company". Nature. 498 (7454): 286–288. Bibcode:2013Natur.498..286J. doi:10.1038/498286a. PMID 23783610.
40. Amy, Matthew; Matteo, Olivia; Gheorghiu, Vlad; Mosca, Michele; Parent, Alex; Schanck, John (November 30, 2016). "Estimating the cost of generic quantum pre-image attacks on SHA-2 and SHA-3". arXiv:1603.09383 [quant-ph].
41. Dyakonov, M. I. (2006-10-14). S. Luryi; J. Xu; A. Zaslavsky (eds.). "Is Fault-Tolerant Quantum Computation Really Possible?". Future Trends in Microelectronics. Up the Nano Creek: 4–18. arXiv:quant-ph/0610117. Bibcode:2006quant.ph.10117D.
42. Freedman, Michael H.; Kitaev, Alexei; Larsen, Michael J.; Wang, Zhenghan (2003). "Topological quantum computation". Bulletin of the American Mathematical Society. 40 (1): 31–38. arXiv:quant-ph/0101025. doi:10.1090/S0273-0979-02-00964-3. MR 1943131.
43. Monroe, Don (2008-10-01). "Anyons: The breakthrough quantum computing needs?". New Scientist.
44. Dyakonov, Mikhail. "The Case Against Quantum Computing". IEEE Spectrum. Retrieved 3 December 2019.
45. Das, A.; Chakrabarti, B. K. (2008). "Quantum Annealing and Analog Quantum Computation". Rev. Mod. Phys. 80 (3): 1061–1081. arXiv:0801.2193. Bibcode:2008RvMP...80.1061D. CiteSeerX 10.1.1.563.9990. doi:10.1103/RevModPhys.80.1061.
46. Nayak, Chetan; Simon, Steven; Stern, Ady; Das Sarma, Sankar (2008). "Nonabelian Anyons and Quantum Computation". Rev Mod Phys. 80 (3): 1083–1159. arXiv:0707.1889. Bibcode:2008RvMP...80.1083N. doi:10.1103/RevModPhys.80.1083.
47. Clarke, John; Wilhelm, Frank (June 19, 2008). "Superconducting quantum bits". Nature. 453 (7198): 1031–1042. Bibcode:2008Natur.453.1031C. doi:10.1038/nature07128. PMID 18563154.
48. Kaminsky, William M (2004). "Scalable Superconducting Architecture for Adiabatic Quantum Computation". arXiv:quant-ph/0403090.
49. Imamoğlu, Atac; Awschalom, D. D.; Burkard, Guido; DiVincenzo, D. P.; Loss, D.; Sherwin, M.; Small, A. (1999). "Quantum information processing using quantum dot spins and cavity-QED". Physical Review Letters. 83 (20): 4204–4207. arXiv:quant-ph/9904096. Bibcode:1999PhRvL..83.4204I. doi:10.1103/PhysRevLett.83.4204.
50. Fedichkin, Leonid; Yanchenko, Maxim; Valiev, Kamil (2000). "Novel coherent quantum bit using spatial quantization levels in semiconductor quantum dot". Quantum Computers and Computing. 1: 58–76. arXiv:quant-ph/0006097. Bibcode:2000quant.ph..6097F. Archived from the original on 2011-08-18.
51. Bertoni, A.; Bordone, P.; Brunetti, R.; Jacoboni, C.; Reggiani, S. (2000-06-19). "Quantum Logic Gates based on Coherent Electron Transport in Quantum Wires" (PDF). Physical Review Letters. 84 (25): 5912–5915. Bibcode:2000PhRvL..84.5912B. doi:10.1103/PhysRevLett.84.5912. hdl:11380/303796. PMID 10991086.
52. Ionicioiu, Radu; Amaratunga, Gehan; Udrea, Florin (2001-01-20). "Quantum Computation with Ballistic Electrons". International Journal of Modern Physics B. 15 (2): 125–133. arXiv:quant-ph/0011051. Bibcode:2001IJMPB..15..125I. doi:10.1142/s0217979201003521. ISSN 0217-9792.
53. Ramamoorthy, A.; Bird, J. P.; Reno, J. L. (2007). "Using split-gate structures to explore the implementation of a coupled-electron-waveguide qubit scheme". Journal of Physics: Condensed Matter. 19 (27): 276205. Bibcode:2007JPCM...19A6205R. doi:10.1088/0953-8984/19/27/276205. ISSN 0953-8984.
54. Leuenberger, MN; Loss, D (Apr 12, 2001). "Quantum computing in molecular magnets". Nature. 410 (6830): 789–93. arXiv:cond-mat/0011415. Bibcode:2001Natur.410..789L. doi:10.1038/35071024. PMID 11298441.
55. Knill, G. J.; Laflamme, R.; Milburn, G. J. (2001). "A scheme for efficient quantum computation with linear optics". Nature. 409 (6816): 46–52. Bibcode:2001Natur.409...46K. doi:10.1038/35051009. PMID 11343107.
56. Nizovtsev, A. P. (August 2005). "A quantum computer based on NV centers in diamond: Optically detected nutations of single electron and nuclear spins". Optics and Spectroscopy. 99 (2): 248–260. Bibcode:2005OptSp..99..233N. doi:10.1134/1.2034610.
57. Gruener, Wolfgang (2007-06-01). "Research indicates diamonds could be key to quantum storage". Archived from the original on 2007-06-04. Retrieved 2007-06-04.
58. Neumann, P.; et al. (June 6, 2008). "Multipartite Entanglement Among Single Spins in Diamond". Science. 320 (5881): 1326–1329. Bibcode:2008Sci...320.1326N. doi:10.1126/science.1157233. PMID 18535240.
59. Millman, Rene (2007-08-03). "Trapped atoms could advance quantum computing". ITPro. Archived from the original on 2007-09-27. Retrieved 2007-07-26.
60. Ohlsson, N.; Mohan, R. K.; Kröll, S. (January 1, 2002). "Quantum computer hardware based on rare-earth-ion-doped inorganic crystals". Opt. Commun. 201 (1–3): 71–77. Bibcode:2002OptCo.201...71O. doi:10.1016/S0030-4018(01)01666-2.
61. Longdell, J. J.; Sellars, M. J.; Manson, N. B. (September 23, 2004). "Demonstration of conditional quantum phase shift between ions in a solid". Phys. Rev. Lett. 93 (13): 130503. arXiv:quant-ph/0404083. Bibcode:2004PhRvL..93m0503L. doi:10.1103/PhysRevLett.93.130503. PMID 15524694.
62. Náfrádi, Bálint; Choucair, Mohammad; Dinse, Klaus-Peter; Forró, László (July 18, 2016). "Room Temperature manipulation of long lifetime spins in metallic-like carbon nanospheres". Nature Communications. 7: 12232. arXiv:1611.07690. Bibcode:2016NatCo...712232N. doi:10.1038/ncomms12232. PMC 4960311. PMID 27426851.
63. Nielsen, p. 42
64. Nielsen, p. 41
65. Bernstein, Ethan; Vazirani, Umesh (1997). "Quantum Complexity Theory". SIAM Journal on Computing. 26 (5): 1411–1473. CiteSeerX 10.1.1.144.7852. doi:10.1137/S0097539796300921.
66. Ozhigov, Yuri (1999). "Quantum Computers Speed Up Classical with Probability Zero". Chaos, Solitons & Fractals. 10 (10): 1707–1714. arXiv:quant-ph/9803064. Bibcode:1998quant.ph..3064O. doi:10.1016/S0960-0779(98)00226-4.
67. Ozhigov, Yuri (1999). "Lower Bounds of Quantum Search for Extreme Point". Proceedings of the London Royal Society. A455 (1986): 2165–2172. arXiv:quant-ph/9806001. Bibcode:1999RSPSA.455.2165O. doi:10.1098/rspa.1999.0397.
68. Aaronson, Scott. "Quantum Computing and Hidden Variables" (PDF).
69. Nielsen, p. 126
70. Scott Aaronson (2005). "NP-complete Problems and Physical Reality". ACM SIGACT News. 2005. arXiv:quant-ph/0502072. Bibcode:2005quant.ph..2072A. See section 7 "Quantum Gravity": "[…] to anyone who wants a test or benchmark for a favorite quantum gravity theory,[author's footnote: That is, one without all the bother of making numerical predictions and comparing them to observation] let me humbly propose the following: can you define Quantum Gravity Polynomial-Time? […] until we can say what it means for a 'user' to specify an 'input' and ‘later' receive an 'output'—there is no such thing as computation, not even theoretically." (emphasis in original)
71. "D-Wave Systems sells its first Quantum Computing System to Lockheed Martin Corporation". D-Wave. 2011-05-25. Retrieved 2011-05-30.