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(1994). Longdell, J. doi:10.1007/978-3-540-88702-7 1. Bill Unruh doubted the practicality of quantum computers in a paper published back in 1994.[77] Paul Davies argued that a 400-qubit computer would even come into conflict with the cosmological information bound implied by the holographic principle.[78] Skeptics like Gil Kalai doubt thatquantum supremacy will ever be achieved.[79][80][81] 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. Chuang, Isaac L.; Gershenfeld, Neil; Kubinec, Markdoi (April 1998)."Simulating the Sycamore quantum supremacy circuits". Quantum Computing : Progress and Prospects (2018). doi:10.1090/S0273-0979-02-00964-3. arXiv:1203.5813 [quant-ph]. SC '21. PMC 6898666. doi:10.1103/RevModPhys.80.1061. Simon, Daniel R. The quantum state vector formalism is usually introduced first because it is conceptually simpler,and because it can be used instead of the density matrix formalism for pure states, where the whole quantum system is known. arXiv:quant-ph/0006097. Zhong, Han-Sen; Wang, Hui; Deng, Yu-Hao; Chen, Ming-Cheng; Peng, Li-Chao; Luo, Yi-Han; Qin, Jian; Wu, Dian; Ding, Xing; Hu, Yi; Hu, Peng (3 December 2020). SIAM Journal on Computing. (23October 2019). Gao, Xun; Anschuetz, Eric R.; Wang, Sheng-Tao; Cirac, J. Mermin, David (28 March 2006). Outeiral, Carlos; Strahm, Martin; Morris, Garrett; Benjamin, Simon; Deane, Charlotte; Shi, Jiye (2021). This vector is viewed as a probability vector and represents the fact that the memory is to be found in a particular state. (14 October2006). 40 (1): 31–38. Fortschritte der Physik. 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. "Physicists in China challenge Google's 'quantum advantage'". "Estimation of effective temperatures in quantum annealers for sampling applications: A case study with possible applications in deep learning". Archived from the original (PDF) on 8 January 2019.The class of problems that can be efficiently solved by a quantum computer with bounded error is called BQP, for "bounded error, quantum, polynomial time". Science News. "Enhancing Generative Models via Quantum Correlations". J.; Manson, N. If the quantum memory is measured to determine whether the state is 0 ⟩ {\textstyle 0\rangle } or 1⟩ {\textstyle 1\rangle } (this is known as a computational basis measurement), the zero state would be observed with probability α 2 {\textstyle \alpha {2}} and the one state with probability β 2 {\textstyle \beta {2}} . Cambridge University Press. "A quantum computer based on NV centers in diamond: Optically detected nutations of singleelectron and nuclear spins". S2CID 122949592. pp. 247–266. Archived from the original on 11 December 2021. 80 (3): 1083–1159. The Washington Post. Ajagekar, Akshay; You, Fengqi (5 December 2020). My answer is simple. arXiv:1611.09347. 410 (6830): 789–793. doi:10.1134/1.2034610. arXiv:2101.03438 [cs.ET]. S2CID 8258191. arXiv:condmat/0011415. MR 1943131. CiteSeerX 10.1.1.242.2165. Bibcode:2020PhRvX.10b1054K. ISBN 978-3-319-08284-4. Given how computational biology is using generic data modeling and storage, its applications to computational biology are expected to arise as well.[44] Computer-aided drug design and generative chemistry Main article: Quantummachine learning Deep generative chemistry models emerge as powerful tools to expedite drug discovery. Benenti, Giuliano (2004). Wired. The numbers α {\textstyle \alpha } and β {\textstyle \beta } are called quantum amplitudes. Leuenberger, Michael N.; Loss, Daniel (April 2001). Table 1 lists switching and dephasing times for various systems. Ramamoorthy, A; Bird, J P; Reno, J L (11 July 2007). 14 (6): 595–600. Vepsäläinen, Antti P.; Karamlou, Amir H.; Orrell, John L.; Dogra, Akshunna S.; Loer, Ben; et al. Cantaloube, N. S2CID 220110562. AMS. arXiv:2108.11644 [quant-ph]. In other words, quantum computers obey the Church–Turing thesis. "The Physical Implementation of QuantumComputation". ISBN 978-0-387-35725-6. "Anyons: The breakthrough quantum computing needs?". Gibney, Elizabeth (23 October 2019). Akama, Seiki (2014). S2CID 231715555.{{cite journal}}: CS1 maint: uses authors parameter (link) DiVincenzo, David P. "Challenges in Reliable Quantum Computing". /276205. It is particularly difficult to maintain qubits' quantum states, as they suffer from quantum decoherence and state fidelity. DSNPR. 99 (2): 248–260. PMID 9912648. Bibcode:2007Sci.316.D. S. "Novel coherent quantum bit using spatial quantization levels in semiconductor quantum dot". PMID 9911677.Bibcode:2006quant.ph.10117D. Vol. 549. Springer International Publishing. Wichert, Andreas (2014). One of the well-known examples would be in computational genomics and how computing has drastically reduced the time to sequence a human genome. B. Further reading Nielsen, Michael; Chuang, Isaac (2000). Bibcode:2000PhRvL.84.5912B.Principles of Quantum Artificial Intelligence. doi:10.1038/s41586-020-2619-8. arXiv:quant-ph/0310130. It is suspected that N P B Q P {\displaystyle {\mathsf {NPsubseteq BQP}}} ; that is, it is believed that there are efficiently checkable problems that are not efficiently solvable by a quantum computer. S2CID 18324734.doi:10.1038/ncomms12232. 549 (7671): 195–202. 93 (13): 130503. S2CID 13403194. 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. PMID 18535240. 10 (1): 5607. {{cite journal}}: Cite uses deprecated parameter lay-url (help) Ohlsson, N.; Mohan, R. Many quantum computers, like those constructed by Google and IBM, need helium-3, a nuclear research byproduct, and special superconducting cables made only by the Japanese company Coax Co.[51] The control of multi-qubit systems requires the generation and coordination of a large number of electricalsignals with tight and deterministic timing resolution. Monroe, Don (1 October 2008). S. Information and Media Technologies. "Room temperature manipulation of long lifetime spins in metallic-like carbon nanospheres". Anderlini, Marco; Lee, Patricia J.; Brown, Benjamin L.; Sebby-Strabley, Jennifer; Phillips, William D.; Porto, J. {\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}}.} The CNOT gate can then be represented using the following matrix: CNOT : ( 1 0 0 0 0 1 0 0 0 0 0 1 0 01 0 ) . Nature Physics. "Dihedral Hidden Subgroup Problem: A Survey". Rinott, Yosef; Shoham, Tomer; Kalai, Gil (13 July 2021). PMID 19905613. a b Nielsen, p. Quantum circuit The Bloch sphere is a representation of a qubit, the fundamental building block of quantum computers. 4 (1): 169–190. Bibcode:2008RvMP.80.1083N. Freedman,Michael H.; Kitaev, Alexei; Larsen, Michael J.; Wang, Zhenghan (2003). One qubit of information is said to be encoded into the quantum memory. "Scalable Superconducting Architecture for Adiabatic Quantum Computation". Pan, Feng; Chen, Keyang; Zhang, Pan (4 November 2021). "The Case Against Quantum Computing". (1 January 2002)."Nonabelian Anyons and Quantum Computation". arXiv:quant-ph/9505011. I. Sibos TV. "Development of Dilution refrigerators – A review". doi:10.1103/PhysRevLett.83.4204. Bousquet, C. Quantum Computing. "Multipartite Entanglement Among Single Spins in Diamond". "On the Power of Quantum Computation". Bibcode:2009PhRvL.103o0502H.Relation to computability and complexity theory Computability theory See also: Computability theory Any computational problem solvable by a classical computer is also solvable by a quantum computer.[18] Intuitively, this is because it is believed that all physical phenomena, including the operation of classical computers, can be described usingquantum mechanics, which underlies the operation of quantum computers. S2CID 13980886. Bibcode:2005quant.ph.2072A. doi:10.1142/S0217979201003521. arXiv:1608.00263. Pan, Feng; Zhang, Pan (4 March 2021). One common such set includes all single-qubit gates as well as the CNOT gate from above. Preskill, John (26 March 2012). Garisto, Daniel. doi:10.1017/cbo9780511976667. S2CID 227254333. Cho, Adrian (23 October 2019). "IBM's Eagle -- 127-Qubit Quantum Processor -- Takes Flight". PMID 28905891. doi:10.1038/498286a. arXiv:1603.09383 [quant-ph]. Bibcode:1995PhRvA.52.3489C. Quantum cryptography could potentially fulfill some of the functions of publickey cryptography. (19 June 2000). S2CID 124545445. As a direct consequence of this belief, it is also suspected that BQP is disjoint from the class of NP-complete problems (if an NP-complete problem were in BQP, then it would follow from NP-hardness that all problems in NP are in BQP).[112] The relationship of BQP to the basic classical complexityclasses can be summarized as follows: P B P P B Q P P P P S P A C E {\displaystyle {\mathsf {P\subseteq BPP\subseteq BQP\subseteq PP\subseteq PSPACE}}} It is also known that BQP is contained in the complexity class # P {\displaystyle \color {Blue}{\mathsf {\#P}}} (or more precisely in the associated class of decision problems P # P{\displaystyle {\mathsf {P {\#P}}}} ),[112] which is a subclass of PSPACE. doi:10.1038/s41567-018-0124-x. doi:10.1103/PhysRevA.51.992. 80 (15): 3408–3411. Kalai, Gil. Definition Main articles: Quantum circuit, Quantum logic gate, and Qubit See also: Quantum state, Density matrix, and Mathematical formulation of quantum mechanics Theprevailing model of quantum computation describes the computation in terms of a network of quantum logic gates.[19] This model can be thought of as an abstract linear-algebraic generalization of a classical circuit. "Solving the sampling problem of the Sycamore quantum supremacy circuits". p. I-5. Since this circuit model obeys quantummechanics, a quantum computer capable of efficiently running these circuits is believed to be physically realizable. V. Springer. Science. {{cite journal}}: Cite journal requires journal (help) Khazali, Mohammadsadegh; Mølmer, Klaus (11 June 2020). A vector representing all memory states thus has 2 n {\textstyle 2 {n}} entries (one for eachstate). PMC 4960311. However, any measurement can be deferred to the end of quantum computation, though this deferment may come at a computational cost, so most quantum circuits depict a network consisting only of quantum logic gates and no measurements. Bibcode:1995PhRvA.51.992U. The devices that perform quantum computations areknown as quantum computers.[1]: I-5 Though current quantum computers are too small to outperform usual (classical) computers for practical applications, they are believed to be capable of solving certain computational problems, such as integer factorization (which underlies RSA encryption), substantially faster than classical computers.[2] Thestudy of quantum computing is a subfield of quantum information science. Scaling these systems to support a growing number of qubits is an additional challenge.[52] Quantum decoherence Main article: Quantum decoherence One of the greatest challenges involved with constructing quantum computers is controlling or removing quantumdecoherence. The mathematics of single qubit gates can be extended to operate on multi-qubit quantum memories in two important ways. Bibcode:2020Natur.588.380B. "Quantum Computing in the NISQ era and beyond". ISSN 0036-8075. New York, NY, USA: Association for Computing Machinery: 1–12. "A cryogenic CMOS chip for generatingcontrol signals for multiple qubits". Dick, G. doi:10.1145/3458817.3487399. We may represent the state of this memory using Dirac notation so that 0 ⟩ : ( 1 0 ) ; 1 ⟩ : ( 0 1 ) {\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 anyquantum superposition ψ ⟩ {\textstyle \psi \rangle } of the two classical states 0 ⟩ {\textstyle 0\rangle } and 1 ⟩ {\textstyle 1\rangle } : ψ ⟩ : α 0 ⟩ β 1 ⟩ ( α β ) ; α 2 β 2 1. 29 Nielsen, Michael A.; Chuang, Isaac L. PMID 28905917. Ambainis, Andris (Spring 2014). S2CID 44098998. MIT Technology Review. 51 (2): 992–997.doi:10.22331/q-2018-08-06-79. PMID 15524694. Bibcode:2017Natur.549.195B. Fedichkin, L.; Yanchenko, M.; Valiev, K. A general class of problems to which Grover's algorithm can be applied[33] is Boolean satisfiability problem, where the database through which the algorithm iterates is that of all possible answers. Luryi; J. S2CID 119389613.interestingengineering.com. "What Can We Do with a Quantum Computer?". However, the use of error correction brings with it the cost of a greatly increased number of required qubits. ISSN 0028-0836. Any quantum computation (which is, in the above formalism, any unitary matrix over n {\displaystyle n} qubits) can be represented as a network ofquantum logic gates from a fairly small family of gates. 303: 117628. Archived from the original (PDF) on 15 November 2012. Archived from the original on 10 May 2013. Norton, Quinn (15 February 2007). (12 March 2004). "The prospects of quantum computing in computational molecular biology". Bibcode:1995Sci.270.255D. arXiv:1801.00862. McEliece, R. doi:10.1002/1521-3978(200009)48:9/113.0.CO;2-E. "The Quantum Imperative: Addressing the Legal Dimension of Quantum Computers". CiteSeerX 10.1.1.563.9990. "Quantum computing based hybrid deep learning for fault diagnosis in electrical power systems". S2CID 119628297. S.; Avchaciov, K.; Fedichev, P.doi:10.1038/nature07128. ISBN 1-4020-8067-0. Trainer, A. 588 (7838): 380. New Jersey: World Scientific. Bibcode:1978DSNPR.44.114M. Bibcode:1996Sci.273.1073L. A choice of gate family that enables this construction is known as a universal gate set, since a computer that can run such circuits is a universal quantum computer. Feynman,Richard (1982). doi:10.1038/nature23474. Physical Review Letters. doi:10.1126/science.273.5278.1073. Bibcode:1982IJTP.21.467F. Kalra, A. S2CID 9816153. ISSN 0362-4331. (2009). (2003). Journal of Physics: Condensed Matter. {\displaystyle \operatorname {CNOT} : \end{pmatrix}}.} As a mathematical consequence of this definition, CNOT 00 ⟩ 00 ⟩ {\textstyle \operatorname {CNOT} 00\rangle 00\rangle } , CNOT 01 ⟩ 01 ⟩ {\textstyle \operatorname {CNOT} 01\rangle 01\rangle } , CNOT 10 ⟩ 11 ⟩ {\textstyle \operatorname{CNOT} 10\rangle 11\rangle } , and CNOT 11 ⟩ 10 ⟩ {\textstyle \operatorname {CNOT} 11\rangle 10\rangle } . K. Another way is to apply the gate to its target only if another part of the memory is in a desired state. PMID 11298441. arXiv:1611.07690. "Quantum machine learning". Rodrigo, Chris Mills (12 February 2020). Because theseproblems are BQP-complete, an equally fast classical algorithm for them would imply that no quantum algorithm gives a super-polynomial speedup, which is believed to be unlikely.[22] Some quantum algorithms, like Grover's algorithm and amplitude amplification, give polynomial speedups over corresponding classical algorithms.[20] Though thesealgorithms give comparably modest quadratic speedup, they are widely applicable and thus give speedups for a wide range of problems.[23] Many examples of provable quantum speedups for query problems are related to Grover's algorithm, including Brassard, Høyer, and Tapp's algorithm for finding collisions in two-to-one functions,[24] which usesGrover's algorithm, and Farhi, Goldstone, and Gutmann's algorithm for evaluating NAND trees,[25] which is a variant of the search problem. doi:10.1103/PhysRevLett.93.130503. Zaslavsky (eds.). A system is placed in the ground state for a simple Hamiltonian, which is slowly evolved to a more complicated Hamiltonian whose ground state representsthe solution to the problem in question. Principles of Quantum Computation and Information Volume 1. Bibcode:2019Natur.574.461G. Physical Review X. However, it is known that P B Q P P S P A C E {\displaystyle {\mathsf {P\subseteq BQP\subseteq PSPACE}}} ; that is, all problems that can be efficiently solved by a deterministic classicalcomputer can also be efficiently solved by a quantum computer, and all problems that can be efficiently solved by a quantum computer can also be solved by a deterministic classical computer with polynomial space resources. Quantum algorithms can be roughly categorized by the type of speedup achieved over corresponding classical algorithms.[20]Quantum algorithms that offer more than a polynomial speedup over the best known classical algorithm include Shor's algorithm for factoring and the related quantum algorithms for computing discrete logarithms, solving Pell's equation, and more generally solving the hidden subgroup problem for abelian finite groups.[20] These algorithms dependon the primitive of the quantum Fourier transform. Jones, Nicola (19 June 2013). arXiv:0707.1889. Problems that can be addressed with Grover's algorithm have the following properties:[citation needed] There is no searchable structure in the collection of possible answers, The number of possible answers to check is the same as the number ofinputs to the algorithm, and There exists a boolean function that 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 scales as the square root of the number of inputs (or elements in the database), as opposed to the linear scalingof classical algorithms. 22 October 2019. Imamog lu, A.; Awschalom, D. For instance, integer factorization and the discrete logarithm problem are known to be in BQP and are suspected to be outside of P. Quantum computers therefore require error correction.[16][17] Any computational problem that can be solved by a classical computer can alsobe solved by a quantum computer.[18] Conversely, any problem that can be solved by a quantum computer can also be solved by a classical computer, at least in principle given enough time. (2008). Biamonte, Jacob; Wittek, Peter; Pancotti, Nicola; Rebentrost, Patrick; Wiebe, Nathan; Lloyd, Seth (September 2017). 574 (7776): 22–24. ScientificAmerican. Bibcode:2000quant.ph.6097F. Davies, Paul. 409 (6816): 46–52. arXiv:2003.00264. "Quantum search algorithms". Rich, Steven; Gellman, Barton (1 February 2014). These two choices can be illustrated using another example. Computation time is about L2 or about 107 steps and at 1 MHz, about 10 seconds.Bibcode:2016PhRvA.94b2308B. "Lunch & Learn: Quantum Computing". Moini, Y. One important gate for both classical and quantum computation is the NOT gate, which can be represented by a matrix X : ( 0 1 1 0 ) . doi:10.1103/PhysRevA.94.022308. S2CID 4410355. PMID 27426851. (26 August 2021). "Quantum computing in molecularmagnets". OCLC 1081001288. 24 April 2020. Lloyd, S. doi:10.1016/S0030-4018(01)01666-2. Retrieved 7 December 2020. Bibcode:1999PhRvL.83.4204I. Mitchell, Ian (1998). ISBN 978-0-309-47969-1. PMID 33273711. "A Public-Key Cryptosystem Based On Algebraic Coding Theory" (PDF). Igeta, K.; Yamamoto, Y. doi:10.1038/35071024. 2 (6):449–472. Retrieved 28 February 2019. "Quantum supremacy using a programmable superconducting processor". Encyclopædia Britannica. Decoherence is irreversible, as it is effectively non-unitary, and is usually something that should be highly controlled, if not avoided. One way is simply to select a qubit and apply that gate to the target qubitwhilst leaving the remainder of the memory unaffected. However, other sources of decoherence also exist. (18 June 2008). ISBN 978-981-238-830-8. 35 (2): 22–35. {{cite journal}}: Cite uses deprecated parameter lay-url (help) Neumann, P.; et al. "Quantum computational advantage using photons". Bibcode:2001Natur.410.789L. Boixo, Sergio;Isakov, Sergei V.; Smelyanskiy, Vadim N.; Babbush, Ryan; Ding, Nan; Jiang, Zhang; Bremner, Michael J.; Martinis, John M.; Neven, Hartmut (2018). K.; Kröll, S. arXiv:2008.05177 [quant-ph]. "Strengths and Weaknesses of Quantum Computing". Other problems, including the simulation of quantum physical processes from chemistry and solid-statephysics, the approximation of certain Jones polynomials, and the quantum algorithm for linear systems of equations have quantum algorithms appearing to give super-polynomial speedups and are BQP-complete. Bulletin of the American Mathematical Society. Journal of Statistical Physics. Jaeger, Gregg (2006). Ignacio; Lukin, Mikhail D.arXiv:2101.08354 [quant-ph]. "Google and NASA Achieve Quantum Supremacy". Institute for Advanced Study. Berlin: Springer. Bernstein and Tanja Lange on cryptography not known to be broken by quantum computing. (August 2005). Quantum computers are naturally good for solving complex quantum many-body problems [45] and thus may beinstrumental in applications involving quantum chemistry. doi:10.1103/PhysRevLett.80.3408. Quantum algorithms Main article: Quantum algorithm Progress in finding quantum algorithms typically focuses on this quantum circuit model, though exceptions like the quantum adiabatic algorithm exist. (23 September 2004). Náfrádi, Bálint; Choucair,Mohammad; Dinse, Klaus-Peter; Forró, László (18 July 2016). 65 (3): 032322. Li, Junde; Topaloglu, Rasit; Ghosh, Swaroop (9 January 2021). 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 andcomparing them to observation] let me humbly propose the following: can you define Quantum Gravity Polynomial-Time? G.; Childress, L.; Jiang, L.; Togan, E.; Maze, J.; Jelezko, F.; Zibrov, A. Franklin, Diana; Chong, Frederic T. Jamie; Yeh, Ping; Zalcman, Adam; Neven, Hartmut; Martinis, John M. Symmetric ciphers such as Triple DES and AES areparticularly vulnerable to this kind of attack.[citation needed] This application of quantum computing is a major interest of government agencies.[34] Simulation of quantum systems Main article: Quantum simulator Since chemistry and nanotechnology rely on understanding quantum systems, and such systems are impossible to simulate in anefficient manner classically, many believe quantum simulation will be one of the most important applications of quantum computing.[35] Quantum simulation could also be used to simulate the behavior of atoms and particles at unusual conditions such as the reactions inside a collider.[36] Quantum simulations might be used to predict future paths ofparticles and protons under superposition in the double-slit experiment.[citation needed] About 2% of the annual global energy output is used for nitrogen fixation to produce ammonia for the Haber process in the agricultural fertilizer industry while naturally occurring organisms also produce ammonia. 320 (5881): 1326–1329. Cuffaro. arXiv:quantph/0011051. I.; Boev, A. S2CID 4362012. Bibcode:2019NatCo.10.5607I. PMID 17540898. arXiv:2001.09190. arXiv:quant-ph/0504012. hdl:11380/303796. arXiv:1910.09534 [quant-ph]. arXiv:quant-ph/0101025. doi:10.1126/science.abe8770. doi:10.1103/PhysRevLett.84.5912. S2CID 41374015. A memory consisting of n {\textstyle n} bits of informationhas 2 n {\textstyle 2 {n}} possible states. 48 (9–11): 771–783. 1 (1): 52–59. 15 (2): 125–133. 2005. [ ] 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) "D-Wave Systems sells its first Quantum Computing System toLockheed Martin Corporation". 4 (4): 64–70. (20 January 2021). Ivády, Viktor; Davidsson, Joel; Delegan, Nazar; Falk, Abram L.; Klimov, Paul V.; Whiteley, Samuel J.; Hruszkewycz, Stephan O.; Holt, Martin V.; Heremans, F. 19 (2/3): 101–128. ISBN 978-0-521-87658-2. Efforts towards building a physical quantum computer focus on technologies suchas transmons, ion traps and topological quantum computers, which aim to create high-quality qubits.[1]: 2–13 These qubits may be designed differently, depending on the full quantum computer's computing model, whether quantum logic gates, quantum annealing, or adiabatic quantum computation. S2CID 30735516. See also pqcrypto.org, abibliography maintained by Daniel J. Quantum mechanical computers with single atom and photon fields. "Fullerene-based electron-spin quantum computer". For instance, it has been shown that a non-local hidden variable quantum computer based on Bohmian Mechanics could implement a search of an N-item database in at most O ( N 3 ){\displaystyle O({\sqrt[{3}]{N}})} steps, a slight speedup over Grover's algorithm, which runs in O ( N ) {\displaystyle O({\sqrt {N}})} steps. "Quantum Register Based on Individual Electronic and Nuclear Spin Qubits in Diamond". Retrieved 9 November 2021. 44: 114–116. David (2007). 84 (25): 5912–5915. C. "Closing the "quantum supremacy"gap: achieving real-time simulation of a random quantum circuit using a new Sunway supercomputer". S2CID 14255125. World Scientific Publishing Co. ISBN 978-981-4566-74-2. J.; Sellars, M. ISBN 978-3-540-33132-2. As a class of probabilistic problems, BQP is the quantum counterpart to BPP ("bounded error, probabilistic, polynomial time"), theclass of problems that can be solved by polynomial-time probabilistic Turing machines with bounded error.[110] It is known that B P P B Q P {\displaystyle {\mathsf {BPP\subseteq BQP}}} and is widely suspected that B Q P B P P {\displaystyle {\mathsf {BQP\subsetneq BPP}}} , which intuitively would mean that quantum computers are morepowerful than classical computers in terms of time complexity.[111] The suspected relationship of BQP to several classical complexity classes.[22] The exact relationship of BQP to P, NP, and PSPACE is not known. 26 (5): 1411–1473. Quantum simulations might be used to understand this process increasing production.[37] Quantum annealing andadiabatic optimization Quantum annealing or Adiabatic quantum computation relies on the adiabatic theorem to undertake calculations. This means that while quantum computers provide no additional advantages over classical computers in terms of computability, quantum algorithms for certain problems have significantly lower time complexitiesthan corresponding known classical algorithms. "Universal Quantum Simulators". arXiv:quant-ph/0403090. Bibcode:2005quant.ph.4012A. arXiv:2110.14502. "Quantum Complexity Theory". Das, R. Abbot, Derek; Doering, Charles R.; Caves, Carlton M.; Lidar, Daniel M.; Brandt, Howard E.; Hamilton, Alexander R.; Ferry, David K.; Gea-Banacloche,Julio; Bezrukov, Sergey M.; Kish, Laszlo B. "Statistical Aspects of the Quantum Supremacy Demonstration". (6 June 2008). 42 Nielsen, p. doi:10.1137/s0097539796300933. doi:10.1038/d41586-019-03213-z. "Light-based Quantum Computer Exceeds Fastest Classical Supercomputers". Retrieved 4 March 2013. Macquarie University. 316 (5829):1312–1316. Retrieved 9 February 2021. A. "Simulating physics with computers". S2CID 122596827. Katwala, Amit (5 March 2020). Physical Review A. Jeutner, Valentin (2021). Harneit, Wolfgang (27 February 2002). Dyakonov, M. "Quantum Annealing and Analog Quantum Computation". hdl:2027.42/45526. arXiv:0708.2073. This usuallymeans isolating the system from its environment as interactions with the external world cause the system to decohere. doi:10.1007/bf01011339. (23 August 1996). Quantum Computers. Retrieved 4 December 2021. Dyakonov, Mikhail (24 March 2020). Proceedings of the International Conference for High Performance Computing, Networking,Storage and Analysis. S. Mermin, N. Retrieved 22 May 2020.[page needed] Clarke, John; Wilhelm, Frank K. International Quantum Electronics Conference. doi:10.1007/BF02650179. 25 May 2011. (2010). Kaminsky, William M.; Lloyd, Seth; Orlando, Terry P. a b Preskill, John (6 August 2018). Decoherence times for candidate systems inparticular, the transverse relaxation time T2 (for NMR and MRI technology, also called the dephasing time), typically range between nanoseconds and seconds at low temperature.[53] Currently, some quantum computers require their qubits to be cooled to 20 millikelvin (usually using a dilution refrigerator[54]) in order to prevent significantdecoherence.[55] A 2020 study argues that ionizing radiation such as cosmic rays can nevertheless cause certain systems to decohere within milliseconds.[56] As a result, time-consuming tasks may render some quantum algorithms inoperable, as maintaining the state

Factoring polynomials maze answer key . 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. PMID 18535240. 10 (1