
#REDIRECT Quantum computer
This should be moved to Quantum computer if there's anything new here.
[[Image:NMR-implementation-alanine.png|thumb|Molecule of alanine used in Nuclear magnetic resonance implementation of error correction. Qubits are implemented by spin (physics) state (physics)s of carbon atoms.]] A quantum computer is any device for computation that makes direct use of distinctively quantum mechanical phenomena, such as quantum superposition and quantum entanglement, to perform operations on data. In a classical (or conventional) computer, data are measured by bits; in a quantum computer the data are measured by qubits. The basic principle of quantum computation is that the quantum properties of particles can be used to represent and structure data, and that devised quantum mechanisms can be used to perform operations with these data. For a generally accessible overview of quantum computing, see [http://www.media.mit.edu/physics/publications/papers/98.06.sciam/0698gershenfeld.html Quantum Computing with Molecules], an article in Scientific American by Neil Gershenfeld and Isaac L. Chuang. Experiments have already been carried out in which quantum computational operations were executed on a very small number of qubits. Research in both theoretical and practical areas continues at a frantic pace; see [http://qist.lanl.gov Quantum Information Science and Technology Roadmap] for a sense of where the research is heading. Many national government and military funding agencies support quantum computing research, to develop quantum computers for both civilian and national security purposes, such as cryptanalysis. See the Nature (journal) article in the references below reporting on work at IBM Almaden Research Center, where scientists implemented a seven qubit computing device that ran Shor's algorithm. It is widely believed that if large-scale quantum computers can be built, they will be able to solve certain problems faster than any classical computer. Quantum computers are different from classical computers such as DNA computers and computers based on transistors, even though these may ultimately use some kind of quantum mechanical effect (for example covalent bonds). Some computing architectures such as optical computers may use classical superposition of electromagnetic waves, but without some specifically quantum mechanical resource such as quantum entanglement, they do not share the potential for computational speed-up of quantum computers. ==The basis of quantum computing== In quantum mechanics, the state of a physical system (such as an electron or a photon) is described by an element of a mathematical object called a Hilbert space. The realization of the Hilbert space depends on the particular system. For instance in the case of a single particle system, the state can be described by a complex-valued function defined on R3 (three-dimensional space) called a wave function. As described in the article on quantum mechanics, this function has a probabilistic interpretation; of particular significance is that quantum states have a property called quantum superposition. A similar realization of the Hilbert space exists for systems of interacting particles. The time evolution of the system state is given by a family {''U''''t''} (with ''t'' denoting time) of unitary transformations of ''H''. Thus if φ is the state at time 0, then ''U''''t'' φ is the state at time ''t''. Note that this is true only if the system is isolated and the phenomenon of decoherence does not occur. A classical computer has a memory made up of bits, where each bit holds either a one or a zero. The device computes by manipulating those bits, i.e. by transporting these bits from memory to (possibly a suite of) logic gates and back. A quantum computer maintains a set of qubits. A qubit can hold a one, or a zero, or a superposition of these. A quantum computer operates by manipulating those qubits, i.e. by transporting these bits from memory to (possibly a suite of) quantum gates and back. Qubits for a quantum computer can be implemented using particles with two spin (physics) states: "up" and "down"; in fact any system, possessing an observable quantity ''A'' which is ''conserved'' under time evolution and such that ''A'' has at least two discrete and sufficiently spaced consecutive eigenvalues, is a suitable candidate for implementing a qubit. For discussion of foundational aspects of quantum computing, see the article on quantum circuits. ==Bits vs qubits == Consider first a classical computer that operates on a 3 bit Processor register. At a given time, the state of the register is determined by a single string of 3 bits, such as "101". This is usually expressed by saying that the register contains a single string of 3 bits. For a quantum computer on the other hand with a 3 qubit register, the state of the register is determined by 8 complex numbers as is illustrated in the following example:
For an ''n'' qubit quantum register, this table would have had 2''n'' rows; for ''n''=300, this is roughly 1090, more rows than there are atoms in the known universe. Note that these values are not all independent, since the probability constraint must be met. One can show the dimension of the set of states of an ''n'' qubit register is
2''n''+1 − 2. See Bloch sphere.
The first column shows all possible states for three bits. Whereas a classical computer can hold only one such pattern at a time, a quantum computer can be in a superposition state of all 8 patterns. The second column shows the "amplitude" for each of the 8 states. These 8 complex numbers are a snapshot of the registers at a given time. In this sense, a 3-qubit quantum computer has far more memory than a 3-bit classical computer.
The third column of the table gives the probability for each possible register configuration. In this example, there is a 14% chance that the returned string will be "000", a 4% chance it will be "001", and so on. Each complex number (a+bi) is called an ''amplitude'', and each probability (a2+b2) is called the ''squared amplitude'', because it equals |a+bi|2. The 8 probabilities add up to 1.
== Initialization, execution and termination ==
In our example, the contents of the qubit registers can be thought of as an 8-dimensional complex vector. An algorithm for a quantum computer must initialize this vector in some specified form (dependent on the design of the quantum computer). On each step of the algorithm, that vector is modified by multiplying it by a unitary matrix_(mathematics). The matrix is determined by the physics of the device. The unitary character of the matrix ensures the matrix is invertible (so each step is reversible).
Upon termination of the algorithm, the 8-dimensional complex vector stored in the register must be somehow read off from the qubit register by a quantum measurement. However, by the laws of quantum mechanics, that measurement will yield a random 3 bit string (and it will destroy the stored state as well). This random string can be used in computing the value of a function because (by design) the probability distribution of the measured output bitstring is skewed in favor of the correct value of the function. By repeated runs of the quantum computer and measurement of the output, the correct value can be determined, to a high probability, by majority polling of the outputs. See quantum circuit for a more precise formulation. In brief, quantum computations are probabilistic.
Let us consider the case of the Nuclear magnetic resonance implementation of a quantum circuit. NMR differs from other implementations on quantum computers in that it uses an ensemble of systems, in this case molecules. The ensemble is initialized to be the thermal equilibrium state (see quantum statistical mechanics). In mathematical parlance, this state is given by the density matrix:
:
where ''H'' is the hamiltonian matrix of an individual molecule and
:
The unitary operations are performed by shooting a short pulse of radiation at the container of molecules. Different types of pulses result in different unitary matrices. The quantum algorithm is implemented by an appropriate sequence of pulses. Note that for a given algorithm, the operations will always be done in exactly the same order. There is no "IF THEN" statement to vary the order, since there is no way to read the state of a molecule before the final measurement.
For more details on the sequences of operations used for various algorithms, see universal quantum computer, Shor's algorithm, Grover's algorithm, Deutsch-Jozsa algorithm, quantum Fourier transform, quantum gate, quantum adiabatic algorithm and quantum error correction.
==The power of quantum computers==
Integer factorization is believed to be practically impossible with an ordinary computer for large numbers that are the product of two prime numbers of roughly equal size (eg. products of two 300-digit primes). By comparison, a quantum computer could solve this problem very quickly. If a number has ''n'' bits (is ''n'' digits long when written in the binary numeral system), then a quantum computer with just over 2''n'' qubits can find its factors. It can also solve a related problem called the discrete log problem. This ability would allow a quantum computer to break many of the cryptography systems in use today. In particular, most of the popular Asymmetric Algorithms could be quickly broken, including forms of RSA, ElGamal and Diffie-Hellman. These are used to protect secure Web pages, encrypted email, and many other types of data. Breaking these would be significant. The only way to make an algorithm like RSA secure would be to make the key size larger than the largest quantum computer that can be built. It seems likely that it will always be possible to build classical computers that have more bits than the number of qubits in the largest quantum computer. If that's true, then algorithms like RSA could be made secure.
Perhaps not as surprisingly, quantum computers could also be useful for running simulations of quantum mechanics. The speedup could be just as large as for factoring. This could be a great boon to physics, chemistry, materials science, nanotechnology, biology and medicine, all of which are limited today by the slow speed of quantum mechanical simulations.
This dramatic advantage of quantum computers is currently known to exist for only those three problems: factoring, discrete log, and quantum physics simulations. However, there is no proof that the advantage is real: an equally fast classical algorithm may still be discovered (though this is considered unlikely). There is one other problem where quantum computers have a smaller, though significant (quadratic) advantage. It is ''quantum database search'', and can be solved by Grover's algorithm. In this case the advantage is provable. This establishes beyond doubt that (ideal) quantum computers are superior to classical computers.
Consider a problem that has these four properties:
* The only way to solve it is to guess answers repeatedly and check them
* There are ''n'' possible answers to check
* Every possible answer takes the same amount of time to check
* There are no clues about which answers might be better. Generating possibilities randomly is just as good as checking them in some special order
An example of this is a password search for an encrypted file.
For problems with all four properties, it will take an average of (''n''+1)/2 guesses to find the answer using a classical computer. The time for a quantum computer to solve this will be proportional to the square root of ''n''. That can be a very large speedup, reducing some problems from years to seconds. It can be used to attack Symmetric Algorithms such as 3DES and AES. But it is also easy to defend against. Just double the size of the key for the cipher. There are also more complicated methods for secure communication, such as using quantum cryptography.
There are currently no other practical problems known where quantum computers give a large speedup over classical computers. Research is continuing, and more problems may yet be found.
==Problems with quantum computing==
One of the major obstacles of quantum computing is the problem of Quantum decoherence, which causes the unitary character (and more specifically, the invertibility) of quantum computational steps to be violated. Decoherence times for candidate systems, in particular the transverse relaxation time T2 (terminology used in Nuclear magnetic resonance and MRI technology), typically range between nanoseconds and seconds at low temperature. Error rates are typically proportional to the ratio of operating time to decoherence time, hence any operation must be completed much quicker than the decoherence time. If the error rate is small enough, it is possible to use quantum error correction, which corrects errors due to decoherence thereby allowing total calculation times longer than the decoherence time. An often cited (but rather arbitrary) figure for required error rate in each gate is 10−4. This implies that each gate must be able to perform its task 10,000 times faster than the decoherence time of the system.
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''4 and ''L''6, where ''L'' is the number of bits in the number to be factored. For a 1000 bit number, this implies a need for 1012 to 1018 qubits. Fabrication and control of this large number of qubits is non-trivial for any of the proposed designs.
==Practical quantum computers==
David DiVincenzo, of IBM, listed the following requirements for a practical quantum computer:
* scalable physically to increase the number of qubits
* qubits can be initialized to arbitrary values
* quantum gates faster than decoherence time
* Turing-complete gate set
* qubits can be read easily
There are a number of practical difficulties in building a quantum computer, and thus far quantum computers have only solved trivial problems. One major problem is keeping the components of the computer in a coherent state as the slightest interaction with the external world would cause the system to quantum decoherence.
===Candidates===
There are a number of quantum computing candidates, among those:
#Superconductor-based quantum computers (including SQUID-based quantum computers)
#Trapped ion quantum computers
#Ion trap-based quantum computers
#"Nuclear magnetic resonance on molecules in solution"-based
#"quantum dot on surface"-based
# "Cavity quantum electrodynamics" (CQED)-based
#molecular magnet-based
#fullerene-based ESR quantum computer
#the solid state NMR Kane quantum computer
==Quantum computing in computational complexity theory==
This section surveys what is currently known mathematically about the power of quantum computers. It describes the known results from computational complexity theory and the theory of computation dealing with quantum computers.
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 randomized algorithms, so BQP on quantum computers is the counterpart of BPP on classical computers. It is defined as the set of problems solvable with a polynomial-time algorithm, whose probability of error is bounded away from one half. A quantum computer is said to "solve" a problem if, for every instance, its answer will be right with high probability. If that solution runs in polynomial time, then that problem is in BQP.
BQP is suspected to be disjoint from NP-complete and a strict superset of complexity classes P and NP, but that is not known. Both integer factorization and discrete logarithm problem 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.
An operator for a quantum computer can be thought of as changing a vector by multiplying it with a particular matrix. Multiplication by a matrix is a linear transformation operation. It has been shown that if a quantum computer could be designed with ''nonlinear'' operators, then it could solve NP-complete problems in polynomial time. It could even do so for Sharp-P-complete problems. It is not yet known whether such a machine is possible.
Although quantum computers are sometimes faster than classical computers, ones of the types described above can't solve any problems that classical computers can't solve, given enough time and memory (albeit possibly an amount that could never practically be brought to bear). 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.
Very recently, some researchers have begun to investigate the possibility of using quantum mechanics for hypercomputation - that is, solving undecidable problems. Such claims have been met with very considerable skepticism as to whether it is even theoretically possible; see the hypercomputation article for more details.
== See also ==
* Timeline of quantum computing
* Rapid single flux quantum
* Reversible computing
* Quantum algorithms
** Shor's algorithm
** Grover's algorithm
** Deutsch-Jozsa algorithm
* D-wave system
== Further information ==
* Quantum computing is part of quantum information processing.
* Companies developing quantum computers
**[http://www.dwavesys.com D-Wave Systems, Vancouver, BC, Canada]
**[http://www.research.ibm.com/quantuminfo IBM]
* Quantum computing related patents
**[http://patft.uspto.gov/netacgi/nph-Parser?Sect1=PTO2&Sect2=HITOFF&u=%2Fnetahtml%2Fsearch-adv.htm&r=0&p=1&f=S&l=50&Query=%22quantum+computing%22+OR+%22quantum+gate%22+OR+%22quantum+computer%22&d=ptxt Some issued quantum computing-related patents]
**[http://appft1.uspto.gov/netacgi/nph-Parser?Sect1=PTO2&Sect2=HITOFF&u=%2Fnetahtml%2FPTO%2Fsearch-adv.html&r=0&p=1&f=S&l=50&Query=%22quantum+computer%22+OR+%22quantum+gate%22+OR+%22quantum+information%22+OR+%22quantum+computing%22&d=PG01 Some published quantum computing-related patents]
* Good general reference:
**[http://www.arxiv.org/ arXiv.org e-Print archive] - a very complete online archive of research papers.
**Centre for Quantum Computation, University of Cambridge [http://cam.qubit.org http://cam.qubit.org]
**[http://www.iqc.ca Institute for Quantum Computing], University of Waterloo
**[http://tph.tuwien.ac.at/~oemer/qcl.html QCL — A Programming Language for Quantum Computers]
* Thermal ensembles
**[http://www.media.mit.edu/physics/projects/spins/home.html Overview] of early developments, with links
**The first two papers ever written on this topic:
*** and
*** ([http://www.media.mit.edu/physics/publications/papers/97.01.science.pdf download])
* Using quantum computers to simulate quantum systems:
**
* Quantum cryptography:
** The first paper ever written on this:
***
***
***
**The first paper ever published on this:
**A listing of a huge number of quantum cryptography papers, with some discussion of them, is at [http://www.cs.mcgill.ca/~crepeau/CRYPTO/Biblio-QC.html http://www.cs.mcgill.ca/~crepeau/CRYPTO/Biblio-QC.html]
* Universal quantum computer and the Church-Turing thesis:
**
* '''Shor's factoring algorithm:'''
** ([http://www.arxiv.org/abs/quant-ph/9508027 copy on Arxiv.org]).
**Jean-Pierre Seifert, "Using fewer Qubits in Shor's Factorization Algorithm via Simultaneous Diophantine Approximation", ([http://eprint.iacr.org/2000/036/ download])
**[http://domino.research.ibm.com/comm/pr.nsf/pages/news.20011219_quantum.html IBM's announcement] of the first actual execution of the algorithm, which also gives the history of the first quantum computers with 2, 3, 5, and 7 qubits. Published in the December 19, 2001 issue of Nature (journal).
* Quantum database search:
** ([http://www.arxiv.org/abs/quant-ph/9605043 copy on Arxiv.org])
* Quantum sorting:
**
** (http://arxiv.org/abs/quant-ph/0102078)
** (http://portal.acm.org/citation.cfm?doid=780542.780553)
* Quantum computer simulators:
**[http://www.enyo.de/libquantum/ libquantum] — A library for quantum computer simulation
**[http://tph.tuwien.ac.at/~oemer/qcl.html QCL] — Simulation of quantum computing with a quantum computing language
**[http://search.cpan.org/search?dist=Quantum-Entanglement Quantum::Entanglement] — Quantum computation module for Perl.
**[http://www.charlesfox.org.uk QCF] — Quantum computation toolbox for MATLAB.
**[http://www.qc.fraunhofer.de Fraunhofer Quantum Computing Simulator] — A free web-based quantum simulator (31 qubits) and a collaborative workspace for the quantum computing community.
* Quantum error correction:
**
** ([http://www.arxiv.org/abs/quant-ph/9512032 copy on Arxiv.org])
** ([http://www.arxiv.org/abs/quant-ph/9605011 copy on Arxiv.org]).
* Quantum error avoidance:
** ([http://arxiv.org/abs/quant-ph/9807004 copy on Arxiv.org])
**D. A. Lidar, K.B. Whaley, "Decoherence-Free Subspaces and Subsystems", in "Irreversible Quantum Dynamics", F. Benatti and R. Floreanini (Eds.), pp. 83-120 (Springer Lecture Notes in Physics vol. 622, Berlin, 2003), [http://arxiv.org/abs/quant-ph/0301032 copy on Arxiv.org].
* Solving NP-complete and #P-complete problems:
** Daniel S. Abrams (1), Seth Lloyd (2) ( (1) Dept. of Physics, MIT, (2) Dept. of Mechanical Engineering, MIT), 1998, "Nonlinear quantum mechanics implies polynomial-time solution for NP-complete and #P problems" ([http://arxiv.org/abs/quant-ph/9801041 copy on Arxiv.org]).
** Phil Gossett, 1998, "NP in BQP with Nonlinearity", ([http://arxiv.org/abs/quant-ph/9804025 copy on Arxiv.org]).
** ([http://www.arxiv.org/abs/quant-ph/9910073 copy on Arxiv.org] or [http://www.worldscinet.com/journals/ijmpb/15/1522/S0217979201007154.html here]).
*For the interested non-expert:
**
** West, J. (2000). [http://www.cs.caltech.edu/~westside/quantum-intro.html The Quantum Computer — An Introduction]. (Easy to understand explanation of quantum computing)
** Hayes, Brian (Jul-Aug 1995). [http://www.americanscientist.org/template/AssetDetail/assetid/24592 The square root of NOT]. ''American Scientist Online''. (Logic gates in a quantum computer)
**
*Similarly named miscellany:
** ''For the online service provider that was previously known as Quantum Computer Services, see America Online.''
Computer science
Cryptography
Information theory
Quantum information science
Quantum mechanics
lv:Kvantu dators
---- "For problems with all four properties, it will take an average of n/2 guesses to find the answer using a classical computer. The time for a quantum computer to solve this will be proportional to the square root of n". Shouldn't that be log(n)? If not, a little explanation on this rather unexpected outcome would be appreciated... Apart from that it's a great piece of work! --MK ---- This quote in the article is nonsense: "Integer factorization is believed to be practically impossible with an ordinary computer for large numbers (eg. 10^600)." While that's a really big number, a high school freshman can factor that big number in his head! 2^600 x 5^600. Gee, that was tough! --User:Seinfeld's_Cat :Thanks for spotting this. I changed the article accordingly. Of course, you could also have changed the article yourself. -- User:Jitse Niesen 15:06, 30 Jan 2005 (UTC) ::No problem. It sometimes seems easier to complain than to just fix it. I like your change (I couldn't come up with a suitable fix; you did it quite well.) I make a better critic than author. -- User:Seinfeld's_Cat 1 Feb 2005 ---- The talk page is as interesting as the main article. Amazing! --User:ReallyNiceGuy ---- Great article! Maybe we could add in the first paragraph that quantum computers are by their very nature probabilistic devices and only probabilistic algorithms can be implemented on them. Also, quantum computers can be simulated by Turing machines and are therefore no attack to the :Church-Turing thesis. --AxelBoldt ---- I've now added both. They're in the complexity theory section, where they seemed to fit the flow best. -user:LC ---- Yes. In the complexity section, it says that BQP is the class of problems that can be solved by quantum computers. These however are only the problems that can be solved by quantum computers in polynomial time. Maybe you could say "the problems that can be solved by quantum computers in reasonable time" or "that can be realistically solved by quantum computers". Then it says that quantum computers probably cannot solve all NP-complete problems. There are two problems with this statement: strictly speaking, a quantum computer only works probabilistically and cannot "solve" any NP-complete problem (or any other decision problem for that matter) in the same sense a Turing machine solves them, with a deterministic and correct Yes/No answer. Furthermore, if we allow probabilistic solutions, then quantum computers can of course solve all NP-complete problems, just like any Turing machine can; it may just take a lot of time to do so... --AxelBoldt ---- It should be clearer now. -user:LC ---- What is this ''#P-Complete'' ? --User:Taw ---- See :Sharp-P-Complete. -user:LC ---- How long does it take to factor an n-digit number with n qubits? --Axel ---- Would Quantum Computing break Elliptic Curve Cryptography ? User:Taw :Apparently so, through a variation on Shor's algorithm. I haven't studied it, though -- User:CYD ---- Some recent work [http://www.eet.com/story/OEG20010924S0101] indicates that if spaced sufficiently closely, quntum entanglement between quantum dots may be possible, so it's possible that in the future a quantum computer could be implemented using quantum dots. Also, it might be useful to mention "reversibility", the haddamard(sp?) transformation and the various types of quantum logic gates (CNOT, etc...). I'm not an expert, so I'll defer to someone who knows about this stuff -- User:Matt Stoker ----- What does it mean to have a "quantum computer with nonlinear operators"? --:Robert Merkel I hope that's clearer now. --user:LC How the heck do you implement a nonlinear operation on qubits? Evolution always proceeds by unitary - therefore linear - operations. -- User:CYD I agree, that seems to be a consequence of Schrodinger's equation, which is pretty basic to QM. --AxelBoldt I agree. The papers proving the result never said it could be done. But it's an interesting result. It hasn't been ruled out that a large linear system could act like a small nonlinear system, and give the desired result. The Shi paper refers to the nonlinearity of the Gross-Pitaevskii equations, but I'm not familiar with them. I would assume the Shi paper is flawed, since it wasn't accepted anywhere, but I'm not aware of any proofs that this sort of thing is inherently impossible. --user:LC Oh. I just checked the site, and the Shi paper has now been accepted in a journal. It hadn't yet been accepted when I first wrote the article. I'll remove the "not peer reviewed" note. Is anyone here familiar with the "Internation Journal of Modern Physics"? Is it generally respectable? --user:LC ---- :''The quantum computer in the above example could be thought of as a black box containing 8 complex numbers. Or, it could be thought of as 8 black boxes, each containing 1 complex number, each sitting in a different alternate universe, and all communicating with each other. These two interpretations correspond to the Copenhagen interpretation and many worlds interpretation, respectively, of quantum mechanics. The choice of interpretation has no effect on the math, or on the behavior of the quantum computer. Either way, it's an 8-element vector that is modified by matrix multiplication'' I don't think this characterization of many worlds is correct. The different universes don't come with complex numbers attached. Instead, the more likely states are exhibited in ''more'' universes. The goal of the matrix manipulations is to bring essentially all universes into the same state. Once you measure the state, the universes stop to communicate and truly split. --AxelBoldt Good point. I've reworded it. --user:LC ----- In the "How they work" section it says: "The square of (a+ib) is (a^2+b^2)." Actually the square of (a+ib) would be (a^2-b^2) + i(2ab). Probably you mean the norm or the mod or something like that? (it's been a while since i used complex numbers, not sure of the name any more). In quantum mechanics, (a+bi) is called the ''amplitude'', and (a2+b2) is called the ''squared amplitude'', even though the latter equals |a+bi|2 rather than the (a+bi)2 that the name might suggest. I suppose the terminology is counterintuitive. I'll reword the article to make it clearer. --user:LC ---- I've removed the following contribution from User:Harry Potter: : ''Quantum computing can also theoretically produce time anomalies. The ability of the Quantum computer to find the correct answer in a factorisation problem can be seen as running all the possibilities simultaneously. However, only the correct answer produces a positive response which is then sent back through time to become the only calculation which in fact needs to be made.'' Quantum computation doesn't involve time travel in any description I've seen. -- User:Oliver Pereira 00:01 9 Jun 2003 (UTC) :Agreed. This sounds purely abstract and theoretical to me, but I am no expert on quantum computing by any means. Perhaps Harry Potter can produce a source in the quantum computing literature that lends some support to this idea? -- User:Wapcaplet 00:10 9 Jun 2003 (UTC) ----- Please specify "... have recently been built" into "... have been built in the early/mid-/late 20th century/21st century". --User:Menchi 08:37 14 Jul 2003 (UTC) ---- I found the first paragraph of the second section, titled "The Power of Quantum Computers", rather unclear. Unfortunately, I cannot tell exactly what makes it unclear - maybe it is too vague; how difficult is factorization (suspected to be) on a classical computer, and what is the speed-up on a quantum computer? I do not know enough about quantum computing to rewrite the paragraph myself ... I liked the rest of the article very much, it contains (almost) everything that I was looking for. However, I am still wondering whether there is an upper bound on the power of quantum computers. The article says that quantum computers cannot solve undecidable problems, but is BQP known/suspected to be a subset of another complexity class, say NP? -- User:Jitse Niesen 11:08, 26 Aug 2003 (UTC) == They CAN solve Turing-unsolvable problems, but... == This is a technicality that should be mentioned: If you define a quantum computer as a Turing Machine with arbitrary complex amplitudes, then the class of machines that you obtain is uncountably infinite, and can easily be shown [ADH97] to contain machines that solve all kinds of Turing-unsolvable problems. But *we* can't build those QC's (or even know one if we see one) because of our inability to know the correct amplitudes required for those tasks. The computability comments in the present version of this article are valid for QC's whose amplitudes are computable numbers. [ADH97] L. Adleman, J. DeMarrais, and M. Huang. Quantum computability, SIAM Journal on Computing 26:1524-1540, 1997. == Key exchange versus encryption == You say: "This ability would allow a quantum computer to break many of the cryptographic systems in use today. In particular, most of the popular public key ciphers could be quickly broken, including forms of RSA, ElGamal and Diffie-Hellman." My understanding is that RSA is the encyrption method and the ElGamal and Diffie-Hellman are key exchange protocols. The references within Wikipedia also say that. Do you mean that the encyrpted key being transmitted is able to be read? BK ---- == Further editing == Some of the section which was there before (and is now called Initialization and termination), though written at a good level I believe contains some innaccuracies, i.e. on how the qubit registers are measured. In fact, only a fixed register is sampled on termination.User:CSTAR 04:27, 6 Jun 2004 (UTC) Oops that's not what I should have said. Anyway I think it's right now. I also split the article up adding the stuff on reversible registers to quantum circuits.User:CSTAR 07:01, 6 Jun 2004 (UTC) == Where did the numbers come from in columns #2 and #3 in the "bits vs qubits" section? == Why is the 3-qubit "000" represented as "0.37 + i 0.04"? Where did the 0.37 and 0.04 come from? Ditto for the rest of the numbers in that table. According to www.qubit.org, a 3-qubit system can represent 8 values. There is no magic in 16 analog values vs 8 complex values, there are just 8 values total. None of the references I have come across mention anything other than an L qubit system representing exactly 2^L possibilities at once, not 2^(L+1) (the 16 you mentioned in this section). In an effort to give the reader some sense of how probability waves can be represented by complex numbers and how complex numbers c1an be manipulated, you've obfuscated the simpler facts of qubits. 1 qubit holds 2 states simultaneously with *equal probability*, 2 qubits hold 4 states simultaneously with *equal probability* and so on. ---- : I didn't write this example, but as an example, I think it's mostly OK (But you're right about the dimension being wrong: see below. When I read it I wasn't thinking) . What the example is trying to express is that fact that in a quantum computer the set of registers is a quantum mechanical object,and the state is described by a wave funbction on it. As far as the number of independent values, the example is meant to suggest the real dimension of the space of states, but you may be right the dimension counting may be wrong, because the relevant space is complex projective space of dimension 2''n''. Thanks. I'll fix it. The set of pure sets is homeomorphic to the compact symmetric space with m = 2''n'' :: : As far as your statement: ::1 qubit holds 2 states simultaneously with *equal probability*, 2 qubits hold 4 states simultaneously with *equal probability* and so on. :I don't believe this statement is correct. User:CSTAR 03:10, 16 Jun 2004 (UTC) P.S. From this log, it wasn't clear who wrote the question and who wrote the reponse, so I went back and edited the past, by separating my answer to the anonymous question. I'm perfectly happy to edit the past. More of us should try it.User:CSTAR 04:48, 16 Jun 2004 (UTC) == other confusing bits == (problem I noted has been fixed -- thanks, everyone. -- User:DavidCary 01:43, 4 Jul 2004 (UTC)) :::Yes, it's rubbish, I rewrote the section. Unfortunately the bit about fault tolerant scaling in Shor's algorithm is a bit hazy, I'm just quoting vague unpublished numbers floating around my research group. -- User:Tim Starling 07:35, Jun 18, 2004 (UTC) ::::Good rewrite! Regarding range for T2, I checked Nielsen & Chuang book (pp. 337) and they claim 7 seconds and 0.3 seconds for proton and carbon in a two-qubit experiment. Should we change "hundreds of microseconds" to "seconds"? Or is there some caveat about Nielsen & Chuang data that I'm not aware of? User:Andris 18:37, Jun 18, 2004 (UTC) :::::Alright. It would be nice to explain at some stage why systems such as chloroform nuclear spins are not scalable, but I guess "nanoseconds to seconds" will do for now. -- User:Tim Starling 01:47, Jun 22, 2004 (UTC) ---- The article currently mentions ''the transverse relaxation time T2 (terminology used in NMR and MRI technology)'' Is there really any difference between NMR quantum computers and MRI quantum computers ? I suspect they're really the same thing. -- User:DavidCary 01:43, 4 Jul 2004 (UTC) :I guess my comments weren't clear -- I was referring to MRI and NMR ''technology'' not computers (MRI technology, the kind used in hospitals) which are essentially the same technology (I think at least they are, since the both involve involve nuclear spin), but seem to have separate wiki articles. Is my making these two references too confusing? Please change it if you can make it clearer. User:CSTAR 01:51, 4 Jul 2004 (UTC) ==Interpretation== :: Now that we're at it, I think we should remove references to ''interprertation'' (Copenhagen, many worlds etc..). Regardless of content, they shouldn't be in this article. I haven't made an effort to understand whether what's writtem in the article makes any sense, so I won't pass judgment on that account_. User:CSTAR 13:19, 18 Jun 2004 (UTC) == Edits of 08/01/2004 == Some of the edits by an anonymous user did increase readability, but I think calling ''entanglement'' or ''superposition'' a process is misleading. Also Qubits don't hold data -- they are measures of the ''size'' of data. The register space is the actual container of data. I tried to keep the spirit of the recent edits, but modifying what seemed misleading. User:CSTAR 02:49, 2 Aug 2004 (UTC) --- So, how the hell are you supposed to make software for these? I assume a normal "if something is something do this, else do this" doesn't work? : Conditionals in the usual sense, no.User:CSTAR 20:08, 16 Aug 2004 (UTC) == Decoherence and Complexity == This article is really great ! Nevertheless, there is one point I found unclear : the relationship between decoherence and complexity. The article state some algortihms are in BQP. Since complexity classes are only meaningful for arbitrarily large calculations, does the definition of the BQP class includes the fact that decoherence may limits the amount of operations that may be performed, or does it assumes we have a way to prevent the effects of decoherence for arbitrarily long calculations (which doesn't seem obvious) ? For example, an old (1995) article by P. Shor and coll. [http://citeseer.ist.psu.edu/chuang95quantum.html "Quantum Computers, Factoring and Decoherence"], calculated a limit to the size of the integers that may be factored faster by a quantum computer than by a classic computer, which to me would contradict the fact that factoring is in BQP. Some clarification of this point would be great ! Thanks ! : That's a good question, but I'm inclined to say that the there is no relationship othre than that which is buit into the model itself. To explain rigorously BQP, you have to formulate a more precise model of quantum computation than is done in this article. The right model is that of quantum circuit, although BQP is not defined there either, although it should be. At some point I'll get around to it. Otherwise look at some of the references such as Kitaev, or Nielsen Chuang. The book by Hirvensalo is also good. I know I have evaded your question but that's the best I can do. Maybe some other wikipedian has a better answer.User:CSTAR 13:57, 1 Oct 2004 (UTC) :The definition of BQP assumes that computation is error-free (and decoherence-free), which is not the case in practice. The limit in Shor's 1995 article is for quantum computers which are subject to decoherence. Thus, there is no contradiction: the 1995 paper essentially says that if there was not decoherence, quantum computers could factor large numbers, but in presence of decoherence, they could not. This is the (pessimistic) view of quantum computation that was common in mid-1990s. This view is mostly obsolete now. Quantum error correction has been invented (in a later work of Shor and colleagues in 1996 and 1997) and it has been shown that arbitrarily large computations can be made fault-tolerant (with "fault" including "decoherence"), as long as fault rate is sufficiently small. User:Andris 00:10, Dec 11, 2004 (UTC) == Cost of quantum computations == The big selling point of quantum computations seems to be that quantum computers with enough qubits can run algorithms much faster than classicalcomputers. That's great, but it's not very useful if, for example, the cost of implementing an n-qubit register is exponential in n. Is there some estimate for the asymptotics of this? Is the apparent advantage of quantum computers due only to the fact that we're measuring computational complexity by the wrong yardstick? (I have no idea, but someone surely knows...) --User:Aarchiba 08:48, Dec 7, 2004 (UTC) ==NP-Complete stuff== "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 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." A teacher claimed that someone has found an quantum algorithm to solve NP-complete problems in polynomial time. Asking for a size estimate, he said that the algorithm description should approximately fit on a page. He's supposed to be teaching everyone the definition of NP-complete problems next week, so he should know what he's talking about. He seemed rather sure. I'd like to see that algorithm... (Haven't had time to search the internet yet.) Would sure make quantum computers (even) more interesting... User:Cyp User talk:Cyp 17:27, 9 Dec 2004 (UTC) :Regarding the previous quote: Are integer factorization and discrete log (suspected to be) outside BPP? I think this is more interesting than whether they are outside P. -- User:Jitse Niesen 19:05, 9 Dec 2004 (UTC) ::Yes, they are thought to be outside BPP. User:Andris 04:32, Dec 10, 2004 (UTC) :Thanks, I updated the article. -- User:Jitse Niesen 13:42, 10 Dec 2004 (UTC) ::Does [http://arxiv.org/PS_cache/quant-ph/pdf/9912/9912100.pdf this] article mean anything? I'm a bit sceptical, since it seems to be saying that because some things can be approximated by a non-linear function, that linearity of quantum mechanics can be ignored... User:Cyp User talk:Cyp 17:15, 16 Dec 2004 (UTC) == Clustered quantum computer ? == Hello, I don't understand a lot about the technics of quantum computers. I would like to know if it is possible (at least in theory) to distributed quantum computing. it is possible to dream about a “quantum grid”, “quantum Beowulf”, “quantum@home?” Thank you User:Vrykolaka 07:31, 25 Jan 2005 (UTC) == Can quantum computers solve undecidable problems? == This article says that quantum computers cannot solve any problems that classical computers cannot solve either. However, after a recent edit, the start of Matiyasevich's theorem reads: :"'''Matiyasevich's theorem''', proven in 1970 by Yuri Matiyasevich, implies that Hilbert's problems is unsolvable (although it has recently been proposed by Tien Kieu [Int.J.Theor.Phys. 42 (2003) 1461-1478] that this result only applies to a deterministic Turing machine, and that the problem could be solved on a quantum computer using a quantum adiabatic algorithm. This is still contraversial.)." How to reconcile both quotes? Does anybody know how contraversial Tien Kieu's work is? -- User:Jitse Niesen 13:36, 10 Feb 2005 (UTC) ::Probabilistic Turing machines or non-deterministic Turing machines don't decide more problems than plain old Turing machines. Neither does the quantum circuit model (and equivalents). Unfortunately, I don't know enough what a quantum adiabatic algorithm is but it possibly doesn't fall under the quantum circuit model of quantum computation. I suppose I should look at the paperUser:CSTAR 14:41, 10 Feb 2005 (UTC) ::: I just looked at one of his papers. It is highly suspect, making claims about probabilistic computation which under any normal interpretation are false. "''Probabilistic computation may be more powerful than Turing Computation provided a suitable probability measure can be defined.''" Now it is well-known, (see Sipster's textbook on computation) that probabilistic Turing machines compute the same functions as normal ones. ::: I would take these claims with lots of salt (and maybe that sentence in Matiyasevich's theorem should be removed).User:CSTAR 18:15, 10 Feb 2005 (UTC) Thank you; I came to the same conclusion. -- User:Jitse Niesen 21:47, 10 Feb 2005 (UTC) == Request for references == Hi, I am working to encourage implementation of the goals of the Wikipedia:Verifiability policy. Part of that is to make sure articles Wikipedia:Cite sources. This is particularly important for featured articles, since they are a prominent part of Wikipedia. Now this article has an extensive list of additional material, but list of further reading is not the same thing as proper references. Further reading could list works about the topic that were not ever consulted by the page authors. If some of the works listed in the that section were used to add or check material in the article, please list them in a references section instead. The Wikipedia:WikiProject Fact and Reference Check has more information. Thank you, and please [http://en.wikipedia.org/w/wiki.phtml?title=User_talk:Taxman&action=edit§ion=new leave me a message] when a few references have been added to the article. - User:Taxman 19:18, Apr 22, 2005 (UTC) == Factorization == "For problems with all four properties, it will take an average of n/2 guesses to find the answer using a classical computer. The time for a quantum computer to solve this will be proportional to the square root of n...But it is also easy to defend against. Just double the size of the key for the cipher." This doesn't sound right to me. If your key length is 2n, a classical computer would take n guesses. A quantum computer would then take time proportional to sqrt(2n), which is asymptotically the same as sqrt(n). I didn't edit the page because I'm not an expert on quantum computing, and feel it's possible that I don't understand (if, for example, you can't do asymptotic analysis on quantium computing algorithms?). As best as I can figure, though, I would expect you'd need to square the key length? Any problem with these four problems will take an average of (''n''+1)/2 guesses to find the answer: Fix the guesses you are going to try. Each guess then has probability 1/''n'' of being correct. Then expected guesses is . :''n'' here is the size of the search space, which for a brute force attack on a symmetric cipher is equal to 2''L'' where ''L'' is key length. So the classical algorithm takes O(2''L'') and the quantum algorithm takes O(2''L''/2). Note that this applies to Grover's algorithm, which is not the usual algorithm used for factorisation. It's a slower algorithm for a more general class of problems. -- User:Tim Starling 19:22, Jun 12, 2005 (UTC)
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Quantum Computer
#REDIRECT Quantum computer
Quantum Computer
This should be moved to Quantum computer if there's anything new here.
Quantum computer
[[Image:NMR-implementation-alanine.png|thumb|Molecule of alanine used in Nuclear magnetic resonance implementation of error correction. Qubits are implemented by spin (physics) state (physics)s of carbon atoms.]] A quantum computer is any device for computation that makes direct use of distinctively quantum mechanical phenomena, such as quantum superposition and quantum entanglement, to perform operations on data. In a classical (or conventional) computer, data are measured by bits; in a quantum computer the data are measured by qubits. The basic principle of quantum computation is that the quantum properties of particles can be used to represent and structure data, and that devised quantum mechanisms can be used to perform operations with these data. For a generally accessible overview of quantum computing, see [http://www.media.mit.edu/physics/publications/papers/98.06.sciam/0698gershenfeld.html Quantum Computing with Molecules], an article in Scientific American by Neil Gershenfeld and Isaac L. Chuang. Experiments have already been carried out in which quantum computational operations were executed on a very small number of qubits. Research in both theoretical and practical areas continues at a frantic pace; see [http://qist.lanl.gov Quantum Information Science and Technology Roadmap] for a sense of where the research is heading. Many national government and military funding agencies support quantum computing research, to develop quantum computers for both civilian and national security purposes, such as cryptanalysis. See the Nature (journal) article in the references below reporting on work at IBM Almaden Research Center, where scientists implemented a seven qubit computing device that ran Shor's algorithm. It is widely believed that if large-scale quantum computers can be built, they will be able to solve certain problems faster than any classical computer. Quantum computers are different from classical computers such as DNA computers and computers based on transistors, even though these may ultimately use some kind of quantum mechanical effect (for example covalent bonds). Some computing architectures such as optical computers may use classical superposition of electromagnetic waves, but without some specifically quantum mechanical resource such as quantum entanglement, they do not share the potential for computational speed-up of quantum computers. ==The basis of quantum computing== In quantum mechanics, the state of a physical system (such as an electron or a photon) is described by an element of a mathematical object called a Hilbert space. The realization of the Hilbert space depends on the particular system. For instance in the case of a single particle system, the state can be described by a complex-valued function defined on R3 (three-dimensional space) called a wave function. As described in the article on quantum mechanics, this function has a probabilistic interpretation; of particular significance is that quantum states have a property called quantum superposition. A similar realization of the Hilbert space exists for systems of interacting particles. The time evolution of the system state is given by a family {''U''''t''} (with ''t'' denoting time) of unitary transformations of ''H''. Thus if φ is the state at time 0, then ''U''''t'' φ is the state at time ''t''. Note that this is true only if the system is isolated and the phenomenon of decoherence does not occur. A classical computer has a memory made up of bits, where each bit holds either a one or a zero. The device computes by manipulating those bits, i.e. by transporting these bits from memory to (possibly a suite of) logic gates and back. A quantum computer maintains a set of qubits. A qubit can hold a one, or a zero, or a superposition of these. A quantum computer operates by manipulating those qubits, i.e. by transporting these bits from memory to (possibly a suite of) quantum gates and back. Qubits for a quantum computer can be implemented using particles with two spin (physics) states: "up" and "down"; in fact any system, possessing an observable quantity ''A'' which is ''conserved'' under time evolution and such that ''A'' has at least two discrete and sufficiently spaced consecutive eigenvalues, is a suitable candidate for implementing a qubit. For discussion of foundational aspects of quantum computing, see the article on quantum circuits. ==Bits vs qubits == Consider first a classical computer that operates on a 3 bit Processor register. At a given time, the state of the register is determined by a single string of 3 bits, such as "101". This is usually expressed by saying that the register contains a single string of 3 bits. For a quantum computer on the other hand with a 3 qubit register, the state of the register is determined by 8 complex numbers as is illustrated in the following example:
| State | Amplitude | Probability |
|---|---|---|
| * | (a+ib) | (a2+b2) |
| 000 | 0.37 + i 0.04 | 0.14 |
| 001 | 0.11 + i 0.18 | 0.04 |
| 010 | 0.09 + i 0.31 | 0.10 |
| 011 | 0.30 + i 0.30 | 0.18 |
| 100 | 0.35 + i 0.43 | 0.31 |
| 101 | 0.40 + i 0.01 | 0.16 |
| 110 | 0.09 + i 0.12 | 0.02 |
| 111 | 0.15 + i 0.16 | 0.05 |
Quantum computer
---- "For problems with all four properties, it will take an average of n/2 guesses to find the answer using a classical computer. The time for a quantum computer to solve this will be proportional to the square root of n". Shouldn't that be log(n)? If not, a little explanation on this rather unexpected outcome would be appreciated... Apart from that it's a great piece of work! --MK ---- This quote in the article is nonsense: "Integer factorization is believed to be practically impossible with an ordinary computer for large numbers (eg. 10^600)." While that's a really big number, a high school freshman can factor that big number in his head! 2^600 x 5^600. Gee, that was tough! --User:Seinfeld's_Cat :Thanks for spotting this. I changed the article accordingly. Of course, you could also have changed the article yourself. -- User:Jitse Niesen 15:06, 30 Jan 2005 (UTC) ::No problem. It sometimes seems easier to complain than to just fix it. I like your change (I couldn't come up with a suitable fix; you did it quite well.) I make a better critic than author. -- User:Seinfeld's_Cat 1 Feb 2005 ---- The talk page is as interesting as the main article. Amazing! --User:ReallyNiceGuy ---- Great article! Maybe we could add in the first paragraph that quantum computers are by their very nature probabilistic devices and only probabilistic algorithms can be implemented on them. Also, quantum computers can be simulated by Turing machines and are therefore no attack to the :Church-Turing thesis. --AxelBoldt ---- I've now added both. They're in the complexity theory section, where they seemed to fit the flow best. -user:LC ---- Yes. In the complexity section, it says that BQP is the class of problems that can be solved by quantum computers. These however are only the problems that can be solved by quantum computers in polynomial time. Maybe you could say "the problems that can be solved by quantum computers in reasonable time" or "that can be realistically solved by quantum computers". Then it says that quantum computers probably cannot solve all NP-complete problems. There are two problems with this statement: strictly speaking, a quantum computer only works probabilistically and cannot "solve" any NP-complete problem (or any other decision problem for that matter) in the same sense a Turing machine solves them, with a deterministic and correct Yes/No answer. Furthermore, if we allow probabilistic solutions, then quantum computers can of course solve all NP-complete problems, just like any Turing machine can; it may just take a lot of time to do so... --AxelBoldt ---- It should be clearer now. -user:LC ---- What is this ''#P-Complete'' ? --User:Taw ---- See :Sharp-P-Complete. -user:LC ---- How long does it take to factor an n-digit number with n qubits? --Axel ---- Would Quantum Computing break Elliptic Curve Cryptography ? User:Taw :Apparently so, through a variation on Shor's algorithm. I haven't studied it, though -- User:CYD ---- Some recent work [http://www.eet.com/story/OEG20010924S0101] indicates that if spaced sufficiently closely, quntum entanglement between quantum dots may be possible, so it's possible that in the future a quantum computer could be implemented using quantum dots. Also, it might be useful to mention "reversibility", the haddamard(sp?) transformation and the various types of quantum logic gates (CNOT, etc...). I'm not an expert, so I'll defer to someone who knows about this stuff -- User:Matt Stoker ----- What does it mean to have a "quantum computer with nonlinear operators"? --:Robert Merkel I hope that's clearer now. --user:LC How the heck do you implement a nonlinear operation on qubits? Evolution always proceeds by unitary - therefore linear - operations. -- User:CYD I agree, that seems to be a consequence of Schrodinger's equation, which is pretty basic to QM. --AxelBoldt I agree. The papers proving the result never said it could be done. But it's an interesting result. It hasn't been ruled out that a large linear system could act like a small nonlinear system, and give the desired result. The Shi paper refers to the nonlinearity of the Gross-Pitaevskii equations, but I'm not familiar with them. I would assume the Shi paper is flawed, since it wasn't accepted anywhere, but I'm not aware of any proofs that this sort of thing is inherently impossible. --user:LC Oh. I just checked the site, and the Shi paper has now been accepted in a journal. It hadn't yet been accepted when I first wrote the article. I'll remove the "not peer reviewed" note. Is anyone here familiar with the "Internation Journal of Modern Physics"? Is it generally respectable? --user:LC ---- :''The quantum computer in the above example could be thought of as a black box containing 8 complex numbers. Or, it could be thought of as 8 black boxes, each containing 1 complex number, each sitting in a different alternate universe, and all communicating with each other. These two interpretations correspond to the Copenhagen interpretation and many worlds interpretation, respectively, of quantum mechanics. The choice of interpretation has no effect on the math, or on the behavior of the quantum computer. Either way, it's an 8-element vector that is modified by matrix multiplication'' I don't think this characterization of many worlds is correct. The different universes don't come with complex numbers attached. Instead, the more likely states are exhibited in ''more'' universes. The goal of the matrix manipulations is to bring essentially all universes into the same state. Once you measure the state, the universes stop to communicate and truly split. --AxelBoldt Good point. I've reworded it. --user:LC ----- In the "How they work" section it says: "The square of (a+ib) is (a^2+b^2)." Actually the square of (a+ib) would be (a^2-b^2) + i(2ab). Probably you mean the norm or the mod or something like that? (it's been a while since i used complex numbers, not sure of the name any more). In quantum mechanics, (a+bi) is called the ''amplitude'', and (a2+b2) is called the ''squared amplitude'', even though the latter equals |a+bi|2 rather than the (a+bi)2 that the name might suggest. I suppose the terminology is counterintuitive. I'll reword the article to make it clearer. --user:LC ---- I've removed the following contribution from User:Harry Potter: : ''Quantum computing can also theoretically produce time anomalies. The ability of the Quantum computer to find the correct answer in a factorisation problem can be seen as running all the possibilities simultaneously. However, only the correct answer produces a positive response which is then sent back through time to become the only calculation which in fact needs to be made.'' Quantum computation doesn't involve time travel in any description I've seen. -- User:Oliver Pereira 00:01 9 Jun 2003 (UTC) :Agreed. This sounds purely abstract and theoretical to me, but I am no expert on quantum computing by any means. Perhaps Harry Potter can produce a source in the quantum computing literature that lends some support to this idea? -- User:Wapcaplet 00:10 9 Jun 2003 (UTC) ----- Please specify "... have recently been built" into "... have been built in the early/mid-/late 20th century/21st century". --User:Menchi 08:37 14 Jul 2003 (UTC) ---- I found the first paragraph of the second section, titled "The Power of Quantum Computers", rather unclear. Unfortunately, I cannot tell exactly what makes it unclear - maybe it is too vague; how difficult is factorization (suspected to be) on a classical computer, and what is the speed-up on a quantum computer? I do not know enough about quantum computing to rewrite the paragraph myself ... I liked the rest of the article very much, it contains (almost) everything that I was looking for. However, I am still wondering whether there is an upper bound on the power of quantum computers. The article says that quantum computers cannot solve undecidable problems, but is BQP known/suspected to be a subset of another complexity class, say NP? -- User:Jitse Niesen 11:08, 26 Aug 2003 (UTC) == They CAN solve Turing-unsolvable problems, but... == This is a technicality that should be mentioned: If you define a quantum computer as a Turing Machine with arbitrary complex amplitudes, then the class of machines that you obtain is uncountably infinite, and can easily be shown [ADH97] to contain machines that solve all kinds of Turing-unsolvable problems. But *we* can't build those QC's (or even know one if we see one) because of our inability to know the correct amplitudes required for those tasks. The computability comments in the present version of this article are valid for QC's whose amplitudes are computable numbers. [ADH97] L. Adleman, J. DeMarrais, and M. Huang. Quantum computability, SIAM Journal on Computing 26:1524-1540, 1997. == Key exchange versus encryption == You say: "This ability would allow a quantum computer to break many of the cryptographic systems in use today. In particular, most of the popular public key ciphers could be quickly broken, including forms of RSA, ElGamal and Diffie-Hellman." My understanding is that RSA is the encyrption method and the ElGamal and Diffie-Hellman are key exchange protocols. The references within Wikipedia also say that. Do you mean that the encyrpted key being transmitted is able to be read? BK ---- == Further editing == Some of the section which was there before (and is now called Initialization and termination), though written at a good level I believe contains some innaccuracies, i.e. on how the qubit registers are measured. In fact, only a fixed register is sampled on termination.User:CSTAR 04:27, 6 Jun 2004 (UTC) Oops that's not what I should have said. Anyway I think it's right now. I also split the article up adding the stuff on reversible registers to quantum circuits.User:CSTAR 07:01, 6 Jun 2004 (UTC) == Where did the numbers come from in columns #2 and #3 in the "bits vs qubits" section? == Why is the 3-qubit "000" represented as "0.37 + i 0.04"? Where did the 0.37 and 0.04 come from? Ditto for the rest of the numbers in that table. According to www.qubit.org, a 3-qubit system can represent 8 values. There is no magic in 16 analog values vs 8 complex values, there are just 8 values total. None of the references I have come across mention anything other than an L qubit system representing exactly 2^L possibilities at once, not 2^(L+1) (the 16 you mentioned in this section). In an effort to give the reader some sense of how probability waves can be represented by complex numbers and how complex numbers c1an be manipulated, you've obfuscated the simpler facts of qubits. 1 qubit holds 2 states simultaneously with *equal probability*, 2 qubits hold 4 states simultaneously with *equal probability* and so on. ---- : I didn't write this example, but as an example, I think it's mostly OK (But you're right about the dimension being wrong: see below. When I read it I wasn't thinking) . What the example is trying to express is that fact that in a quantum computer the set of registers is a quantum mechanical object,and the state is described by a wave funbction on it. As far as the number of independent values, the example is meant to suggest the real dimension of the space of states, but you may be right the dimension counting may be wrong, because the relevant space is complex projective space of dimension 2''n''. Thanks. I'll fix it. The set of pure sets is homeomorphic to the compact symmetric space with m = 2''n'' :: : As far as your statement: ::1 qubit holds 2 states simultaneously with *equal probability*, 2 qubits hold 4 states simultaneously with *equal probability* and so on. :I don't believe this statement is correct. User:CSTAR 03:10, 16 Jun 2004 (UTC) P.S. From this log, it wasn't clear who wrote the question and who wrote the reponse, so I went back and edited the past, by separating my answer to the anonymous question. I'm perfectly happy to edit the past. More of us should try it.User:CSTAR 04:48, 16 Jun 2004 (UTC) == other confusing bits == (problem I noted has been fixed -- thanks, everyone. -- User:DavidCary 01:43, 4 Jul 2004 (UTC)) :::Yes, it's rubbish, I rewrote the section. Unfortunately the bit about fault tolerant scaling in Shor's algorithm is a bit hazy, I'm just quoting vague unpublished numbers floating around my research group. -- User:Tim Starling 07:35, Jun 18, 2004 (UTC) ::::Good rewrite! Regarding range for T2, I checked Nielsen & Chuang book (pp. 337) and they claim 7 seconds and 0.3 seconds for proton and carbon in a two-qubit experiment. Should we change "hundreds of microseconds" to "seconds"? Or is there some caveat about Nielsen & Chuang data that I'm not aware of? User:Andris 18:37, Jun 18, 2004 (UTC) :::::Alright. It would be nice to explain at some stage why systems such as chloroform nuclear spins are not scalable, but I guess "nanoseconds to seconds" will do for now. -- User:Tim Starling 01:47, Jun 22, 2004 (UTC) ---- The article currently mentions ''the transverse relaxation time T2 (terminology used in NMR and MRI technology)'' Is there really any difference between NMR quantum computers and MRI quantum computers ? I suspect they're really the same thing. -- User:DavidCary 01:43, 4 Jul 2004 (UTC) :I guess my comments weren't clear -- I was referring to MRI and NMR ''technology'' not computers (MRI technology, the kind used in hospitals) which are essentially the same technology (I think at least they are, since the both involve involve nuclear spin), but seem to have separate wiki articles. Is my making these two references too confusing? Please change it if you can make it clearer. User:CSTAR 01:51, 4 Jul 2004 (UTC) ==Interpretation== :: Now that we're at it, I think we should remove references to ''interprertation'' (Copenhagen, many worlds etc..). Regardless of content, they shouldn't be in this article. I haven't made an effort to understand whether what's writtem in the article makes any sense, so I won't pass judgment on that account_. User:CSTAR 13:19, 18 Jun 2004 (UTC) == Edits of 08/01/2004 == Some of the edits by an anonymous user did increase readability, but I think calling ''entanglement'' or ''superposition'' a process is misleading. Also Qubits don't hold data -- they are measures of the ''size'' of data. The register space is the actual container of data. I tried to keep the spirit of the recent edits, but modifying what seemed misleading. User:CSTAR 02:49, 2 Aug 2004 (UTC) --- So, how the hell are you supposed to make software for these? I assume a normal "if something is something do this, else do this" doesn't work? : Conditionals in the usual sense, no.User:CSTAR 20:08, 16 Aug 2004 (UTC) == Decoherence and Complexity == This article is really great ! Nevertheless, there is one point I found unclear : the relationship between decoherence and complexity. The article state some algortihms are in BQP. Since complexity classes are only meaningful for arbitrarily large calculations, does the definition of the BQP class includes the fact that decoherence may limits the amount of operations that may be performed, or does it assumes we have a way to prevent the effects of decoherence for arbitrarily long calculations (which doesn't seem obvious) ? For example, an old (1995) article by P. Shor and coll. [http://citeseer.ist.psu.edu/chuang95quantum.html "Quantum Computers, Factoring and Decoherence"], calculated a limit to the size of the integers that may be factored faster by a quantum computer than by a classic computer, which to me would contradict the fact that factoring is in BQP. Some clarification of this point would be great ! Thanks ! : That's a good question, but I'm inclined to say that the there is no relationship othre than that which is buit into the model itself. To explain rigorously BQP, you have to formulate a more precise model of quantum computation than is done in this article. The right model is that of quantum circuit, although BQP is not defined there either, although it should be. At some point I'll get around to it. Otherwise look at some of the references such as Kitaev, or Nielsen Chuang. The book by Hirvensalo is also good. I know I have evaded your question but that's the best I can do. Maybe some other wikipedian has a better answer.User:CSTAR 13:57, 1 Oct 2004 (UTC) :The definition of BQP assumes that computation is error-free (and decoherence-free), which is not the case in practice. The limit in Shor's 1995 article is for quantum computers which are subject to decoherence. Thus, there is no contradiction: the 1995 paper essentially says that if there was not decoherence, quantum computers could factor large numbers, but in presence of decoherence, they could not. This is the (pessimistic) view of quantum computation that was common in mid-1990s. This view is mostly obsolete now. Quantum error correction has been invented (in a later work of Shor and colleagues in 1996 and 1997) and it has been shown that arbitrarily large computations can be made fault-tolerant (with "fault" including "decoherence"), as long as fault rate is sufficiently small. User:Andris 00:10, Dec 11, 2004 (UTC) == Cost of quantum computations == The big selling point of quantum computations seems to be that quantum computers with enough qubits can run algorithms much faster than classicalcomputers. That's great, but it's not very useful if, for example, the cost of implementing an n-qubit register is exponential in n. Is there some estimate for the asymptotics of this? Is the apparent advantage of quantum computers due only to the fact that we're measuring computational complexity by the wrong yardstick? (I have no idea, but someone surely knows...) --User:Aarchiba 08:48, Dec 7, 2004 (UTC) ==NP-Complete stuff== "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 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." A teacher claimed that someone has found an quantum algorithm to solve NP-complete problems in polynomial time. Asking for a size estimate, he said that the algorithm description should approximately fit on a page. He's supposed to be teaching everyone the definition of NP-complete problems next week, so he should know what he's talking about. He seemed rather sure. I'd like to see that algorithm... (Haven't had time to search the internet yet.) Would sure make quantum computers (even) more interesting... User:Cyp User talk:Cyp 17:27, 9 Dec 2004 (UTC) :Regarding the previous quote: Are integer factorization and discrete log (suspected to be) outside BPP? I think this is more interesting than whether they are outside P. -- User:Jitse Niesen 19:05, 9 Dec 2004 (UTC) ::Yes, they are thought to be outside BPP. User:Andris 04:32, Dec 10, 2004 (UTC) :Thanks, I updated the article. -- User:Jitse Niesen 13:42, 10 Dec 2004 (UTC) ::Does [http://arxiv.org/PS_cache/quant-ph/pdf/9912/9912100.pdf this] article mean anything? I'm a bit sceptical, since it seems to be saying that because some things can be approximated by a non-linear function, that linearity of quantum mechanics can be ignored... User:Cyp User talk:Cyp 17:15, 16 Dec 2004 (UTC) == Clustered quantum computer ? == Hello, I don't understand a lot about the technics of quantum computers. I would like to know if it is possible (at least in theory) to distributed quantum computing. it is possible to dream about a “quantum grid”, “quantum Beowulf”, “quantum@home?” Thank you User:Vrykolaka 07:31, 25 Jan 2005 (UTC) == Can quantum computers solve undecidable problems? == This article says that quantum computers cannot solve any problems that classical computers cannot solve either. However, after a recent edit, the start of Matiyasevich's theorem reads: :"'''Matiyasevich's theorem''', proven in 1970 by Yuri Matiyasevich, implies that Hilbert's problems is unsolvable (although it has recently been proposed by Tien Kieu [Int.J.Theor.Phys. 42 (2003) 1461-1478] that this result only applies to a deterministic Turing machine, and that the problem could be solved on a quantum computer using a quantum adiabatic algorithm. This is still contraversial.)." How to reconcile both quotes? Does anybody know how contraversial Tien Kieu's work is? -- User:Jitse Niesen 13:36, 10 Feb 2005 (UTC) ::Probabilistic Turing machines or non-deterministic Turing machines don't decide more problems than plain old Turing machines. Neither does the quantum circuit model (and equivalents). Unfortunately, I don't know enough what a quantum adiabatic algorithm is but it possibly doesn't fall under the quantum circuit model of quantum computation. I suppose I should look at the paperUser:CSTAR 14:41, 10 Feb 2005 (UTC) ::: I just looked at one of his papers. It is highly suspect, making claims about probabilistic computation which under any normal interpretation are false. "''Probabilistic computation may be more powerful than Turing Computation provided a suitable probability measure can be defined.''" Now it is well-known, (see Sipster's textbook on computation) that probabilistic Turing machines compute the same functions as normal ones. ::: I would take these claims with lots of salt (and maybe that sentence in Matiyasevich's theorem should be removed).User:CSTAR 18:15, 10 Feb 2005 (UTC) Thank you; I came to the same conclusion. -- User:Jitse Niesen 21:47, 10 Feb 2005 (UTC) == Request for references == Hi, I am working to encourage implementation of the goals of the Wikipedia:Verifiability policy. Part of that is to make sure articles Wikipedia:Cite sources. This is particularly important for featured articles, since they are a prominent part of Wikipedia. Now this article has an extensive list of additional material, but list of further reading is not the same thing as proper references. Further reading could list works about the topic that were not ever consulted by the page authors. If some of the works listed in the that section were used to add or check material in the article, please list them in a references section instead. The Wikipedia:WikiProject Fact and Reference Check has more information. Thank you, and please [http://en.wikipedia.org/w/wiki.phtml?title=User_talk:Taxman&action=edit§ion=new leave me a message] when a few references have been added to the article. - User:Taxman 19:18, Apr 22, 2005 (UTC) == Factorization == "For problems with all four properties, it will take an average of n/2 guesses to find the answer using a classical computer. The time for a quantum computer to solve this will be proportional to the square root of n...But it is also easy to defend against. Just double the size of the key for the cipher." This doesn't sound right to me. If your key length is 2n, a classical computer would take n guesses. A quantum computer would then take time proportional to sqrt(2n), which is asymptotically the same as sqrt(n). I didn't edit the page because I'm not an expert on quantum computing, and feel it's possible that I don't understand (if, for example, you can't do asymptotic analysis on quantium computing algorithms?). As best as I can figure, though, I would expect you'd need to square the key length? Any problem with these four problems will take an average of (''n''+1)/2 guesses to find the answer: Fix the guesses you are going to try. Each guess then has probability 1/''n'' of being correct. Then expected guesses is . :''n'' here is the size of the search space, which for a brute force attack on a symmetric cipher is equal to 2''L'' where ''L'' is key length. So the classical algorithm takes O(2''L'') and the quantum algorithm takes O(2''L''/2). Note that this applies to Grover's algorithm, which is not the usual algorithm used for factorisation. It's a slower algorithm for a more general class of problems. -- User:Tim Starling 19:22, Jun 12, 2005 (UTC)
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