Quantum computer

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A quantum computer is any device for computation that makes direct use of distinctively quantum mechanical phenomena, such as superposition and 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 Quantum Computing with Molecules (http://www.media.mit.edu/physics/publications/papers/98.06.sciam/0698gershenfeld.html), 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 Quantum Information Science and Technology Roadmap (http://qist.lanl.gov) 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 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 factorization algorithm.
It is widely believed that if largescale 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 entanglement, they do not share the potential for computational speedup of quantum computers.
Contents 
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 complexvalued function defined on R^{3} (threedimensional 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 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 logic gates and back.
Qubits for a quantum computer can be implemented using particles with two spin 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 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)  (a^{2}+b^{2}) 
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 
For an n qubit quantum register, this table would have had 2^{n} rows; for n=300, this is roughly 10^{90}, 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 3qubit quantum computer has far more memory than a 3bit 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 (a^{2}+b^{2}) 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 8dimensional 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. 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 8dimensional 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 NMR 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:
 <math> \rho = \frac{e^{ \beta H}}{\operatorname{Tr}(e^{ \beta H})}, <math>
where H is the hamiltonian matrix of an individual molecule and
 <math> \beta = (\mbox{Boltzmann constant } \times \mbox{ temperature })^{ 1}<math>
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, DeutschJozsa 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 300digit 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 2n 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 cryptographic systems in use today. In particular, most of the popular public key ciphers could be quickly broken, including forms of RSA, ElGamal and DiffieHellman. 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 ciphers 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 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 T_{2} (terminology used in NMR 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 10^{12} to 10^{18} qubits. Fabrication and control of this large number of qubits is nontrivial 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
 Turingcomplete 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 decohere.
Candidates
There are a number of quantum computing candidates, among those:
 Superconductorbased quantum computers (including SQUIDbased quantum computers)
 Trapped ion quantum computers
 Ion trapbased quantum computers
 "Nuclear magnetic resonance on molecules in solution"based
 "quantum dot on surface"based
 "Cavity quantum electrodynamics" (CQED)based
 molecular magnetbased
 fullerenebased 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 polynomialtime 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 NPcomplete and a strict superset of P, but that is not known. Both integer factorization and discrete log are in BQP. Both of these problems are NP problems suspected to be outside BPP, and hence outside P. Both are suspected to not be NPcomplete. There is a common misconception that quantum computers can solve NPcomplete 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 operation. It has been shown that if a quantum computer could be designed with nonlinear operators, then it could solve NPcomplete problems in polynomial time. It could even do so for #Pcomplete 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 ChurchTuring 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
 Dwave system
Further information
 Quantum computing is part of quantum information processing.
 Companies developing quantum computers
 DWave Systems, Vancouver, BC, Canada (http://www.dwavesys.com)
 IBM (http://www.research.ibm.com/quantuminfo)
 Quantum computing related patents
 Some issued quantum computingrelated patents (http://patft.uspto.gov/netacgi/nphParser?Sect1=PTO2&Sect2=HITOFF&u=%2Fnetahtml%2Fsearchadv.htm&r=0&p=1&f=S&l=50&Query=%22quantum+computing%22+OR+%22quantum+gate%22+OR+%22quantum+computer%22&d=ptxt)
 Some published quantum computingrelated patents (http://appft1.uspto.gov/netacgi/nphParser?Sect1=PTO2&Sect2=HITOFF&u=%2Fnetahtml%2FPTO%2Fsearchadv.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)
 Good general reference:
 arXiv.org ePrint archive (http://www.arxiv.org/)  a very complete online archive of research papers.
 Centre for Quantum Computation, University of Cambridge http://cam.qubit.org
 Institute for Quantum Computing (http://www.iqc.ca), University of Waterloo
 QCL — A Programming Language for Quantum Computers (http://tph.tuwien.ac.at/~oemer/qcl.html)
 Thermal ensembles
 Overview (http://www.media.mit.edu/physics/projects/spins/home.html) of early developments, with links
 The first two papers ever written on this topic:
 Template:Journal reference and
 Template:Journal reference (download (http://www.media.mit.edu/physics/publications/papers/97.01.science.pdf))
 Using quantum computers to simulate quantum systems:
 Quantum cryptography:
 The first paper ever written on this:
 The first paper ever published on this: Template:Conference reference
 A listing of a huge number of quantum cryptography papers, with some discussion of them, is at http://www.cs.mcgill.ca/~crepeau/CRYPTO/BiblioQC.html
 Universal quantum computer and the ChurchTuring thesis:
 Shor's factoring algorithm:
 Template:Conference reference (copy on Arxiv.org (http://www.arxiv.org/abs/quantph/9508027)).
 JeanPierre Seifert, "Using fewer Qubits in Shor's Factorization Algorithm via Simultaneous Diophantine Approximation", (download (http://eprint.iacr.org/2000/036/))
 IBM's announcement (http://domino.research.ibm.com/comm/pr.nsf/pages/news.20011219_quantum.html) 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.
 Quantum database search:
 Template:Conference reference (copy on Arxiv.org (http://www.arxiv.org/abs/quantph/9605043))
 Quantum sorting:
 Quantum computer simulators:
 libquantum (http://www.enyo.de/libquantum/) — A library for quantum computer simulation
 QCL (http://tph.tuwien.ac.at/~oemer/qcl.html) — Simulation of quantum computing with a quantum computing language
 Quantum::Entanglement (http://search.cpan.org/search?dist=QuantumEntanglement) — Quantum computation module for Perl.
 QCF (http://www.charlesfox.org.uk) — Quantum computation toolbox for MATLAB.
 Fraunhofer Quantum Computing Simulator (http://www.qc.fraunhofer.de) — A free webbased quantum simulator (31 qubits) and a collaborative workspace for the quantum computing community.
 Quantum error correction:
 Template:Journal reference
 Template:Journal reference (copy on Arxiv.org (http://www.arxiv.org/abs/quantph/9512032))
 Template:Conference reference (copy on Arxiv.org (http://www.arxiv.org/abs/quantph/9605011)).
 Quantum error avoidance:
 Template:Journal reference (copy on Arxiv.org (http://arxiv.org/abs/quantph/9807004))
 D. A. Lidar, K.B. Whaley, "DecoherenceFree Subspaces and Subsystems", in "Irreversible Quantum Dynamics", F. Benatti and R. Floreanini (Eds.), pp. 83120 (Springer Lecture Notes in Physics vol. 622, Berlin, 2003), copy on Arxiv.org (http://arxiv.org/abs/quantph/0301032).
 Solving NPcomplete and #Pcomplete problems:
 Daniel S. Abrams (1), Seth Lloyd (2) ( (1) Dept. of Physics, MIT, (2) Dept. of Mechanical Engineering, MIT), 1998, "Nonlinear quantum mechanics implies polynomialtime solution for NPcomplete and #P problems" (copy on Arxiv.org (http://arxiv.org/abs/quantph/9801041)).
 Phil Gossett, 1998, "NP in BQP with Nonlinearity", (copy on Arxiv.org (http://arxiv.org/abs/quantph/9804025)).
 Template:Journal reference (copy on Arxiv.org (http://www.arxiv.org/abs/quantph/9910073) or here (http://www.worldscinet.com/journals/ijmpb/15/1522/S0217979201007154.html)).
 For the interested nonexpert:
 Template:Book reference
 West, J. (2000). The Quantum Computer — An Introduction (http://www.cs.caltech.edu/~westside/quantumintro.html). (Easy to understand explanation of quantum computing)
 Hayes, Brian (JulAug 1995). The square root of NOT (http://www.americanscientist.org/template/AssetDetail/assetid/24592). American Scientist Online. (Logic gates in a quantum computer)
 Template:Book reference
 Similarly named miscellany:
 For the online service provider that was previously known as Quantum Computer Services, see America Online.
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