A Quantum Approach

 A quantum computer is a device for computation that makes direct use of quantum mechanical phenomena, such as superposition and entanglement, to perform operations on data. Quantum computers are different from digital computers based on transistors. Whereas digital computers require data to be encoded into binary digits, quantum computation utilizes quantum properties to represent data and perform operations on these data. A theoretical model is the quantum Turing machine, also known as the universal quantum computer. Quantum computers share theoretical similarities with non-deterministic and probabilistic computers, like the ability to be in more than one state simultaneously. The field of quantum computing was first introduced by Richard Feynman in 1982.

Although quantum computing is still in its infancy, experiments have been carried out in which quantum computational operations were executed on a very small number of qubits . Both practical and theoretical research continues, and 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.

Large-scale quantum computers could be able to solve certain problems much faster than any classical computer by using the best currently known algorithms, like integer factorization using Shor's algorithm or the simulation of quantum many-body systems. There exist quantum algorithms, such as Simon's algorithm, which run faster than any possible probabilistic classical algorithm. Given unlimited resources, a classical computer can simulate an arbitrary quantum algorithm so quantum computation does not violate the Church–Turing thesis. However, in practice infinite resources are never available and the computational basis of 500 qubits, for example, would already be too large to be represented on a classical computer because it would require 2500 complex values to be stored. 
Nielsen and Chuang point out that "Trying to store all these complex numbers would not be possible on any conceivable classical computer."

If you measure the three qubits, you will observe a three-bit string. The probability of measuring a given string is the squared magnitude of that string's coefficient . Thus, measuring a quantum state described by complex coefficients  gives the classical probability distribution  and we say that the quantum state "collapses" to a classical state as a result of making the measurement.
Note that an eight-dimensional vector can be specified in many different ways depending on what basis is chosen for the space. The basis of bit strings  is known as the computational basis. Other possible bases are unit-length, orthogonal vectors and the eigenvectors of the Pauli-x operator. Ket notation is often used to make the choice of basis explicit. For example, the state  in the computational basis can be written as:

The computational basis for a single qubit  is  0\rangle   \left and  1\rangle   \left.
Using the eigenvectors of the Pauli-x operator, a single qubit is  +\rangle   \tfrac \left.

Operation 

While a classical three-bit state and a quantum three-qubit state are both eight-dimensional vectors, they are manipulated quite differently for classical or quantum computation. For computing in either case, the system must be initialized, for example into the all-zeros string,  000\rangle, corresponding to the vector . In classical randomized computation, the system evolves according to the application of stochastic matrices, which preserve that the probabilities add up to one . In quantum computation, on the other hand, allowed operations are unitary matrices, which are effectively rotations .  Consequently, since rotations can be undone by rotating backward, quantum computations are reversible.
Finally, upon termination of the algorithm, the result needs to be read off. In the case of a classical computer, we sample from the probability distribution on the three-bit register to obtain one definite three-bit string, say 000. Quantum mechanically, we measure the three-qubit state, which is equivalent to collapsing the quantum state down to a classical distribution, followed by sampling from that distribution. Note that this destroys the original quantum state. Many algorithms will only give the correct answer with a certain probability. However, by repeatedly initializing, running and measuring the quantum computer, the probability of getting the correct answer can be increased.
For more details on the sequences of operations used for various quantum algorithms, see universal quantum computer, Shor's algorithm, Grover's algorithm, Deutsch-Jozsa algorithm, amplitude amplification, quantum Fourier transform, quantum gate, quantum adiabatic algorithm and quantum error correction.

Potential 

Integer factorization is believed to be computationally infeasible with an ordinary computer for large integers if they are the product of few prime numbers . By comparison, a quantum computer could efficiently solve this problem using Shor's algorithm to find its factors. This ability would allow a quantum computer to decrypt many of the cryptographic systems in use today, in the sense that there would be a polynomial time  algorithm for solving the problem. In particular, most of the popular public key ciphers are based on the difficulty of factoring integers, including forms of RSA. These are used to protect secure Web pages, encrypted email, and many other types of data. Breaking these would have significant ramifications for electronic privacy and security.
However, other existing cryptographic algorithms do not appear to be broken by these algorithms. Some public-key algorithms are based on problems other than the integer factorization and discrete logarithm problems to which Shor's algorithm applies, like the McEliece cryptosystem based on a problem in coding theory. Lattice-based cryptosystems are also not known to be broken by quantum computers, and finding a polynomial time algorithm for solving the dihedral hidden subgroup problem, which would break many lattice based cryptosystems, is a well-studied open problem. It has been proven that applying Grover's algorithm to break a symmetric  algorithm by brute force requires roughly 2n/2 invocations of the underlying cryptographic algorithm, compared with roughly 2n in the classical case, meaning that symmetric key lengths are effectively halved: AES-256 would have the same security against an attack using Grover's algorithm that AES-128 has against classical brute-force search . Quantum cryptography could potentially fulfill some of the functions of public key cryptography.

Besides factorization and discrete logarithms, quantum algorithms offering a more than polynomial speedup over the best known classical algorithm have been found for several problems, including the simulation of quantum physical processes from chemistry and solid state physics, the approximation of Jones polynomials, and solving Pell's equation. No mathematical proof has been found that shows that an equally fast classical algorithm cannot be discovered, although this is considered unlikely. For some problems, quantum computers offer a polynomial speedup. The most well-known example of this is quantum database search, which can be solved by Grover's algorithm using quadratically fewer queries to the database than are required by classical algorithms. In this case the advantage is provable. Several other examples of provable quantum speedups for query problems have subsequently been discovered, such as for finding collisions in two-to-one functions and evaluating NAND trees.

Consider a problem that has these four properties:
#The only way to solve it is to guess answers repeatedly and check them,
#The number of possible answers to check is the same as the number of inputs,
#Every possible answer takes the same amount of time to check, and
#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 cracker that attempts to guess the password for an encrypted file .
For problems with all four properties, the time for a quantum computer to solve this will be proportional to the square root of the number of inputs. That can be a very large speedup, reducing some problems from years to seconds. It can be used to attack symmetric ciphers such as Triple DES and AES by attempting to guess the secret key.

Grover's algorithm can also be used to obtain a quadratic speed-up over a brute-force search for a class of problems known as NP-complete.

Since chemistry and nanotechnology rely on understanding quantum systems, and such systems are impossible to simulate in an efficient manner classically, many believe quantum simulation will be one of the most important applications of quantum computing.
There are a number of technical challenges in building a large-scale quantum computer, and thus far quantum computers have yet to solve a problem faster than a classical computer. David DiVincenzo, of IBM, listed the following requirements for a practical quantum computer:
scalable physically to increase the number of qubits;
qubits can be initialized to arbitrary values;
quantum gates faster than decoherence time;
universal gate set;
qubits can be read easily.

Quantum decoherence  

One of the greatest challenges is controlling or removing quantum decoherence. This usually means isolating the system from its environment as interactions with the external world causes the system to decohere. This effect is irreversible, as it is non-unitary, and is usually something that should be highly controlled, if not avoided. Decoherence times for candidate systems, in particular the transverse relaxation time T2, typically range between nanoseconds and seconds at low temperature. With error correction, the figure would rise to about 107 qubits. Note that computation time is about L^2 or about 10^7 steps and on 1 MHz, about 10 seconds.
A very different approach to the stability-decoherence problem is to create a topological quantum computer with anyons, quasi-particles used as threads and relying on braid theory to form stable logic gates.

Developments 

There are a number of quantum computing models, distinguished by the basic elements in which the computation is decomposed. The four main models of practical importance are
the quantum gate array,
the one-way quantum computer,
the adiabatic quantum computer,
and the topological quantum computer

The Quantum Turing machine is theoretically important but direct implementation of this model is not pursued. All four models of computation have been shown to be equivalent to each other in the sense that each can simulate the other with no more than polynomial overhead.