# Quantum computer - Essay Example

Quantum computers are deferent from digital superposition states. A theoretical model is the quantum Turing machine, also was first introduced by Yuri Main in 1980[2] and Richard Funnyman in 1982. 3][4] A quantum space-time in 1 of 2014 quantum computing is still in its infancy were executed on a very small number of cubits. [6]Both practical and theoretical civilian and national security purposes, such as quantum 1 OFF imputer using the best currently known algorithms, like integer factorization using Shore’s algorithm or the simulation of quantum many-body systems. There exist quantum algorithms, such as Simony’s algorithm, which run faster than any possible probabilistic classical algorithm. 8] Given sufficient computational resources, however, a classical computer could be made to simulate any quantum algorithm; quantum computation does not violate the Church-Turing thesis. [9] A classical computer has a memory made up of bits, where each bit represents either a one or a zero. A quantum computer maintains a sequence of cubits. A single cubit can represent a one, a zero, or any quantum superposition of these two cubit states; moreover, a pair of cubits can be in any quantum superposition of 4 states, and three cubits in any superposition of 8.

In general, a quantum computer with n cubits can be in an arbitrary superposition of up to Nan different states simultaneously (this compares to a normal computer that can only be in one of these Nan states at any one time). A quantum computer operates by setting the cubits in a controlled initial state that represents the problem at hand and by manipulating those cubits with a fixed sequence of quantum logic gates. The sequence of gates to be applied is called a quantum algorithm.

The calculation ends with a measurement, collapsing the system of cubits into one of the Nan pure states, where each cubit is purely zero or one. The outcome can therefore be at most n classical bits of information. Quantum algorithms are often non-deterministic, in that they provide the correct solution only with a certain known probability. An example of an implementation of cubits for a quantum computer could start with the use of particles with two spin states: “down” and “up” (typically written l{downwards}
ange and l{arrow}
ange, or I

O{
ange} and I But in fact any system possessing an observable quantity A, which is conserved under time evolution such that A has at least two discrete and sufficiently spaced consecutive sunglasses, is a suitable candidate for implementing a cubit. This is true because any such system can be mapped onto an effective spin-1/2 system. Bits vs.. cubits A quantum computer with a given number of cubits is fundamentally different from a classical computer composed of the same number of classical bits.

For example, to represent the state of an n-cubit system on a classical computer would require the outrage of an complex coefficients. Although this fact may seem to indicate that cubits can hold exponentially more information than their classical counterparts, care must be taken not to overlook the fact that the cubits are only in a probabilistic superposition of all of their states. This means that when the final state of the cubits is measured, they will only be found in one of the possible configurations they were in before measurement.

Moreover, it is incorrect to think of the cubits as only being in one particular state before measurement since the fact that they were in a preposition of states before the measurement was made directly affects the possible outcomes of the computation. Cubits are made up of controlled particles and the means of control (e. G. Devices that trap particles and switch them from one state to another). [10] For example: Consider first a classical computer that operates on a three-bit register. The state of the strings 000, 001, 010, 011, 100, 101, 110, 111.

If it is a deterministic computer, then it is in exactly one of these states with probability 1. However, if it is a probabilistic computer, then there is a possibility of it being in any one of a number of different dates. We can describe this probabilistic state by eight nonnegative numbers A,B,C,D,E,F,G,H (where A = probability computer is in state 000, B = probability computer is in state 001, etc. ). There is a restriction that these probabilities sum to 1 . The state of a three-cubit quantum computer is similarly described by an eight- dimensional vector (a,b,c,d,e,f,g,h), called a get.

Here, however, the coefficients can have complex values, and it is the sum of the squares of the coefficients’ magnitudes, la IA+lb IA+… +la IA, that must equal 1. These square magnitudes represent the probability amplitudes of given states. However, because a complex number encodes not Just a magnitude but also a direction in the complex plane, the phase difference between any two coefficients (states) represents a meaningful parameter. This is a fundamental difference between quantum computing and probabilistic classical computing. 11] If you measure the three cubits, you will observe a three-bit string. The probability of measuring a given string is the squared magnitude of that string’s coefficient (I. E. , the probability of measuring 000= Allah, the probability of measuring 001 = Bible, etc.. ). Thus, measuring a quantum state described by employ coefficients (a,b,… ,h) gives the classical probability distribution (la IA, lb I AH, … , I h IA) 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 (e. G. , 000, 001, … , 1 1 1) is known as the computational basis. Other possible bases are unit-length, orthogonal vectors and the eigenvectors of the Pauli-x operator. Get notation is often used to make the choice of basis explicit. For example, the state (a,b,c,d,e,f,g,h) in the computational basis can be written as: OHO
ange +
ange + +
ange + + 101
ange + 110
ange + 1 11
ange where, e. G. 1010
ange = The computational basis for a single cubit (two dimensions) is II
ange = left(l ,O
ight) and I l
ange =
ight). Using the eigenvectors of the Pauli-x operator, a single cubit is l+
ange = , 1
ight) and I-
ange – left(l
ight). Operation List of unsolved problems in physics Is a universal quantum computer sufficient to efficiently simulate an arbitrary physical system? While a classical three-bit state and a quantum three-cubit 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, 1000
ange, corresponding to the vector (1 In classical randomized computation, the system evolves according to the application of stochastic matrices, which preserve that the probabilities add up to one (I. E. , preserve the Al norm). In quantum computation, on the other hand, allowed operations are unitary matrices, which are effectively tuitions (they preserve that the sum of the squares add up to one, the Euclidean or quantum device. Consequently, since rotations can be undone by rotating backward, quantum computations are reversible. (Technically, quantum operations can be probabilistic combinations of Unitarian, so quantum computation really does generalize classical computation. See quantum circuit for a more precise formulation. ) 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-cubit state, which is equivalent to collapsing the quantum state down to a classical distribution (with the coefficients in the classical state being the squared magnitudes of the coefficients for the quantum state, as described above), 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, Shore’s algorithm, Grove’s algorithm, Deutsche-Joss 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 (e. G. , products of two 300-digit primes). [12] By comparison, a quantum computer could efficiently solve this problem using Shore’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 (in the number of digits of the integer) algorithm for solving the problem. In particular, most of the popular public key ciphers are based on the difficulty of factoring integers or the discrete logarithm problem, which can both be solved by Shore’s algorithm. In particular the RASA, Defile-Hellman, and Elliptic curve Defile-Hellman algorithms could be broken. These are used to protect secure Web pages, encrypted email, and many other types of data.

Breaking these would have significant ramifications for electronic privacy and security. However, other cryptographic algorithms do not appear to be broken by these Some public-key algorithms are based on problems other than the integer factorization and discrete logarithm problems to which Shore’s algorithm applies, like the Micelle cryptologist based on a problem in coding 5] Lattice-based cryptograms 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 cryptograms, is a well-studied open problem. 6] It has been proven that applying Grove’s algorithm to break a symmetric (secret key) algorithm by brute force requires time equal to roughly an/2 invocations of the underlying cryptographic algorithm, compared with roughly an in the classical case,[17] meaning that symmetric key lengths are effectively halved: AES-256 would have the same security against an attack using Grove’s algorithm that AES-128 has against classical brute- the functions of public key cryptography.

Besides factorization and discrete logarithms, quantum algorithms offering a more than polynomial speedup over the est. known classical algorithm have been found for several problems,[18] including the simulation of quantum physical processes from chemistry and solid state physics, the approximation of Jones polynomials, and solving Peal’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 Grove’s algorithm using quadrilateral 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 AND 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 (assuming that the password has a maximum possible length). 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. It can be used to attack symmetric ciphers such as Triple DES and AES by attempting to guess the secret key. [19] Grove’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 aunt 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. [20] 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 Divergence, of MM, listed the following requirements for a practical quantum computer:[21] scalable physically to increase the number of cubits; cubits can be initialized to arbitrary values; quantum gates faster than despondence time; universal gate set; bits can be read easily. Quantum despondence One of the greatest challenges is controlling or removing quantum despondence. This usually means isolating the system from its environment as interactions with the external world cause the system to decoder.

However, other sources of despondence also exist. Examples include the quantum gates, and the lattice vibrations and background nuclear spin of the physical system used to implement the cubits. Despondence is irreversible, as it is non-unitary, and is usually something that should particular the transverse relaxation time TO (for NORM and MR. technology, also called he depending time), typically range between nanoseconds and seconds at low temperature. 11] These issues are more difficult for optical approaches as the timescales are orders of magnitude shorter and an often-cited approach to overcoming them is optical pulse shaping. Error rates are typically proportional to the ratio of operating time to despondence time, hence any operation must be completed much more quickly than the despondence time. If the error rate is small enough, it is thought to be possible to use quantum error correction, which corrects errors due to coherence, thereby allowing the total calculation time to be longer than the despondence time.

An often cited figure for required error rate in each gate is 10-4. This implies that each gate must be able to perform its task in one 10,10th of the despondence 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 cubits. The number required to factor integers using Shore’s algorithm is still polynomial, and thought to be between L and

LA, where L is the number of bits in the number to be factored; error correction algorithms would inflate this figure by an additional factor of L. For a 1000-bit number, this implies a need for about 104 cubits without error correction. [22] With error correction, the figure would rise to about 107 cubits. Note that computation time is about LA or about 107 steps and on 1 Mesh, about 10 seconds. A very different approach to the stability-despondence problem is to create a topological quantum computer with anions, quasi-particles used as threads and relying on braid theory to form stable logic gates.