QUANTUM COMPUTER

In these we are going to discuss about the topic on
• Qubits
• DNA computing
• Quantum decoherence
• Quantum cryptography
• Power of quantum computing
• Quantum error correcting codes
• Advantages of quantum computing



Introduction
Quantum computing was first theorized just 20 years ago, by a physicist at the Argonne National Laboratory. Paul Benioff is credited with first applying quantum theory to computers in 1981. Benioff theorized about creating a quantum Turing machine. Most digital computers, like the one you are using to read this article, are based on the Turing Theory. Today's computers, like a Turing machine, work by manipulating bits that exist in one of two states: a 0 or a 1. Quantum computers aren't limited to two states; they encode information as quantum bits, or qubits. A qubit can be a 1 or a 0, or it can exist in a superposition that is simultaneously both 1 and 0 or somewhere in between. Qubits represent atoms that are working together to act as computer memory and a processor. Because a quantum computer can contain these multiple states simultaneously, it has the potential to be millions of times more powerful than today's most powerful supercomputer



So what is a ‘Quantum Computer’?
A Quantum Computer is a computer that harnesses the power of atoms and molecules to perform memory and processing tasks. It has the potential to perform certain calculations billions of times faster than any silicon-based computer. The classical desktop computer works by manipulating bits, digits that are binary i.e., which can either represent a zero or a one. Everything from numbers and letters to the status of your modem or mouse are all represented by a collection of bits in combinations of ones and zeros. These bits correspond very nicely with the way classical physics represents the world. Electrical switches can be on or off, objects are in one place or they're not, etc. Quantum computers aren't limited by the binary nature of the classical physical world, however they depend on observing the state of quantum bits or qubits that might represent a one or a zero, might represent a combination of the two or might represent a number expressing that the state of the qubit is somewhere between 1 and 0.
How does a quantum computer work?
In the classical model of a computer, the most fundamental building block - the bit, can only exist in one of two distinct states, a '0' or a '1'. In a quantum computer the rules are changed. Not only can a qubit, exist in the classical '0' and '1' states, but it can also be in a superposition of both! In this coherent state, the bit exists as a '0' and a '1' in a particular manner. Let's consider a register of three classical bits: it would be possible to use this register to represent any one of the numbers from 0 to 7 at any one time. If we then consider a register of three qubits, we can see that if each bit is in the superposition or coherent state, the register can represent all the numbers from 0 to 7 simultaneously!
A processor that can use registers of qubits will in effect be able to perform calculations using all the possible values of the input registers simultaneously. This phenomenon is called quantum parallelism, and is the motivating force behind the research being carried out in quantum computing.
Today's Quantum Computers: Quantum computers could one day replace silicon chips, just like the transistor once replaced the vacuum tube. But for now, the technology required to develop such a quantum computer is beyond our reach. Most research in quantum computing is still very theoretical. The most advanced quantum computers have not gone beyond manipulating more than 7 qubits, meaning that they are still at the "1 + 1" stage. However, the potential remains that quantum computers one day could perform, quickly and easily, calculations that are incredibly time-consuming on conventional computers.
Comparision of digital computers with quantum computers
Today’s digital supercomputers would take billions of years to find the prime factors of a number that is a few hundred digits long, whereas large-scale quantum computers, if they can eventually be built, might perform that task in just seconds.
• A classical computer requires a time proportional to N to search for a particular item in a list of N items, whereas a quantum computer can perform the search in a time proportional to the square root of N.
• If quantum information rather than classical information is exchanged between processors, then the amount of communication required to perform certain distributed computing tasks can be drastically reduced.
• A quantum computer could efficiently and accurately simulate the evolution of quantum many-body systems and quantum field theories that cannot be simulated on classical computers without making unjustified approximations
Quantum decoherence
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. This effect 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 NMR and MRI technology, also called the dephasing time), typically range between nanoseconds and seconds at low temperature. The issue for optical approaches are more difficult as these timescales are orders of magnitude lower and an often cited approach to overcome it uses optical pulse shaping approach. 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 the total calculation time to be 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 L4 and L6, 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.
Quantum cryptography Quantum cryptography uses quantum mechanics for secure communications. Unlike traditional cryptography, which depends on the computational complexity of mathematical techniques to restrict the possibility that eavesdroppers might learn the contents of encrypted messages, quantum cryptography depends on the fact that naive attempts to read quantum information will destroy the information (ie. there is no way to copy unknown quantum states). For the message participants, a combination of quantum and classical techniques is used to produce a key which can be proven to be secure —that is, a produced hidden key cannot have been read by any other than the intended participants.
In quantum information, eavesdropping can be viewed as measurements on a physical object in this case the carrier of the information. Using quantum phenomena such as quantum superpositions or quantum entanglement one can design and implement a communication system which can always detect eavesdropping. This is because measurements on the quantum carrier of information disturbs it and therefore leaves traces.
qubit
A qubit is not to be confused with a cubit, which is an ancient measure of length.
A quantum bit, or qubit is a unit of quantum information. That information is described by a state vector in a two-level quantum mechanical system which is formally equivalent to a two-dimensional vector space over the complex numbers.
Benjamin Schumacher discovered a way of interpreting quantum states as information. He came up with a way of compressing the information in a state, and storing the information on a smaller number of states.This is now known as Schumacher compression. Schumacher is also credited with inventing the term qubit.
Bit versus qubit A bit is the base of computer information. Regardless of its physical representation, it is always read as either a 0 or a 1. An analogy to this is a light switch - the down position can represent 0 (normally equated to off) and the up position can represent 1 (normally equated to on).A qubit has some similarities to a classical bit, but is overall very different. Like a bit, a qubit can have only two possible values - normally a 0 or a 1. The difference is that whereas a bit must be either 0 or 1, a qubit can be 0, 1, or a superposition of both.
Representation
The states a qubit may be measured in are known as basis states (or vectors). As is the tradition with any sort of quantum states, Dirac, or bra-ket notation is used to represent them.This means that the two computational basis states are conventionally written as and (pronounced: 'ket 0' and 'ket 1').

DNA computers have the potential to take computing to new levels, picking up where Moore's Law leaves off. There are several advantages to using DNA instead of silicon:
As long as there are cellular organisms, there will always be a supply of DNA.
• The large supply of DNA makes it a cheap resource.
• Unlike the toxic materials used to make traditional microprocessors, DNA biochips can be made cleanly.
• DNA computers are many times smaller than today's computers.
DNA's key advantage is that it will make computers smaller than any computer that has come before them, while at the same time holding more data. One pound of DNA has the capacity to store more information than all the electronic computers ever built; and the computing power of a teardrop-sized DNA computer, using the DNA logic gates, will be more powerful than the world's most powerful supercomputer. More than 10 trillion DNA molecules can fit into an area no larger than 1 cubic centimeter (0.06 cubic inches). With this small amount of DNA, a computer would be able to hold 10 terabytes of data, and perform 10 trillion calculations at a time. By adding more DNA, more calculations could be performed.
Unlike conventional computers, DNA computers perform calculations parallel to other calculations. Conventional computers operate linearly, taking on tasks one at a time. It is parallel computing that allows DNA to solve complex mathematical problems in hours, whereas it might take electrical computers hundreds of years to complete them.
The first DNA computers are unlikely to feature word processing, e-mailing and solitaire programs. Instead, their powerful computing power will be used by national governments for cracking secret codes, or by airlines wanting to map more efficient routes. Studying DNA computers may also lead us to a better understanding of a more complex computer -- the human brain.
The power of quantum computers
Integer factorization is believed to be computationally infeasible with an ordinary computer for large numbers that are the product of two prime numbers of roughly equal size (e.g., products of two 300-digit primes). By comparison, a quantum computer could solve this problem relatively easily. 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 use Shor's algorithm to find its factors. It can also solve a related problem called the discrete logarithm problem. This ability would allow a quantum computer to "break" many of the cryptographic systems in use today, in the sense that there would be a relatively fast (polynomial time in n) algorithm for solving the problem. In particular, most of the popular public key ciphers could be much more quickly broken, including forms of RSA and ElGamal. 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. The only way to increase the security of an algorithm like RSA would be to increase the key size and hope that an adversary does not have the resources to build and use a powerful enough quantum computer. It seems plausible 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 by ensuring that keylengths exceed the storage capacities of quantum computers, but at the cost of an extreme penalty in computational time.
There are some digital signature schemes that are believed to be secure against quantum computers. See for instance Lamport signatures.
Perhaps not as surprisingly, quantum computers could also be useful for running simulations of quantum mechanics. This idea goes back to Richard Feynman (1982) who observed that there is no known algorithm for simulating quantum systems on a classical computer and suggested to study the use of quantum computer for this purpose. The speedup achieved by quantum computers 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. For example, some modern simulations that are taking IBM's Blue Gene supercomputer years, would take a quantum computer only a matter of seconds.
This dramatic advantage of quantum computers is currently known to exist for only those three problems: factoring, discrete logarithm, and quantum physics simulations. However, there is no proof that the advantage is real: an equally fast classical algorithm may still be discovered (though some consider this 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 for at least one problem.
Consider a problem that has these four properties:
1. The only way to solve it is to guess answers repeatedly and check them,
2. There are n possible answers to check,
3. 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.
Quantum Error-correcting Codes
One of the most surprising recent developments in quantum information science, and one of the most important, is the discovery that unknown quantum states, if properly encoded, can be protected from errors . Since the complex states that arise at intermediate stages of a quantum computation are extraordinarily fragile, quantum error correction will be essential to prevent large scale quantum computers from crashing.
The state of a quantum computer can be viewed as a vector in an abstract space of very high dimension. On first acquaintance, it sounds strange that a vector that takes values in a continuum (in contrast to the discrete values assumed by a classical bit string) can be protected against damage. How will we know if the vector drifts slightly in an unexpected direction? The secret of quantum error correction is to encode a quantum state in a cleverly selected subspace of a larger vector space. Errors that move the vector in a direction perpendicular to the code subspace can easily be detected and reversed, while errors parallel to the code subspace cause trouble. But if the code subspace is carefully chosen, typical errors will have only a very small component along the code subspace, and encoded information will be well protected.
The advantages of Quantum Computing:
There are several reasons that researchers are working so hard to develop a practical quantum computer. First, atoms change energy states very quickly -- much more quickly than even the fastest computer processors. Next, given the right type of problem, each qubit can take the place of an entire processor -- meaning that 1,000 ions of say, barium, could take the place of a 1,000-processor computer. The key is finding the sort of problem a quantum computer is able to solve.
If functional quantum computers can be built, they will be valuable in factoring large numbers, and therefore extremely useful for decoding and encoding secret information. If one were to be built today, no information on the Internet would be safe. Our current methods of encryption are simple compared to the complicated methods possible in quantum computers. Quantum computers could also be used to search large databases in a fraction of the time that it would take a conventional computer.
It has been shown in theory that a quantum computer will be able to perform any task that a classical computer can. However, this does not necessarily mean that a quantum computer will outperform a classical computer for all types of task. If we use our classical algorithms on a quantum computer, it will simply perform the calculation in a similar manner to a classical computer. In order for a quantum computer to show its superiority it needs to use new algorithms which can exploit the phenomenon of quantum parallelism. The implications of the theories involved in quantum computation reach further than just making faster computers.
Conclusion:
Although the future of quantum computing looks promising, we have only just taken our first steps to actually realizing a quantum computer. There are many hurdles, which need to be overcome before we can begin to appreciate the benefits they may deliver. Researchers around the world are racing to be the first to achieve a practical system, a task, which some scientists think, is futile. David Deutsch - one of the groundbreaking scientists in the world of quantum computing - himself said, "Perhaps, their most profound effect may prove to be theoretical".
Can we really build a useful quantum computer? Who knows; in a quantum world, anything is possible!