Everything about Quantum Computer totally explained
A
quantum computer is a hypothetical 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, information is stored as
bits; in a quantum computer, it's stored as
qubits (
quantum
binary digi
ts). The basic principle of quantum computation is that the quantum properties can be used to represent and structure data, and that quantum mechanisms can be devised and built to perform
operations with this data.
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. Research in both theoretical and practical areas continues at a frantic pace, 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.
If large-scale quantum computers can be built, that'll be able to solve certain problems much faster than any of our current classical computers (for example
Shor's algorithm). Quantum computers are different from other
computers such as
DNA computers and traditional computers based on
transistors. Some computing architectures such as
optical computers may use classical superposition of electromagnetic waves, but without some specifically quantum mechanical resources such as
entanglement, they don't have the same computational speed-up as quantum computers.
The basis of quantum computing
A classical computer has a memory made up of
bits, where each bit holds either a one or a zero. A quantum computer maintains a sequence of
qubits. A single qubit can hold a one, a zero, or, crucially, a
quantum superposition of these; moreover, a pair of qubits can be in a quantum superposition of 4 states, and three qubits in a superposition of 8. In general a quantum computer with n qubits can be in up to
different states simultaneously (this compares to a normal computer that can only be in
one of these
states at any one time). A quantum computer operates by manipulating those qubits with a fixed sequence of
quantum logic gates. The sequence of gates to be applied is called a
quantum algorithm.
An example of an implementation of qubits for a quantum computer could start with the use of particles with two
spin states: "up" and "down" (typically written
and
). But 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. This is true because any such system can be mapped onto an effective
spin-1/2 system.
Quantum Hardware
Before talking about quantum hardware, you must know the basis of the computer hardware and physics.
As a small introduction to quantum hardware, you should know about the two positions of the hardware: 0 and 1;
The 0 position or Ђ is the stage where, for example the Hardisk, is turned off and no electricity passes through it,
and the 1 position or ђ is the "turned on".
The atom has the same positions (Ђ and ђ) but it's located between them, called superposition.
Now, to use the quantum hardware the scientists are trying to make an atom conserve information without losing data. For better understanding of
superposition see
Mach-Zehnder interferometer.
Bits vs. Qubits
Consider first a classical computer that operates on a 3-bit
register. At any given time, the bits in the register are in a definite state, such as 101. In a quantum computer, however, the qubits can be in a superposition of all the classically allowed states. In fact, the register is described by a
wavefunction:
»
where the coefficients a, b, c,..., h are
complex numbers whose amplitudes squared are the probabilities to measure the qubits in each state- for example,
is the probability to measure the register in the state 010. It is important that these numbers are complex, because the
phases of the numbers can constructively and destructively interfere with one another; this is an important feature for quantum algorithms. The basis made up of 0 and 1 (true and false) is called the computational basis, but other bases can be used. Another common basis, used for example in measurement based quantum computation is the Hadamard basis of
and
. Any two
orthogonal vectors can be used as a basis.
Recording the state of a
quantum register requires an exponential number of complex numbers (the 3-qubit register above requires
complex numbers). The number of classical bits required even to estimate the complex numbers of some quantum state grows exponentially with the number of qubits. For a 300-qubit quantum register, somewhere on the order of
classical registers are required, more than there are atoms in the
observable universe.
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). In 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 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'll 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 one particular value: 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. In brief, quantum computations are probabilistic; see
quantum circuit for a more precise formulation.
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. Also refer to the growing field of
quantum programming.
The power of quantum computers
Integer factorization is believed to be computationally infeasible with an ordinary computer for large integers that are the product of only a few
prime numbers (for example, products of two 300-digit primes). 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 "break" many of the
cryptographic systems in use today, in the sense that there would be a
polynomial time (in the number of bits 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 related
discrete logarithm problem which can also be solved by Shor's algorithm), 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. The only way to increase the security of an algorithm like
RSA would be to increase the key size and hope that an adversary doesn't have the resources to build and use a powerful enough quantum computer.
A way out of this dilemma would be to use some kind of
quantum cryptography.
There are also some
digital signature schemes that are believed to be secure against quantum computers. See for instance
Lamport signatures.
This dramatic advantage of quantum computers has only been discovered for factorization and discrete logarithms
so far. However, there's no proof that the advantage is real: an equally fast classical algorithm may still be discovered. 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:
- The only way to solve it's 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, 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
n (it would take an average of (
n + 1)/2 guesses to find the answer using a classical computer.) 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.
Regardless of whether any of these problems can be shown to have an advantage on a quantum computer, they nonetheless will always have the advantage of being an excellent tool for studying quantum mechanical interactions, which of itself is an enormous value to the scientific community.
Grover's algorithm can also be used to obtain a quadratic speed-up [overa brute-force search] for a class of problems known as
NP-complete.
Problems and practicality issues
There are a number of practical difficulties in building a quantum computer, and thus far quantum computers have only solved trivial problems. 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
To summarize the problems from the perspective of an engineer, one needs to solve the challenge of building a system which is isolated from everything except the measurement and manipulation mechanism. Furthermore, one needs to be able to turn off the coupling of the qubits to the measurement so as to not decohere the qubits while performing operations on them.
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. With error correction, the figure would rise to about 107 qubits. Note that computation time is about or about 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.
Candidates
There are a number of quantum computing candidates, among those:
Superconductor-based quantum computers (including SQUID-based quantum computers)
Trapped ion quantum computer
Electrons on helium quantum computers
Nuclear magnetic resonance on molecules in solution"-based
Quantum dot on surface (for example the Loss-DiVincenzo quantum computer)
Cavity quantum electrodynamics (CQED)
Molecular magnet
Fullerene-based ESR quantum computer
Solid state NMR Kane quantum computers
Optic-based quantum computers (Quantum optics)
Topological quantum computer
Spin-based quantum computer
Adiabatic quantum computation
Diamond-based quantum computer
Bose–Einstein condensate-based quantum computer
Transistor-based quantum computer - string quantum computers with entrainment of positive holes using a electrostatic trap
The large number of candidates shows explicitly that the topic, in spite of rapid progress, is still in its infancy. But at the same time there's also a vast amount of flexibility.
In 2005, researchers at the University of Michigan built a semiconductor chip which functioned as an ion trap. Such devices, produced by standard lithography techniques, may point the way to scalable quantum computing tools. An improved version was made in 2006.
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 probabilistic 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 quarter., which is a subclass of PSPACE.
BQP is suspected to be disjoint from NP-complete and a strict superset of P, but that isn't 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 NP-complete. There is a common misconception that quantum computers can solve NP-complete problems in polynomial time. That isn't known to be true, and is generally suspected to be false. [
Quantum gates may be viewed as linear transformations. Daniel S. Abrams and Seth Lloyd have shown that if nonlinear transformations are permitted, then NP-complete problems could be solved in polynomial time. It could even do so for #P-complete problems. They don't believe that such a machine is possible.]
Although quantum computers may be faster than classical computers, those 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 doesn't disprove the Church-Turing thesis.[Further Information]
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