Quantum computing
Quantum computing studies theoretical computation systems (quantum computers) that make direct use of mechanical Phenomena such as superposition and entanglement, to perform operations on data.
Quantum computers are different from binary
digital based on transistors. Whereas common digital computing
requires that the data be encoded into binary digits (bits),
each of which is always in one of two definite states (0 or 1), quantum
computation uses quantum bits,
which can be in super positions of states. A quantum Turing machine is a theoretical model of such a
computer, and is also known as the universal quantum computer. The field of
quantum computing was initiated by the work of Paul Ben
off and Yuri Manin in 1980, Richard Feynman in 1982,and David Deutsch in 1985.A quantum computer with
spins as quantum bits was also formulated for use as a quantum space–time in 1968.
 |
|
As of 2017, the development of actual quantum computers is still
in its infancy, but experiments have been carried out in which quantum
computational operations were executed on a very small number of quantum bits. Both
practical and theoretical research continues, and many national governments and
military agencies are funding quantum computing research in an effort to
develop quantum computers for civilian, business, trade,
environmental and national security purposes, such as cryptanalysis.
 |
|
Large-scale quantum computers would theoretically be able to solve
certain problems much quicker than any classical computers that use even 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, that run
faster than any possible probabilistic classical algorithm. A classical
computer could in principle (with exponential
resources) simulate a quantum algorithm, as quantum computation does not
violate the Church–Turing thesis. On the other hand, quantum computers
may be able to efficiently solve problems which are not practically feasible on classical computers.
Basis:
A classical computer has a memory made up of bits, where each bit is represented by either a
one or a zero. A quantum computer maintains a sequence of qubits. A single qubit can
represent a one, a zero, or any quantum
superposition of
those two qubit states; a pair of qubits can be in any
quantum superposition of 4 states, and three qubits in any superposition of 8
states. In general, a quantum computer with {\displaystyle n}qubits can be in an arbitrary
superposition of up to different states simultaneously (this compares to a
normal computer that can only be in one of these {\displaystyle 2^{n}}states at any one
time). A quantum computer operates by setting the qubits in a perfect drift
that represents the problem at hand and by manipulating those qubits 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 qubits into
one of the {\displaystyle 2^{n}}pure
states, where each qubit is zero or one, decomposing into a classical state.
The outcome can therefore be at most {\displaystyle n}classical bits of information. Quantum algorithms
are often probabilistic, in that they provide the correct solution only with a
certain known probability. Note that the term non-deterministic computing must
not be used in that case to mean probabilistic (computing), because the term non-deterministic has a different meaning in
computer science.
 |
|
An example of an implementation of qubits of a
quantum computer could start with the use of particles with two spin states: "down" and
"up"{\displaystyle |1{\rangle
}}). 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 Eigen values,
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.
A quantum computer with a given number of
qubits is fundamentally different from a classical computer composed of the
same number of classical bits. For example, representing the state of an n-qubit
system on a classical computer requires the storage of 2n complex coefficients, while to
characterize the state of a classical n-bit system it is sufficient
to provide the values of the n bits, that is, only n numbers.
Although this fact may seem to indicate that qubits can hold exponentially more
information than their classical counterparts, care must be taken not to
overlook the fact that the qubits are only in a probabilistic superposition of
all of their states. This means that when the final state of the qubits is
measured, they will only be found in one of the possible configurations they
were in before the measurement. It is in general incorrect to think of a system
of qubits as being in one particular state before the measurement, since the
fact that they were in a superposition of states before the measurement was
made directly affects the possible outcomes of the computation.
To better understand this point, consider a
classical computer that operates on a three-bit register. If the exact state of the register
at a given time is not known, it can be described as a probability distribution
over the {\displaystyle 2^{3}=8}different
three-bit strings 000, 001, 010, 011, 100, 101, 110, and 111. If
there is no uncertainty over its state, then it is in exactly one of these
states with probability 1. However, if it is a probabilisticcomputer,
then there is a possibility of it being in any one of a number
of different states.
Principles of operation:
A quantum computer with a given number of
qubits is fundamentally different from a classical computer composed of the
same number of classical bits. For example, representing the state of an n-qubit
system on a classical computer requires the storage of 2n complex coefficients, while to
characterize the state of a classical n-bit system it is sufficient
to provide the values of the n bits, that is, only n numbers.
Although this fact may seem to indicate that qubits can hold exponentially more
information than their classical counterparts, care must be taken not to
overlook the fact that the qubits are only in a probabilistic superposition of
all of their states. This means that when the final state of the qubits is
measured, they will only be found in one of the possible configurations they
were in before the measurement. It is in general incorrect to think of a system
of qubits as being in one particular state before the measurement, since the fact
that they were in a superposition of states before the measurement was made
directly affects the possible outcomes of the computation.
 |
|
To
better understand this point, consider a classical computer that operates on a
three-bit register. If the exact state of the register
at a given time is not known, it can be described as a probability distribution
over the {\displaystyle 2^{3}=8}different
three-bit strings 000, 001, 010, 011, 100, 101, 110, and 111. If
there is no uncertainty over its state, 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
states.
Quantum Coherence:
There are
two formulations of mechanics: classical mechanics and quantum
mechanics. Classical mechanics was the formulation of the laws of
nature until 1900. After 1900, quantum mechanics was discovered/invented as a
way to explain numerous anomalies that couldn't be explained by classical
mechanics. It was quickly shown that many aspects of classical mechanics
could be derived from quantum mechanics.
Having quantum mechanics supersede classical mechanics is necessary to avoid
the contradictions between the two frameworks.
The primary challenge was the transition from quantum realm where there are
super positions of states (i.e. a particle can be "here and there", rather than only
"here or there" as classical mechanics
dictates). There were ad hoc rules that were created to make the
transition from one realm to the other and the loose concept of
"measurement" was introduced to dictate when a quantum system becomes
classical. This led to many conceptual problems such as Schrodinger Cat.
It took a surprisingly long time to construct a non-ad hoc process to describe
the change -- if you think about it, you could expect that the quantum realm
and classical realm might not be a dichotomy, but instead something that can be
interpolated between (i.e. a system might have lost part of its quantumness,
but not all of it).
Quantum decoherence is the physical process that describes this transition from
the quantum realm to the classical realm. Somewhat shockingly, quantum
decoherence follows directly from taking quantum mechanics itself
seriously. No real, extension of quantum mechanics was necessary.
The details of quantum decoherence are as follows and are somewhat technical.
In quantum mechanics, a system is described by a state and if you have multiple
independent components, you combine the states by concatenating the multiple
states together.
To see quantum mechanical behavior, a system must separated into two different
states and then be brought back together. I'll describe this as a
process: Start in a state, split that state into two components (a coherent
superposition), and bright the states back together
[math] |\Psi_1\rangle \rightarrow_{\text{split}}
(|\Psi_1\rangle + |\Psi_2\rangle)/\sqrt {2} \rightarrow_{\text{combine}}
|\Psi_1\rangle[/math]
The key realization is that you can't ignore the rest of the Universe. If
the two parts of the quantum wave function interact with the rest of the
Universe while they were apart, they change the Universe. If this
happens, then when you try to bring the states back together, the Universe has
changed (and you can't unchange the Universe simply, it's like unscrambling an
egg).
Mathematically
[math] |\Psi_1\rangle|U\rangle[/math]
[math].\qquad\rightarrow_{\text{split}}[/math]
[math].\qquad\qquad (|\Psi_1\rangle|U\rangle +
|\Psi_2\rangle |U'\rangle)/\sqrt{2} [/math]
[math].\qquad\rightarrow_{\text{combine}} [/math]
[math].\qquad\qquad(|\Psi_1\rangle|U\rangle+
|\Psi_1\rangle |U'\rangle)\sqrt{2}[/math].
Where the second part of the "state" is the rest of the
Universe. Therefore, you can't observe quantum mechanical interference,
the critical aspect that differentiates the quantum realm from the classical
realm. The degree that the Universe has changed between the two halves of the
wave function is the degree to which the quantumness has been lost. {\displaystyle
|-\rangle ={\tfrac {1}{\sqrt {2}}}\left(1,-1\right)}
