A quantum computer is a computation device
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,
each of which is always in one of two definite
states, quantum computation uses qubits, which
can be in superpositions of states. 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;
one example is the ability to be in more than
one state simultaneously. The field of quantum
computing was first introduced by Yuri Manin
in 1980 and Richard Feynman in 1982. A quantum
computer with spins as quantum bits was also
formulated for use as a quantum space–time
in 1969.
As of 2014 quantum computing is still in its
infancy but 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 governments 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 will be able
to solve certain problems much quicker than
any classical computer 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 sufficient computational
resources, however, a classical computer could
be made to simulate any quantum algorithm;
quantum computation does not violate the Church–Turing
thesis.
Basis
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 qubits. A single qubit can represent
a one, a zero, or any quantum superposition
of these two qubit states; moreover, a pair
of qubits can be in any quantum superposition
of 4 states, and three qubits in any superposition
of 8. In general, a quantum computer with
qubits can be in an arbitrary superposition
of up to different states simultaneously.
A quantum computer operates by setting the
qubits in a controlled initial state 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 pure states, where
each qubit is purely zero or one. The outcome
can therefore be at most 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 qubits
for a quantum computer could start with the
use of particles with two spin states: "down"
and "up". 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
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.
Bits vs. qubits
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, to represent the state
of an n-qubit system on a classical computer
would require the storage of 2n complex coefficients.
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 measurement.
Moreover, it is incorrect to think of the
qubits as only being in one particular state
before 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.
For example: Consider first a classical computer
that operates on a three-bit register. The
state of the computer at any time is a probability
distribution over the different three-bit
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
states. We can describe this probabilistic
state by eight nonnegative numbers A,B,C,D,E,F,G,H.
There is a restriction that these probabilities
sum to 1.
The state of a three-qubit quantum computer
is similarly described by an eight-dimensional
vector, called a ket. Here, however, the coefficients
can have complex values, and it is the sum
of the squares of the coefficients' magnitudes,
, 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 represents a
meaningful parameter. This is a fundamental
difference between quantum computing and probabilistic
classical computing.
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:
where, e.g.,
The computational basis for a single qubit
is and .
Using the eigenvectors of the Pauli-x operator,
a single qubit is and .
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, , 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 or the discrete logarithm
problem, which can both be solved by Shor's
algorithm. In particular the RSA, Diffie-Hellman,
and Elliptic curve Diffie-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 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 time equal to 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. 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 cause
the system to decohere. However, other sources
of decoherence also exist. Examples include
the quantum gates, and the lattice vibrations
and background nuclear spin of the physical
system used to implement the qubits. Decoherence
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.
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 decoherence
time, hence any operation must be completed
much more quickly than the decoherence time.
If the error rate is small enough, it is thought
to be 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 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 10,000th of 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 L and L2, 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 qubits without error
correction. With error correction, the figure
would rise to about 107 qubits. Note that
computation time is about L2 or about 107
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:
Quantum gate array
One-way quantum computer
Adiabatic quantum computer or computer based
on Quantum annealing
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.
For physically implementing a quantum computer,
many different candidates are being pursued,
among them:
Superconductor-based quantum computers)
Trapped ion quantum computer
Optical lattices
Electrically defined or self-assembled quantum
dots
Quantum dot charge based semiconductor quantum
computer
Nuclear magnetic resonance on molecules in
solution
Solid-state NMR Kane quantum computers
Electrons-on-helium quantum computers
Cavity quantum electrodynamics
Molecular magnet
Fullerene-based ESR quantum computer
Linear optical quantum computer
Diamond-based quantum computer
Bose–Einstein condensate-based quantum computer
Transistor-based quantum computer – string
quantum computers with entrainment of positive
holes using an electrostatic trap
Rare-earth-metal-ion-doped inorganic crystal
based quantum computers
The large number of candidates demonstrates
that the topic, in spite of rapid progress,
is still in its infancy. But at the same time,
there is also a vast amount of flexibility.
Timeline
In 2001, researchers were able to demonstrate
Shor's algorithm to factor the number 15 using
a 7-qubit NMR computer.
In 2005, researchers at the University of
Michigan built a semiconductor chip that 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.
In 2009, researchers at Yale University created
the first rudimentary solid-state quantum
processor. The two-qubit superconducting chip
was able to run elementary algorithms. Each
of the two artificial atoms were made up of
a billion aluminum atoms but they acted like
a single one that could occupy two different
energy states.
Another team, working at the University of
Bristol, also created a silicon-based quantum
computing chip, based on quantum optics. The
team was able to run Shor's algorithm on the
chip. Further developments were made in 2010.
Springer publishes a journal devoted to the
subject.
In April 2011, a team of scientists from Australia
and Japan made a breakthrough in quantum teleportation.
They successfully transferred a complex set
of quantum data with full transmission integrity
achieved. Also the qubits being destroyed
in one place but instantaneously resurrected
in another, without affecting their superpositions.
In 2011, D-Wave Systems announced the first
commercial quantum annealer on the market
by the name D-Wave One. The company claims
this system uses a 128 qubit processor chipset.
On May 25, 2011 D-Wave announced that Lockheed
Martin Corporation entered into an agreement
to purchase a D-Wave One system. Lockheed
Martin and the University of Southern California
reached an agreement to house the D-Wave One
Adiabatic Quantum Computer at the newly formed
USC Lockheed Martin Quantum Computing Center,
part of USC's Information Sciences Institute
campus in Marina del Rey. D-Wave's engineers
use an empirical approach when designing their
quantum chips, focusing on whether the chips
are able to solve particular problems rather
than designing based on a thorough understanding
of the quantum principles involved. This approach
was liked by investors more than by some academic
critics, who said that D-Wave had not yet
sufficiently demonstrated that they really
had a quantum computer. Such criticism softened
once D-Wave published a paper in Nature giving
details, which critics said proved that the
company's chips did have some of the quantum
mechanical properties needed for quantum computing.
During the same year, researchers working
at the University of Bristol created an all-bulk
optics system able to run an iterative version
of Shor's algorithm. They successfully factored
21.
In September 2011 researchers also proved
that a quantum computer can be made with a
Von Neumann architecture.
In November 2011 researchers factorized 143
using 4 qubits.
In February 2012 IBM scientists said that
they had made several breakthroughs in quantum
computing with superconducting integrated
circuits that put them "on the cusp of building
systems that will take computing to a whole
new level."
In April 2012 a multinational team of researchers
from the University of Southern California,
Delft University of Technology, the Iowa State
University of Science and Technology, and
the University of California, Santa Barbara,
constructed a two-qubit quantum computer on
a crystal of diamond doped with some manner
of impurity, that can easily be scaled up
in size and functionality at room temperature.
Two logical qubit directions of electron spin
and nitrogen kernels spin were used. A system
which formed an impulse of microwave radiation
of certain duration and the form was developed
for maintenance of protection against decoherence.
By means of this computer Grover's algorithm
for four variants of search has generated
the right answer from the first try in 95%
of cases.
In September 2012, Australian researchers
at the University of New South Wales said
the world's first quantum computer was just
5 to 10 years away, after announcing a global
breakthrough enabling manufacture of its memory
building blocks. A research team led by Australian
engineers created the first working "quantum
bit" based on a single atom in silicon, invoking
the same technological platform that forms
the building blocks of modern day computers,
laptops and phones.
In October 2012, Nobel Prizes were presented
to David J. Wineland and Serge Haroche for
their basic work on understanding the quantum
world—work which may eventually help make
quantum computing possible.
In November 2012, the first quantum teleportation
from one macroscopic object to another was
reported.
In December 2012, the first dedicated quantum
computing software company, 1QBit was founded
in Vancouver, BC. 1QBit is the first company
to focus exclusively on commercializing software
applications for commercially available quantum
computers, including the D-Wave Two processor.
1QBit's research demonstrated the ability
of superconducting quantum annealing processors
to solve real-world problems.
In February 2013, a new technique, boson sampling,
was reported by two groups using photons in
an optical lattice that is not a universal
quantum computer but which may be good enough
for practical problems. Science Feb 15, 2013
In May 2013, Google Inc announced that it
was launching the Quantum Artificial Intelligence
Lab, to be hosted by NASA's Ames Research
Center. The lab will house a 512-qubit quantum
computer from D-Wave Systems, and the USRA
will invite researchers from around the world
to share time on it. The goal is to study
how quantum computing might advance machine
learning.
In early 2014 it was reported, based on documents
provided by former NSA contractor Edward Snowden,
that the U.S. National Security Agency is
running a $79.7 million research program with
the aim of developing a quantum computer capable
of breaking encryption vulnerable to quantum
computers. The same year, a group of researchers
from ETH Zürich, USC, Google, Microsoft published
a report how to define quantum speedup, and
reported that they were not able to measure
quantum speedup with the D-Wave Two device.
But, they did explicitly not rule out that
quantum speedups might be possible and might
depend on the question posed.
Relation to computational complexity theory
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 half. A quantum computer is
said to "solve" a problem if, for every instance,
its answer will be right with high probability.
If that solution runs in polynomial time,
then that problem is in BQP.
BQP is contained in the complexity class #P,
which is a subclass of PSPACE.
BQP is suspected to be disjoint from NP-complete
and a strict superset of P, but that is not
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 is not known
to be true, and is generally suspected to
be false.
The capacity of a quantum computer to accelerate
classical algorithms has rigid limits—upper
bounds of quantum computation's complexity.
The overwhelming part of classical calculations
cannot be accelerated on a quantum computer.
A similar fact takes place for particular
computational tasks, like the search problem,
for which Grover's algorithm is optimal.
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.
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 does not disprove the Church–Turing
thesis. It has been speculated that theories
of quantum gravity, such as M-theory or loop
quantum gravity, may allow even faster computers
to be built. Currently, defining computation
in such theories is an open problem due to
the problem of time, i.e., there currently
exists no obvious way to describe what it
means for an observer to submit input to a
computer and later receive output.
See also
Chemical computer
DNA computer
Electronic quantum holography
List of emerging technologies
Natural computing
Normal mode
Photonic computing
Post-quantum cryptography
Quantum annealing
Quantum bus
Quantum cognition
Quantum gate
Quantum threshold theorem
Soliton
Timeline of quantum computing
Topological quantum computer
Valleytronics
References
Bibliography
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External links
Stanford Encyclopedia of Philosophy: "Quantum
Computing" by Amit Hagar.
Quantiki – Wiki and portal with free-content
related to quantum information science.
Scott Aaronson's blog, which features informative
and critical commentary on developments in
the field
D-Wave thinks it has built the world's first
commercial quantum computer. Mother Nature
has other ideas, in the January 2014 issue
of Inc. magazine
Quantum Annealing and Computation: A Brief
Documentary Note, A. Ghosh and S. Mukherjee
Maryland University Laboratory for Physical
Sciences: conducts researches for the quantum
computer-based project led by the NSA, named
'Penetrating Hard Target'.
Lectures
Quantum Mechanics and Quantum Computation
— Coursera course by Umesh Vazirani
Quantum computing for the determined — 22
video lectures by Michael Nielsen
Video Lectures by David Deutsch
Lectures at the Institut Henri Poincaré
Online lecture on An Introduction to Quantum
Computing, Edward Gerjuoy
Quantum Computing research by Mikko Möttönen
at Aalto University on YouTube
