Quantum computer
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 more quickly
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 (this compares
to a normal computer that can only be in one
of these states at any one time).
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" (typically written and, or and ). 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
(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-qubit quantum computer
is similarly described by an eight-dimensional
vector (a,b,c,d,e,f,g,h), 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 (states) 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 (i.e., the probability of measuring
000 =, the probability of measuring 001 =, etc..).
Thus, measuring a quantum state described
by complex coefficients (a,b,...,h) 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 (e.g., 000, 001,...,
111) 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 (a,b,c,d,e,f,g,h) in
the computational basis can be written as:
The computational basis for a single qubit
(two dimensions) 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 (1,0,0,0,0,0,0,0).
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 L1 norm).
In quantum computation, on the other hand,
allowed operations are unitary matrices, which
are effectively rotations (they preserve that
the sum of the squares add up to one, the
Euclidean or L2 norm).
(Exactly what unitaries can be applied depend
on the physics of the quantum device.)
Consequently, since rotations can be undone
by rotating backward, quantum computations
are reversible.
(Technically, quantum operations can be probabilistic
combinations of unitaries, 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-qubit
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, 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 (e.g., 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 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 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 (secret key)
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 (see
Key size).
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 (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.
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 (for NMR and MRI technology, also called
the dephasing time), 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 (computation decomposed
into sequence of few-qubit quantum gates)
One-way quantum computer (computation decomposed
into sequence of one-qubit measurements applied
to a highly entangled initial state or cluster
state)
Adiabatic quantum computer or computer based
on Quantum annealing (computation decomposed
into a slow continuous transformation of an
initial Hamiltonian into a final Hamiltonian,
whose ground states contains the solution)
Topological quantum computer (computation
decomposed into the braiding of anyons in
a 2D lattice)
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 (distinguished by the physical
system used to realize the qubits):
Superconductor-based quantum computers (including
SQUID-based quantum computers) (qubit implemented
by the state of small superconducting circuits
(Josephson junctions))
Trapped ion quantum computer (qubit implemented
by the internal state of trapped ions)
Optical lattices (qubit implemented by internal
states of neutral atoms trapped in an optical
lattice)
Electrically defined or self-assembled quantum
dots (e.g. the Loss-DiVincenzo quantum computer
or) (qubit given by the spin states of an
electron trapped in the quantum dot)
Quantum dot charge based semiconductor quantum
computer (qubit is the position of an electron
inside a double quantum dot)
Nuclear magnetic resonance on molecules in
solution (liquid-state NMR) (qubit provided
by nuclear spins within the dissolved molecule)
Solid-state NMR Kane quantum computers (qubit
realized by the nuclear spin state of phosphorus
donors in silicon)
Electrons-on-helium quantum computers (qubit
is the electron spin)
Cavity quantum electrodynamics (CQED) (qubit
provided by the internal state of atoms trapped
in and coupled to high-finesse cavities)
Molecular magnet
Fullerene-based ESR quantum computer (qubit
based on the electronic spin of atoms or molecules
encased in fullerene structures)
Linear optical quantum computer (qubits realized
by processing appropriate states of different
modes of the electromagnetic field through
linear optics elements such as mirrors, beam
splitters and phase shifters, e.g.)
Diamond-based quantum computer (qubit realized
by the electronic or nuclear spin of nitrogen-vacancy
centers in diamond)
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 (qubit realized by
the internal electronic state of dopants in
optical fibers)
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.
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 (or qubits)
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 ("Quantum Information
Processing") 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 (USC) 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 (separation of RAM).
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 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 (Universities
Space Research Association) 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 (NSA)
is running a $79.7 million research program
(titled "Penetrating Hard Targets") with the
aim of developing a quantum computer capable
of breaking encryption vulnerable to quantum
computers.
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 ("bounded error, probabilistic, polynomial
time") 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
(or more precisely in the associated class
of decision problems P#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
(however, those amounts might be practically
infeasible).
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.
