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Already scientists have built rudimentary quantum system and its time evolution. Classical computers
computers in the research laboratory to run quan- are being used for quantum simulation in design-
tumalgorithmsandperformcertaincalculations.In- ing novel molecules and creating innovative nano-
tensive research efforts are under way around the products. Quantum computers built upon quantum
world to investigate a number of technologies that systems may excel in simulating naturally occurring
could lead to more powerful and more prevalent quantumsystems,whilelargequantumsystemsmay
quantumcomputersinthenearfuture.Itisbelieved be impossible to simulate in an efficient manner by
that quantum information and quantum bits are to classical computers. A quantum system with b dis-
lead toa21stcentury technological revolution much tinct components may bedescribed with b quantum
as classic information and classic bits did to the bits in a quantum computer, while a classical com-
20th century. Since the theory of quantum mechan- puter requires 2b bits of memory to store its quan-
ics is fundamentally stochastic, randomnessand un- tum state. This advantage allows quantum comput-
certaintyaredeeplyrootedinquantumcomputation ers to efficiently simulate general quantum systems
and quantum information. As a result, quantum al- that are not efficiently simulatable on classical com-
gorithms are of random nature in the sense that puters.
they yield correct solutions only with some prob- In this article we review the concepts of quantum
abilities, and Monte Carlo methods are widely em- computationandintroducequantumalgorithmsand
ployed in quantum simulation. Thus statistics has quantum simulation. The quantum algorithms are
an important role to play in quantum computation, known to be much faster than the available classi-
quantum simulation and quantum information. On cal algorithms. Statistical analyses of quantum al-
theotherhand,quantumcomputationandquantum gorithms and quantum simulation are provided. We
simulation have tremendous potential to revolution- give a brief description on quantum information.
ize computational statistics. Thearticlesectionsstartwithpresentationsinbroad
A quantum system is generally described by its brushstrokes, followed by specific discussions along
state, and the state is mathematically defined to be withsomemathematicalderivationsifnecessary.The
a unit vector in some complex Hilbert space. The intention is to give each topic first an overview and
numberofcomplexnumbersrequiredtocharacterize then a general description and a precise characteri-
the quantum state usually grows exponentially with zation.Itisrecommendedtofocusonthequalitative
thesizeofthesystem,ratherthanlinearly,asoccurs discussions but skip the derivations for the readers
in classical systems. As a consequence, it takes an who would like to get a quick picture of quantum
exponential number of bits of memory on a classical computation and quantum simulation.
computer to store the quantum state, which puts The rest of the paper proceeds as follows. Sec-
classical computers in a difficult position to simu- tion 2 briefly introduces quantum mechanics, quan-
late a quantum system. On the other hand, nature tum probability and quantum statistics. Section 3
quantum systems are able to store and keep track reviewsbasicconceptsofquantumcomputation and
of an exponential number of complex numbers and entanglement.Section4illustratessomewidelyknown
3
QUANTUMCOMPUTATIONAND QUANTUMINFORMATION
quantumalgorithmsandprovidesastatisticalframe- tummechanics,Holevo(1982),Parthasarathy(1992)
workforthestudyofquantumalgorithms.Section 5 andWang(1994)forquantumprobabilityandquan-
presents quantum simulation and discusses its sta- tumstochastic processes,andArtiles, GillandGu¸t˘a
tistical analysis. Section 6 gives a short description (2005) and Barndorff-Nielsen, Gill and Jupp (2003)
on quantum information theory. Section 7 features for quantum statistics.
concluding remarks and lists some open research
2.1 Hilbert Space and Operator
problems.
For the sake of simplicity we choose to work with
2. BRIEF BACKGROUND REVIEW ON comparativelyeasyfinite-dimensionalsituations.De-
QUANTUM THEORY note by C the set of all complex numbers. We start
with vector space in linear algebra. A simple exam-
Quantum mechanics has been applied to every-
ple of vector space is Ck consisting of all k-tuples
thing under and inside the Sun,from chemical reac-
tion and superconductor to the structure of DNA of complex numbers (z 1,...,z k). The elements of
and nuclear fusion in stars. Although the signifi- a vector space are called vectors. As in quantum
cant difference between classical physics and quan- mechanics and quantum computation, we use Dirac
tum physics lies in the quantum prediction of phys- notations (which is called ket) and (which is
|·i h·|
ical entity when the scale of observations becomes called bra) to indicate that the objects are column
comparabletotheatomic orsub-atomicscale,many vectors or row vectors in the vector space, respec-
macroscopic properties of systems can only be fully tively. Denote by superscripts , and the conju-
∗ ′ †
explained and understoodby quantum physics. The gate of a complex number, the transpose of a vec-
quantum world is extremely strange, and quantum tor or matrix, and conjugate transpose operation,
theory is completely counterintuitive. Light waves respectively. We define an inner producton the vec-
behave like particles and particles behave like waves tor space to be a function that takes as input two
(wave particle duality);matter cangofromonespot vectors from the vector space and produces a com-
to another without moving through the intermedi- plex number as output. For u and v in the vec-
| i | i
ate space (quantum tunneling); information can be tor space, we denote their inner product by uv .
h | i
moved across a vast distance without transmitting The inner product must satisfy (i) conjugate sym-
it through the intervening space (quantum telepor- metry, uv =( v u ) ∗; (ii) linearity in the second
h | i h | i
tation). Quantum theory provides a mathematical argument, uv +w = uv + uw ; (iii) positive-
h | i h | i h | i
description of wave particle duality and interaction definiteness, uu 0 with equality only for u=0.
example,hC| i h≥
of matter and energy. It describes the time evolu- For k as a natural inner product
tions of physical systems via wave functions. The
k
wave functions encapsulate the probabilities that uv = u v =(u ,...,u )(v ,...,v ),
∗j j ∗1 ∗k 1 k ′
particles are to be found in a given state at a given h | i
Xj=1
time. For example, the probability of finding a pho-
where u =(u ,...,u ) and v =(v ,...,v ). An
ton in some region is the square of the modulus 1 k 1 k ′
h | | i
inner product induces a norm u = uu , and
of a wave function, and, since at some point the
k k h | i
a distance u v between u and pv . For the
sum of two wave functions can be zero but neither
k − k | i | i
finite-dimensional case, a Hilbert space is simply
wave function is zero, probabilities appear to can-
H
a vector space with an inner product.
cel out each other in a way totally unexpected from