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Beforeperforming anybraiding, it isessential to know 2016;Hakonenetal.,1982),superconductors(Abrikosov,
what braids are necessary to perform the desired opera- 2004; Essmann and Träuble, 1967), Bose–Einstein con-
tion. In quantum computation, the quantum algorithms densates (Abo-Shaeer et al., 2001; Fetter, 2009; Madison
arecomposedofseveralquantumgates,whicheachenact et al., 2000; Matthews et al., 1999) and superfluid Fermi
a predetermined operation. It is necessary to determine gases (Zwierlein et al., 2005). The particular types of
what braid enacts the required gates to within a desired non-Abelian anyons that may be realised depend on the
accuracy, and this is performed using classical computa- physical details of the vortices. For example, in chiral
tion with a combination of exhaustive search (Bonesteel p-wave Fermi systems the vortices may host Majorana
et al., 2005) and iterative methods (Burrello et al., 2011; zero modes (Gurarie and Radzihovsky, 2007; Mizushima
Dawson and Nielsen, 2006; Kitaev, 1997; Kliuchnikov et al., 2008; Volovik, 1999), the topological properties of
et al., 2014). Once the braid corresponding to a given which correspond to the Majorana zero modes found in
gate has been determined, that braid can be recorded solidstatesystemsleadingtoIsinganyons(Sarmaet al.,
5
0 H H
| i
FIG. 3 A weave of non-Abelian anyons approximating the
(cid:0) (cid:1)
Hadamardgate,H= 1 1 1 ,withanerrorof0.003. Time
j
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StatisticalScience
2012,Vol.27,No.3,373–394
DOI:10.1214/11-STS378
(cid:13)c InstituteofMathematicalStatistics,2012
Quantum Computation and Quantum
Information
Yazhen Wang
2102 Abstract. Quantumcomputationandquantuminformationareofgreat
current interest in computer science, mathematics, physical sciences
and engineering. They will likely lead to a new wave of technological
tcO innovations in communication, computation and cryptography. As the
theoryofquantumphysicsisfundamentallystochastic,randomnessand
uncertaintyaredeeplyrootedinquantumcomputation,quantumsimu-
2 lationandquantuminformation.Consequentlyquantumalgorithmsare
random in nature, and quantum simulation utilizes Monte Carlo tech-
]EM.tats[
niques extensively. Thus statistics can play an important role in quan-
tum computation and quantum simulation, which in turn offer great
potential to revolutionize computational statistics. While only pseudo-
random numbers can be generated by classical computers, quantum
computers are able to produce genuine random numbers; quantum
computers can exponentially or quadratically speed up median eval-
1v6370.0121:viXra uation, Monte Carlo integration and Markov chain simulation. This
paper gives a brief review on quantum computation, quantum simu-
lation and quantum information. We introduce the basic concepts of
quantum computation and quantum simulation and present quantum
algorithms that are known to be much faster than the available clas-
sic algorithms. We provide a statistical framework for the analysis of
quantum algorithms and quantum simulation.
Key words and phrases: Quantum algorithm, quantum bit (qubit),
quantum Fourier transform, quantum information, quantum mechan-
ics, quantum Monte Carlo, quantum probability, quantum simulation,
quantum statistics.
1. INTRODUCTION ventional computer technology, this dream run is
ending. The conventional approaches to the fabrica-
Fordecadescomputerhardwarehasgrowninpow-
tion of computer technology are to make electronic
er approximately according to Moore’s law, which
devices smaller and smaller in order to increase the
states that the computer power doubles for con-
computer power. As the sizes of the electronic de-
stant cost roughly once every two years. However,
vices get close to the atomic scale, quantum effects
becauseofthefundamentaldifficultiesofsizeincon-
are starting to interfere in their functioning, and
thustheconventional approaches runupagainstthe
Yazhen Wang is Professor, Department of Statistics,
size limit. One possible way to get around the dif-
University of Wisconsin–Madison, Madison, Wisconsin
ficulties is to move to a new computing paradigm
53706, USA e-mail: yzwang@stat.wisc.edu.
providedbyquantuminformationscience.Quantum
This is an electronic reprint of the original article information science is based on the idea of using
published by the Institute of Mathematical Statistics in quantum devices to perform computation and ma-
Statistical Science, 2012, Vol. 27, No. 3, 373–394. This nipulate and transmit information, instead of elec-
reprint differs from the original in pagination and tronic devices following the laws of classical physics,
typographic detail. see Deutsch (1985), DiVincenzo (1995), Feynman
1
2
Y.WANG
(1981/82).Quantummechanicsandinformationthe- perform data manipulations and calculations as the
ory are two of the great scientific developments and systems evolve. Quantum information science is to
technological revolutions in the 20th century, and grapple with understanding how to take advantage
quantum information science is to marry the two of the enormous information hidden in the quan-
previously disparate fields and form a single unify- tum systems and to harness the immense potential
ingviewpoint.Quantuminformation science studies computationalpowerofatoms andmolecules forthe
the preparation and control of the quantum states purpose of performing computation and processing
of physical systems for the purposes of information information. Already it has been shown that quan-
transmissionandmanipulation.Itincludesquantum tum algorithms like Grover’s search algorithm and
computation, quantum communication and quan- Shor’s factoring algorithm provide great advantage
tum cryptography. This revolutionary field will en- over known classical algorithms.
able a range of exotic new devices to be possible. Contemporary scientific studies often rely on un-
There is now a general agreement that quantum in- derstandingcomplexquantumsystems,suchasthose
formation science will likely lead to the creation of in biochemistry and nanotechnology for the design
a quantum computer to solve problems that could of biomolecules and nano-materials. Quantum sim-
not be efficiently solved on a classical computer. ulation is to use computers to simulate a quantum