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quantum

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is an infinite matrix and then use nonparametric
after observing measurement outcome x we have
a
methods to estimate the density matrix. For details
the following ensemble of states: the quantum sys-
tem is in pure state ψa with probability P(j a), seeArtiles,GillandGu¸t˘a(2005),Barndorff-Nielsen,
| ji | Gill and Jupp (2003), Butucea, Gu¸t˘a and Artiles
where Bayes’s theorem shows
(2007) and Nussbaum and Szkol a (2009).
P(j a)=p P(a j)/P(a).
j
| |
3. QUANTUM COMPUTING CONCEPTS
Thus after measurement x the density operator for
a
the ensemble state is given by Unlike classical computers using transistors to
crunch the ones and zeroes individually, quantum
J
ρ = P(j a)ψa ψa computers can handle both one and zero simulta-
a Xj=1 | | jih j| neously via what are known as superposition quan-
6
Y.WANG
tum states. A superposition state is a state of mat- eiθα 2 = α 2, where x=0,1, i=√ 1, and θ is
x x
| | | | −
ter which we may think of as both one and zero at areal number,from theviewpoint of the qubitmea-
the same time. Quantum computers use the strange surements, states eiθ ψ and ψ are identical. That
| i | i
superposition states and quantum entanglements to is, multiplying a qubit state by a global phase fac-
dothetrick of performingsimultaneous calculations tor eiθ bears no observational consequence.
and extracting the calculated results. The spooky Note the distinction between superposition states
phenomena of quantum entanglement and superpo- andprobabilitymixtures(orensembleofpurestates
sition are the key that enables quantum computers definedinSection2.1).Considersuperposition(0 +
| i
tobesuperfastandvastly outperformclassical com- 1 )/√2 as a pure state. Its density matrix is given
| i
puters. by
3.1 Quantum Bit 1(0 + 1 )( 0 + 1)
2 | i | i h | h |
Analogous to the fundamental concept of bit in = 1(0 0 + 1 1)+ 1(0 1 + 1 0),
classical computation and classical information, we 2 | ih | | ih | 2 | ih | | ih |
while the first term on the right-hand side of the
haveits counterpart,quantumbit,inquantumcom-
above equation corresponds to the ensemble of pure
putation and quantum information. Quantum bit
states 0 and 1 , that is, a probabilistic mixture of
is called qubit for short. Just like a classical bit
| i | i
states 0 and 1 with equal probability.
with state either 0 or 1, a qubit has states 0 and
| i | i
| i Similar to classic bits, we can define multiple qu-
1 . However, the real difference between a bit and
| i bits. The states of b qubits are unit vectors in a 2b-
a qubit is that besides states 0 and 1 , a qubit
| i | i dimensional complex vector space with 2b computa-
may take the superposition states,
tional basis states of the form x x x , x =0
1 2 b j
ψ =α 0 +α 1 , | ··· i
| i 0 | i 1 | i or 1, j = 1,...,b. For example, the states of two
where α and α are complex numbers and called qubits are unit vectors in a four-dimensional com-
0 1
amplitudes satisfying α 2+ α 2=1. That is, the plex vector space, with four computational basis
0 1
| | | |
statesofaqubitareunitvectorsinatwo-dimensional states labeled by 00 , 01 , 10 and 11 . The com-
| i | i | i | i
complex vector space, and states 0 and 1 consist putational basis states 00 , 01 , 10 and 11 gen-
| i | i | i | i | i | i
of an orthonormal basis for the space and are often eratethefour-dimensionalcomplexvectorspace,and
referredto as computational basis states. For a clas- the superposition states are all unit vector in the
sical bit we can examine it to determine whether it space with the forms
isinthestate 0or 1.However, foraqubitwecannot
ψ =α 00 +α 01 +α 10 +α 11 ,
determine its state and find the values of α 0 and α 1 | i 00 | i 01 | i 10 | i 11 | i
by examining it. The stochastic nature of quantum where amplitudes α are complex numbers satisfy-
x
theory shows that we can measure a qubit and ob- ing α 2 + α 2 + α 2 + α 2 = 1. As in the
00 01 10 11
| | | | | | | |
tain either the result 0, with probability α 2, or single qubit case, when two qubits are measured we
0
| |
the result 1, with probability α 2. Physical exper- get result x being one of 00,01,10,11, with prob-
1
iments have realized qubits a| s p| hysical objects in ability α x 2. Moreover, we may measure just the
| |
different physical systems, such as the two states first qubit of the two-qubit system and obtain ei-
of an electron orbiting a single atom, the two dif- ther the result 0, with probability α 00 2 + α 01 2,
| | | |
ferent polarizations of a photon, or the alignment or the result 1, with probability α 2+ α 2. As
10 11
| | | |
of a nuclear spin in a uniform magnetic field. Con- quantum measuring changes the quantum state, if
sider the case of atom model by corresponding 0 the measurement result on the first qubit is 0, after
| i
and 1 with the so-called “ground” and “excited” the measurement the qubits are in the state
| i
states of the electron, respectively. As the atom is