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α 00 +α 01
00 01
shinedbylightwithsuitableenergyandforaproper (5) | i | i.
α 2+ α 2
amount of time, the electron can bemoved from the | 00 | | 01 |
p
0 state to the 1 state and vice versa. Further- Aqubitisthesimplestquantumsystem.Thequan-
m| i ore, by shorten| ini g the length of time shining the tumsystemofbqubitsisdescribedbya2b-dimension-
light on the atom, we may move the electron ini- alcomplexvectorspacewitheachsuperpositionstate
tially in the state 0 to “halfway” between 0 and specifiedby 2b amplitudes.As 2b increases exponen-
| i | i
1 , say, into a state (0 + 1 )/√2. tially in b, it is very easy for such a system to have
| i | i | i
Note that for qubit state ψ the only measurable an enormously big vector space. A quantum sys-
quantities aretheprobabilit| iesi α 2 and α 2;since tem with even a few dozens of “qubits” will strain
0 1
| | | |
7
QUANTUMCOMPUTATIONAND QUANTUMINFORMATION
the resources of even the largest supercomputers. qubit is the 2 2 unitary matrix that realizes the
×
Consider a quantum system of 50 qubits. 250 1015 following transformation:
≈
complex amplitudes are needed to specify its quan- 0 + 1 0 1
tum states. With 128 bits of precision, it requires 0 | i | i, 1 | i−| i.
| i→ √2 | i→ √2
approximately 32thousandterabytes ofinformation
to store all 1015 complex amplitudes. Such storage Consideranotherimportantgateontwoqubitswhich
is called control-NOT gate. It takes the two input
capacity may beavailable infuturesupercomputers.
qubits as control qubit and target qubit, respec-
For a quantum system with b=500 qubits we need
tively, and the output target qubit of the gate re-
to specify 2500 complex amplitudes for its states. It
tains the input target qubit if the control qubit is
is unimaginable to store all 2500 complex numbers
0 and is flipped if the control qubit is 1 , that is,
in any classical computers. In principle, a quantum | i | i
system with only a few hundred atoms can manage 00 00 , 01 01 ,
| i→| i | i→| i
such an enormous amount of data and execute cal-
10 11 , 11 10 .
culations as the system evolves. Quantum computa- | i→| i | i→| i
Generally for any single qubit unitary operation U,
tion and quantum information are to find ways to
a control-U gate is a two-qubit gate, with one con-
utilize the immense potential computational power
trol qubit and one target qubit. If the control qubit
in quantum systems.
is 1 , U is applied to the target qubit; if the control
3.2 Quantum Circuit Model | i
qubit is 0 , the target qubit is left alone, that is,
| i
As a classical computer is built from an electri- 0 0 0 0 , 0 1 0 1 ,
| i| i→| i| i | i| i→| i| i
cal circuit consisting of wires for carrying informa-
1 0 1 U0 , 1 1 1 U1 .
tion around the circuit and logic gates for perform- | i| i→| i | i | i| i→| i | i
ing simple computational tasks, a quantum com- If f(x) maps 0,1 b onto 0,1 , we define a unitary
{ } { }
puter can be created from a quantum circuit with transformationU f thatoperatesonb+1qubitstate
quantum gates to perform quantum computation
(6) x,y x,y f(x) ,
and manipulate quantum information. A number of | i→| ⊕ i
where x=x x with x =0 or 1 is the data reg-
physical systems are being investigated for build- 1 ··· b j
ister, y=0 or 1 is the target register, denotes ad-
ing quantum computers. These include optical pho- ⊕
ditional modulo 2. If y=0, after the transformation
ton, optical cavity quantum electrodynamics, ion
U , the state of the last qubit is the value of f(x).
traps, nuclear magnetic resonance with molecules, f
quantum dots, and superconductors (Nielsen and 3.3 Entanglement
Chuang (2000)). In fact, primitive solid-state quan-
Quantum entanglement is one of the most mind-
tum processors have been created in research lab-
bending creatures known to science. It is referred
oratories to run quantum algorithms (DiCarlo et
to as the phenomenon that two qubits behave like
al. (2009); Johnson et al. (2011); Mariantoni et al.
twins that are connected by an invisible wave to
(2011); Sayrin et al. (2011)). The circuit model is
share each other’s properties.
particularlyimportantinquantumcomputationand
quantum information, and a quantum computer is 3.3.1 Bellstates Consideraquantumgateontwo-
often synonymous with the quantum circuit model. qubit basis states 00 , 01 , 10 and 11 that is
| i | i | i | i
A quantum circuit operates on b qubits for some in- composedofaHadamardgate onthefirstqubitand
teger b. The state takes a form of x x , with then is followed by a control-NOT gate. The output
1 b
state space being a 2b-dimensional c| omp·· le· x Hi ilbert states of the gate are as follows:
space. When x =0 or 1, states x x are the 00 + 11 01 + 10
i 1 b
| ··· i 00 | i | i, 01 | i | i,
computationalbasisstatesofthequantumcomputer | i→ √2 | i→ √2
and often written as x , where x is the integer with
| i 00 11 01 10
binary representation x 1 x b. 10 | i−| i, 11 | i−| i.
Asaclassicallogicgate· c· o· nvertsclassicalbitsfrom | i→ √2 | i→ √2
oneformtoanothersuchas0 1and1 0,aquan- Physicists Bell, Einstein, Podolsky and Rosen dis-