PROC. 28th INTERNATIONAL CONFERENCE ON MICROELECTRONICS (MIEL 2012), NIŠ, SERBIA, 13-16 MAY, 2012
Quantum Computers: Registers, Gates and Algorithms
Paul Isaac Hagouel and Ioannis G. Karafyllidis
Abstract - Quantum computing is a process that incorporates
interacting physical systems that represent quantum bits and
quantum gates. We present the quantum bit (qubit), the quantum
register and the quantum gates. The qubit is described as a vector
in a two-dimensional Hilbert space and the quantum register,
which comprises a number of qubits, as a vector in a
multidimensional Hilbert space. Quantum gates are Hilbert space
operators that rotate the qubit or the quantum register vectors.
Quantum computations are modeled and described using a
quantum circuit model. We also present a quantum computer
simulator based on the circuit model of quantum computation.
In this model quantum computations and quantum algorithms are
represented by circuits, which comprise quantum gates and
quantum registers. The well-known Deutsch’s algorithm is
described and the corresponding quantum circuit is presented.
Possible applications of quantum computers are be presented and
discussed.
is the probability of measuring each one of the possible
quantum register basis states and the phase of each state at
each computation step.
We will also present a quantum computer simulator
based on the circuit model of quantum computation [14].
In this model quantum computations and quantum
algorithms are represented by circuits, which comprise
quantum gates and quantum registers. This simulator has
been used for the visualization of the quantum Fourier
transform [15], the study of cellular quantum computer
architecture [16], the simulation of entanglement
generation and variation in quantum computations [17],
and the definition and evolution of quantum cellular
automata [18]. The complete simulator comprises a
graphical user interface used for entering the inputs, which
are the initial states of the qubits that form a quantum
register and the quantum gates applied at each computation
step. The quantum computation is visualized using two
graphical outputs showing the probability of measuring
each one of the possible quantum register basis states and
the phase of each state at each computation step.
The well-known Deutsch’s algorithm was the first
algorithm that showed that a quantum computer can
outperform a classical one. The problem posed by Deutsch
is: “Given a binary function f(x) find out if the function is
constant ( i.e. f(0)=f(1) ) or balanced ( i.e. f(0) ≠ f(1) ) by
performing the minimum number of computations.” On a
classical computer one has to compute the value of f(x)
twice and compare the results, but, as Deutsch showed, on
a quantum computer one computation is enough [19].
This algorithm will be described and the corresponding
quantum circuit will be presented. We will also present
analytically the execution of this algorithm on the quantum
computer simulator. Finally, quantum algorithms and
possible applications of quantum computers will be
presented and discussed.
I. INTRODUCTION
Quantum computation is one of the major challenges
for computer science and engineering in the 21st century
[1, 2]. Quantum computers are far more efficient than their
classical counterparts in factoring large numbers [3],
searching databases [4–6] and simulating physical systems
[7]. All the aspects of quantum computing are currently
under intensive study and, as a result, the first quantum
algorithms appeared before the actual fabrication of the
first quantum gate [8-10].
Recently, several implementations of quantum
computing circuits have been presented. Trapped ions
[11], superconductors [12] and photonics [13].
Here we present the quantum bit (qubit), the quantum
register and the quantum gates. The qubit will be described
as a vector in a two-dimensional Hilbert space and the
quantum register, which comprises a number of qubits, as a
vector in a multidimensional Hilbert space. Quantum gates
are operators of the Hilbert space that rotate the qubit or the
quantum register vectors.
Quantum computations will be modeled and described
using a circuit model in which the inputs are the initial
states of the qubits that form a quantum register and the
computation is broken down in successive steps in which
quantum gates act on the state of the quantum register.
The output of the circuit, i.e. the result of the computation,
II. THE QUBIT AND THE QUANTUM REGISTER
A. The qubit
The unit of quantum information is the quantum bit or
qubit. As its classical analog, the qubit can be in two basis
states which are symbolized as
P. I. Hagouel is with Optelec, 11 Chrysostomou Smyrnis
Street,
GR–54622
Thessaloniki,
GREECE,
E–mail:
hagouel@eecs.berkeley.edu
I. G. Karafyllidis is with the Department of Electrical and
Computer Engineering, Democritus University of Thrace,
GR-67100 Xanthi, GREECE, E-mail: ykar@ee.duth.gr
978-1-4673-0238-8/12/$31.00 © 2012 IEEE
0 and 1 . While a
classical bit can be only 0 or 1, a qubit
states
can be in
0 or 1 , or in any linear combination of these
two states, called a superposition:
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ψ
ψ =a 0 +b1
,
where a and b are complex numbers and are called
probability amplitudes. When measuring the state of a
qubit, which is in a superposition of basis states, the
measurement outcome is 0 with probability a
2
q R = c 0 0 0 + c 1 01 + c 2 10 + c 3 11
The probability of measuring each basis state must be equal
to one:
and 1 with
2
probability b . Since probabilities must sum up to one:
c
a 2 + b 2 = 1 . The qubit state is a vector in a two–
dimensional Hilbert space and can be described as a vector
in the Bloch sphere, shown in Figure 1, and is usually
written as:
ψ = cos
ϑ
2
iϕ
0 + e sin
ϑ
2
1
(3)
0
2
+ c1
2
+ c
2
2
+ c3
2
=1
(4)
Replacing the binary state representation with decimal
numbers, the state of a quantum register with n qubits is
given by:
qR =
(1)
2 n −1
∑c
i
i
(5)
i =0
with
where ω and φ are angles that determine the position of the
qubit vector direction in the Bloch sphere. The phase angle
φ is related to quantum mechanical interference.
2 n −1
∑
i =0
ci
2
=1
(6)
III. QUANTUM GATES
Quantum gates are operators of the Hilbert space that
act on qubits and quantum registers and change their states
by rotating their corresponding state vectors. Quantum
gates can act on one, two or three qubits. The action of a
quantum gate G on a qubit or on a quantum register is
described by:
G q I = qR
where q I is the state of the qubit or the quantum register
Fig. 1. The qubit vector in the Bloch sphere.
before the action of the gate and q R is the new state after
The qubit can be represented by several two-state
quantum systems such as the electron spin or the
polarization of the photon.
the action of the gate. The most important quantum gates
are presented below.
The Hadamard gate, symbolized by H, acts on one
qubit. If the qubit is in a basis state the H gate brings it to a
state superposition in which the probability to measure both
basis states equals to 0.5. The circuit symbol and the action
of the H gate are shown in Figure 2.
The phase shift gate, Φ, acts on one qubit and changes
its phase angle φ as shown in Figure 3.
The controlled–NOT (CNOT) gate acts on a two-qubit
quantum resister and is shown in Figure 4. The upper
B. The quantum register
A quantum register is a physical system that
comprises more than one qubits. The state of a quantum
register is also a vector in a multi-dimensional Hilbert
space and is given by the tensor product of the states of its
qubits. In the case of a quantum register that comprises
two qubits q 1 and
q 0 its state q R
(7)
qubit, c , is the control qubit and the lower qubit, t , is
is given by:
q R = q1 ⊗ q0 = q1 q0 = q1 q0
the target qubit. The gate acts on the target qubit and
reverses its state only if the control qubit is in state 1 .
(2)
The states of the quantum register before and after the
where ⊗ is the tensor product symbol. For example the
state of a four-qubit quantum register is written as:
action of the CNOT gate are:
respectively.
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cI t I
and
cO tO
The controlled phase gate, CΦ, shown in Figure 5,
acts on a two-qubit register and changes the phase angle of
the target qubit by an angle φ if the state of the control
qubit is 1 .
qI
qO
1
2
1
2
0
1
1
2
1
2
(
0 + 1
)
(
0 − 1
)
(
0 + 1
)
0
(
0 − 1
)
1
Fig. 2. The circuit symbol and the action of the H gate.
ci t i
co t o
00
00
01
01
10
10
11
e i ϕ 11
Fig. 5. The circuit symbol and the action of the CΦ gate.
qI
qO
0
0
1
eiϕ 1
a 0 +b 1
a 0 + eiϕ b 1
Fig. 3. The circuit symbol and the action of the Φ gate.
c 2 i c 1i t i
c 2 o c 1o t o
000
000
0 01
0 01
010
010
ci t i
co t o
011
011
00
00
10 0
10 0
01
01
101
101
10
11
110
111
11
10
111
110
Fig. 4. The circuit symbol and the action of the CNOT gate.
Fig. 6. The circuit symbol and the action of the CCNOT gate.
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The controlled-controlled-NOT gate, CCNOT, acts on
a three-qubit quantum register. The two upper qubits
The gates H, Φ and CNOT form a universal quantum
gate set and any quantum computation can be carried out
using only these gates.
t , is the target qubit. The gate acts on the target qubit
IV. QUANTUM COMPUTING
c2 and c1 are the control qubits and the lower qubit,
and reverses its state only if the control qubits are both in
In the circuit model of quantum computation the input
is the initial state of the quantum register. A sequence of
computation steps is executed and at the end the final state
of the quantum register, which is the computation result, is
measured. In each computation step a number of quantum
gates acts on the quantum register. The combined action of
the gates in each step is given by the tensor product of the
corresponding Hilbert space operators. If the state of the
quantum register at the end of computation step n-1 is
state 1 . The states of the quantum register before and
after the action of the CCNOT gate are: c2 I c1I t I
c2O c1O tO
respectively.
and
The circuit symbol and the
action of the CCNOT gate are shown in Figure 6.
q Rn −1 , the new state at the end of step n is given by:
q Rn = G n q Rn −1
8)
We developed a quantum computer simulator (QCS)
to simulate quantum computations on classical computers
[14]. The inputs to the simulator are the number of the
qubits in the quantum register, the number of computation
steps, the initial state of the qubits that form a quantum
register and the gates applied at each computation step.
All inputs are entered through a graphical user interface.
The pseudocode of the quantum computer simulator is:
c i t 2 i t 1i
c o t 2 o t 1o
000
000
0 01
0 01
010
010
(2). Read the number of qubits in the quantum register and
the number of computation steps.
011
011
(3). Read the initial state of the quantum register.
10 0
10 0
101
110
(4). Read the quantum gate configuration at each
computation step.
110
101
111
111
(1). Start
(5). Calculate the tensor product of the initial state of the
quantum register qubits.
(6). Take the first computation step ( i=1 ).
(7). Calculate the tensor product of the quantum gate
matrices applied at the ith computation step.
Fig. 7. The circuit symbol and the action of the F gate.
(8). Calculate the new state of the quantum register.
The Fredkin gate, F, acts on a three-qubit quantum
register. The upper qubit, c , is the control qubit and the
(9). Calculate the phase of the quantum register state.
two lower qubits, t 2 and t1 , are the target qubits. The
(10). If the number of computation steps is less that or equal
to the number defined by the user set i=i+1 and go to
step (7) else, continue.
F gate swaps the states of the two target qubits if the state
of the control qubit is 1 . The states of the quantum
(11). Write the probability amplitudes and probability
distribution at each computation step.
register before and after the action of the F gate are:
c I t 2 I t1I
and
cO t 2O t1O
respectively.
The circuit
(12). Write the final quantum register state.
symbol and the action of the F gate are shown in Figure 7.
(13). Produce graphical output.
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The quantum computation shown in Figure 8
comprises two qubits and five steps. The first step is the
initial state of the quantum register, 11 . In the second
step, an H gate is applied to the upper (second) qubit and
no gate to the lower (first) qubit. In the third step a CNOT
gate is applied with the second qubit as control and the first
as target. The same gate is applied in the fourth step.
In the fifth step an H gate is applied to the second qubit and
no gate to the first qubit. The circuit that describes this
quantum computation is shown in Figure 8(a). This circuit
entangles and disentangles the two qubits in the quantum
register [10].
In Figure 8(b) a matrix is shown. The columns of this
matrix represent the computation steps and the rows the
possible base states of the quantum register, which are also
written in decimal form. The first computation step is the
initial state of the quantum register, which in this case
is 11 . A measurement of the quantum register at this step
will give the state
11 with probability equal to 1.
All matrix elements in the first column are zero except the
element that corresponds to state 3 (in decimal), which is 1.
At the second step an H gate is applied to the second qubit
and sets its state in basis-state superposition.
A measurement of the quantum register state at the end of
this step will give states
01 and 11 with probability
0.5 and the matrix elements that correspond to states 1 and
3 (in decimal) are 0.5. A CNOT gate is applied at the third
step. A measurement at the end of this step will give states
01 and 10 (1 and 2 in decimal) with probability 0.5
(d)
Fig. 8. A quantum computation. (a) The quantum circuit. (b) The
matrix showing the probability of measuring each base state of the
quantum register at each computation step. (c) The QCS graphical
output showing the probability distribution at each computation
step. (d) The QCS graphical output showing the phase variation
during the quantum computation.
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each, and so on. A graphical representation of this matrix is
shown in Figure 8(c).
The x-axis represents the
computation steps (matrix columns) and the y-axis the
possible quantum register base states in decimal form
(matrix rows).
Each matrix element corresponds to a rectangle and its
value, i.e. the probability to measure this state at this
computation step, is represented as a gray-scale image in
which probability 1 is represented by black and probability
0 by white. Figure 8(d) shows the phase of each quantum
register state at each compotation step. An arrow with its
initial point located at the point (i, j) corresponds to the
th
th
phase of the j quantum register state (in decimal) at the i
computation step. The direction of the arrow indicates the
value of the phase, which can take any value between 0o
and 360o. An arrow parallel to the x-axis pointing to the
right corresponds to a phase angle of 0o. If the state j has
zero probability to be measured at step i, there exist no
arrow with initial point at (i, j).
According to the hidden variable theorem no classical
system can simulate completely a quantum system.
Therefore, the complete visualization of the quantum
computer operation using a classical computer is
impossible. The visualization shown in Figures 8(c) and (d)
is an effective visualization that depicts the most important
aspects of quantum computations.
It is well-known that the computational complexity of
any algorithm simulating quantum systems with two base
states is O(2n), where n is the number of quantum systems.
This simulator is no exception to that and suffers from
exponential slowdown as the number of qubits increases.
Matrices representing qubits and quantum gates contain a
large number of zeros and the computation complexity was
somewhat improved using sparse matrix techniques.
V. DEUTSCH’S QUANTUM ALGORITHM
The first algorithm proving that quantum computers
can outperform classical ones was developed by
D. Deutsch [19]. The problem posed by Deutsch is:
“Given a binary function f(x) find out if the function is
constant ( i.e. f(0)=f(1) ) or balanced ( i.e. f(0) ≠ f(1) ) by
performing the minimum number of computations.” On a
classical computer one has to compute the value of f(x)
twice and compare the results, but, as Deutsch showed, on
a quantum computer one computation is enough [10, 19].
⎧
⎪±
⎪
⎪⎪
⎡0 + 1 ⎤ ⎡0 − 1 ⎤
H
⎥ ⎯⎯→ ⎨
⎢
⎥ ⎢
2
2
⎦
⎣
⎦ ⎣
⎪
⎪
⎪±
⎪⎩
⎡0 + 1 ⎤ ⎡0 − 1 ⎤
⎢
⎥ ⎢
⎥
2 ⎦ ⎣
2 ⎦
⎣
if f (0 ) = f (1)
⎡0 − 1 ⎤ ⎡0 − 1 ⎤
⎢
⎥ ⎢
⎥
2 ⎦ ⎣
2 ⎦
⎣
if f (0 ) ≠ f (1)
(11)
Figures 10(b) and 10(c) show the simulation of Deutsch’s
algorithm in the case where f(x) = x. In this case the circuit
of Figure 9 is a CNOT gate.
(a)
Fig. 9. A quantum circuit used in Deutsch’s algorithm.
The key element in Deutsch’s algorithm is the
quantum circuit shown in Figure 9, which acts on a twoqubit quantum register x
a and leaves the x qubit in
its previous state, but takes the other qubit to a state which
is the modulo 2 sum of a and f(x). The action of this circuit
is given by:
U
x a ⎯⎯→
x a ⊕ f (x )
(b)
(9)
The quantum circuit of Deutsch’s algorithm is shown
in Figure 10(a). The initial state of the two qubits is
0 1 . This is considered as the first computation step.
At the second step two H quantum gates are applied to the
qubits changing their states according to:
⎡0 +1⎤ ⎡0 −1⎤
H
0 1 ⎯⎯→
⎥
⎢
⎥ ⎢
2 ⎦ ⎣
2 ⎦
⎣
(10)
At the third step the quantum circuit of Figure 9 is
applied and the states of the two qubits are taken to [10]:
20
(c)
Fig.10. (a) The quantum circuit of Deutsch’s algorithm. (b) and
(c) Simulation of Deutsch’s algorithm in the case of a CNOT gate.
(b) The probability distribution at each computation step. (c) The
phase variation during the computation.
We have f(0) = 0 and f(1) = 1, therefore
[6] C. P. Williams, “Quantum search algorithms in science and
engineering”, IEEE Computing in Science and Engineering,
vol. 3, no. 2, pp. 44-51, 2001.
f (0) ≠ f (1) and the upper qubit should be 1 . Figure
10(b) shows that at the end of the fourth step two quantum
register base states are possible each with probability 0.5:
the state
1 0
(decimal 2) and the state
1 1
[7]
S. Lloyd, “Universal quantum simulators”, Science, vol.
273, issue 5278, pp. 1073-1078, 1996.
[8]
C. P. Williams and S. H. Clearwater, Explorations in
Quantum Computing, New York: Springer-Verlag, Telos,
1998.
[9]
G. P. Berman, G. D. Doolen, R. Mainieri and V. I.
Tsifrinovich, Introduction to Quantum Computers, London:
World Scientific, 1998.
(decimal 3). In both states the upper qubit is in state 1 , as
expected. Figure 10(c) shows the phase variation during
the quantum computation. Note Figure 10(b) shows the
same data for steps two and three, in which all four
quantum register base states are equally probable, but
Figure 10(c) shows differed phases for these two steps,
which is very important in Deutsch’s algorithm.
[10] M. A. Nielsen and I. L. Chuang, Quantum Computation and
Quantum Information, Cambridge: Cambridge University
Press, 2000.
[11] B. P. Lanyon et al., “Universal digital quantum simulator
using trapped ions, Science, vol. 334, pp. 57-61, 2011.
VI. CONCLUSIONS
[12] M. Mariantoni et al., “Implementing the quantum von
Neumann architecture with superconducting circuits”,
Science, vol. 334, 61- 65, 2011.
We presented the qubit, the quantum register and the
most important quantum gates. We presented the circuit
model for quantum computation and a quantum computer
simulator, which we used to present an analytical example
of quantum computation. We presented analytically the
Deutsch’s quantum algorithm. The quantum Fourier
transform is an important computation and is a key
subroutine of quantum algorithms for factoring and
simulation and can be simulated using QCS. Two wellknown quantum algorithms are the Grover’s quantum
algorithm which is used for searching unstructured
databases [20] and Shor’s quantum algorithm for factoring
large number, which among others can be used in
cryptanalysis [21].
[13] N. K. Langford et al., “Efficient quantum computing using
coherent photon conversion”, Nature, vol. 487, pp. 360363, 2011.
[14] I. G. Karafyllidis, “Quantum Computer Simulator based on
the Circuit Model of Quantum Computation”, IEEE
Transactions on Circuits and Systems I, vol. 52, pp. 15901596, 2005.
[15] I. G. Karafyllidis, “Visualization of the quantum Fourier
transform using a quantum computer simulator”, Quantum
Information Processing, vol. 2, no. 4, pp. 271-288, 2003.
[16] I. G. Karafyllidis, “Cellular quantum computer
architecture”, Physics Letters A, vol. 320, no. 1, pp. 35-38,
2003.
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