Quantum Computation: An optical approach
E. Çabej, F. Fontana, C. Conti
Introduction
It is known that the dimensions of the physical system needed to encode a bit of information are
becoming smaller and smaller, going towards the dimensions of a single atom. At atomic scale the
laws of quantum mechanics govern the behaviour of physical systems. It turns out that quantum
systems used to encode a bit of quantum information can support new modes of computation that do
not have nothing in common with classical analogues.
The classical unit of information is a bit, which can take one of the two values 0 and 1. Thus any
macroscopic system, that can take two well-distinguished values is a physical realization of a bit. So
an n-bit classical memory register can exist in any of the 2n logical states labelled 00…0 to 11…1.
The quantum unit of information is a qubit (quantum bit). Any quantum mechanical system can be
used as a qubit providing that it is possible to define one of his states as |0 and another as |1. It is
practical to have a fixed pair of reliably distinguishable states of the qubit (for example horizontal and
vertical photon polarization). More generally two quantum states are reliably distinguishable if and
only if their vector representation are orthogonal [1].
Quantum mechanics tells us that if a qubit can exist in one or other of two distinguishable states, than
it can also exist in a coherent superposition of these states. In these new states, which generally have
no classical analogues, the microscopic system represents both values at the same time. So an n-bit
quantum memory register can exist in a superposition of 2n qubits at the same time.
The difference between a classical and a quantum register is that the classical register encodes one of
the 2n logical states at time, while the quantum register encodes all the 2n logical states
simultaneously, being so in a superposition of all possible classical states.
Thus if it is performed a mathematical operation on the quantum register, being this register in a
superposition state, the same computation is carried out on 2n numbers in a single step, and the result
will be a superposition of all corresponding outputs. In other words the Quantum Computer performs
a massive parallel computation.
Even though the quantum computer can store all the outcomes of 2n computations, the laws of
quantum mechanics tell us that it is possible to have only one of the outputs. It is the quantum
interference, which plays an important role to give the right output.
Another requirement imposed from the quantum mechanics laws is that the evolution of quantum
systems must be unitary. So the quantum gates that transform the qubits must be unitary.
1
Till now, it is introduced that Quantum Computation is based on typical quantum phenomena:
superposition of quantum states, entanglement and quantum interference.
Quantum computation might be achieved in these fundamental steps:

the input is evolved in a superposition of some preselected quantum basis states. This
superposition is achieved sending the input through unitary transformations (quantum gates).

through other unitary transformation (that encode the classical function to be evaluated) are
introduced non-local correlation on this superposition state (entanglement between different
qubits). As long as the mutual coherence among a set of qubits is preserved, they can
simultaneously take on more than one value giving rise to a useful effect known as quantum
parallelism.

Finally through another unitary transformation is achieved multi-particle quantum
interference (among different computational paths). The quantum interference amplifies the
correct outcomes and suppress the incorrect outcomes of computations. So the final step gives
the correct outcomes when the measurement is carried out.
As we noted above, quantum computation is achieved through actions of unitary transformations. The
unitary transformations can be constructed with a finite number of 4x4 matrices, that is, using only
one and two bit quantum gates, which are universal in quantum computation [2]. From this it can be
deduced that quantum gates can be constructed with a finite number of 2x2 and 4x4 matrices. The
optical realization for any NxN unitary matrix has also been demonstrated by Reck et al. [3]. So it is
possible to construct optical quantum gates. Furthermore it is shown in ref. [4,5] the simulation of
universal quantum gates using linear optics for building simple quantum circuits This method of
constructing quantum optical circuits is based on non-local superposition of “eventualities” rather
than physical objects. “Qubits” are considered as “which path” eventualities in linear optics,
implementable on standard optical benches. The wave function at the exit of the optical circuit can be
made to coincide arbitrarily well with the outcome of the anticipated computation, thus implementing
the quantum circuit. It provides an excellent means for testing small circuits for quantum error
correction or quantum algorithms [5].
The purpose of this survey paper is to introduce Optical Quantum Computing for educational
purposes.
The paper is arranged according to the following items:

Optical devices and Boolean logic

Optical Hadamard gate

Optical Controlled-NOT gate

Teleportation as quantum computation

Teleportation as an optical quantum circuit.

Possible applications to quantum optical computing of new concepts coming from
advanced components technologies.
In particular it is shown a correspondence between quantum networks and systems of linear optical
devices, such as beamsplitters and phase shifters, building on an equivalence between them. This
equivalence is inspired by the standard two-slit experiment of quantum mechanics, in which a single
quantum can interfere with itself to produce fringes on a screen. The idea is to find protocols for
translating any quantum circuit diagram into linear optical networks [4,5].
2
Optical devices and Boolean logic
It is known that the phase shift between the transmitted and the reflected optical fields of a lossless
beamsplitter is  / 2 . We will try to explain it in few words (for more details one can see ref. [6-8]).
M
BS
A
N
Fig.1 Scheme of the Michelson interferometer. The
electric field vectors vibrate perpendicularly to
the plane of drawing. M and N are totally
reflecting mirrors, BS is a lossless beamsplitter.
D
Consider a Michelson interferometer (Fig 1), which is composed of a beamsplitter BS and two totally
reflecting mirrors M, N, illuminated by a linearly polarized plane wave whose electric field has
complex amplitude E0 . The electric field Et of the output beam is the superposition of two
contributions Et originated by the path A(BS)N(BS)D and Et originated by the path A(BS)M(BS)D.
The interferometer produces also a reflected field Er which is the superposition of two contributions
Er and Er following the two paths A(BS)N(BS)A and A(BS)M(BS)A. If R is the phase shift due to
reflecting on the beamsplitter BS and T the phase shift due to transmitting on the beamsplitter BS and
if we assume that no absorption is present in the interferometer (lossless beamsplitter) [6], from
energy conservation condition derives:
 R  T   2
It is seen from the last expression that the lossless symmetric beamsplitter introduces a phase shift of
 / 2 between the transmitted and reflected field: the reflected one is in a delay of λ/4 respect to the
transmitted one. So we can write the reflected field as e
i

2 
multiplying the transmitted field.
As it is mentioned above the electric field that impinges on the beamsplitter BS is divided into two
beams propagating along the two orthogonal directions (BS)M and (BS)N.
Consider a single photon that impinges on BS. It will be registered with equal probability by two
photodetectors situated behind the BS [Fig. 2]. The photon doesn’t split in two, it takes both paths at
once. The photon will be in a quantistic superposition of its being in two orthogonal directions, so in a
superposition of different spatial location. [16].

π
i
π
π
e 2  cos  i sin  i
2
2
3
D1
Fig.2
BS
A
D2
A single photon impinges on a
beamsplitter and it is detected with
an equal probability by the two
photodetectors D1 and D2.
Let now consider the two input ports of the beamsplitter. In the reference [4,5] is introduced a singlephoton representation of several quantum bits (qubits), in order to build on an equivalence between
traditional linear optics elements (such as beamsplitters or phase shifters) and one-bit quantum gates.
It is established a kind of correspondence between a photon and the qubits describing its state. For
example one qubit is involved in the description of the beamsplitter in terms of a quantum circuit: the
"location" qubit, which characterizes the information about "which path" is taken by the photon [4,5].
It is used 0 and 1 to represent the two input modes entering the beamsplitter [5,12].
|1’
|0
|0’
BS
A
Fig.3 Scheme of the symmetric beamsplitter.
The two input modes entering the
beamsplitter BS1 are labelled |0 and
|1, while |0’ and |1’ are the exit
modes.
|1
B
The quantum state of the photon leaving the beamsplitter depends on its input mode. So if its input
mode is 0 , its quantum state after leaving BS1 is:

i

1 
1 

 
2
0 
0  1 (cos  isin )
0 e 1  

2
2 
2
2

1
 0 i1 

2
'
Furthermore if its input mode is 1 , its quantum state after exiting the beamsplitter BS is:

i

1 
1 

 
2
1

e
0
1  0 (cos  isin )


2
2 
2
2

1
 1 i 0  1  1 i 0 

2
2
1' 
4
In matrix form:
 0' 

  1 1 i   0 


 1' 
2  i 1  1 


This optical symmetric beamsplitter acts as a quantum
NOT gate. This gate converts a classical bit
into a qubit, namely in a coherent superposition of 0 and 1 [15,16].
It can be easily seen that:
NOT NOT 
 0 1
1 1 i  1 1 i  1  0 2i 



  
  i 

2  i 1 2  i 1 2  2i 0 
 1 0
And the matrix operator
 0 1

NOT  
 1 0
(1)
corresponds to the quantum NOT gate. The conclusion is that a sequence of two symmetric
beamsplitters must implement the quantum NOT gate [16]. If we analyse such sequence will have:
0 '' 

1'' 

0
2
1

 i 1' 
'
1  1
 0  i 1   i 1  1  i 0 

2 2
2

1
 0 i1 i1  0  i1
2
1
2
i 0
'

 1' 
1  1
 0  i 1  1  1  i 0
 i
2 2
2


1
i 0  1  1  i 0   i 0
2
In matrix form:
 0 ''   0 1   0 

  i
 
 1''   1 0   1 


(2)
The expression (2) is the same as (1). If it is not considered the phase factor i=eiπ/2 (which can be
removed by locating two phase shifters of –π/2 at each exit port), it is clear that the quantum gate
NOT can be implemented optically by a sequence of two symmetric beamsplitters [Fig.4].
Output 0
Output 1
Fig.4 The Mach-Zehnder interferometer as NOT gate,
implemented by two concatenated
Input 0
Input 1
5
NOT .
Optical Hadamard gate
As it is known the Hadamard gate sends:
0 
1 

1
2
1

2
0 1
0 1
H
a
(3)
x
In matrix form:
0 
1 1 1   0 
 

  
1 
2 1  1  1 
 
In the language of Quantum Computer we can say that the Hadamard gate is used to initialise it,
because it transforms the register of n-qubits in a superposition of equally balanced quantum states.
We can have an optical Hadamard gate very simply: placing two phase shifters of value - /2 at the
input and output ports of a beamsplitter [Fig.5].
|1'
-/2

|0
|0'
loc

H
loc
-/2
Fig.5 Hadamard gate on a “location” qubit, obtained by a
lossless symmetric beamsplitter with the addition of
two phase shifters of -/2.
|1
Indeed:
0 
'

2
( 0  ie
i

2
1)
1 

  1
( 0  1)
 0  i 1 (cos  isin )  
2
2 
2
2
1'  e

1
i

2
1 

i
 i 
1 
 i 0  e 2 1 e 2

2 

1 


1

( 0  1)
 i 0 (cos  isin )  1 (cos   isin  )  
2
2
2
2

6
So it transforms:
0 
1 
1
2
1
2
( 0  1)
(4)
( 0  1)
The transformations in (3) and (4) are the same and the conclusion is that it is possible to have an
optical Hadamard gate.
The combination of two Hadamard gates yields a balanced Mach-Zehnder interferometer [Fig.6].
|0''
-/2
-/2
|1''
H2
|1'
H
-/2
|0
H
|0'
H1
-/2
Fig.6 The balanced Mach-Zehnder Interferometer in terms
of two Hadamard gates.
|1
1
2
0 '' 
1

1

2

1  1
1

( 0  1)
( 0  1 )

2 2
2

0  1  0  1  0
 0' 
'
 1
2

1  1
1

( 0  1)
( 0  1 )

2
2
2

1
  0  1  0  1   1
2
1'' 
1
'
 0' 
In matrix form:
 0 ''  1 0  0 
 

  

 1''   0 1   1 


Thus, a balanced Mach Zehnder interferometer, can be translated in two Hadamard gates one after the
other (Fig. 6).
Since H2 =1, it is not a surprise that the location qubit returns to the initial basis state after two
beamsplitters. This simple quantum circuit describes the fact that the contributions of the two paths
interfere destructively in one of the output ports, so that the photon always leaves the interferometer in
the same direction as it entered [4,5].
7
Optical Controlled-NOT gate
Let’s consider now the same interferometer using horizontally polarized photons at the input. If none
of the devices acts on the polarization, the photon exits the interferometer with the same polarization
as it entered. In the quantum circuit language, this corresponds to introducing a "polarization" qubit
 pol after the "location" qubit  loc and the two qubits remain in a product state (  loc  pol ).
So:
0
1
loc
0
1
 stands for entering the BS horizontally
loc
 stands for entering the BS vertically
pol
pol
 stands for horizontal polarization
 stands for vertical polarization
|1'

|0

|0'

loc
pol


loc
loc
 
pol
Fig.7 Controlled-NOT gate using a Polarization Rotator.
|1
The introduction of a polarization rotator in the path 1 flipping the photon polarization from
horizontal to vertical, yields a Controlled-NOT gate, with 

pol
qubit acting as target one [Fig.7].
Indeed this behaviour is described by the following relations :
0
0
1
1
loc
loc
loc
loc
0
1
0
1
pol
pol
pol
pol
 0
 0
1
1
0
loc
loc
loc
loc
1
1
0
pol
pol
pol
pol
8
loc
qubit acting as control qubit and
In matrix form:







 1
 
0 loc 1 pol   0

1 loc 0 pol   0

1 loc 1 pol   0

0
loc
0
0
1
0
0
pol
0  

0 

1 

0  
0
0
0
1


0 loc 1 pol 

1 loc 1 pol 
1 loc 0 pol 

0
loc
0
pol
Now returning to the balanced Mach-Zehnder interferometer: a polarization rotator is placed in one of
the arms of the interferometer, flipping the photon polarization from horizontal 0 pol to vertical 1 pol ,
in the arm of the interferometer where it is placed. As it is discussed above this corresponds to placing
a 2-bit Controlled-NOT gate between the two Hadamard gates [Fig.8], where the location qubit is the
control bit and the polarization is the target bit [4].
PR -/2
-/2
H2
|1'
H
H
-/2
|0
|0'
H1
-/2
Fig.8 The balanced Mach-Zehnder Interferometer in terms
of two Hadamard gates with conditional operation on
the polarization. PR is a polarization Rotator.
|1
0
loc
0
pol
H1

0
2
1
loc

0
pol
1
1  1
0 loc 0

2 2
1

0 loc 0 pol  1
2
H2


pol
loc
loc
0
1
0
pol
 12  0
PR
pol
loc
0
 0
pol
loc
 12  0
1
pol
1
loc
loc
loc
0
1
1
pol
pol
pol

1
1
loc
loc
1
1
pol
pol



In this expression we note that after leaving the polarization rotator (P) and before entering the
second beamsplitter (H2) each path is "tagged" with a particular polarization
0 loc 0 pol  1 loc 1 pol . So the path 0 has the polarization 0 , while the path 1 has the


9
polarization 1 . In this way it is achieved an entanglement between the location qubit and the
polarization qubit and it is reflected to the disappearance of interference. One can also say that the
polarization of photon is flipped conditionally on its location.
Here it is introduced another basic quantum optical gate using a polarizing beamsplitter. In this case it
is achieved a Controlled-NOT gate, where the location qubit is the target qubit and the polarization
qubit is the control qubit [Fig.9]. A polarizing beamsplitter leaves an horizontally polarized photon (in
polarizing state 0 ) unchanged, while the vertical polarization (in polarizing state 1 ) is reflected.
0
1
0
1
pol
pol
pol
pol
0
0
1
1
loc
loc
loc
loc
 0
1
 0
1
0
pol
pol
pol
pol
1
1
0
loc
loc
loc
loc
|1'


|0'
|0
loc
pol



pol
 
loc
pol
Fig. 9 Controlled-NOT gate using a polarizing beam splitter,
so the control and target qubits are interchanged.
|1
Thus the behaviour of the location qubit depends on his state of polarization, (it is reflected or not
conditionally on his state of polarization)
In principle a universal quantum computation can be simulated using these optical substitutes for the
universal quantum gates. The optical set-up is constructed straightforwardly by inspection of the
quantum circuit [5].
So far are introduced three optical quantum gates. In this way it becomes possible to describe
quantum circuits using these optical gates. On the other hand the optical gates are obtained by nothing
more than the use of linear optical devices.
A circuit involving n qubits will require in general n successive splitting stages of the incoming beam,
that is 2n optical paths are obtained via 2n-1 beamsplitters. In this way can be simulated quantum
networks with only a small number of qubits, because there is an exponential increase between
number of qubits and number of devices involved [4,5].
10
Teleportation as Quantum Computation
It is shown [18] that “an unknown quantum state can be disassembled into, then later reconstructed
from purely classical information and purely nonclassical Einsten-Podolsky-Rosen (EPR)
correlations. To do so the sender, Alice and the receiver, Bob, must prearrange the sharing of an EPRcorrelated pair of particles. Alice makes a joint measurement on her EPR particle and the unknown
quantum system, and sends Bob the classical result of this measurement. Knowing this, bob can
convert the state of his EPR particle into an exact replica of the unknown state which Alice
destroyed.” They called this process teleportation.
In this section it is shown how to achieve quantum teleportation in terms of quantum computation.
The quantum teleportation circuit is not really different in principle from the original idea since it uses
quantum computation to create and measure nonlocal states [13].
We shall need two basic ingredients: the quantum exclusive-or gate (CNOT), which acts on two
qubits and the Walsh-Hadamard gate, which acts on a single qubit.
Consider the following quantum circuit, which is equivalent to the circuit described in Ref.13 [4].

a |
Alice
H1
b |0
Bob
H3
C1



u
|0 + |1
v
|0 + |1
x
y
C2
c |0
H2



H4
C3
A
B
D
Fig. 10 Quantum circuit for teleportation from [4,5]. The initial state


|
z
C4
E
F
G
of qubit a is teleported to the state of
qubit c. The dashed line divides Alice from Bob.
Alice receives a mystery qubit in the unknown state  , which can be written as:    0   1
so at input a she sets the arbitrary one-qubit state:, while for inputs b and c she prepares two qubits in
state 0 . We will see that the state  , will be transferred at the lower output z, whereas both other
outputs x and y will come out in state: 0  1 with some normalization coefficient.
Alice sends the unknown state  through a Hadamard gate:
11
H1   H1  0   1    H1 0   H1 1
1

2
1

2
  0  1 
1
2
 0  1 
    0      1 
We denote this state by  :
 
1
2
    0      1 
Alice pushes also the third qubit through a Hadamard gate:
0
1
Hc H0 
2
1
We denote this state by  :
1
CNOT  b  CNOT
1

2
2
 00
0
 1 0
 11    
The state that Alice will have after the Controlled NOT C1 will be   . These qubits are not
denoted by separate kets, because they are not individual pure states, they are in an entangled state.
Alice keeps the upper qubit  in quantum memory and send the other  to Bob.
So at the point A the circuit state is written as:
A
1
   


2
000 
2
    0      1  1  00
2
 
2
100 

2
011 
 11 
 
2
111
where the triplets corresponds respectively to the input state at a, b, c.
In order to teleport the state 
to Bob, she pushes the entangled qubit that she kept (  ) and the
unknown state  through C2 and H3. The action of C2 is:
C2 A 

2


We denote this state by
C 2 000 

2

2
 
2
C 2 011 
000 
 
 
2
C 2 100
2
C 2 111
110 

2
011 
 
2
101
B , and at this point of circuit Alice sends the first qubit through a
Hadamard gate:
12
H3 B 

H 3 000 
 
H 3 110 

H 3 011 
 
H 3 101
2
2
2

 

 

H 3 0 00 
H 3 1 10 
H 3 0 11 
H 3 1 01
2
2
2
2
1   
 

 


 2  0  1  00  2  0  1  10  2  0  1  11  2  0  1  01 
2

1   
 

 


 2  000  100   2  010  110   2  011  111   2  001  101 
2

2
At this point Alice measures the qubits u and v. The measurement will lead in two random classical
bits, which being classical are necessarily unentangled with the third qubit. She communicates u and v
to Bob via a classical channel. These two classical bits can be thought of as basis quantum states u
and v , respectively with the value of their classical output, so for the rest of the circuit they can be
used as states u and v . Since these two qubits participate in the rest of computation as control bits
of Controlled-NOT gates, (we know that the control bits doesn't change their value, but only flip the
target bit when the value of control is one) they remain unchanged for the rest of computation, so
measuring them at this point, doesn't affect the final outcome of the computation [13].
The measurement performed by Alice yields the following values of classical bits:
(u v) = (00); (01); (10); (11).
For this respective values of the pair (u v) she will have the von Neuman precipitation [13,19]:
u v
0 0
0 1
1
0
1 1
precipitat ion
 
000 
2
2


010 
2
2

 
100 
2
2



110 
2
2

001
011
101
111
Alice communicate u and v to Bob by way of classical communication. According to the value Bob
receives from Alice, he creates the quantum states u and v releases the qubit  he has kept in
quantum memory and pushes all three qubits into his part of the circuit (on the right of the dashed
line) [13].
We need to demonstrate that Bob’s output z is in state  . At the point D of the circuit Bob will
have:
13
u v
  
00 


01 


10 


00 

0 0
0 1
1 0
1 1
D
 

1
2
2


 
0 
1
2
2


  
0 
1
2
2

 
 
0 
1
2
2

0 
The qubits  and v are pushed through the CNOT C3
u v
0 0
0 1
1 0
1 1

E
000 

001
2
2


011 
010
2
2


100 
101
2
2



111 
110
2
2
After C3 the third qubit is pushed through the Hadamard gate C4:
u v
0 0
0 1
1 0
1 1

F
 
 000   001 
2
2
2


1
 010   011 
01 H 1 
01 H 0 
2
2
2


1
 101   100 
10 H 0 
10 H 1 
2
2
2
 

1
 111   110 

11 H 1 
11 H 0 
2
2
2
00 H 0 
00 H 1 
1
Finally the CNOT C4 acts on the second and third qubit:
u v
0 0
0 1
1
0
1 1
1
2
1
2
1
2
1
2
G

000   001  

010   011  
 100
  101  
 110
  111  
1
2
1
2
1
2
1
2
00  0   1 
01  0   1 
10  0   1 
11  0   1 
As it can be noted from the last expression the third qubit is in the unknown state  , while the
qubits x and y, if measured at the end of the circuit yield two classical bits that are uniformly
14
distributed. In this way the circuit has teletransported the state of qubit 
from the upper input of
the circuit to the lowest one. So it can be asserted that the experiments of quantum mechanics are
subroutines for a quantum computer.
Teleportation as an optical quantum circuit
As reported in [ 4,5] we will try to translate the quantum computing teleportation circuit into an
optical quantum circuit.
01
C4
00
C4
01
H4
00
10
11
-/2
00
C4
C3
11
01
10
01
C1
11
H2'
C3
11
H3
C2
11
-/2
00
10
H4
10
10
H3
-/2
-/2
-
ac = 00
10
01
00
H1
00
H2
Fig. 11 Optical realization of the quantum circuit for teleportation using polarized photons from [4,5]. The
ocation qubit a characterizes the “which-arm” information at the first beam splitter, while qubit c
stands for “which-path” information at the second level of splitting. The initial location qubit a is
teleported to qubit c and probed via the interference pattern observed at the upper or lower (a = 0,1)
final beam splitter for both polarization states (b = 0,1) of the detected photon.
In this optical realization of the teleportation circuit qubit a corresponds to the location of the photon
at the first splitting level and c corresponds to the location of the photon at the second splitting level
(while a single photon is involved), while b stands for the polarization qubit (the initial polarization
state of the photons is horizontal, i.e. 0 ).
For convenience we depict the teleportation of state   0 so the incident photon enters the first
beamsplitter in the input port labelled 0 (we have said that 0 corresponds to entering the
beamsplitter horizontally). The state of our photon is represented by the qubits ac = 00.
15
The first Hadamard gate H1 is implemented by the first beamsplitter, and it changes the state of the
first qubit a which is also our unknown state, and leaves unchanged the state of the second qubit c,
after leaving the first beamsplitter we will have ac = 00 and 10.
The second Hadamard gate H2 corresponds to the second level of splitting and for convenience this
gate is implemented by two beamsplitters H2 and H2’. H2’ is obtained from H2 by interchanging the
output ports. H2 and H2’ act on the qubit c. As we have previously seen after leaving the second
splitting level, will have ac = 00, 01, 10 and 11. At this point we have four paths, which label the four
components of the state vector characterizing qubits a and c.
The action of Controlled-NOT gate is implemented by C, which is a polarization rotator and it flips
the polarization state of the photon (qubit  ) conditionally on the parity of a+c (mod 2)
As we can see the polarization is flipped on the paths 01 or 10 and it remains unchanged on the paths
00 and 11. So on 01 and 10 paths we will have vertical polarization for the photon ( 1 ), and on the
paths 00 and 11 we will have horizontal polarization for the photon ( 0 )
If we get a look at the state labelled B :
B 

2
000 
 
2
110 

2
011 
 
2
101
on the paths 00 and 11 (first and third qubit) we have the polarization horizontal (second qubit = 0),
while on the paths 01 and 10 we have vertical polarization (second qubit = 1).
We have seen that the third Hadamard gate H3 acts only on the first qubit a, and it does nothing on the
qubit c, so if we group in pairs the paths that have the same value of c (crossing the paths) and use two
beamsplitters for each value of c we will effect a Hadamard transformation on a.
Then we have a Controlled-NOT gate C3 acting on the qubit c, which is implemented by the use of
two polarizing beamsplitters C3. As we have previously seen a polarizing beamsplitter leaves an
horizontally polarized photon unchanged, while the vertical polarized photon is reflected. So on the
upper arm, where the value of a = 0, we have on the path 01 vertical polarization ( 1 ), the photon is
reflected on the path 00 where the polarization is horizontal ( 0 ). We can make the same analysis for
the lower arm , where the value of qubit a = 1. Before entering the polarizing beamsplitter C3 on the
path 11 the polarization is horizontal ( 0 ), while on the path 10 the polarization is vertical ( 1 ), so
the vertical polarization is reflected on the path 11. So we have both polarizations on the path 00 of
the upper arm and on the path 11 of the lower arm.
The Hadamard Gate H4 is implemented by the two beamsplitters H4.
The Controlled-NOT C4 is implemented very simply. We see that it flips the value of qubit c and
leaves unchanged the value of qubit a. So on the upper arm (a = 0) the C4 does not do anything, while
on the arm (a = 1) the Controlled-NOT C4 is achieved by crossing the paths (b = 0, 1).
Hence, it is implemented the teleportation circuit in an optical circuit using linear optical devices. We
will try to interpret it in the following way:
A photon which was initially horizontally polarized can reach one of two “light” detectors (the solid
line in Fig. 10). The polarization of this photon as we have shown will be horizontal or vertical. The
final measurement, yields the value of qubit a and b, at this point two classical bits (random). So the
upper or lower arm indicates the value of qubit a, (1 or 0) while the vertical or horizontal polarization
indicates the value of qubit b (1 or 0). The entire set-up forms a simple balanced Mach-Zehnder
interferometer, (because there are exactly two indistinguishable paths leading to each of the eight
possible outcomes, coming out from the four detectors and two possible polarizations. These paths
interfere pair wise, just as in a standard Mach-Zehnder interferometer, explaining the fact that the
photon always reaches the “light detector” (in both a = 0 and a = 1 and for both polarization).The
third qubit c contains the teleported quantum bit, that is the initial arbitrary state of a. Since the
location state of the photon is initially 0 , it impinges the beamsplitter on its left side (“which-arm”),
it always exits in the same direction as it entered, so it reaches always the “light side”, it never reaches
16
the “dark side” (dashed lines). In this sense the initial “which-arm” qubit a has been teleported to the
final “which-path” qubit c.
A limit of this technique is the exponential increase of the optical devices with the size of the circuit,
but it can be easily constructed when the number of qubits involved is not too large.
Possible applications to quantum optical computing of new concepts
coming from advanced components technologies.
Among the several possible approaches to a practical embodiment of quantum gates and processors,
the optical route appears to be particularly interesting because of the unique properties of
electromagnetic beams, whose full quantum behaviour can be wisely exploited to implement
fundamental functions (basic logic gates, etc.).
Either some preliminary experiments of this kind appear in the literature [20] and new recent ideas apt
to overcome the limits encountered in other technological solution to Quantum Computing (as NMR,
etc.) have drawn the attention of the specialized community because of their inherent elegance and
simplicity.
One of the main drawbacks of optical Quantum Computing has been so far the limited viability of
optical integration. The current technologies widely used, e.g. in the telecommunications domain,
have made real the use of several devices, like modulators in lithium niobate crystals, which have
become popular in high speed transmission systems. However, when cascading of several simple
devices is required, severe limits hamper the possibility of manufacturing a complex “optical chip”.
Although the process for fabricating a waveguide in optical materials is rather well understood, the
main difficulty is related to the constraints on the minimum radius of curvature that can be accepted
before bending losses of the waveguide become too high. For instance, in a branching junction/splitter
in a lithium niobate waveguide the output paths are tilted with respect to the input section by an angle
that cannot be greater than 5/10 degrees. It is easy to understand that even a simple structure like a
Mach-Zehnder interferometer requires a section of the original wafer whose overall length cannot be
reduced under 15-20 mm. Even when other technologies were exploited, the bending losses [21]
which arise from the unavoidable coupling between the guided mode and the radiation modes in the
curved waveguide impose rather strict limits on the total number of elementary optical gates that
could be contained on the same chip. We could therefore conclude that standard techniques for optical
integration are not amenable to the production of quantum processor.
In order to improve the scenario of possible applications of optics to Quantum Computing, new ways
have to be explored.
A first point to be taken into consideration is that, according to recent researchers [22] the quantum
“core” of a real computer should not contain millions of quantum gates. It is sufficient to concentrate
in the nucleus of the system, few tens/hundreds of purely quantistic functions, interfaced towards an
external environment of classical electronic processors. Even in this reduced architecture, fully
quantum performance can be retained. This medium/low level of integration can be successfully
approached by the new concept of photonic crystals. Under this name a new family of optical
materials have become recently popular because of their unique optical properties related to the
presence of a multidimensional periodic structure which is able to affect strongly the behaviour of
electromagnetic field propagation. It is therefore possible to extend some solutions previously
endeavoured in monodimensional structures, creating the photonic equivalent of “bandgaps “ well
known to solid state physicists. In particular, photonic crystal waveguide can be easily designed by
removing a row of “defects” in an otherwise ordered and perfect periodic array [Fig 12]. Despite the
apparent similitude to a integrated optics waveguide (the pits fabricated in the slab reduce its
refractive index and therefore can be viewed as the equivalent of a cladding), the presence of a
17
photonic bandgap material outside the waveguide wipes out the possibility of electromagnetic field
propagation outside the guide itself. In other words, electromagnetic fields cannot be radiated from
the waveguide because no travelling wave propagation in the “cladding” is allowed. Exploiting this
effect, the radius of curvature of the waveguide can be reduced down to levels unconceivable in the
domain of the present technology. A well known result of this new approach is the possibility of
making a 90° angle in a waveguide, where the light is forced to flow in much the same way as a
liquid flowing in a channel [23].
The new design methodology made possible by the introduction of photonic crystals, can also be
expanded to the design of birefringent waveguides, phase retarders and polarizers. Beam splitters,
which appears to be a fundamental building block in any kind of optical quantum processor, can be
easily implemented by a T-shaped defect [Fig 13]. As the periodical structure fabricated in the crystal
can modify its birefringent properties, also retardation plates could have an integrated equivalent.
It is therefore possible to envisage a photonic crystal replication of some architectures originally
studied as a succession of bulk optical components. Photonic crystals appear to have additional
advantages in terms of waveguide attenuation and/or manufacturability of active sections of the
optical circuit natively integrated in the optical component. Active functions like modulation, etc.,
could provide the designer with additional degrees of freedom and even advanced quantum operations
like the generation of entangled states (usually done in bulk nonlinear crystals in preliminary
experiments) could be performed in semiconductor photonic crystals.
Accordingly, we can foresee an extensive use of these nanoengineered materials in optical Quantum
Computing, as soon as the manufacturing technology will be able to produce them at a moderate cost
and in high volumes. This achievement could probably be eased by the large application of photonic
crystals to the domain of telecommunications and signal routing.
Fig.12 A photonic crystal waveguide realized by a defect in a hexagonal lattice of hole in
Gallium Arsenide. The pseudo-color map of the modulus of the electric field is reported.
18
Fig. 13 Beamsplitter realized by a T-shaped defect in a rectangular lattice. The pseudo-color map
of the real part of the electric field is reported.
Acknowledgements
This note is supported by Pirelli Cavi e Sistemi Spa and the Didactic and Research Pool of Crema.
We are grateful to Prof. Degli Antoni for his continued encouragements and Prof. Fornili for the
discussions and collaboration.
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20
Teleportation
as
quantum
Computation