Success Probability Evaluation of Quantum Circuits

based on Probabilistic CNOT-Gates

Amor Gueddana, Rihab Chatta and Noureddine Boudriga

Communication Network and Reserach Laboratory (CNAS), Engineering School of Communication of Tunis (SUP’COM),

Ghazala Technopark, 2083, Ariana, Tunisia

Keywords:

CNOT, C

NOT, Abstract Probabilistic CNOT, Quantum CNOT-based Circuit.

Abstract:

In this paper, we study the effect of non deterministic CNOT gates on the success probability of Quantum

CNOT-based circuits. Based on physical implementation, we deﬁne an abstract probabilistic model of the

CNOT gate that takes into consideration error sources and realizability constraints. Using the proposed model,

we simulate a three-qubit quantum adder and show the evolution of the probability of realizing correctly the

SUM operation depending on the success probability and errors of the CNOT gates.

1 INTRODUCTION

Controlled-NOT gates associated with single qubits

operation are universal for building quantum circuits

(Nakahara and Ohmi, 2008). Quantum CNOT gates

based on linear optics still presents some conceptual

and realization problems. It has been shown that the

use of linear components doesn’t permit to reach de-

terministic gates. Several works proposed non deter-

ministic CNOT gate functioning at least with a suc-

cess probability of 1/4. Some of these gates were

physically realized and the expected result is quite

consistent with theoretical modeling, this is due es-

sentially to unexpected errors caused by the imper-

fection of linear components. We believe that stud-

ies concerning errors affecting the functioning of the

CNOT gate is missing modeling.

Quantum circuits based on CNOT gates were sim-

ply treated in the ideal case where the gate works per-

fectly. All what has been said about the use of non de-

terministic gates is that the success probability of re-

alizing a function will exponentially decrease depend-

ing on the number of gates used. To our knowledge,

no detailed study were achieved to show the behavior

of quantum circuit against non deterministic gates.

Our contribution in this work is three fold: ﬁrst,

we propose an error control model of an abstract

probabilistic CNOT gate, while taking into consider-

ation physical implementation constraints. Second,

we identify errors affecting the success probability of

the gate at the implementation level and we model

errors related to the basic quantum linear compo-

nents. Third, based on physical implementation of

the TC.Ralph CNOT model, we deﬁne a set ofCNOT

gates having the form of an abstract CNOTgate that

are physically realizable and extend our results to the

probabilistic algorithms.

This paper is organized around ﬁve sections. Sec-

tion 2 introduces the universality of CNOT gates and

illustrates brieﬂy several steps used to get CNOT de-

composition of C

NOT gate. In section 3, we present

ﬁrst a model of an abstract probabilistic CNOT gate

and based on the TC.Ralph model, a subspace of re-

alizable probabilistic gate is presented, second, we

study the errors caused by linear components and

model their effect at the implementation level. Sec-

tion 4 presents in a ﬁrst hand, a scheme for model-

ing probabilistic CNOT-based quantum circuits and

in a second hand, the CNOT based three qubit Min-

imized Quantum Ripple Carry Adder is treated as a

case study. Finally some numerical experimentation

are illustrated.

2 QUANTUM CNOT-BASED

CIRCUITS

2.1 Quantum C

NOT Gate

In the general form, a single qubit quantum gate has a

unitary 2×2 matrix representation denoted by u and

having the following expression:

378

Gueddana A., Chatta R. and Boudriga N..

Success Probability Evaluation of Quantum Circuits based on Probabilistic CNOT-Gates.

DOI: 10.5220/0004058503780387

In Proceedings of the International Conference on Data Communication Networking, e-Business and Optical Communication Systems (OPTICS-2012),

pages 378-387

ISBN: 978-989-8565-23-5

 2012 SCITEPRESS (Science and Technology Publications, Lda.)

u =





(1)

where u

, u

and u

∈C and describing the

amplitude probability of being in a speciﬁc quantum

state (Figure 1a).

Figure 1: General form of a quantum gate.

We consider two major single qubit operations for

building quantum circuits:

= |0ih0|+ |1ih1| =



1 0

0 1



(2)

NOT

= |1ih0|+ |0ih1| =



0 1

1 0



(3)

A C

u gate, where k ∈ N

∗

, is a gate acting on

(k+ 1) qubits. It inverts the state of the last qubit if

all of the k qubits are set to |1i, the k qubits left un-

changed are the control qubits.

The matrix representation of C

u (Figure 1b) is

denoted by U

and obtained as follows:



k+1

−2

O u



(4)

Where I

k+1

−2

is a



k+1

−2





k+1

−2



iden-

tity matrix.

For u = NOT, C

NOT gates are universal for

building quantum circuits. For k = 1, the C

NOT

is called controlled −NOT gate and denoted simply

CNOT.

2.2 CNOT based Implementation of

NOT Gate

In this paragraph, we focus our study on a detailed

decomposition of the C

NOT gates, since k = 2 is the

highest k value among all gates constituting the adder

circuit to be studied later.

Decomposition of C

NOT is obtained across the

following steps:

Step 1: We determine a ﬁrst decomposition of

the C

NOT gate (Figure 2a) to a circuit composed of

CNOT, ν and ν

†

gates as depicted by Figure 2b.

Where ν and ν

†

are determined such that ν



0 1

1 0



and ν

†

is the transpose conjugate of

ν. Transfer matrix of ν and ν

†

are given as follows:

Figure 2: First decomposition of the C

NOT gate.

ν =



1+ i 1−i

1−i 1+ i



; ν

†



1−i 1+ i

1+ i 1−i



(5)

Step 2: Apply a control qubit to ν (cν) and

†

(cν

†

). Equation 5 becomes:

cν =







1 0 0 0

0 1 0 0

0 0

1+i

1−i

0 0

1−i

1+i







(6)

cν

†

is the transpose conjugate of cν.

Determine equivalent decomposition of cν and

cν

†

to a set of single qubit and CNOT gates as de-

picted by Figure 3(a) and Figure 3(b), respectively.

(a) Decomposition of cν. (b) Decomposition of cν

†

Figure 3: Decomposition of cν and cν

†

Where A, B, C and D are given as follows:

A =

√



cos





(1−i) sin





(1−i)

−sin





(1+ i) cos





(1+ i)



B =



cos





sin



−



−sin



−



cos







C =

√



(1+ i) 0

0 (1−i)



E =



1 0

√

(1+ i)



∗

, B

∗

, C

∗

and E

∗

are the conjugate matrix of A, B,

C and E, respectively.

Step 3: Reassemble the equivalent parts of the cir-

cuit to obtain a ﬁnal equivalent implementation as de-

picted by Figure 4.

Figure 4: Final decomposition of the C

NOT.

According to (Nakaharaand Ohmi, 2008; Barenco

et al., 1995), for k ≥3 , the decomposition of C

NOT

SuccessProbabilityEvaluationofQuantumCircuitsbasedonProbabilisticCNOT-Gates

379

follows the same steps and all what differs from the

NOT decomposition is that ν and ν

†

transforms

changes.

2.3 Modeling and Implementing CNOT

Gate

During the last decade, large set of works have been

addressed modeling and implementing CNOT gate.

We consider in the following those based on linear

optical components.

Early model have been proposed since 2001 by

T.B.Pittman et al (Pittman et al., 2001), the construc-

tion for a probabilisticCNOT gate, using linear optics

and auxiliary photon pair, was achieved by the com-

bining of quantum encoder and a destructive CNOT.

The desired CNOT gate was deﬁned to work with

a success probability of 1/16. This model has been

optimized and the success probability raised to 1/4.

T.B.Pittman presented an improvement of this model

in 2003 (Pittman et al., 2003) and instead of using

auxiliary entangled photon pair, a single auxiliary

photon was used. The success probability remained

equal to 1/4 and a physical realization including un-

expected errors was presented.

A third model developed during 2002 is related to

T.C.Ralph et al (Ralph et al., 2002), the model showed

that the CNOT gate operates in the coincidence ba-

sis and the success probability is 1/9. This model

presented some weaknesses related to path interfer-

ence, to avoid this problem, a fourth model comes

with the use of three Partially Polarizing Beam Split-

ter (PPBS). This model, known under the name “com-

pact CNOT gate”, was proposed by Ryo Okamoto et

al (Okamoto et al., 2005) and kept same success prob-

ability value (1/9).

Another experimentation related to the third cited

model was proposed by J.L.O.Brien et al in 2003

(Brien et al., 2003). The success probability obtained

presented some errors comparing to the model.

Based on the TC.Ralph model theoretically pro-

posed in (Ralph et al., 2002) and implemented in

(Brien et al., 2003), we aim in this paper to model

errors affecting the success probability of the gate at

the experimentation level.

3 QUANTUM PROBABILISTIC

GATE

3.1 Abstract Probabilistic CNOT

Transform

Let |ci and |ti, be vectors from a two dimensional

real vector space spanned by the basis {|0i,|1i},

representing control and target qubits of a CNOT

gate. The system’s quantum state is a vector in

the four dimensional real vector space spanned by

the basis {|00i,|01i, |10i, |11i}, representing the col-

umn vectors



1 0 0 0





0 1 0 0





0 0 1 0



and



0 0 0 1



, respec-

tively.

A probabilistic CNOT gate realizes the function

CNOT

: |c,ti → |c,t ⊕ci in a non deterministic way.

In the sens that, for i, j, k ∈ N

∗

∃p = (p

)

i≤4

∈ [−1,1]

∃ε =





j≤4

∈ [−1,1]

∃χ = (χ

)

k≤4

∈ [−1,1]

(7)

Satisfying:

+ |ε

+ |χ

= 1

+ |ε

+ |χ

= 1

+ |ε

+ |χ

= 1

+ |ε

+ |χ

= 1 (8)

Such that:

CNOT











|00i → p

|00i+ ε

|01i+ε

|10i

+ε

|11i+ χ

|ψ

|01i → ε

|00i+ p

|01i+ε

|10i

+ε

|11i+ χ

|ψ

|10i → ε

|00i+ε

|01i+ε

|10i

|11i+ χ

|ψ

|11i → ε

|00i+ε

|01i+ p

|10i

+ε

|11i+χ

|ψ

(9)

When the input of the CNOT is the basis state

|00i, p

represents the amplitude probability of re-

alizing correctly the function f

CNOT

, yielding to the

correct output |00i. ε

, ε

and ε

are the amplitude

probabilities of ending in the erroneous output basis

state |01i, |10i and |11i, respectively, χ

is an ampli-

tude probability that appears, when auxiliary qubits

are used by the CNOT gate, and assigned to all states

|ψ

i that takes the system out of the basis states.

Following the same considerations for the rest of

CNOT input states |01i, |10i and |11i, |ψ

i, |ψ

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380

and |ψ

i denotes the states out of the system basis,

respectively.

We call probabilistic CNOT transform the matrix

associated to the CNOT function given by equation 9

and denoted by U

CNOT

We deﬁne P

CNOT

to be the probability matrix de-

scribing theoretical probability of ending in a ba-

sis state after measure. P

CNOT

components are ob-

tained directly from the module of U

CNOT

compo-

nents squared.

Implementation of the quantum probabilistic

CNOT gate gives a circuit that should be able to

produce after measure P

CNOT

or something close.

However, implementation and measuring errors will

only allow the determination of an estimated matrix

denoted by P

Imp

CNOT

Deﬁnition.

An abstract probabilistic transform is denoted by

p,ε

, satisfying properties of equations 7 and 8, and

has the following form:

p,ε













(10)

Where p = (p

)

1≤i≤4

and ε = (ε

)

1≤j≤12

for i, j∈

∗

It’s worth to notice that a probabilistic CNOT

transform is an abstract probabilistic transform, but

reciprocal way is not necessary checked. Therefore,

there must be a technique capable of implementing

the abstract probabilistic transform. We assign to the

feasibility of implementation the concept of realiz-

ability.

p,ε

is a realizable matrix if there exist a quan-

tum CNOT circuit whose physical parametrization

permits to compute theoretically it’s transfer matrix

CNOT

and verifying the equality U

CNOT

= A

p,ε

is α-realizable, for α ∈ R

, α > 1, if A

p,ε

realizable and the following condition is satisﬁed:

| > α



(11)

Under condition of equation 11, we don’t know at

which level it’s possible to determine p and ε to get

CNOT

having the form of A

p,ε

. For this purpose, we

study in the following the Ralph CNOT model (Ralph

et al., 2002).

3.2 Realizable Abstract Probabilistic

CNOT Transform based on the

Ralph Model

A generalization of the Ralph CNOT model is the

central componentillustrated by stage 3 of Figure 5. It

includes ﬁve Beam Splitters (BS), denoted BS

, BS

and BS

, characterized by ﬁve reﬂectivity co-

efﬁcients η

, η

and η

, respectively. We de-

note the generalized CNOT Ralph model by CR(H),

where H = (η

, η

) ∈ ]−1,1[

. The asso-

ciatedCNOT transfer matrix obtained from the circuit

is denoted by U

CR(H)

The encoding and decoding modules contains four

Polarizing Beam Splitter (PBS) and four Half Wave

Plate (HWP).

Figure 5: Generalization of the CNOT gate of TC.Ralph.

CR(H)operates on the dual rail coding to realize

the CNOT function.

Recall that in their work, TC.Ralph et al (Ralph

et al., 2002) used reﬂectivity coefﬁcient η = η

= η

/3 and η

′

= η

/2 and showed that

the CNOT gate operates with a success probability of

/9.

In the reality, BS imperfection of realization can’t

produce the values η =

/3, η

′

/2 but only values

that are closed to them. Since (

/3,

/2) are supposed

to be the ideal values, CR(H) is proposed.

Proposition.

p,ε

is realizable by CR(H), for H = (η

, η

) ∈ ]0,1[

, if the following equalities are satis-

ﬁed:

√

(1−η

)(1−η

)

√

(1−η

)(1−η

)

= (1−2η

)

(1−η

)η

(1−η

)

= (1−2η

)

(1−η

) +

(1−η

)η

(1−η

)η

−

(1−η

)

(1−η

) −

(1−η

)η

SuccessProbabilityEvaluationofQuantumCircuitsbasedonProbabilisticCNOT-Gates

381

= (1−2η

)

√

−

(1−η

)(1−η

)

= (1−2η

)

(1−η

)(1−η

) −

√

(1−η

)(1−η

)





1≤j≤12, j6={1,4,9,12}

= 0 (12)

Moreover, A

p,ε

is α-realizable ∀α ≥ 1 by CR(H),

where H = (η, η, η, η

′

, η

′

) ∈ ]0, 1[

, if η =

/3 and

′

/2.

Proof.

We consider η

∈]0,1[ the reﬂectivity coefﬁcient

of a BS. Let a

, b

be the two incoming photons

of the BS and a

out

, b

out

the outgoing photons. The

Heisenberg equation relating outputs-inputs are illus-

trated by Figure 6 (reﬂection upon dashed lines intro-

duces a π phase shift).

Figure 6: Heisenberg equation of the BS.

We consider the Heisenberg equations relating the

control (c

, c

) and target (t

, t

) inputs photons to

their corresponding outputs, depending on η

, η

and η

(Figure 5). After excluding auxiliary inputs

, ν

and outputs ν

, ν

, these equations are given

by the following:

√

(1−η

= −

√

(1−η

)η

(1−η

)(1−η

√

(1−η

)(1−η

)

(1−η

)η

−

(1−η

)

(1−η

)η

(1−η

)(1−η

(1−η

) −

(1−η

)η

√

(1−η

)(1−η

)

(1−η

)(1−η

−

(1−η

)η

(13)

For H = (η

, η

) and s, t∈ N

∗

, these

equations permits to determine the transfer matrix

CR(H)

= (u

s,t

)

s,t≤4

as follows:

The input state |00i is represented by a presence

of a photon in |c

i and |t

i, the amplitude probability

of having the correct output |00i, meaning a simulta-

neous detection (coincidence basis) in |c

i and |t

is given by the product of amplitude probabilities of

having a photon in |c

i and |t

i, when |c

i = |1i

and |t

i = |1i. Therefore, the resulting probability

amplitude u

1,1

is expressed as:

1,1

√

(1−η

)(1−η

)

The amplitude probability of having the erroneous

output |01i, |10i and |11i, meaning a simultaneous

detection on |c

i and |t

i, |c

i and |t

i , |c

i and

i, are u

2,1

, u

3,1

and u

4,1

, respectively, expressed as:

2,1

(1−η

)η

−

(1−η

)

3,1

= 0, u

4,1

= 0

Following the same manner, the input state |01i

gives u

2,2

= p

, u

1,2

= ε

, u

3,2

= ε

, u

4,2

= ε

, the

input state |10i gives u

4,3

= p

, u

1,3

= ε

, u

2,3

= ε

3,3

= ε

and the input states |11i gives u

3,4

= p

1,4

= ε

, u

2,4

= ε

, u

4,4

= ε

, where (p

)

1≤i≤4

and

(ε

)

1≤j≤12

are expressed by equation 12.

We consider p = (p

, p

) and ε =

(ε

,0,0, ε

) a set of amplitude prob-

abilities depending on η

, η

and η

. U

CR(H)

deﬁnes a set of abstract probabilistic CNOT matrix

having the following form:

p,ε







0 0

0 0 ε

0 0 p







(14)

Where A

p,ε

= U

CR(H)

We suppose that A

p,ε

is α-realizable ∀α ≥ 1 and

as requested by Ralph, η=η

=η

, η

′

=η

. En-

coding and decoding parts are supposed to operate

perfectly. According to these considerations, U

CR(H)

becomes:

CR(H)







η 0 0 0

0 η 0 0

0 0 −η+ η

′

(1−η) (1−η)

(1−η

′

)η

′

0 0 (1−η)

(1−η

′

)η

′

−η+ (1−η)(1−η

′

)







(15)

Moreover, by substituting these considerations

into equation 12, we deduce that p and ε becomes:

= p

= η

= p

= (1−η)

(1−η

′

)η

′

= ε

−η+η

′

(1−η); ε

= −η+(1−η)



1−η

′







1≤j≤11, j6=9

= 0 (16)

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382

∀α ≥ 1, A

p,ε

of equation 16 is α-realizable if ε

0 and ε

= 0. Under these conditions, we deduce that

η =

/3 and η

′

/2.

4 ERRORS OF THE CNOT

RALPH MODEL

4.1 Internal Errors

We consider in the following errors affecting all BSs

composing stage 3 of ﬁgure 5.

BS reﬂectivity coefﬁcient presents some uncer-

tainties with current BS technology. Work presented

in (Ralph et al., 2002) predicted an error of about

0.007 on BS reﬂectivity coefﬁcient and it concluded

that errors below 0.01 are realistic. In the sequel, we

assume this error lower than 0.05.

We study in the following the inﬂuence of the

BSs errors on the α-realizability of A

p,ε

First Case:

For the ideal Ralph model, meaning η = η

= η

/3 and η

′

= η

/2, we suppose that com-

mon error ξ ∈[−0.05,0.05] affects BS1, BS2 and BS3

and ξ

′

∈ [−0.05,0.05] affects BS4 and BS5, meaning

that η =

/3 + ξ and η

′

/2 + ξ

′

. Under these suppo-

sitions, p and ε of equation 16 changes as follows:

/3 + ξ

= (

/3 −ξ)

(

/2 −ξ

′

)(

/2 + ξ

′

)

= −

ξ+

′

−ξξ

′

;ε

= −

ξ−

′

+ ξξ

′

(ε

)

1≤j≤11, j6=9

= 0 (17)

According to equations 17, a set of α-realizable

p,ε

transforms is deﬁned for |p

| > α|ε

|, |p

| >

α|ε

|, |p

| > α|ε

| and |p

| > α|ε

We vary ξ in [−0.05,0.05] and α in

{1.5, 2,10, 50}. The delimited area illustrated

by Figure 7(a), 7(b), 7(c) and 7(d), gives a represen-

tation of the parameters p and ε, for which A

p,ε

α-realizable by CR

(H)

According to Figure 7, α-realizability of A

p,ε

deﬁned by the ranges of ξ and ξ

′

inside the intersec-

tion. Table 1 shows the range of the smallest rect-

angle containing the surfaces of interest that allows

α-realizability.

Second Case:

Even in the case where same technology is used

to construct BS1, BS2, BS3, BS4 and BS5, different

(a) α=1.5. (b) α=2.

Figure 7: α-realizability of A

p,ε

depending on BSs errors.

Table 1: ξ and ξ

′

ranges deﬁning α-realizable A

p,ε

α ξ ξ

′

1.5 [−0.05,0.05] [−0.05,0.05]

2 [−0.05,0.05] [−0.05,0.05]

10 [−0.021,0.023] [−0.05,0.05]

50 [−0.005,0.005] [−0.01,0.01]

errors occurs independently on η

, η

and η

respectively.

We consider (ξ

,ξ

) ∈ ]−0.05,0.05[

the errors affecting optimal values (

/3,

/2) as:

(η

/3 +ξ

)

1≤i≤3

;



/2 +ξ



1≤j≤2

(18)

By substituting equations 18 into equations 12, we

obtain a set of A

p,ε

that are α-realizable and has the

form of equation 14.

Similarly to the process applied to common errors

(ξ and ξ

′

), one can use numerical simulation to build

Table 2 that illustrates the ranges of ξ

, ξ

and

, yielding to the smallest area permitting to get a set

of α-realizable A

p,ε

It’s worth to notice from this study that if we want

that A

p,ε

be α-realizable for high α values, then errors

should be minimal.

4.2 Input-output Errors

Encoding module in Figure 5 is composed of two

PBSs and two HWPs, this permits to move from po-

larization to dual rail encoding where the presence of

the single photon on the upper or the lower arms de-

ﬁnes the |0i and |1i states, respectively. The transfer

matrix of the encoding part is denoted by U

end

Decoding module of Figure 5 realizes the inverted

process and has a transfer matrix denoted by U

dec

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383

Table 2: ξ

, ξ

and ξ

ranges deﬁning α-realizable A

p,ε

transform.

α ξ

1.5 [−0.05,0.05] [−0.05,0.05] [−0.05,0.05] [−0.05,0.05] [−0.05,0.05]

2 [−0.05,0.05] [−0.05,0.05] [−0.05,0.05] [−0.05,0.05] [−0.05,0.05]

10 [−0.05,0.05] [−0.03,0.03] [−0.05,0.05] [−0.05, 0.05] [−0.05,0.05]

50 [−0.05,0.05] [−0.001,0.001] [−0.02,0.02] [−0.01,0.01] [−0.01,0.01]

The encoding and decoding parts associated with

CR(H) previously studied, constitutes a polarization

encoding CNOT gate that is used to construct proba-

bilistic CNOT-based circuits.

The total transform of the CNOT gate, including

encoding and decoding part, is denoted by U

enc,dec

CR(H)

and obtained as follows:

enc,dec

CR(H)

= U

dec

CR(H)

enc

(19)

(H)

of Figure 5 uses encoding-decoding mod-

ules, these latter may introduce errors due to imper-

fect PBS (Tyan et al., 1996). In our study, we neglect

errors that may be introduced by HWP since it does

not affect the logic function of the gate but rather it’s

second one, which is entangled photons state genera-

tion.

We denote a

PBS

the incoming photon of the PBS

(Figure 8) and a

PBS

out

, b

PBS

out

the outgoing photons.

Figure 8: Polarizing Beam Splitter with error.

The error introduced by the PBS is modeled by

ς ∈ [0,1], the PBS acts on the incident Horizontal (H)

and vertical (V) photons as follows:

PBS

in,H

→

1−ςa

PBS

out,H

√

ςb

PBS

out,H

(20)

PBS

in,V

→

√

ςa

PBS

out,V

1−ςb

PBS

out,V

In the two dimensional real vector space spanned

by the basis {|0i,|1i} with components |0i =



1 0



and |1i =



0 1



, the function of the

PBS is given as:

PBS



|0i →

√

1−ς|0i+

√

ς|1i

|1i →

√

ς|0i+

√

1−ς|1i

(21)

The matrix transform describing the PBS function

with error ς ∈ [0, 1], for h0| and h1| representing the

bras vectors and having matrix expression (10) and

(01), respectively, is denoted by U

PBS

and given as:

PBS



1−ς|0i+

√

ς|1i



h0|



√

ς|0i+

1−ς|1i



h1|



√

1−ς

√

1−ς



(22)

We consider ς

, ς

and ς

, the error introduced

by PBS1, PBS2, PBS3 and PBS4, respectively. By

considering parallel combining of PBS1 and PBS2,

parallel combining of PBS3 and PBS4, U

enc

and U

dec

are obtained as:

enc

= U

PBS1

⊗U

PBS2

; U

dec

= U

PBS3

⊗U

PBS4

Using the expression of U

enc

and U

dec

, one can

deduce U

enc,dec

CR(H)

by simple computation.

Let us show know that the transfer matrix pro-

vided experimentally by J.L.O.Brein (Brien et al.,

2003) can be computed with U

enc,dec

CR(H)

using speciﬁc

values for the errors. Since the values are hardly com-

plicated to obtain, we only show that we can approx-

imate closely the matrix P

Imp

CNOT

by selecting a series

of values. For example, if we take η

/3 −0.005,

/3 + 0.015, η

/3 −0.02, η

/2 + 0.04,

/2 + 0.05, ς

= 10

−3.2

, ς

= 10

−2

, ς

= 10

−2

and ς

= 10

−2

, then a direct computation of U

enc,dec

CR(H)

is obtained and the associated probability matrix, de-

noted by P

enc,dec

CR(H)

is given as:

enc,dec

CR(H)







0.1091 0.0051 0.0003 0.0011

0.0061 0.1080 0.0011 0.0001

0.0012 0.0002 0.0060 0.0970

0.002 0.0011 0.0969 0.0005







Knowing the expression of P

Imp

CNOT

(Brien et al.,

2003) which is equal to:

Imp

CNOT







0.1056 0.0034 0.0006 0.0012

0.0026 0.1044 0.0012 0.0001

0.0027 0.0002 0.0256 0.08

0.0001 0.0024 0.0833 0.0289







One can deduce that the approximation is in the

order of 10

−2

. A similar computation for other errors

values could show that P

enc,dec

CR(H)

is close to P

Imp

CNOT

lower order.

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It’s worth to mention that in their implementation,

J.L.O.Brein et al (Brien et al., 2003)used as a Sin-

gle Photon Source (SPS) a pairs of energy degen-

erate photons generated through beam-like sponta-

neous parametric down-conversion and collected into

single-mode optical ﬁbers (stage 1 of Figure 5), at the

output level (stage 5 of Figure 5), C

out

and t

out

are an-

alyzed by a system ending with a single photon count-

ing module (SPCM).

Let us ﬁnally notice that SPS and SPCM, accord-

ing to (Brida et al., 2006; Eiseman et al., 2011), do

introduce some extra errors that are not under investi-

gation in this work.

5 TOWARDS QUANTUM

ALGORITHM SIMULATION

5.1 Computation Scheme

A quantum algorithm whose circuit is acting on a

set of n qubits is a collection of binary functions

: {0,1}

→ {0,1}

, j = [1..x] where x ∈ N

∗

. The

quantum circuit realizing the algorithm which we de-

note by Qc

alg

, is composed by serial and parallel com-

bining of circuits realizing f

, denoted by Qc

. We

assume that Qc

is based on C

NOT gates.

Using the techniques developed in section 2.2, an

equivalent single qubit and CNOT gate based circuit

denoted by QC

CNOT

may be obtained. QC

alg

and

CNOT

compute the same transfer matrix U

alg

. We

describe brieﬂy in the following, several techniques

used to determine U

alg

An abstract probabilistic CNOT gate, acting on

two qubits is represented by Figure 9a. Study of

probabilistic CNOT-based quantum circuits requires

description of the abstract probabilistic CNOT trans-

form in multiple qubits system (Figure 9b) composed

of m + 2 qubits, where m ∈ N. To this end, A

p,ε

will

have the equivalent block matrix representation:

p,ε

≡



(1,1)

(1,2)

(2,1)

(2,2)



(23)

(1,1)





, A

(1,2)





(2,1)





and A

(2,2)





For m qubits between the control and the target ,

the effect on the ﬁnal transform, depending on m, is

denoted by A

p,ε

(m) and obtained as:

p,ε

(m) =



⊗m

⊗A

(1,1)

⊗m

⊗A

(1,2)

⊗m

⊗A

(2,1)

⊗m

⊗A

(2,2)



(24)

Figure 9: CNOT gate used with m+ 2 qubits.

Using equation 24 and methods presented in

(Chakrabarti and Kolay, 2008; Shende et al., 2003),

we can use serial and parallel combining to determine

alg

by using identical CR(H) in all the circuit.

alg

is a function of nine errors, they are ξ

, ξ

affecting BSs and ς

, ς

affecting

PBSs. A control of the errors may provide a better

approximation of the algorithm function. We consider

this in more details in the next paragraph.

5.2 Case Study

Several proposal of Quantum adder circuits were pro-

posed in (Nakahara and Ohmi, 2008; Bannerjee and

Pathak, 2009; Kaye, 2004; Florio and Picca, 2004;

Vedral et al., 1996).The system used for our study

is the three qubits Minimized Quantum Ripple Carry

Adder (MQRCA) (Chakrabarti and Kolay, 2008). The

3-qubits MQRCA circuit is presented by Figure 10,

it computes the SUM of two numbers A and B,

represented by three qubits each as |a

i and

i, respectively.

Figure 10: 3-qubits CNOT based MQRCA.

The total number of CNOT gates composing the

MQRCA is 9 ×8+ 3 = 75. The result of MQRCA is

given by |c

, a

+ b

, a

+ b

, a

+ b

We present simulation results describing the errors

effect on the success probability when realizing the

SUM of |Ai = |4i and |Bi = |7i.

Deterministic CNOT gate realizes the addition

with certainty as illustrated by Figure 11(a).

When using CR(H), in one hand, we vary only

BSs errors for ﬁxed (ς

,ς

) = (0,0, 0,0) as il-

lustrated by Table 3, in the other hand, we vary

PBSs errors for ﬁxed values (ξ

,ξ

) =

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385

Table 3: Varying BSs errors.

α η η

′

6.55

0.05 -0.05 0.04 0.01 -0.01 0 0 0 0 4.45×10

−48

20.94

0.03 -0.01 -0.02 0.015 0.01 0 0 0 0 4.51×10

−52

40.65

-0.01 0.001 -0.02 -0.001 0.007 0 0 0 0 4.76×10

−52

∞

0 0 0 0 0 0 0 0 0 2.96×10

−52

Table 4: Varying PBSs errors.

α η η

′

6.01

0.05 -0.05 0.04 0.01 -0.01 10

−4.1

−4

−4.5

−3.8

5.15×10

−48

5.8

0.05 -0.05 0.04 0.01 -0.01 10

−3

−3.2

−3.4

−3.5

8.41×10

−48

4.66

0.05 -0.05 0.04 0.01 -0.01 10

−2

−2.2

−2.4

−2.5

1.8×10

−44

(a) Ideal CNOT. (b) α = ∞.

(f) α = 6.01. (g) α = 5.8. (h) α = 4.66.

Figure 11: Success probability of (4+ 7).

(0.05,−0.05, 0.04,0.01, −0.01) as illustrated by Ta-

ble 4.

CR(H)

associated to (ξ

, ξ

)=(0.05,

−0.05, 0.04, 0.01, −0.01) and (ς

, ς

)=(0, 0,

0, 0) is given as follows:

CR(H)

= U

enc,dec

CR(H)







0.3539 −0.0173 0 0

−0.0314 0.3539 0 0

0 0 0.054 0.3804

0 0 0.3782 0.054







(25)

According to equation 25, α =

0.3539

/0.054 =6.55.

For different α values, the resulting success probabil-

ity of realizing correctly the SUM 4+7, denoted by

, is illustrated by Figure 11.

The correct output is obtained for probability P

around 10

−52

, which is signiﬁcant comparing to the

other outputs



−54



, but non interesting for realiz-

ing arithmetic operations.

We notice that this probability is very low since

the success probability of the used model is around

/9, the success probability decreases exponentially

depending on the number of probabilistic CNOT

gates used (=75).

Figures 11(b), 11(c), 11(d) and 11(e) shows the re-

sult of the SUM for α = [∞,6.55,20.94,40,65]. This

ﬁgure shows that the higher the α value, the higher is

the GAP between P

and non signiﬁcant results, but

the lower is P

Figure 11(f), 11(g) and 11(h) illustrate the impact

of the encoding and decoding parts. ς

, ς

and ς

contribute to decrease α value and push non signiﬁ-

cant results to be closer to P

. An upper bound to

keep detection possible in our case is approximated

to a PBS error around ζ = 10

−3

6 CONCLUSIONS

In this work, we have deﬁned an abstract probabilis-

tic CNOT model, we identiﬁed and modeled errors

occurring in the success probability in the case of

T.C.Ralph CNOT based implementation. We also

studied the effect of the errors occurring in the imple-

mentation of quantum algorithm when it uses identi-

calCNOT called generalized RalphCNOT model and

abbreviated CR(H). The work we have performed

here, for CR(H) based technology can be used with

other technologies. We omitted in this paper to dis-

cuss the other technologies because of the lack of

space and the redundancy of results. We believe that

the study of implementations based on linear compo-

nents will highlight a large range of α-realizable ab-

stract probabilistic CNOT. Our future work address

this issue.

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