We consider the realization of quantum Fredkin gate with only linear optics
and single photons. First we construct a heralded Fredkin gate using four
heralded controlled-not (CNOT) gates. Then we simplify this method to a
post-selected one utilizing only two CNOT gates. We also give a possible
realization of this method which is feasible with current experimental
technology. Another post-selected scheme requires time entanglement of the
input photons but needs no ancillary photons.Comment: 5 pages, 5 figure

Guang-Can Guo

M. A. Nielsen

Timothy C. Ralph

Yan-Xiao Gong

English

arXiv

We consider the realization of a quantum Fredkin gate with only linear optics and single photons. First we construct a heralded Fredkin gate using four heralded controlled-NOT (CNOT) gates. Then we simplify this method to a post-selected one utilizing only two CNOT gates. We also give a possible realization of this method which is feasible with current experimental technology. Another post-selected scheme requires time entanglement of the input photons but needs no ancillary photons

Gong, Yan-Xiao

Guo, Guang-Can

Ralph, Timothy C.

University of Queensland eSpace

Methods for a linear optical quantum Fredkin gate
Yan-Xiao Gong,1,2,* Guang-Can Guo,2 and Timothy C. Ralph1
1Centre for Quantum Computer Technology, Department of Physics, University of Queensland, St Lucia 4072, Australia
2Key Laboratory of Quantum Information, University of Science and Technology of China, CAS,
Hefei, 230026, People’s Republic of China
Received 7 April 2008; published 2 July 2008
We consider the realization of a quantum Fredkin gate with only linear optics and single photons. First we
construct a heralded Fredkin gate using four heralded controlled-NOT CNOT gates. Then we simplify this
method to a post-selected one utilizing only two CNOT gates. We also give a possible realization of this method
which is feasible with current experimental technology. Another post-selected scheme requires time entangle-
ment of the input photons but needs no ancillary photons.
DOI: 10.1103/PhysRevA.78.012305 PACS numbers: 03.67.Lx, 42.50.Dv
I. INTRODUCTION
Quantum computing 1, due to its potential to solve prob-
lems far beyond classical computers, has attracted great at-
tention in recent years. Many physical systems have been
considered for a quantum computer 2. One promising sys-
tem is to use single photons, showing benefits such as low
decoherence and easy single-qubit manipulation. However,
such systems suffer a major disadvantage—the lack of inter-
action between individual photon qubits, which is needed for
implementing nontrivial multiqubit gates. Surprisingly, Knill,
Laflamme, and Milburn demonstrated that scalable quantum
computing was possible using linear optical elements, single
photons and photon detection 3. Since then there has been
considerable progress in improving the original scheme and
demonstrating its basic elements 4.
Here we focus on the implementation of a linear optical
Fredkin gate 5. The Fredkin gate plays an important role in
both classical computing and quantum computing 1. The
Fredkin gate is a three-qubit controlled-swap gate, that is, if
the control qubit is in state 1, the two target qubits swap
their states, otherwise they remain in their initial states. In
the context of universal quantum computer, multiqubit gates
are usually thought to be built by a combination of single-
and two-qubit gates. Smolin and DiVincenzo have shown
that five two-qubit gates two controlled-NOT CNOT and
three controlled-square-root-NOT CSRNOT gates are suffi-
cient to implement the Fredkin gate 6. The most efficient
known CNOT implementation requires two ancillary photons
and has a probability of success of 1/4 7, while the CS-
RNOT also requires two ancillary photons but has a probabil-
ity of success of 1/8 8. Hence the Smolin, Divincenzo gate
needs ten ancillary photons and the total success probability
is 2−131.210−4. Therefore, their scheme is too difficult to
be realized with current experimental technology. Recently,
another scheme was proposed in Ref. 9 by simulating the
Kerr medium in Milburn’s optical Fredkin gate 10 with
linear optical elements. It needs only six ancillary photons
with the success probability 4.110−3.
Recently, the complexity of the Toffoli gate was greatly
reduced and the success probability was improved by ex-
ploiting additional photonic degrees of freedom 11. In this
paper, we wish to see if a similar effect can be achieved by
applying those techniques to the Fredkin gate. We propose
some methods for implementing the Fredkin gate with linear
optics and single photons. The qubits in our schemes are all
encoded in polarization states of single photons, so that 0
H and 1V, where H V denotes the horizontal
vertical polarization state. The rest of the paper is orga-
nized as follows. In the next section we propose a heralded
Fredkin gate using four heralded CNOT gates. In Sec. III we
give a post-selected Fredkin gate, i.e., working in the coin-
cidence basis, and we also present a possible optical realiza-
tion which is feasible with existing technology. In Sec. IV we
replace the four heralded CNOT gates in the heralded scheme
with four post-selected CNOT gates assisted by time entangle-
ment but without ancillary photons. We conclude in Sec. V.
II. HERALDED FREDKIN GATE
Our heralded Fredkin gate is built up from four CNOT
gates. The schematic structure is shown in Fig. 1. To show
how the scheme works, we consider an arbitrary input state
written as
a1HcinHt1inHt2in + a2HcinHt1inVt2in
+ a3HcinVt1inHt2in + a4HcinVt1inVt2in
+ a5VcinHt1inHt2in + a6VcinHt1inVt2in
+ a7VcinVt1inHt2in + a8VcinVt1inVt2in, 1
where aii=1,2 , . . . ,8 is an arbitrary complex number sat-
isfying normalization condition.
First the polarizing beam splitter PBS1 PBS2 transmits
the horizontally polarized photons to beam 1 4 and verti-
cally polarized photons to beam 2 3. Then the photons in
each of the beams 1, 2, 3, and 4 undergo a CNOT gate con-
trolled by the control state. Therefore, the input state be-
comes
a1HcH1H4 + a2HcH1V3 + a3HcV2H4
+ a4HcV2V3 + a5VcV1V4 + a6VcV1H3
+ a7VcH2V4 + a8VcH2H3. 2*yxgong@mail.ustc.edu.cn
PHYSICAL REVIEW A 78, 012305 2008
1050-2947/2008/781/0123055 ©2008 The American Physical Society012305-1
Next the photons in modes 2 1 and 3 4 are mixed at PBS3
PBS4, followed by half-wave plates HWPs. Of these,
HWP1 and HWP3 oriented at 67.5° induce the transforma-
tions
H→ 1	2 − H + V , 3
V→ 1	2 H + V , 4
while HWP2 and HWP4 set to 22.5° result in
H→ 1	2 H + V , 5
V→ 1	2 H − V . 6
Finally, PBS5 PBS6 combines the photons in modes 5 4
and 8 7. Thus, conditioned on a simultaneous zero detec-
tion in each of the modes t1 and t2 we can obtain the suc-
cessful output state in modes c, t1, and t2,
a1HcHt1Ht2 + a2HcHt1Vt2 + a3HcVt1Ht2
+ a4HcVt1Vt2 + a5VcHt1Ht2 + a6VcVt1Ht2
+ a7VcHt1Vt2 + a8VcVt1Vt2. 7
The success probability for vacuum detections at detectors
D1 and D2 is 1/4. If we use the heralded CNOT gate proposed
by Pittman et al. 7, we need eight ancillary photons and the
success probability is 4−51.010−3. Compared with the
scheme by Smolin and DiVincenzo 6, our scheme has
higher success probability and needs less ancillary photons.
However, our scheme is not as efficient as Fiurášek’s scheme
9, as we need more ancillary photons and have lower suc-
cess probability. However, as we shall see, our scheme can
be simplified to a post-selected gate using only two ancillary
photons with higher success probability, which may be fea-
sible with existing experimental technology.
III. POST-SELECTED FREDKIN GATE
USING TWO CNOT GATES
We now consider the construction of a post-selected gate.
By this we mean that a gate succeeds conditioned on simul-
taneous successful detection of exactly one photon for each
qubit, so-called coincidence detection. Figure 2 is the sche-
matic of a post-selected Fredkin gate. Comparing this
scheme with the heralded one shown in Fig. 1, we can see
that the simplification is replacing the two CNOT gates imple-
menting on the photons in beams 3 and 4 controlled by the
photon in beam c by HWP5 67.5° and HWP6 22.5° with
the transformations given by Eqs. 4 and 5, respectively.
Therefore, for the input state given by Eq. 1, the state
before PBS3 and PBS4 is
a1HcH1 + a3HcV2 + a5VcV1
+ a7VcH2 
1
	2 H4 + V4
+ a2HcH1 + a4HcV2 + a6VcV1
+ a8VcH2 
1
	2 H3 + V3 . 8
Then through the analogous analysis in Sec. II and in the
case of coincidence detection of the output modes c, t1, and
t2, we can obtain the success output state the same as Eq. 7.
Figure 3 shows a possible optical realization of this
scheme. We utilize a heralded CNOT gate proposed in Ref.
7, with the success probability 1/4. Another CNOT gate need
not be heralded and a post-selected CNOT gate given in Ref.
12 can work with the success probability 1/9. However, as
FIG. 1. Schematic of the heralded linear optical Fredkin gate
comprising four heralded CNOT gates. Lowercase letters and num-
bers label the beams. The input control qubit is in beam cin and the
two input target qubits are in beams t1in and t2in. Polarizing beam
splitters PBS transmit horizontally polarized photons and reflect
vertically polarized photons. Half-wave plates HWP1 and HWP3
are oriented at 67.5°. HWP2 and HWP4 are set to 22.5°. The gate
succeeds if the two photon number resolving detectors D1 and D2
detect no photons. The output control and target states lie in modes
c, t1, and t2.
FIG. 2. Schematic of the post-selected linear optical Fredkin
gate using two controlled-NOT CNOT gates. Lowercase letters and
numbers label the beams. The control and two target qubits are
input in beams cin, t1in, and t2in. Polarizing beam splitters PBSs
transmit horizontally polarized photons and reflect vertically polar-
ized photons. Half-wave plates HWP1, HWP3, and HWP5 are ori-
ented at 67.5°. HWP2, HWP4, and HWP6 are set to 22.5°. This gate
succeeds conditioned on exactly one photon in each of the output
beams c, t1, and t2. Symbols “” denote the discarded outputs.
GONG, GUO, AND RALPH PHYSICAL REVIEW A 78, 012305 2008
012305-2
in our scheme the target state of the second CNOT gate is
known V or vacuum, it turns out that the gate can be
optimized for maximum success probability 1/6 11,13 see
the gate in the dashed box. Therefore in the case of fivefold
coincidence, i.e., detection of exactly one photon in each of
the output modes c, t1, and t2 and successful detection at D1
and D2, the gate succeeds with a total probability of success
1 /41 /61 /8=1 /1925.210−3. Note that the success
probability for coincidence detection in Figs. 2 and 3 is re-
duced to 1/8 compared with that of 1/4 in Fig. 1 due to
HWP5 and HWP6. However, this is more than offset by the
halving of the number of CNOTs required. As an ancillary
Bell state is needed, to implement this scheme requires at
least a five-photon source, which is available at present
14–16, and therefore our scheme is feasible with current
technology. However, the low success probability of our
scheme would make the experiment more difficult and
longer time detection would be needed.
IV. POST-SELECTED FREDKIN GATE ASSISTED
BY TIME ENTANGLEMENT
In this section we introduce another post-selected Fredkin
gate assisted by time entanglement. Let us first remind the
reader of the CNOT gate presented by Sanaka et al. 17 see
Fig. 4.
The control and target photons are a photon pair generated
by spontaneous parametric down-conversion pumped by a
continuous wave laser. Such a source is said to be time-
energy entangled 18 as the photon pair is in a superposition
of many possible emission times. The control photon is split
along the short cS or long cL path at PBS1 and combined
again in the same path at PBS2. The target photon is split
along the short tS or long tL path at the first beam splitter
BS1 and combined again in the same path at BS2. A HWP
oriented at 45° rotates the polarization state of the photon
taking the long path by 90°. The path-length difference L of
cL and cS is the same as that of tL and tS and satisfies the
condition
lSPDCL lpump, 9
where lSPDC is the coherence length of the down-converted
photon and lpump is the spectral width of the pump laser.
Conditioned on coincidence of detection with the time win-
dow of the coincidence counter satisfying TL /c, we
can write the evolution of an arbitrary input state as
b1HcinHtin + b2HcinVtin + b3VcinHtin + b4VcinVtin
→ b1HScoutHStout + b2HScoutVStout + b3VLcoutVLtout
+ b4VLcoutH
Ltout, 10
where bii=1,2 ,3 ,4 is an arbitrary complex number satis-
fying normalization condition, and the superscript S L de-
notes the photon passing the short long path. Here the co-
incidence counting has post-selected out unwanted state
components in which the control and target photons followed
paths of different lengths. Because of the time-energy en-
tanglement, paths of the same length are indistinguishable
and so add coherently. The success probability is 1/4.
Figure 5 shows an optical realization of a post-selected
Fredkin gate by replacing the four CNOT gates in Fig. 1 with
the CNOT gates we have just introduced. Based on the analy-
sis above, in the case of the input state given by Eq. 1, the
successful output state can be found to be
FIG. 3. Optical realization of a post-selected Fredkin gate. Low-
ercase letters and numbers label the beams. Photons in beams cin,
t1in, and t2in are the control and two target qubits. Polarizing beam
splitters PBSs transmit horizontally polarized H photons and
reflect vertically polarized V photons. A heralded controlled-NOT
CNOT gate proposed in Ref. 7 lies in the dotted box. EPR is an
ancillary Bell state source. RPBS, transmitting 45° polarized + 
photons and reflecting −45° polarized V photons, can be realized
by inserting one half-wave plate HWP oriented at 22.5° in each of
the two inputs and two outputs of a PBS. D1 and D2 are photon
detectors detecting + /− basis and H /V basis, respectively. A post-
selected CNOT gate suggested in Ref. 12 is enclosed by the dashed
box. Beam splitters are represented as black lines with their reflec-
tivity indicated aside and dotted line indicates the surface from
which a sign change occurs upon reflection. HWP1, HWP3, and
HWP5 are oriented at 67.5°. HWP2, HWP4, and HWP6 are set to
22.5°. HWP7 is set to 45°. This gate succeeds conditioned on the
coincidence of the successful detection at D1 and D2 and exactly
one photon in each of the output beams c, t1, and t2. Symbols “”
denote the discarded outputs.
FIG. 4. Schematic of the controlled-NOT CNOT gate demon-
strated in Ref. 17. Lowercase letters label the beams. Polarizing
beam splitters PBS transmit horizontally polarized photons and
reflect vertically polarized photons. The control target photon is
input in mode cin tin and output is in mode cout tout. BS1 and BS2
are both balanced beam splitters. HWP is a half-wave plate set to
45°.
METHODS FOR A LINEAR OPTICAL QUANTUM FREDKIN GATE PHYSICAL REVIEW A 78, 012305 2008
012305-3
a1HScHSt1H
St2 + a2H
ScHSt1V
St2
+ a3HScVSt1H
St2 + a4H
ScVSt1V
St2
+ a5VLcHLt1H
Lt2 + a6V
LcVLt1H
Lt2
+ a7VLcHLt1V
Lt2 + a8V
LcVLt1V
Lt2. 11
From Eqs. 10 and 11, we can see that to make the output
state entangled the three input photons need to be time-
entangled in the two time bins, “S” and “L.” This scheme
needs no ancillary photons and the probability of success is
1/64. Three-qubit time entangled states of the type required,
i.e., in which a triple coincidence is in a superposition of
many times, have been described in Refs. 19–21, however,
an experimental demonstration of such states has not yet
been made.
V. CONCLUSIONS
We have discussed the implementation of the Fredkin gate
with linear optics and single photons. We have presented a
heralded method using four heralded CNOT gates. Our
method needs eight ancillary photons which is less than that
of ten in Ref. 6, but more than that of six in Fiurášek’s
scheme 9. The success probability of our scheme is 4−5
1.010−3, which is higher than that of 1.210−4 in Ref.
6, but is less than that of 4.110−3 in Ref. 9. We have
also simplified the heralded scheme to a post-selected one by
replacing two CNOT gates with two HWPs. This scheme
needs only two ancillary photons, and therefore is feasible
with existing technology. However, the low success probabil-
ity of 1 /1925.210−3 would make the experiment very
difficult. The other post-selected Fredkin gate we have pro-
posed is assisted by time entanglement. Although this
scheme needs no ancillary photons and has higher success
probability of 1/64, the three-photon time entangled source
required is not available at present.
It should be noted that since the post-selected schemes
work in the coincidence basis, such schemes could not be
scalable unless photon-number quantum nondemolition
QND detectors were added to each output beam, neverthe-
less they open the door to experimental tests of an optical
Fredkin gate and would make its application possible. We
hope our proposals will stimulate such investigations of the
Fredkin gate.
ACKNOWLEDGMENTS
Y.X.G. thanks Yun-Feng Huang for useful discussions.
Y.X.G. and G.C.G. were funded by National Fundamental
Research Program Grant No. 2006CB921907, National
Natural Science Foundation of China Grants No. 60121503
and No. 60621064, Innovation Funds from Chinese Acad-
emy of Sciences, International Cooperate Program from
CAS, and Ministry of Science & Technology of China.
T.C.R. was supported by the DTO-funded U.S. Army Re-
search Office Contract No. W911NF-05-0397 and the Aus-
tralian Research Council.
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Methods for a linear optical quantum Fredkin gate

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