∎

11institutetext: Hai-Rui Wei 22institutetext: 1 School of Mathematics and
Physics, University of Science and Technology Beijing, Beijing 100083, China
Wen-Qiang Liu 33institutetext: 1 School of Mathematics and Physics, University
of Science and Technology Beijing, Beijing 100083, China
Leong-Chuan Kwek 44institutetext: 2Centre for Quantum Technologies, National
University of Singapore, Singapore 117543, Singapore
3 MajuLab, CNRS-UNS-NUS-NTU International Joint Research Unit, Singapore UMI
3654, Singapore
4 National Institute of Education and Institute of Advanced Studies, Nanyang
Technological University, Singapore 637616, Singapore
44email<EMAIL_ADDRESS>
# Efficient Fusion of Photonic $W$-states with Nonunitary Partial-swap Gates

Hai-Rui Wei1∗ Wen-Qiang Liu1 and Leong-Chuan Kwek2,3,4

(Received: date / Accepted: date)

###### Abstract

We introduce a nonunitary partial-swap gate for fusing arbitrary small-sized
photonic $W$-states into a large-scale entangled network of $W$-state
efficiently without ancillary photons. A partial-swap gate is designed in an
optical architecture based on linear optics elements. By introducing auxiliary
degree of freedom, this gate provides a higher success probability with less
cost. Our implementation can create a larger target state with a simpler set-
up than previous proposals for $W$-state fusion. Also all “garbage” states are
recyclable, i.e., there is no complete failure output in our scheme in
principle.

Keywords: $W$-state fusion, Multipartite entanglement, Quantum gate

##  1 Introduction

Quantum entanglement is a key resource in many quantum information processing
(QIP) tasks book , such as quantum teleportation teleportation1 ;
teleportation2 , quantum superdense coding superdense , quantum key
distribution distribution , quantum algorithms algorithm , and measurement-
based quantum computation one-way . In particular, depending on the
requirement of a specific task, the preparation and the manipulation of
multiqubit entangled states with different multipartite features (for
instance, the Greenberger–Horne–Zeilinger GHZ , Dicke Dick , $W$ W , and
cluster states cluster ) are needed, and these different types of entangled
states cannot be converted into each other by using local operations and
classical communications. However, there are still theoretical and
experimental challenges in study of multipartite entanglement due to the more
complex mathematical form and rapidly growing resource overhead with the
number of particles increasing. In recent years, the weblike property of the
$W$-state, due to its robustness against particle loss and decoherence
effects, has rendered it to be a useful resource for quantum communication
fujii2011robust . Indeed, the $W$-state has been shown to be relevant for many
schemes and applications ranging from its use in the foundation of quantum
mechanics test ; mattar2017experimental ; wu2014robust , in anonymous quantum
networks network , in quantum telecloning and teleportation telecloning , in
quantum computation cloning-machine ; Ai , in quantum memories
choi2010entanglement  and as a probe for reading information guha2013reading .

So far, many theoretical proposals and realistic experiments for generating
small-size $W$ states have been proposed exp-W1 ; exp-W2 ; exp-W3 . Currently,
there are two efficient ways to generate large-scale photonic $W$ states:
expansion and fusion. Both schemes have now been achieved in a wide range of
physical settings optics1 ; optics2 ; ion ; NMR . In 2008, Tashima _et al_.
optics1  proposed a scheme for locally expanding any polarization-based
$n$-photon $W$ ($|W_{n}\rangle$) state to an $(n+2)$-photon $W$
($|W_{n+2}\rangle$) state by accessing just one of the qubits with a success
probability of $(n+2)/(16n)$. This scheme was subsequently demonstrated
experimentally in 2010 demonstrated . Schemes for expanding $|W_{n}\rangle$
locally to $|W_{n+1}\rangle$ with a success probability of $(n+1)/(5n)$ were
also proposed in 2009 expand2009 , and even one for $|W_{n}\rangle$ to
$|W_{n+k}\rangle$ was proposed in 2011 expand2011 . Notably, the success
probability of the expansion from $|W\rangle_{n}$ to $|W_{n+k}\rangle$
decreases with an approximately exponential dependence with increasing $n$.

Fusion, on the other hand, was first proposed in 2011 by Özdemir _et al_.
Ozdemir . The idea was to join $|W_{n}\rangle$ and $|W_{m}\rangle$ to give the
$|W_{n+m-2}\rangle$ with a success probability of $(n+m-2)/(nm)$. In 2013,
enhancement the $W$-state fusion process was proposed through the use of a
Fredkin gate, Bugu _et al_. Bugu  then proposed a mechanism to fuse
$|W_{n}\rangle$ and $|W_{m}\rangle$ with one ancillary single photon into
$|W_{n+m-1}\rangle$ with a success probability of $(n+m-1)/(mn)$. In 2014,
Ozaydin _et al_. Ozaydin  generalized the setup for fusing three $W$ states:
$|W_{n}\rangle$, $|W_{m}\rangle$, and $|W_{t}\rangle$ states and one ancillary
single photon were joined into $|W_{n+m+t-3}\rangle$ with a success
probability of $(n+m+t-3)/(mnt)$ with a Fredkin gate. Using three CNOT gates
and one Toffoli gate, Yesilyurt _et al_. Yesilyurt  further generalized the
scheme for creating $|W_{n+m+t+z-4}\rangle$ from $|W_{n}\rangle$,
$|W_{m}\rangle$, $|W_{z}\rangle$, and $|W_{t}\rangle$ states with a success
probability of $(n+m+t+z-4)/(mntz)$. However, the success probabilities of the
required CNOT KLM ; 1/9 ; 1/41 ; 1/42 , Toffoli Toffoli0 ; Toffoli1 ; Toffoli
; Toffoli2 ; Toffoli3 , and Fredkin Fredkin1 ; Fredkin ; Fredkin2 ; Fredkin3 ;
Fredkin4  gates with linear optical elements were generally not considered.
Currently, nonlinear fusion schemes for fusing $|W_{n}\rangle$ and
$|W_{m}\rangle$ into $|W_{n+m}\rangle$ without qubit loss have also been
proposed loss ; Gaoting .

In this paper, we propose a protocol for fusing $W$ states of arbitrary size
into a larger one via nonunitary partial-swap gates. By introducing auxiliary
spatial degrees of freedom and using $(n-1)$ partial-swap gates, a
$|W_{nm-n+1}\rangle$ state can be created from $n$ arbitrary-sized
$|W_{m}\rangle$ states. All the “garbage” states are recyclable, and our
scheme avoids failed outcomes. Moreover, additional ancillary photon is not
required for our scheme. The length (cost or complexity) of our scheme
(measured by the number of the two-qubit entangling gates needed to construct
the scheme) is much less than the Fredkin- and CNOT-Toffoli-based schemes Bugu
; Yesilyurt .

Figure 1: Schematic diagram of the proposed scheme for fusing two
$|W_{3}\rangle$ states into a larger one $|W_{5}\rangle$. The fusion gate
operates on the two qubits in the dashed blue rectangle.

##  2 Simplifying a fusion-based $W$ state with nonunitary partial-swap gate

###  2.1 Fusion of $|W_{n}\rangle$ and $|W_{m}\rangle$ to give
$|W_{n+m-1}\rangle$

Suppose Alice and Bob possess $n$\- and $m$-partite polarization encoded $W$
states, $|W_{n}\rangle_{A}$ and $|W_{m}\rangle_{B}$, respectively, and they
wish to fuse their states together. A schematic example for the fusion process
of two three-partite $W$-states is depicted in Fig. 1. The entangled
$W$-states of Alice ($|W_{n}\rangle_{A}$) and Bob ($|W_{m}\rangle_{B}$) can be
written as

$\displaystyle\begin{split}|W_{n}\rangle_{A}=(|(n-1)_{H}\rangle_{a}|1_{V}\rangle_{1}+\sqrt{n-1}|W_{n-1}\rangle_{a}|1_{H}\rangle_{1})/\sqrt{n},\end{split}$
(1)
$\displaystyle\begin{split}|W_{m}\rangle_{B}=(|(m-1)_{H}\rangle_{b}|1_{V}\rangle_{2}+\sqrt{m-1}|W_{m-1}\rangle_{b}|1_{H}\rangle_{2})/\sqrt{m},\end{split}$
(2)

where $|(N-k)_{H}\rangle_{i}|k_{V}\rangle_{j}$ represents the superposition of
all possible permutations of $N-k$ photons with a horizontal polarization
($H$) in mode $i$ and $k$ photons with a vertical polarization ($V$) in mode
$j$. Captial letters $A$ and $B$ label the $W$ states are held by Alice and
Bob, respectively. Therefore, the initial state of the system composed of
Alice and Bob can be written as

$\displaystyle\begin{split}|W_{n}\rangle_{A}\otimes|W_{m}\rangle_{B}=&\frac{1}{\sqrt{nm}}|(n-1)_{H}\rangle_{a}|(m-1)_{H}\rangle_{b}|1_{V}\rangle_{1}|1_{V}\rangle_{2}\\\
&+\frac{\sqrt{m-1}}{\sqrt{nm}}|(n-1)_{H}\rangle_{a}|W_{m-1}\rangle_{b}|1_{V}\rangle_{1}|1_{H}\rangle_{2}\\\
&+\frac{\sqrt{n-1}}{\sqrt{nm}}|W_{n-1}\rangle_{a}|(m-1)_{H}\rangle_{b}|1_{H}\rangle_{1}|1_{V}\rangle_{2}\\\
&+\frac{\sqrt{(n-1)(m-1)}}{\sqrt{nm}}|W_{n-1}\rangle_{a}|W_{m-1}\rangle_{b}|1_{H}\rangle_{1}|1_{H}\rangle_{2}.\end{split}$
(3)

Figure 2: Fusion gate for fusing two $W$ states of arbitrary size to obtain a larger $W$ state. The circle “$\circ$” denotes $|H\rangle$, signifying that the states of the two photons in modes 1 and 2 will be exchanged if the photon in mode 1 is the $H$-polarized states, and has no effect otherwise. $D_{1}$ and $D_{2}$ are single-photon detectors. Table 1: Truth table of the polarization partial-swap gate. $x_{1}x_{2}$ | $\rightarrow$ | $y_{1}y_{2}$ |
---|---|---|---
$H_{1}H_{2}$ |  | $H_{1}H_{2}$ |
$H_{1}V_{2}$ |  | $V_{1}H_{2}$ |
$V_{1}H_{2}$ |  | $V_{1}H_{2}$ |
$V_{1}V_{2}$ |  | $V_{1}V_{2}$ |

As shown in Fig. 2, the fusion of $|W_{m}\rangle_{A}$ and $|W_{n}\rangle_{B}$
states into a larger $W$ state is achieved by sending photons in mode 1 (2),
i.e., $|1_{H}\rangle_{1}$ and $|1_{V}\rangle_{1}$ ($|1_{H}\rangle_{2}$ and
$|1_{V}\rangle_{2}$), into the partial-swap gate and those in mode a (b) are
kept intact at Alice’s (Bob’s) site. Note that the partial-swap gate swaps the
states of the two photons if the first photon is $H$ polarization, and has no
effect otherwise (see Table 1). That is, the action of this gate on the four
input states yields

$\displaystyle\begin{split}&|1_{H}\rangle_{1}|1_{H}\rangle_{2}\xrightarrow{\text{p-swap}}|1_{H}\rangle_{1}|1_{H}\rangle_{2},\quad|1_{H}\rangle_{1}|1_{V}\rangle_{2}\xrightarrow{\text{p-swap}}|1_{V}\rangle_{1}|1_{H}\rangle_{2},\\\
&|1_{V}\rangle_{1}|1_{H}\rangle_{2}\xrightarrow{\text{p-swap}}|1_{V}\rangle_{1}|1_{H}\rangle_{2},\quad|1_{V}\rangle_{1}|1_{V}\rangle_{2}\xrightarrow{\text{p-swap}}|1_{V}\rangle_{1}|1_{V}\rangle_{2}.\end{split}$
(4)

Therefore, such partial-swap gate completes the transformation

$\displaystyle\begin{split}|W_{n}\rangle_{A}\otimes|W_{m}\rangle_{B}\rightarrow&\frac{1}{\sqrt{nm}}|(n-1)_{H}\rangle_{a}|(m-1)_{H}\rangle_{b}|1_{V}\rangle_{1}|1_{V}\rangle_{2}\\\
&+\frac{\sqrt{m-1}}{\sqrt{nm}}|(n-1)_{H}\rangle_{a}|W_{m-1}\rangle_{b}|1_{V}\rangle_{1}|1_{H}\rangle_{2}\\\
&+\frac{\sqrt{n-1}}{\sqrt{nm}}|W_{n-1}\rangle_{a}|(m-1)_{H}\rangle_{b}|1_{V}\rangle_{1}|1_{H}\rangle_{2}\\\
&+\frac{\sqrt{(n-1)(m-1)}}{\sqrt{nm}}|W_{n-1}\rangle_{a}|W_{m-1}\rangle_{b}|1_{H}\rangle_{1}|1_{H}\rangle_{2}\\\
=\;&\frac{\sqrt{n+m-1}}{\sqrt{nm}}|W_{n+m-1}\rangle_{a,b,2}|1_{V}\rangle_{1}+\frac{\sqrt{(n-1)(m-1)}}{\sqrt{nm}}\\\
&\times|W_{n-1}\rangle_{a}|W_{m-1}\rangle_{b}|1_{H}\rangle_{1}|1_{H}\rangle_{2}.\end{split}$
(5)

The photon in mode 1 is then measured in the $\\{|H\rangle,\;|V\rangle\\}$
basis. From Eq. (5), one sees that (i) When the photon in mode 1 is
$V$-polarized and detector $D_{1}$ clicks, the scheme is successful with
probability (success probability) of $(n+m-1)/(nm)$, and the system collapses
into the desired state

$\displaystyle\begin{split}\frac{\sqrt{n+m-1}}{\sqrt{nm}}|W_{n+m-1}\rangle_{a,b,2}.\end{split}$
(6)

(ii) When the photon in mode 1 is $H$-polarized state and detector $D_{2}$
clicks, then the remaining photon collapses into state
$|W_{n-1}\rangle_{a}\otimes|W_{m-1}\rangle_{b}\otimes|1_{H}\rangle_{2}$ with
probability (recycle probability) of $(n-1)(m-1)/(nm)$. It is interesting to
see that the “garbage” state $|W_{n-1}\rangle_{a}\otimes|W_{m-1}\rangle_{b}$
of Alice and Bob remains a $W$ state but with a reduced number of qubits, and
therefore this state can be recycled, much like a repeat-until-success scheme
lim2005repeat . Remarkably, the fail probability of the designed scheme is
zero in principle as the system can not collapse into the failure states, such
as $|(n-1)_{H}\rangle_{a}|(m-1)_{H}\rangle_{b}|1_{V}\rangle_{2}$.

In Table 2, we compare our scheme with previous protocols. Here the success
probabilities of the Fredkin, Toffoli, CNOT, and partial-swap gates are
disregarded. The linear optical entangling gates are inherently probabilistic.
The optimal cost of a Fredkin or Toffoli gate is five two-qubit gates Optimal1
; Optimal2 , therefore, the complexity of our partial-swap-based scheme is
much lower than the Fredkin-based one Bugu  and the Toffoli-CNOT-based one
Fu-T . Moreover, extra ancillary photon is necessary for the schemes presented
in Refs. Bugu ; Fu-T , and is not required in our scheme. Remarkably, our
protocol is less complex with a higher success probability than any of the
other protocols for the generation of a larger $W$ state with the same size
Ozdemir ; Bugu ; Fu-T .

Table 2: Quantum resource required and success probability of various protocols for creating a larger $W$ state. $H$ is an extra ancillary $H$-polarized photon required for the fusion program. Proposed | Initial | Success | Success | Recycle | Fail
---|---|---|---|---|---
protocol | state | result | probability | probability | probability
with $I$ Ozdemir  | $W_{m},W_{n}$ | $W_{m+n-2}$ | $\frac{m+n-2}{mn}$ | $\frac{(m-1)(n-1)}{mn}$ | $\frac{1}{mn}$
with 1 Fredkin Bugu  | $W_{m},W_{n},H$ | $W_{m+n-1}$ | $\frac{m+n-1}{mn}$ | $\frac{(m-1)(n-1)}{mn}$ | 0
with 1 Toffoli, 1 CNOT Fu-T  | $W_{m},W_{n},H$ | $W_{m+n-1}$ | $\frac{m+n-1}{mn}$ | $\frac{(m-1)(n-1)}{mn}$ | 0
ours with 1 partial-swap | $W_{m},W_{n}$ | $W_{m+n-1}$ | $\frac{m+n-1}{mn}$ | $\frac{(m-1)(n-1)}{mn}$ | 0

###  2.2 Fusing $n$ arbitrary-sized $|W_{m}\rangle$ states into a large-
scalable $|W_{nm-n+1}\rangle$ state

Fig. 3 displays a scheme for fusing $|W_{n}\rangle_{A}$, $|W_{m}\rangle_{B}$,
and $|W_{t}\rangle_{C}$ states into $|W_{n+m+t-2}\rangle$ by using two
partial-swap gates. We denote polarization-based entangled $W$ states of
Alice, Bob, and Charlie as

$\displaystyle\begin{split}|W_{n}\rangle_{A}=(|(n-1)_{H}\rangle_{a}|1_{V}\rangle_{1}+\sqrt{n-1}|W_{n-1}\rangle_{a}|1_{H}\rangle_{1})/\sqrt{n},\end{split}$
(7)
$\displaystyle\begin{split}|W_{m}\rangle_{B}=(|(m-1)_{H}\rangle_{b}|1_{V}\rangle_{2}+\sqrt{m-1}|W_{m-1}\rangle_{b}|1_{H}\rangle_{2})/\sqrt{m},\end{split}$
(8)
$\displaystyle\begin{split}|W_{t}\rangle_{C}=(|(t-1)_{H}\rangle_{c}|1_{V}\rangle_{3}+\sqrt{t-1}|W_{t-1}\rangle_{c}|1_{H}\rangle_{3})/\sqrt{t}.\end{split}$
(9)

As shown Fig. 3, after Alice, Bob, and Charlie send one of their photons
($|1_{H}\rangle_{1}$ and $|1_{V}\rangle_{1}$, $|1_{H}\rangle_{2}$ and
$|1_{V}\rangle_{2}$, $|1_{H}\rangle_{3}$ and $|1_{V}\rangle_{3}$) to the two
partial-swap gates through modes 1, 2, and 3, respectively, the two partial-
swap gates lead to the following transformations:

$\displaystyle\begin{split}|W_{n}\rangle\otimes&|W_{m}\rangle\otimes|W_{t}\rangle\\\
\rightarrow&\frac{1}{\sqrt{nmt}}|(n-1)_{H}\rangle_{a}|(m-1)_{H}\rangle_{b}|(t-1)_{H}\rangle_{c}|1_{V}\rangle_{1}|1_{V}\rangle_{2}|1_{V}\rangle_{3}\\\
&+\frac{\sqrt{n-1}}{\sqrt{nmt}}|W_{n-1}\rangle_{a}|(m-1)_{H}\rangle_{b}|(t-1)_{H}\rangle_{c}|1_{V}\rangle_{1}|1_{V}\rangle_{2}|1_{H}\rangle_{3}\\\
&+\frac{\sqrt{m-1}}{\sqrt{nmt}}|(n-1)_{H}\rangle_{a}|W_{m-1}\rangle_{b}|(t-1)_{H}\rangle_{c}|1_{V}\rangle_{1}|1_{V}\rangle_{2}|1_{H}\rangle_{3}\\\
&+\frac{\sqrt{t-1}}{\sqrt{nmt}}|(n-1)_{H}\rangle_{a}|(m-1)_{H}\rangle_{b}|W_{t-1}\rangle_{c}|1_{V}\rangle_{1}|1_{V}\rangle_{2}|1_{H}\rangle_{3}\\\
&+\frac{\sqrt{(n-1)(m-1)}}{\sqrt{nmt}}|W_{n-1}\rangle_{a}|W_{m-1}\rangle_{b}|(t-1)_{H}\rangle_{c}|1_{H}\rangle_{1}|1_{V}\rangle_{2}|1_{H}\rangle_{3}\\\
&+\frac{\sqrt{(n-1)(t-1)}}{\sqrt{nmt}}|W_{n-1}\rangle_{a}|(m-1)_{H}\rangle_{b}|W_{t-1}\rangle_{c}|1_{V}\rangle_{1}|1_{H}\rangle_{2}|1_{H}\rangle_{3}\\\
&+\frac{\sqrt{(m-1)(t-1)}}{\sqrt{nmt}}|(n-1)_{H}\rangle_{a}|W_{m-1}\rangle_{b}|W_{t-1}\rangle_{c}|1_{V}\rangle_{1}|1_{H}\rangle_{2}|1_{H}\rangle_{3}\\\
&+\frac{\sqrt{(n-1)(m-1)(t-1)}}{\sqrt{nmt}}|W_{n-1}\rangle_{a}|W_{m-1}\rangle_{b}|W_{t-1}\rangle_{c}|1_{H}\rangle_{1}|1_{H}\rangle_{2}|1_{H}\rangle_{3}.\end{split}$
(10)

Figure 3: Fusion gate for fusing three $W$ states of arbitrary size to obtain
a larger $W$ state.

Eq. (10) implies the following four possible outcomes:

(i) When the photon in mode 1 is $V$-polarized state, the photon in mode 2 is
also $V$-polarized state and detectors $D_{1}$ and $D_{3}$ click, the system
collapses into the successful state $|W_{n+m+t-2}\rangle$

$\displaystyle\begin{split}|W_{n+m+t-2}\rangle=&\frac{1}{\sqrt{nmt}}|(n-1)_{H}\rangle_{a}|(m-1)_{H}\rangle_{b}|(t-1)_{H}\rangle_{c}|1_{V}\rangle_{3}\\\
&+\frac{\sqrt{n-1}}{\sqrt{nmt}}|W_{n-1}\rangle_{a}|(m-1)_{H}\rangle_{b}|(t-1)_{H}\rangle_{c}|1_{H}\rangle_{3}\\\
&+\frac{\sqrt{m-1}}{\sqrt{nmt}}|(n-1)_{H}\rangle_{a}|W_{m-1}\rangle_{b}|(t-1)_{H}\rangle_{c}|1_{H}\rangle_{3}\\\
&+\frac{\sqrt{t-1}}{\sqrt{nmt}}|(n-1)_{H}\rangle_{a}|(m-1)_{H}\rangle_{b}|W_{t-1}\rangle_{c}|1_{H}\rangle_{3}.\end{split}$
(11)

(ii) When the photon in mode 1 is $H$-polarized state, the photon in mode 2 is
also $H$-polarized state and detectors $D_{2}$ and $D_{4}$ click, the system
collapses into the recyclable state

$\displaystyle\frac{\sqrt{(n-1)(m-1)(t-1)}}{\sqrt{nmt}}|W_{n-1}\rangle_{a}|W_{m-1}\rangle_{b}|W_{t-1}\rangle_{c}|1_{H}\rangle_{3}.$
(12)

(iii) When the photon in mode 1 is $H$-polarized state, the photon in mode 2
is $V$-polarized state and detectors $D_{2}$ and $D_{3}$ click, the system
collapses into the “garbage” state

$\displaystyle\frac{\sqrt{(n-1)(m-1)}}{\sqrt{nmt}}|W_{n-1}\rangle_{a}|W_{m-1}\rangle_{b}|(t-1)_{H}\rangle_{c}|1_{H}\rangle_{3}.$
(13)

We call this case “a partial recyclable” because the states between Alice and
Bob remain a $W$-state but Charlie needs to prepare a new $W$ state for the
subsequent round of the fusion process.

(iv) When the photon in mode 1 is $V$-polarized state, the photon in mode 2 is
$H$-polarized state and detectors $D_{1}$ and $D_{4}$ click, the system
collapses into the “garbage” state

$\displaystyle\begin{split}\frac{\sqrt{(t-1)(m+n-2)}}{\sqrt{nmt}}|W_{n+m-2}\rangle_{a,b}|W_{t-1}\rangle_{c}|1_{H}\rangle_{3}.\end{split}$
(14)

We call this case “a partial success” because the state between Alice and Bob
has been fused but not Charlie.

Fig. 4 shows a scheme for fusing multiple $W$ states of arbitrary size
simultaneously. Table 3 compares the success and failure probabilities and an
estimation of the required quantum resources for our proposal against previous
schemes. Compared with other proposals for generating a $W$ state of given
size, our proposal scores a higher success probability and a lower failure
probability, with a simpler network.

Table 3: Quantum resources required and success probabilities of various protocols for fusing multiple $W$ states into a larger one simultaneously. $F=((5-m-n-z-t)-(n-1)(m-1)(t-1)(z-1))/(mntz)$. Proposed | Initial | Achieved | Success | Fail
---|---|---|---|---
protocol | state | state | probability | probability
with 1 Fredkin Bugu  | $W_{m},W_{n},W_{t},H$ | $W_{m+n+t-3}$ | $\frac{m+n+t-3}{mnt}$ | $\frac{(t-1)(m+n-2)+1}{mnt}$
ours with 2 partial-swaps | $W_{m},W_{n},W_{t},$ | $W_{m+n+t-2}$ | $\frac{m+n+t-2}{mnt}$ | 0
with 1 Toffoli, 3 CNOTs Yesilyurt  | $W_{m},W_{n},W_{t},W_{z}$ | $W_{m+n+t+z-4}$ | $\frac{m+n+t+z-4}{mntz}$ | $F$
ours with 3 partial-swaps | $W_{m},W_{n},W_{t},W_{z}$ | $W_{m+n+t+z-3}$ | $\frac{m+n+z+t-3}{mntz}$ | 0
Figure 4: Fusion gate for fusing $n$ arbitrary-sized $W$ states
simultaneously.

##  3 Linear-optics fusion-based $W$ state using auxiliary spatial degrees of
freedom

Figure 5: Linear-optical post-selected partial-swap gate. HWP${}^{45^{\circ}}$
is a half-wave plate (HWP) rotated by 45∘ to induce the transformation
$|H\rangle\leftrightarrow|V\rangle$. Setting HWP${}^{22.5^{\circ}}$
(HWP${}^{67.5^{\circ}}$) to 22.5∘ (67.5∘) completes
$|H\rangle\leftrightarrow(|H\rangle+|V\rangle)/\sqrt{2}$ and
$|V\rangle\leftrightarrow(|H\rangle-|V\rangle)/\sqrt{2}$
($|H\rangle\leftrightarrow(-|H\rangle+|V\rangle)/\sqrt{2}$ and
$|V\rangle\leftrightarrow(|H\rangle+|V\rangle)/\sqrt{2}$).

Based on Sec. 2, one can see that the key component of our fusion gates is the
partial-swap gate described by Eq. (4). The matrix form of this partial-swap
gate in the basis $\\{|1_{H}\rangle|1_{H}\rangle$,
$|1_{H}\rangle|1_{V}\rangle$, $|1_{V}\rangle|1_{H}\rangle$,
$|1_{V}\rangle|1_{V}\rangle\\}$ can be written as

$\displaystyle N_{\text{p-swap}}=\left(\begin{array}[]{cccc}1&0&0&0\\\
0&0&0&0\\\ 0&1&1&0\\\ 0&0&0&1\\\ \end{array}\right).$ (19)

Obviously, this operation is not a unitary one due to $NN^{\dagger}\neq
N^{\dagger}N\neq I$, with $I$ being an identity matrix.

The nonunitary gate can be implemented by utilizing the framework of quantum
measurement, or by expanding the state space to a larger one, and then
performing a proper unitary operation and an orthogonal measurement in the
enlarged space in succession. Here, we employ auxiliary spatial degrees of
freedom introduced by polarizing beam splitters PBS1 and PBS2 (see Fig. 5) to
implement the nonunitary polarization partial-swap gate. Next, we provide a
step-by-step description of our protocol for implementing this partial-swap
gate.

We consider photons 1 and 2 as being initially prepared in an arbitrary two-
qubit polarization-encoded state

$\displaystyle\begin{split}|\psi_{0}\rangle=&\alpha_{1}|1_{H}\rangle_{in}|1_{H}\rangle_{in^{\prime}}+\alpha_{2}|1_{H}\rangle_{in}|1_{V}\rangle_{in^{\prime}}\\\
&+\alpha_{3}|1_{V}\rangle_{in}|1_{H}\rangle_{in^{\prime}}+\alpha_{4}|1_{V}\rangle_{in}|1_{V}\rangle_{in^{\prime}}.\end{split}$
(20)

In the first step, as shown in Fig. 5, photons 1 and 2 pass through PBS1 and
PBS2, respectively. Next, photons in modes 1, 3, and 4 interact with half-wave
plates (HWP) oriented at $45^{\circ}$ (HWP${}^{45^{\circ}}$), $67.5^{\circ}$
(HWP${}^{67.5^{\circ}}$), and $22.5^{\circ}$ (HWP${}^{22.5^{\circ}}$),
respectively. The PBSs and HWPs cause the state to evolve from
$|\psi\rangle_{0}$ to

$\displaystyle\begin{split}|\psi_{1}\rangle=&\frac{1}{\sqrt{2}}(\alpha_{1}|1_{V}\rangle_{1}(|1_{H}\rangle_{4}+|1_{V}\rangle_{4})+\alpha_{2}|1_{V}\rangle_{1}(|1_{H}\rangle_{3}+|1_{V}\rangle_{3})\\\
&+\alpha_{3}|1_{V}\rangle_{2}(|1_{H}\rangle_{4}+|1_{V}\rangle_{4})+\alpha_{4}|1_{V}\rangle_{2}(|1_{H}\rangle_{3}+|1_{V}\rangle_{3})).\end{split}$
(21)

A PBS transmits the $H$-polarized and reflects the $V$-polarized components.
Therefore, PBS1 and PBS2 impart the spatial degrees of freedom of the incident
photon. The HWPs oriented at $45^{\circ}$ (denoted HWP${}^{45^{\circ}}$)
induce the qubit-flip operation $|1_{H}\rangle\leftrightarrow|1_{V}\rangle$
while the HWP${}^{67.5^{\circ}}$ results in

$\displaystyle\begin{split}&|1_{H}\rangle\leftrightarrow\frac{1}{\sqrt{2}}(-|1_{H}\rangle+|1_{V}\rangle),\quad|1_{V}\rangle\leftrightarrow\frac{1}{\sqrt{2}}(|1_{H}\rangle+|1_{V}\rangle).\end{split}$
(22)

Finally, the HWP${}^{22.5^{\circ}}$ completes the transformation

$\displaystyle\begin{split}&|1_{H}\rangle\leftrightarrow\frac{1}{\sqrt{2}}(|1_{H}\rangle+|1_{V}\rangle),\qquad|1_{V}\rangle\leftrightarrow\frac{1}{\sqrt{2}}(|1_{H}\rangle-|1_{V}\rangle).\end{split}$
(23)

In the second step, the photons in modes 2 and 3 are then mixed at PBS3 before
going through HWP${}^{67.5^{\circ}}$ while the photons in modes 1 and 4 are
mixed at PBS4 before going through HWP${}^{22.5^{\circ}}$. The completion of
these operations leads to the joint state

$\displaystyle\begin{split}|\psi_{2}\rangle=&\frac{1}{2\sqrt{2}}(\alpha_{1}(|1_{H}\rangle_{7}-|1_{V}\rangle_{7})(|1_{H}\rangle_{7}+|1_{V}\rangle_{7}+|1_{H}\rangle_{8}-|1_{V}\rangle_{8})\\\
&+\alpha_{2}(|1_{H}\rangle_{7}-|1_{V}\rangle_{7})(-|1_{H}\rangle_{5}+|1_{V}\rangle_{5}+|1_{H}\rangle_{6}+|1_{V}\rangle_{6})\\\
&+\alpha_{3}(|1_{H}\rangle_{5}+|1_{V}\rangle_{5})(|1_{H}\rangle_{7}+|1_{V}\rangle_{7}+|1_{H}\rangle_{8}-|1_{V}\rangle_{8})\\\
&+\alpha_{4}(|1_{H}\rangle_{5}+|1_{V}\rangle_{5})(-|1_{H}\rangle_{5}+|1_{V}\rangle_{5}+|1_{H}\rangle_{6}+|1_{V}\rangle_{6})).\end{split}$
(24)

In the third step, the photons in modes 5 and 8 (6 and 7) are combined at PBS5
(PBS6), and the photon in mode 10 (12) passes through HWP${}^{45^{\circ}}$
(HWP${}^{45^{\circ}}$). The operations
$\text{PBS}_{5}\rightarrow\text{HWP}^{45^{\circ}}$ and
$\text{PBS}_{6}\rightarrow\text{HWP}^{45^{\circ}}$ make $|\psi\rangle_{2}$
become

$\displaystyle\begin{split}|\psi_{3}\rangle=&\frac{1}{2\sqrt{2}}(\alpha_{1}(|1_{H}\rangle_{11}-|1_{H}\rangle_{12})(|1_{H}\rangle_{11}+|1_{H}\rangle_{12}+|1_{H}\rangle_{9}-|1_{H}\rangle_{10})\\\
&+\alpha_{2}(|1_{H}\rangle_{11}-|1_{H}\rangle_{12})(-|1_{V}\rangle_{10}+|1_{V}\rangle_{9}+|1_{V}\rangle_{12}+|1_{V}\rangle_{11})\\\
&+\alpha_{3}(|1_{V}\rangle_{10}+|1_{V}\rangle_{9})(|1_{H}\rangle_{11}+|1_{H}\rangle_{12}+|1_{H}\rangle_{9}-|1_{H}\rangle_{10})\\\
&+\alpha_{4}(|1_{V}\rangle_{10}+|1_{V}\rangle_{9})(-|1_{V}\rangle_{10}+|1_{V}\rangle_{9}+|1_{V}\rangle_{12}+|1_{V}\rangle_{11})).\end{split}$
(25)

Eq. (25) indicates that the two-qubit partial-swap operation (i.e., exchanges
the information of the two photons, conditional on the first photon being
$H$-polarized) is completed when a coincidence is observed between modes 9 and
11 (10 and 12). Table 4 lists the photon count rates in modes 9 and 11 (10 and
12) for computing basis inputs.

Table 4: Coincident expectation values calculated for the four logic basis inputs.  | $\langle n_{|1_{H}\rangle_{9}}n_{|1_{H}\rangle_{11}}\rangle$ | $\langle n_{|1_{H}\rangle_{9}}n_{|1_{V}\rangle_{11}}\rangle$ | $\langle n_{|1_{V}\rangle_{9}}n_{|1_{H}\rangle_{11}}\rangle$ | $\langle n_{|1_{V}\rangle_{9}}n_{|1_{V}\rangle_{11}}\rangle$
---|---|---|---|---
Input | $\langle n_{|1_{H}\rangle_{10}}n_{|1_{H}\rangle_{12}}\rangle$ | $\langle n_{|1_{H}\rangle_{10}}n_{|1_{V}\rangle_{12}}\rangle$ | $\langle n_{|1_{V}\rangle_{10}}n_{|1_{H}\rangle_{12}}\rangle$ | $\langle n_{|1_{V}\rangle_{10}}n_{|1_{V}\rangle_{12}}\rangle$
$|1_{H}\rangle_{in}|1_{H}\rangle_{in^{\prime}}$ | 1/8 | 0 | 0 | 0
$|1_{H}\rangle_{in}|1_{V}\rangle_{in^{\prime}}$ | 0 | 0 | 1/8 | 0
$|1_{V}\rangle_{in}|1_{H}\rangle_{in^{\prime}}$ | 0 | 0 | 1/8 | 0
$|1_{V}\rangle_{in}|1_{V}\rangle_{in^{\prime}}$ | 0 | 0 | 0 | 1/8

##  4 Discussion and Conclusion

In this paper, we have proposed an effective scheme for fusing $|W_{n}\rangle$
and $|W_{m}\rangle$ states into a large-size $|W_{n+m-1}\rangle$ state by
using a partial-swap gate. We have also designed a scheme for fusing multiple
$W$ states of arbitrary size simultaneously (see Fig. 4). By exploiting the
spatial degrees of freedom of single-photons introduced by the PBSs, the
partial-swap gate was implemented using an optically polarized architecture
designed with linear-optical elements.

As shown in Table 2, our scheme outperforms previous ones in fusing two $W$
states of arbitrary size into a large-sized $W$ state. An ancillary photon,
which is necessary in the Fredkin- and Toffoli-based schemes Bugu ; Fu-T  to
create a $|W_{n+m-1}\rangle$ state, is not required in our scheme. Moreover,
our scheme minimizes failure outcomes. From Table 3, one can see that, if the
gate (Fredkin, Toffoli, CNOT, and partial-swap gates) operations are
considered, the fail probability in the presented scheme is lower than that
with the Fredkin- and Toffoli-based schemes Bugu ; Yesilyurt .

Our presented scheme has the further advantage of reducing cost in terms of
the number of two-qubit gates. In previous studies Bugu ; Fu-T , the fusion of
two $W$ states required either one Fredkin gate, or one Toffoli and one CNOT
gate. Our presented approach requires just one partial-swap gate. Notably, the
optimal cost of an unconstructed Toffoli or Fredkin gate is five two-qubit
gates Optimal1 ; Optimal2 . If we impose a further condition of using only
CNOT gates, at least six CNOT gates are required to synthesize a Toffoli and
at least eight for a Fredkin gate Optimal3 . In contrast, the current
proposals based on partial-swap gates surpass the Fredkin-gate scheme Bugu
and Toffoli-CNOT-scheme Fu-T , and also surpass the scheme based on one
Toffoli gate and three CNOT gates Yesilyurt  (see Table 2 and Table 3).

Another important advantage of the our scheme is its increased success
probability. It is known that entangling quantum gates can be implemented only
probabilistically using linear-optical elements. With a polynomial quantum
resource, a linear-optical CNOT gate can be implemented with a maximal success
probability of 3/4 KLM , and a post-selected CNOT gate with a success
probability of 1/9 1/9 . The most efficient scheme for a CNOT gate with a
success probability of 1/4 is achieved with the help of a maximally entangled
photon pair 1/41 ; 1/42 . Moreover, the ideal success probability of a Toffoli
gate is 1/9 under a linear-optics setup Toffoli . At present, the optimal
success probability of a linear optical Fredkin gate is 1/64 Fredkin . In
contrast, the proposed partial-swap gate with a success probability of 1/4 is
achievable. To sum up, our partial-swap-based fusion schemes outperform the
Fredkin-based Bugu  and Toffoli-CNOT-based schemes Yesilyurt ; Fu-T  in term
of the cost and success probability.

## Acknowledgment

The work was supported by the National Natural Science Foundation of China
under Grant No. 11604012, and the Fundamental Research Funds for the Central
Universities under Grant Nos. FRF-BR-17-004B and 230201506500024, and a grant
from the China Scholarship Council. KLC is supported by the Ministry of
Education and the National Reserach Foundation Singapore.

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