T}\Big{[}\|\phi_{1}-\phi_{2}\|_{C_{\Phi}((-\infty,-T)\times(0,\infty))}+\|h_{1}-h_{2}\|_{\Lambda_{T}}\Big{]},$

where we choose $T$ large enough. Thus we can obtain the estimate (4.5).

##  5\. The reduction procedure: choosing the parameter $h$

As we stated in Section 2.2, the left job is to choose suitable $h(t)$, i.e.
$\rho(t)=\gamma_{n}(t)+h(t)$, such that the function $c(t)$ vanishes in
equation (2.16).

###  5.1. Deriving the reduced equation involving $h(t)$

According to (2.21), the relation $c(t)=0$ is equivalent to

$\displaystyle 0=$
$\displaystyle\int_{0}^{\infty}\Big{[}\omega^{\prime\prime\prime}\big{(}r-\rho(t)\big{)}+\frac{n-1}{r}\omega^{\prime\prime}\big{(}r-\rho(t)\big{)}-W^{\prime\prime}\big{(}z(t,r)\big{)}\omega^{\prime}\big{(}r-\rho(t)\big{)}\Big{]}$
$\displaystyle\qquad\qquad\times\left(-\phi_{rr}-\frac{n-1}{r}\phi_{r}+W^{\prime\prime}\big{(}z(t,r)\big{)}\phi\right)r^{n-1}{\mathrm{d}}r$
$\displaystyle+\int_{0}^{\infty}\left[\partial_{rr}z(t,r)+\frac{n-1}{r}\partial_{r}z(t,r)-W^{\prime}\big{(}z(t,r)\big{)}\right]\phi\omega^{\prime}\big{(}r-\rho(t)\big{)}r^{n-1}{\mathrm{d}}r$
$\displaystyle+\int_{0}^{\infty}\phi(t,r)\partial_{t}\big{[}\omega^{\prime}\big{(}r-\rho(t)\big{)}\big{]}r^{n-1}{\mathrm{d}}r\,+\,\int_{0}^{\infty}\big{(}E(t,r)+N(\phi)\big{)}\omega^{\prime}\big{(}r-\rho(t)\big{)}r^{n-1}{\mathrm{d}}r,$
(5.1)

where the error term $E(t,r)$ and the nonlinear term $N(\phi)$ are defined by
(2.18)-(2.19), $z(t,r)$ is given by (2.14) and $\phi,\phi_{r},\phi_{rr}\in
C_{\Phi}((-\infty,-T)\times(0,+\infty))$ defined by (3.18). The tedious
computations of all terms in (5.1) will be given in the following parts.

#### 5.1.1.

We first estimate the projection of the error term $E(t,r)$. According to
(2.24) and (2.25), we have that

$\displaystyle\int_{0}^{+\infty}E(t,r)\omega^{\prime}\big{(}r-\rho(t)\big{)}r^{n-1}{\mathrm{d}}r=$
$\displaystyle\int_{0}^{\delta_{0}}E(t,r)\omega^{\prime}\big{(}r-\rho(t)\big{)}r^{n-1}{\mathrm{d}}r$
$\displaystyle+\sum_{l=1}^{5}\int_{\delta_{0}}^{+\infty}\widetilde{E}_{l}(t,r)\omega^{\prime}\big{(}r-\rho(t)\big{)}r^{n-1}{\mathrm{d}}r,$
(5.2)

where $\gamma_{n}(t)$ is given by (2.8) with fixed small positive number
$\delta_{0}$, and the functions
$\widetilde{E}_{1}(t,r),\cdots,\widetilde{E}_{5}(t,r)$ are defined as follows

$\displaystyle\widetilde{E}_{1}(t,r)$
$\displaystyle:=\omega^{\prime}\big{(}r-\rho(t)\big{)}\left(\rho^{\prime}(t)+\frac{(n-3)(n-1)^{2}}{2r^{3}}\right),$
(5.3) $\displaystyle\widetilde{E}_{2}(t,r)$
$\displaystyle:=-\partial_{rr}\Big{(}\partial_{rr}\widehat{\omega}(t,r)-W^{\prime}\big{(}\widehat{\omega}(t,r)\big{)}\Big{)}+W^{\prime\prime}\big{(}\widehat{\omega}(t,r)\big{)}\Big{(}\partial_{rr}\widehat{\omega}(t,r)-W^{\prime}\big{(}\widehat{\omega}(t,r)\big{)}\Big{)},$
$\displaystyle\widetilde{E}_{3}(t,r)$
$\displaystyle:=\frac{2(n-1)}{r}\Big{[}W^{\prime\prime}\big{(}\widehat{\omega}(t,r)\big{)}-W^{\prime\prime}\big{(}\omega\big{(}r-\rho(t)\big{)}\big{)}\Big{]}\omega^{\prime}\big{(}r-\rho(t)\big{)},$
$\displaystyle\widetilde{E}_{4}(t,r)$
$\displaystyle:=F^{\prime}\big{(}\widehat{\omega}(t,r)\big{)}\big{[}\widetilde{z}(t,r)\big{]}-\frac{(n-1)(n-3)}{r^{2}}\left[\frac{(n-3)}{2r}\omega^{\prime}\big{(}r-\rho(t)\big{)}+\omega^{\prime\prime}\big{(}r-\rho(t)\big{)}\right],$
$\displaystyle\widetilde{E}_{5}(t,r)$
$\displaystyle:=F^{\prime\prime}\big{(}\widehat{\omega}(t,r)+\theta\widetilde{z}(t,r)\big{)}\big{[}\widetilde{z}(t,r),\widetilde{z}(t,r)\big{]}-\frac{\partial\widetilde{z}(t,r)}{\partial
t},$

where the functions $\widehat{\omega}(t,r)$ and $\widetilde{z}(t,r)$ are
defined by (2.6) and (2.22) respectively.

For the first term in the right hand side of (5.2), by Lemma 2.3, we have that

$\displaystyle\left|\int_{0}^{\delta_{0}}E(t,r)\partial_{r}\big{[}\omega\big{(}r-\rho(t)\big{)}\big{]}r^{n-1}{\mathrm{d}}r\right|$
$\displaystyle\leq
C\int_{0}^{\delta_{0}}\Phi(t,r)\omega^{\prime}\big{(}r-\rho(t)\big{)}r^{n-1}{\mathrm{d}}r$
(5.4) $\displaystyle\leq
C\frac{\,\big{[}\rho(t)\big{]}^{n-1}\log\left|t\right|}{\left|t\right|^{1/2}\,}\exp\Bigg{\\{}-\frac{\gamma_{n}(t)}{2}\Bigg{\\}},$

where $\gamma_{n}(t)$ is defined by (2.8) and $C>0$ only depends on $n$ and
$\alpha=\sqrt{W^{\prime\prime}(1)}$.

Next we will estimate other terms in the right hand side of (5.2). Using the
definition of $\widehat{\omega}(t,r)$ in (2.6) and equation in (1.11), we have
that

$\widetilde{E}_{2}(t,r)=\widetilde{E}_{3}(t,r)=0,\qquad\text{for all}\
r>\delta_{0}.$

For $\widetilde{E}_{1}(t,r)$ in (5.3), by Lemma 2.1, we have that

$\displaystyle\int_{\delta_{0}}^{\infty}\widetilde{E}_{1}(t,r)\omega^{\prime}\big{(}r-\rho(t)\big{)}r^{n-1}{\mathrm{d}}r$
$\displaystyle=\int_{\delta_{0}}^{\infty}\omega^{\prime}\big{(}r-\rho(t)\big{)}\left(\rho^{\prime}(t)+\frac{(n-3)(n-1)^{2}}{2r^{3}}\right)\omega^{\prime}\big{(}r-\rho(t)\big{)}r^{n-1}{\mathrm{d}}r$
$\displaystyle=\left(\rho^{\prime}(t)+\frac{(n-3)(n-1)^{2}}{2\rho^{3}(t)}\right)\big{[}\rho(t)\big{]}^{n-1}\int_{\mathbb{R}}\big{[}\omega^{\prime}(x)\big{]}^{2}{\mathrm{d}}x\,+\,O\left(\frac{\big{[}\rho(t)\big{]}^{n-1}\log\left|t\right|}{\left|t\right|^{5/4}}\right),$
(5.5)

where we have used the fact that

$\int_{\mathbb{R}}[\omega^{\prime}(y)]^{2}y{\mathrm{d}}y=0,$

in the last equality.

Next we estimate the terms $\widetilde{E}_{4}(t,r)$ given in (5.3), by the
definition of $\widetilde{z}(t,r)$ in (2.22) and the same arguments in proof
of Lemma 2.3, (2.20), for all $r>\delta_{0}$, we have that

$\displaystyle\widetilde{E}_{4}(t,r)$
$\displaystyle=\frac{(n-1)(n-3)}{r^{3}}\Bigg{\\{}4\partial_{rrr}\widetilde{\omega}\big{(}r-\rho(t)\big{)}-4W^{\prime\prime}\big{(}\omega\big{(}r-\rho(t)\big{)}\big{)}\partial_{r}\widetilde{\omega}\big{(}r-\rho(t)\big{)}-\frac{(n-3)}{2}\omega^{\prime}\big{(}r-\rho(t)\big{)}$
$\displaystyle\qquad\qquad\qquad-2(n-3)\Big{[}\partial_{rrr}\widetilde{\omega}\big{(}r-\rho(t)\big{)}-W^{\prime\prime}\big{(}\omega\big{(}r-\rho(t)\big{)}\big{)}\partial_{r}\widetilde{\omega}\big{(}r-\rho(t)\big{)}\omega^{\prime}\big{(}r-\rho(t)\big{)}$
$\displaystyle\qquad\qquad\qquad\qquad-W^{\prime\prime\prime}\big{(}\omega\big{(}r-\rho(t)\big{)}\big{)}\widetilde{\omega}\big{(}r-\rho(t)\big{)}\Big{]}\Bigg{\\}}$
$\displaystyle\quad+\frac{1}{r^{4}}\Bigg{\\{}\Big{[}12(n-4)-(n-1)(n-3)\Big{]}\partial_{rr}\widetilde{\omega}\big{(}r-\rho(t)\big{)}-4(n-4)W^{\prime\prime}\big{(}\omega\big{(}r-\rho(t)\big{)}\big{)}\widetilde{\omega}\big{(}r-\rho(t)\big{)}\Bigg{\\}}$
$\displaystyle\quad+\rho^{\prime}(t)\frac{(n-1)(n-3)}{r^{2}}\widetilde{\omega}^{\prime}\big{(}r-\rho(t)\big{)}\,+\,O\left(\frac{1}{r^{5}}\sum_{j=1}^{k}e^{-\frac{\alpha}{2}}\left|r-\rho(t)\right|\right),$

where we have used that $\widetilde{\omega}(x)$ in (2.11) and its derivatives
have exponential decay.

Thus, by same arguments in above estimate of the projection of
$\widetilde{E}_{2}(t,r)$ and the equalities in (2.12), (2.11) and (1.11),
integrating by parts, we have that

$\displaystyle\int_{\delta_{0}}^{\infty}\widetilde{E}_{4}(t,r)\omega^{\prime}\big{(}r-\rho(t)\big{)}r^{n-1}{\mathrm{d}}r$
$\displaystyle=4(n-3)(n-1)\int^{\infty}_{\frac{\gamma_{n}}{2}-\rho(t)}\omega^{\prime}(y)\Big{[}\widetilde{\omega}^{\prime\prime\prime}(y)-W^{\prime\prime}\big{(}\omega(y)\big{)}\widetilde{\omega}^{\prime}(y)\Big{]}\big{[}y+\rho(t)\big{]}^{n-4}{\mathrm{d}}y$
$\displaystyle\quad+\frac{(n-3)^{2}(n-1)}{2}\int^{\infty}_{\delta_{0}-\rho(t)}\omega^{\prime}(y)\Big{[}2y\omega^{\prime\prime}(y)+\omega^{\prime}(y)\Big{]}\big{[}y+\rho(t)\big{]}^{n-4}{\mathrm{d}}y\,+\,\int^{\infty}_{\delta_{0}-\rho}\frac{\omega^{\prime}(y)\big{[}y+\rho(t)\big{]}^{n-6}}{\exp\Big{\\{}\alpha\frac{\left|y\right|}{2}\Big{\\}}}{\mathrm{d}}y$
$\displaystyle\quad\,+\,\int^{\infty}_{\delta_{0}-\rho(t)}\Bigg{[}\Big{(}12(n-4)-4(n-1)(n-3)\Big{)}\widetilde{\omega}^{\prime\prime}(y)-4(n-4)W^{\prime\prime}\big{(}\omega(y)\big{)}\widetilde{\omega}(y)\Bigg{]}\omega^{\prime}(y)\big{[}y+\rho(t)\big{]}^{n-4}{\mathrm{d}}y$
$\displaystyle\quad+\frac{\rho^{\prime}(t)}{\big{[}\rho(t)\big{]}^{2}}\int^{\infty}_{\delta_{0}-\rho(t)}\widetilde{\omega}^{\prime}(y)\omega^{\prime}(y)dy$
$\displaystyle=O\left(\frac{\big{[}\rho(t)\big{]}^{n-1}}{\left|t\right|^{5/4}}\right),$

where we have used (2.10), (2.12), (2.13) and the fact that

$\int_{\mathbb{R}}\omega^{\prime}(y)\Big{[}2y\omega^{\prime\prime}(y)+\omega^{\prime}(y)\Big{]}{\mathrm{d}}y=0.$

For the term $\widetilde{E}_{5}(t,r)$ in (5.3), by the definitions in (2.22)
and (2.23), the properties of $W$ in (1.2) and the oddness of
$\widetilde{\omega}$ in (2.11) respectively, and the equalities in (2.12), we
get that

$\displaystyle\int_{\delta_{0}}^{\infty}\widetilde{E}_{5}(t,r)\omega^{\prime}\big{(}r-\rho(t)\big{)}r^{n-1}{\mathrm{d}}r$
$\displaystyle=\int_{\delta_{0}}^{\infty}\Bigg{\\{}\partial_{rr}\left[W^{\prime\prime\prime}\big{(}\omega\big{(}r-\rho(t)\big{)}\big{)}\big{(}\omega\big{(}r-\rho(t)\big{)}\big{)}^{2}\right]-W^{\prime\prime\prime}\big{(}\omega\big{(}r-\rho(t)\big{)}\big{)}W^{\prime\prime}\big{(}\omega\big{(}r-\rho(t)\big{)}\big{)}\big{(}\omega\big{(}r-\rho(t)\big{)}\big{)}^{2}$
$\displaystyle\quad+2\Big{[}\partial_{rr}\widetilde{\omega}\big{(}r-\rho(t)\big{)}-W^{\prime\prime}\big{(}\omega\big{(}r-\rho(t)\big{)}\big{)}\widetilde{\omega}\big{(}r-\rho(t)\big{)}\Big{]}W^{\prime\prime\prime}\big{(}\omega(r-\rho(t)\big{)}\widetilde{\omega}\big{(}r-\rho(t)\big{)}\Bigg{\\}}\omega^{\prime}\big{(}r-\rho(t)\big{)}r^{n-5}{\mathrm{d}}r$
$\displaystyle\quad+O\left(\int^{\infty}_{\delta_{0}-\rho(t)}\frac{\omega^{\prime}(y)\big{[}y+\rho(t)\big{]}^{n-6}}{\exp\Big{\\{}\alpha\frac{\left|y\right|}{2}\Big{\\}}}{\mathrm{d}}y\right)$
$\displaystyle=O\left(\frac{\big{[}\rho(t)\big{]}^{n-1}}{\left|t\right|^{5/4}}\right).$

####  5.1.2.

We next estimate other terms in the right hand side of (5.1). By $\phi\in
C_{\Phi}((-\infty,-T)\times(0,+\infty))$ and the definitions of $\Phi(t,r)$
and $\rho(t)$ in (2.26) and (2.7) respectively, we have that

$\displaystyle\int_{0}^{\infty}\left|N(\phi)\right|\omega^{\prime}\big{(}r-\rho(t)\big{)}r^{n-1}{\mathrm{d}}r$
$\displaystyle\leq
C\int_{0}^{\infty}\big{(}\Phi(t,r)\big{)}^{2}\omega^{\prime}\big{(}r-\rho(t)\big{)}r^{n-1}{\mathrm{d}}r$
$\displaystyle\leq
C\frac{\big{[}\log\left|t\right|\big{]}^{2}}{\left|t\right|}\int^{+\infty}_{\frac{\gamma_{n}(t)}{4}-\rho(t)}\left(1+\left|x+\rho(t)-\frac{\alpha}{3}\log\left|t\right|\right|\right)^{-2p}\big{[}x+\rho(t)\big{]}^{n-1}\omega^{\prime}(x){\mathrm{d}}x$
$\displaystyle\quad+C\frac{\big{[}\log\left|t\right|\big{]}^{2}}{\left|t\right|}\big{[}\rho(t)\big{]}^{n-1}\exp\Bigg{\\{}-\frac{\gamma_{n}(t)}{2}\Bigg{\\}}.$

Thus, by Lemma 2.1 and direct computations, we have that

$\int_{0}^{\infty}\left|N(\phi)\right|\omega^{\prime}\big{(}r-\rho(t)\big{)}r^{n-1}{\mathrm{d}}r\leq
C\frac{\big{[}\rho(t)\big{]}^{n-1}}{\left|t\right|\big{[}\log\left|t\right|\big{]}^{2(p-1)}},$
(5.6)

where $C$ only depends on $\alpha,n$ and $\|h\|_{L^{\infty}}$.

We estimate the first and second term in (5.1). According to Taylor expansion
and (1.11), the same argument as the proof of Lemma 3.6, we have that

$\displaystyle\Bigg{|}$
$\displaystyle\int_{0}^{\infty}\Big{[}-\omega^{\prime\prime\prime}\big{(}r-\rho(t)\big{)}-\frac{n-1}{r}\omega^{\prime\prime}\big{(}r-\rho(t)\big{)}+W^{\prime\prime}\big{(}z(t,r)\big{)}\omega^{\prime}\big{(}r-\rho(t)\big{)}\Big{]}$
$\displaystyle\qquad\qquad\times\left[\phi_{rr}+\frac{n-1}{r}\phi_{r}-W^{\prime\prime}\big{(}z(t,r)\big{)}\phi\right]r^{n-1}{\mathrm{d}}r$
$\displaystyle\
+\int_{0}^{\infty}\left[\partial_{rr}z(t,r)+\frac{n-1}{r}\partial_{r}z(t,r)-W^{\prime}\big{(}z(t,r)\big{)}\right]W^{\prime\prime\prime}\big{(}z(t,r)\big{)}\phi\omega^{\prime}\big{(}r-\rho(t)\big{)}r^{n-1}{\mathrm{d}}r\Bigg{|}$
$\displaystyle\leq
C\int_{-\rho(t)}^{\infty}\left|\left[\omega^{(4)}(r)-W^{\prime\prime}\big{(}\omega(r)\big{)}\omega^{\prime\prime}(r)-W^{\prime\prime\prime}\big{(}\omega(r)\big{)}[\omega^{\prime}(r)]^{2}\right]\phi\big{(}t,r+\rho(t)\big{)}\right|\big{[}r+\rho(t)\big{]}^{n-1}{\mathrm{d}}r$
$\displaystyle\leq
C\frac{\big{[}\rho(t)\big{]}^{n-1}}{\left|t\right|\big{[}\log\left|t\right|\big{]}^{p-2}}.$

By Lemma 2.1 and the definition of $\rho$ in (2.7), similar arguments as in
(5.6), we have

$\displaystyle\left|\rho^{\prime}(t)\int_{0}^{\infty}\phi(t,r)\omega^{\prime\prime}\big{(}r-\rho(t)\big{)}r^{n-1}{\mathrm{d}}r\right|$
$\displaystyle\leq\frac{C}{\left|t\right|^{3/4}}\left|\int_{0}^{\infty}\Phi(t,r)\omega^{\prime\prime}\big{(}r-\rho(t)\big{)}r^{n-1}{\mathrm{d}}r\right|$
$\displaystyle\leq C\frac{\big{[}\rho(t)\big{]}^{n-1}}{\left|t\right|^{5/4}}.$

Combining the above estimates, we can obtain that (5.1) is equivalent to the
following ODE:

$\displaystyle\rho^{\prime}(t)+\frac{(n-3)(n-1)^{2}}{2\rho^{3}(t)}=Q\big{(}\rho(t),\rho^{\prime}(t)\big{)},$
(5.7)

for all $t<-T$, where we recall that $\rho(t)=\gamma_{n}(t)+h(t)$, and
$\gamma_{n}(t)$ is given by (2.8). The function $h(t)$ belongs to the set
$\Lambda_{T}$ with $T>T_{2}$, where $T_{2}$ is given by Proposition 4.1 and we
also recall the following two definitions in (4.2)-(4.3)

$\Lambda_{T}=\left\\{h(t):h\in
C^{1}(-\infty,-T]\quad\text{and}\quad\|h\|_{\Lambda_{T}}<1\right\\},$

where

$\|h\|_{\Lambda_{T}}=\sup_{t\leq-T}\left|h(t)\right|+\sup_{t\leq-T}\left[\frac{\left|t\right|}{\log\left|t\right|}\left|h^{\prime}(t)\right|\right].$
(5.8)

According to the above arguments, Proposition 4.1 and Lemma 2.3, we have that

######  Proposition 5.1.

Let $p\in(n,n+1]$ and
$P(h(t),h^{\prime}(t)):=Q\big{(}\rho(t),\rho^{\prime}(t)\big{)}$ in (5.7).
Then for all $h,h_{1},h_{2}\in\Lambda_{T}$, there hold that

$\left|P(h,h^{\prime})\right|\leq\frac{C(n,\alpha)}{\left|t\right|\big{[}\log\left|t\right|\big{]}^{2(p-1)}}$

and

$\left|P\big{(}{h}_{1},(h_{1})^{\prime}\big{)}-P\big{(}h_{2},(h_{2})^{\prime}\big{)}\right|\leq\frac{C(n,\alpha)}{\left|t\right|\big{[}\log\left|t\right|\big{]}^{2(p-1)}}\|h_{1}-h_{2}\|_{\Lambda_{T}},$

where we recall that $\alpha=\sqrt{W^{\prime\prime}(1)}>0$ and $C(n,\alpha)$
is an uniform constant depending on $n$ and $\alpha$.

###  5.2. Solving the reduced equation involving $h$

In the rest of this section we will study the ODE in (5.7). We look for
solutions of (5.7) of the form $\rho(t)=\gamma_{n}(t)+h(t)$, then $h(t)$
satisfies that

$h^{\prime}(t)+\gamma^{\prime}_{n}(t)+\frac{(n-3)(n-1)^{2}}{2\big{[}\gamma_{n}(t)+h(t)\big{]}^{3}}=P\big{(}h(t),h^{\prime}(t)\big{)}\qquad\text{in}\
(-\infty,-\widehat{T}_{0}],$

where $\widehat{T}_{0}>T_{2}$, where $T_{2}$ is given by Proposition 4.1. By
the definition of $\gamma_{n}(t)$ in (2.8), the above equation is equivalent
to

$h^{\prime}(t)+\frac{3h(t)}{4t}=\widetilde{P}\big{(}h(t),h^{\prime}(t)\big{)}\qquad\text{in}\
(-\infty,-\widehat{T}_{0}],$ (5.9)

where

$\widetilde{P}\big{(}h(t),h^{\prime}(t)\big{)}:=P\big{(}h(t),h^{\prime}(t)\big{)}+\frac{(n-3)(n-1)^{2}}{2}\left[\frac{1}{\big{(}\gamma_{n}(t)\big{)}^{3}}-\frac{1}{\big{(}\gamma_{n}(t)+h(t)\big{)}^{3}}-\frac{3h(t)}{\big{(}\gamma_{n}(t)\big{)}^{4}}\right].$
(5.10)

We will solve equation (5.9) by applying the fixed-point theorem in a suitable
space with $h(-\widehat{T}_{0})=0$. It is easily to check that if $h(t)$ is a
solution of (5.9) with initial data $0$, then it has the form

$h(t)=-\frac{1}{{(-t)}^{\frac{3}{4}}}\int^{-T_{0}}_{t}{(-s)}^{\frac{3}{4}}\widetilde{P}\big{(}h(s),h^{\prime}(s)\big{)}\mathrm{d}s,$
(5.11)

with $t\leq-\widehat{T}_{0}$.

Let us define two operators as following

$\mathcal{P}(h(t)):=-\frac{1}{{(-t)}^{\frac{3}{4}}}\int^{-\widehat{T}_{0}}_{t}{(-s)}^{\frac{3}{4}}\widetilde{P}\big{(}h(s),h^{\prime}(s)\big{)}\mathrm{d}s\qquad\text{and}\qquad\mathcal{P}^{\prime}(h(t)):=\partial_{t}\mathcal{P}(h(t)).$

Then using Proposition 5.1 and (5.10), we have that

$\left|\mathcal{P}(0)\right|\leq\frac{\widetilde{C}(n,\alpha)}{\log\widehat{T}_{0}}\quad\text{and}\quad\frac{\left|t\right|}{\log\left|t\right|}\left|\mathcal{P}^{\prime}(0)\right|\leq\frac{\widetilde{C}(n,\alpha)}{\log\widehat{T}_{0}},$
(5.12)

with $\widehat{T}_{0}>e^{2}$, where $\widetilde{C}(n,\alpha)$ is a positive
constant depending on $n$ and $\alpha$. We consider the domain

$Y:=\left\\{h(t)\in C^{1}(-\infty,-\widehat{T}_{0}]\ :\
\|h(t)\|_{\Lambda_{\widehat{T}_{0}}}\leq\frac{2\widetilde{C}(n,\alpha)}{\log\widehat{T}_{0}}\right\\},$

where the norm $\|\cdot\|_{\Lambda_{T}}$ is given by (5.8) and
$\widetilde{C}(n,\alpha)$ is a positive constant in (5.12).

According to Proposition 5.1 and (5.10), we get that for all $h_{1},h_{2}\in
Y$,

$\left|\mathcal{P}(h_{1})-\mathcal{P}(h_{2})\right|\leq\frac{C(n,\alpha)}{\log\widehat{T}_{0}}\|h_{1}(t)-h_{2}(t)\|_{\Lambda_{\widehat{T}_{0}}}$

and

$\left|\mathcal{P}^{\prime}(h_{1})-\mathcal{P}^{\prime}(h_{2})\right|\leq\frac{C(n,\alpha)}{\log\widehat{T}_{0}}\|h_{1}(t)-h_{2}(t)\|_{\Lambda_{\widehat{T}_{0}}},$

where $C(n,\alpha)>0$ only depends on $n$ and $\alpha$. Thus by Banach fix
point theorem, there exists $h(t)\in Y$ such that $\mathcal{P}(h(t))=h(t)$, if
we choose $\widehat{T}_{0}$ big enough. Thus we proved that the ODE in (5.7)
is solvable. Furthermore, by the formula in (5.11), we get that

$\left|h(t)\right|\leq\frac{C}{\log\left|t\right|},\qquad\text{as}\
t\rightarrow-\infty.$

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# Qubit-based distributed frame synchronization for quantum key distribution

Ye Chen,1 Chunfeng Huang1, Shuyi Huang1, Zhenrong Zhang2 and Kejin Wei1,∗
1Guangxi Key Laboratory for Relativistic Astrophysics, School of Physical
Science and Technology, Guangxi University, Nanning 530004, China
2Guangxi Key Laboratory of Multimedia Communications and Network Technology,
School of Computer Electronics and Information, Guangxi University, Nanning
530004, China
<EMAIL_ADDRESS>
###### Abstract

Quantum key distribution (QKD) is a method that enables two remote parties to
share a secure key string. Clock synchronization between two parties is a
crucial step in the normal operation of QKD. Qubit-based synchronization can
achieve clock synchronization by transmitting quantum states between two
remote parties, eliminating the necessity for hardware synchronization and
thereby greatly reducing the hardware requirements of a QKD system.
Nonetheless, classical qubit-based synchronization exhibits poor performance
in continuous and high-loss systems, hindering its wide applicability in
various scenarios. We propose a qubit-based distributed frame synchronization
method that can achieve time recovery in a continuously running system and
resist higher losses. Experimental results show that the proposed method
outperforms the advanced qubit-based synchronization method Qubit4Sync in a
continuously running system. We believe our method is applicable to a broad
range of QKD scenarios, including drone-based QKD and quantum network
construction.

##  I Introduction

Quantum key distribution (QKD) is a secure protocol that enables two
communicating parties to share a secret key over an insecure channel. Since
the proposal of the first QKD protocol in 1984 Bennett and Brassard (1984),
QKD has achieved significant advancements in fiber links Chen  _et al._
(2010); Wang  _et al._ (2012); Yuan  _et al._ (2018); Boaron  _et al._ (2018);
Zhou  _et al._ (2021); Ma  _et al._ (2021); Tang  _et al._ (2022), free-space
links Takenaka  _et al._ (2017); Liao  _et al._ (2017); Chen  _et al._ (2020);
Liu  _et al._ (2020), and network structures Dynes  _et al._ (2019); Chen  _et
al._ (2021); Fan-Yuan  _et al._ (2022); Wang  _et al._ (2014); Sasaki  _et
al._ (2011). Remarkably, long-distance communication over a 1000 km fiber
spool Liu  _et al._ (2023) was achieved using twin-field QKD Lucamarini  _et
al._ (2018); Wang  _et al._ (2018).

Clock synchronization is a crucial step in the normal operation of QKD
systems. Existing QKD demonstrations generally synchronize their clocks using
the global navigation satellite system Schmitt-Manderbach  _et al._ (2007);
Vallone  _et al._ (2015); Avesani  _et al._ (2021a), electrical cables Wei
_et al._ (2013); Islam  _et al._ (2017); Ioannou  _et al._ (2022); Wei  _et
al._ (2020); Gu  _et al._ (2022); Lu  _et al._ (2023), or classical light
Avesani  _et al._ (2022); Li  _et al._ (2023); Grünenfelder  _et al._ (2023).
All of these methods require additional hardware, increasing the complexity of
the QKD system.

In 2020, Calderaro et al. Calderaro  _et al._ (2020) introduced the Qubit4Sync
method, wherein Alice transmits an initial public synchronization string in
the first state, and time synchronization is accomplished by Bob, by post-
processing the detection events. Consequently, synchronization can be achieved
by transmitting quantum states between Alice and Bob, thus eliminating the
necessity for hardware synchronization. This method has been demonstrated in
realistic QKD systems Agnesi  _et al._ (2020); Avesani  _et al._ (2021b), and
several variants have been reported Wang  _et al._ (2021); Cochran and
Gauthier (2021); Wang  _et al._ (2022).

However, Qubit4Sync has limitations that hinder its wide applicability. First,
Qubit4Sync encodes a synchronization string at the beginning of the quantum
states. The inherent jitter of the system clock may cause the clock recovery
process to fail. When clock synchronization fails, Alice must reestablish the
qubit-based synchronization process and becomes unable to distribute key bits
with Bob continuously. Second, to compensate for the channel loss, the length
of the synchronization string must increase with higher losses to achieve
successful synchronization. For example, at a channel loss of approximately 15
dB, a synchronization string of length $10^{6}$ is required, whereas at
channel losses exceeding 34 dB, the length is increased to $10^{7}$ Agnesi
_et al._ (2020).

Figure 1: Schematic of QKD based on qubit-based distributed frame
synchronization. Alice divides the synchronization string into individual bits
and inserts them into random strings to form synchronization frames, and uses
the synchronization frames to encode light pulses. The red and blue squares
represent synchronization and random bits, respectively. After the encoded
light pulse is transmitted through the channel, it is first decoded by Bob
into four quantum states and record time of detections. Bob recovers the
period $\tau^{B}$ of the pulse through these detections, and separates the
consecutive detections according to $\tau^{B}$ to form Bob’s receive string.
We use yellow squares to represent bits of the received string because Bob
cannot confirm the source of the received bits. Bob joins the portions of the
received string according to the structure of the synchronization frame
constructed by Alice. The separated detections encoded by the synchronization
string are used for clock synchronization, and the random string detections
are required for the QKD protocol.

Inspired by the distributed frame synchronization method used in classical
communication de Lind van Wijngaarden and Willink (2000), we propose a qubit-
based distributed frame synchronization method that can overcome the above
limitations in Qubit4Sync. In our method, Alice periodically inserts a segment
within the synchronization string into quantum states. This allows Bob to
execute the qubit-based synchronization algorithm and distill the secure key
bits continuously and simultaneously, even in the presence of the inherent
jitter of the system clock.

The proposed qubit-based distributed frame synchronization method was tested
in a polarization-based BB84 system with a frequency of 50 MHz. Experimental
results demonstrate that the system can establish synchronization using the
proposed method with a synchronization string of length $L=10^{5}$ under a
maximum transmittance loss of 29.7 dB. Furthermore, we compared our
distributed frame synchronization method with Qubit4Sync in the same system,
operating in continuously running mode. Experimental results show that when
the system operation time exceeds 80 s, our method maintains a low quantum bit
error rate and hence extracts secure key bits, whereas Qubit4Sync fails under
these conditions. In addition, we demonstrate that our method can execute time
offset recovery successfully without lengthening $L$ at a larger channel loss
via post-processing.

The remainder of this paper is organized as follows: In Sections II and III,
we describe our synchronization algorithm in detail. Section IV presents our
experimental results. Finally, we summarize our work in Section V.

##  II Description of our qubit-based synchronization method

Fig. 1 shows the schematic of a QKD system based on our proposed qubit-based
distributed frame synchronization method. In conventional QKD systems Alice
only encodes random bits into the quantum states for distilling secure key
bits and the time synchronization requires additional hardware. Conversely, in
our setup, Alice must encode the quantum states with a public synchronization
string for synchronization and random bits for key bit distillation. The time
between two consecutive qubits $\tau^{A}$ is set by Alice’s clock. The encoded
quantum states are then sent to Bob over a lossy channel.

Owing to channel loss, Bob receives some of the quantum states and analyzes
the received qubits, measuring the arrival time using his clock. Bob’s main
goal is to determine the positions of the detected qubits in Alice’s random
bits. This crucial step is essential for correctly generating the raw key
bits, and subsequently, the sifted key. To do so, Bob must accomplish three
tasks: $(i)$ Recover the period $\tau^{B}$ from the detections. $(ii)$
Separate the detection of the synchronization string from the random string.
$(iii)$ Calculate the time delay between the sent and measured strings. Below,
we describe how these three tasks can be accomplished using only the qubits
exchanged during the QKD protocol, without requiring additional hardware.

###  II.1 Generation of qubits for synchronization and key distribution

We begin by describing how Alice generates a frame for synchronization and key
distribution. As shown in Fig. 1, Alice generates a synchronization string
$s^{A}$ (the red string) with a total length $L$ and a random bit string (the
blue string) with a length of $M\times L$ where $M$ represents the ratio of
random bits to the number of synchronization bits in one synchronization
frame. $s^{A}$ is public and known to Bob; however, the random bit string is
used to generate secure key bits, which are kept secret from Bob. Alice
disperses the synchronization string of length $L$ into individual bits and
evenly inserts them into the random string to form a frame of length
$N_{f}=(M+1)L$. If Alice must produce several frames, she reuses $s_{A}$ and
refreshes the random string to meet security demands.

Similar to the method used in Qubit4Sync Calderaro  _et al._ (2020), $s^{A}$
is a particular string in which each bit $s_{i}^{A}$ has a value of $+1$ or
$-1$ and the total string has an autocorrelation $x^{AA}_{m}$ satisfying

$\displaystyle x^{AA}_{0}=1,$ (1) $\displaystyle x^{AA}_{jL_{1}}\simeq
c_{0},~{}for~{}j>0,$ $\displaystyle x^{AA}_{u+jL_{1}}\simeq 0,~{}for~{}u>0,$

where $1>c_{0}>0$, $u=0,1,...,L_{1}-1$ and $j=0,1,...,N_{1}-1$. $N_{1}$ is the
number of periodic peaks and $L_{1}N_{1}=L$. $s^{A}$ can be generated using
the following equation:

$s^{A}_{u+jL_{1}}=2\Theta(y_{u,j}-\lambda x_{u})-1,$ (2)

where $\Theta(x)$ is the Heaviside function and $x_{u}$ and $y_{u,j}$ are
random numbers uniformly distributed between 0 and 1. The parameter $\lambda$
is used to tune the value of $c_{0}$. If $\lambda\leq 1$, we have
$c_{0}=\lambda^{2}/3$, whereas if $\lambda>1$, $c_{0}=1-2\lambda/3$.

Alice encodes her quantum states based on the previously generated strings. If
the individual bits are part of $s^{A}$, Alice assigns the values $+1$ and
$-1$ to H and V polarizations in $Z$ bases. If the bits are part of the random
string, Alice encodes the four polarizations based on the BB84 protocol.
Finally, the encoded quantum states are sent to Bob via a lossy quantum
channel.

Upon arrival at Bob’s location, the qubits are analyzed using a decoder, and
their arrival time is measured using Bob’s clock. Although Bob receives some
of the quantum states, channel loss and detector dark counts prevent him from
determining whether the detected signals originate from the synchronization
string $s^{A}$ or a random string. Initially, Bob cannot accurately determine
which detected qubits belong to Alice’s random bits. Next, we describe in
detail how Bob can accomplish this by establishing clock synchronization based
solely on the detections, making use of the autocorrelation properties of the
synchronization string. That is, Bob must first recover the period $\tau^{B}$
from the detections, then separate the detection of the synchronization string
from the random string, and finally calculate the time delay between the sent
and measured strings using the autocorrelation properties of the
synchronization string.

###  II.2 Period recovery and time compensation

In this subsection, we describe the process of recovering the period
$\tau^{B}$ and compensating for the constant time delay caused by detectors in
the detections. It is important to note that Bob’s period recovery plays a
crucial role in clock synchronization between Alice and Bob. Knowing the exact
value of Bob’s period $\tau^{B}$ enables him to accurately reconstruct the
separations in the raw key between consecutive detections.

To describe the synchronization algorithm clearly, we define $t^{m}_{a}$ as
the measured arrival time of qubits according to Bob’s clock, where $a\geq 1$
enumerates the obtained detections by Bob. Bob predicts $t^{e}_{a}$ as the
expected arrival time, expressed as

$t^{e}_{a}=t_{0}+n_{a}\tau^{B}+\epsilon_{a},\quad n_{a}\in{\mathbb{N}},$ (3)

where $t_{0}$ is the initial time offset and $n_{a}$ denotes the position of
the sent qubit in Alice’s raw key. $\epsilon_{a}$ is a normal random variable
with variance $\sigma^{2}$ and zero mean. Bob’s goal is to obtain $t_{0}$ and
$\tau^{B}$, which can be calculated from Bob’s detections and the exposed
synchronization string.

We first describe how Bob recovers the period $\tau^{B}$ from the detections.
Following an approach similar to that of Qubit4Sync Calderaro  _et al._
(2020), Bob samples the arrival times of photons and applies a fast Fourier
transform to the sampled data to obtain an initial estimate $\tau^{B}_{0}$ for
$\tau^{B}$. The sampling rate is $4/\tau^{A}$, where $\tau^{A}$ represents the
time at which Alice sends the light pulse. To expedite computation, the number
of samples is limited to $N_{s}=10^{6}$, resulting in an error between
$\tau^{B}_{0}$ and Bob’s actual period $\tau^{B}$ of approximately
$4/\tau^{A}N_{s}$. In general, $\tau^{B}_{0}$ cannot satisfy the accuracy
requirements for period calculation. To obtain a more accurate value, Bob
applies a least-trimmed-squares algorithm of mod${\tau_{0}^{B}}(t{a+b}^{m})$
as a function of $t_{a+b}^{m}$ and obtains the slope $k$ of the linear model.
The more accurate period for Bob, denoted by $\tau^{B}$, is given by
$\tau^{B}=\tau^{B}_{0}/(1-k)$.

The precisely calculated $\tau^{B}$ allows us to predict the measured time
$t^{m}_{a}$ more accurately, which satisfies

$\displaystyle\frac{1}{D}\sum_{b=1}^{D}|\mathcal{E}^{I}_{a}(b)|^{2}\simeq\sigma_{g}^{2}.$
(4)

where $D\in\mathbb{N}$ is the number of pulses detected by Bob,
$\mathcal{E}^{I}_{a}(b)$ is the time-interval error between two different
detections $a$ and $a+b$ and
$\mathcal{E}^{I}_{a}(b)=(t^{m}_{a+b}-t^{e}_{a+b})-(t^{m}_{a}-t^{e}_{a})$.
$\sigma_{g}$ is the gate width for post-processing, and all detection events
outside the gate are discarded.

When Bob obtains an accurate estimation of $\tau^{B}$, he performs post-
processing to compensate for the time deviation caused by photons arriving at
different detectors via various paths. We remark that this step is crucial for
implementing qubit-based synchronization in real-life systems, but has not
received sufficient consideration in previous studies. As shown in Fig. 1, in
a typical QKD system, the photons are detected by four detectors. Owing to
inherent electronics or path differences, the click times of the four
detectors are not exactly the same, even when using the same trigger. This
discrepancy can result in incorrect clock synchronization. If the absolute
value of the time deviation between two detectors is greater than
$\sigma_{g}$, they cannot satisfy Eq. 4 simultaneously. If it exceeds
$\tau^{B}/2$, the two detectors cannot recover the time offset simultaneously.

To compensate for electronics and path differences in the system, we can
calculate the time deviations of the four detectors based on Bob’s detections.
We denote the time of detection events corresponding to the four polarizations
as $t_{a}^{m,P}$ for $P\in{H,V,D,A}$. We then perform linear regression of mod
${}_{\tau^{B}}(t_{a}^{m,P})$. The intercept of the linear model for each
polarization on the vertical axis corresponds to the time deviations
introduced by the different paths. We use $t_{d}^{P}$ to represent the time
deviations, and define $t_{d}^{H}$ as 0. The time deviations $t_{d}^{P}$ for
the other polarizations can be calculated using the difference in the
intercept between the corresponding linear model and the linear model of
polarization $H$. We can compensate for the time deviations in $t^{m}_{a}$
directly through post-processing.

###  II.3 Synchronization string separation and time offset recovery

After successfully recovering the period $\tau^{B}$, Bob can correctly
separate consecutive detections and assign the values $+1$ and $-1$ to the two
orthogonal $Z$-basis states. In the event of $X$ bases or no detection, he
assigns the value 0. Finally, Bob can produce a string $s^{B}$ of length
$N_{f}$ with values 0, $-1$, or $+1$.

However, owing to channel loss and detector dark counts, Bob cannot
immediately determine which detections come from the synchronization string
and which are from the random string. This makes it impossible for Bob to
calculate the time offset by performing a cross-correlation operation
directly. Therefore, to calculate the time offset, Bob must first determine
which detections come from the synchronization string and reconstruct the
synchronization string $s^{B}_{syn}$ from these detections, and then calculate
the time offset using the reconstructed synchronization string $s^{B}_{syn}$.

We first describe how Bob identifies the detections of the synchronization
string. As shown in Fig. 2 (a), Bob first separates strings of $M+1$ adjacent
bits into blocks, producing $L$ blocks of length $M+1$ which are fitted into
an $(M+1)\times L$ matrix. Each row of the matrix constructs a string
$s^{B}_{i}$ of length $L$, where $i\in\\{1,2,3...(M+1)\\}$. Here, we use
yellow squares to represent the elements of $s^{B}$, because Bob cannot
confirm whether the detections are from the synchronization string or random
bit string. Based on the method described in Sections II.1, one row of the
constructed matrix has a maximal correlation with $s^{A}$; however, Bob does
not know which row that is. Bob’s next goal is to determine the optimal row
and use it to calculate the time offset.

Figure 2: Schematic of synchronization string separation and time offset
recovery. Yellow squares represent undetermined bits, and red squares
represent the synchronization string and detections from the synchronization
string. (a) Bob selects a received string $s^{B}$ of length $N_{f}$, and
extracts $M+1$ separated strings $s^{B}_{i}$ from it according to the
structure of the synchronization frame. (b) Bob rearranges the synchronization
string $s^{A}$ and all $s^{B}_{i}$ into a two-dimensional matrix of order
$L_{1}\times N_{1}$ and applies a fast Fourier transform to each row, denoted
as $S^{A}$ and $S^{B}_{i}$. (c) Cross correlation is made between the first
column of all $S^{B}_{i}$ and the first column of $S_{A}$, and $i_{opt}$ and
$u_{opt}$ corresponding to the maximum cross correlation value are recorded.
(d) According to $i_{opt}$ and $u_{opt}$, the remaining ($N_{1}$-1) columns of
$S^{B}_{i_{opt}}$ and $S_{A}$ are multiplied and combined with $X_{u_{opt},0}$
obtained from step (b) to form an array $X_{u_{opt},j}$. The row cross
correlation $X_{u_{opt}+jL_{1}}$ is obtained by inverse fast Fourier transform
(FFT-1), and the corresponding $j_{opt}$ when the cross-correlation value was
maximum is recorded.

Next we describe how Bob finds the optimal row, which has index $i_{opt}$ and
contains the string $s_{syn}$. Bob first rearranges the synchronization string
as shown in Fig. 2 (b), and applies a fast Fourier transform to each row of
all the rearranged matrices. The number of rows in the synchronization matrix
$S^{A}$ is doubled to reserve space for the lookup of $i_{opt}$ and offset
calculations.

To calculate $i_{opt}$ quickly, Bob cross-correlates the first column of
$S^{A}$ and all $S^{B}_{i}$ and denotes the result as $X_{u,0}$, as shown in
Fig. 2 (c). The matrix $S^{B}_{i}$ corresponding to the maximum cross-
correlation peak value $X_{u_{opt},0}$ is $S^{B}_{i_{opt}}$. Bob records
$i_{opt}$ and $u_{opt}$ for calculation of the offset. Finally, Bob conducts a
horizontal cross-correlation operation between $S^{B}_{i_{opt}}$ and $S_{A}$
according to the optimal row $u_{opt}$, and the position corresponding to the
peak of the cross-correlation is the optimal column $j_{opt}$.

The time offset can be calculated as
$\tau^{B}[i_{opt}+u_{opt}(M+1)+j_{opt}L_{1}(M+1)]$, based on the construction
coefficient $M$ of the synchronization frame, the period $\tau^{B}$ and the
position $i_{opt}$, $u_{opt}$, $j_{opt}$ corresponding to the optimal value of
the cross-correlation.

##  III Distributed frame synchronization resisting high loss

From Section II, we note that calculating the time offset is a key step that
requires sufficient detections by Bob to ensure that the maximum correlation
peak is distinguished from the others. When the overall transmittance loss is
excessively large, the time offset calculation may fail. Calderaro et al.
showed that given a synchronization string of length $L=10^{6}$, Qubit4Sync
can cope with a total loss of 40 dB in ideal conditions with no errors or dark
counts Calderaro  _et al._ (2020). The value of $L$ should increase as the
loss increases, which sacrifices the final secure bit rate.

Unlike Qubit4Sync, our distributed frame synchronization method generates each
frame using an identical synchronization string, although the synchronization
string $s_{syn_{i}}^{B}$ reconstructed from $K$ frames would differ owing to
the probabilistic loss of photons in a lossy channel, where the subscript
$i\in\\{1,2,...K\\}$. Because they all originate from $s^{A}$, Bob can
accumulate all the information from $s_{syn_{i}}^{B}$, forming a new string
$s_{all}^{B}$, to resist channel loss without increasing the length of the
synchronization string.

The detailed method is explained as follows. First, Bob extracts $K$ frames of
length $N_{f}$ continuously from his detections, which all contain complete
synchronization frame detections. Because these strings are contiguous, the
synchronization string is in the same position in all synchronization frames.
This allows Bob to supplement missing detections with detections of the same
location in other synchronization frames. In the case of multiple detections
occurring at the same location, the results with the highest frequency are
used. The time offset between Bob and Alice is estimated by calculating the
correlation between the reconstructed strings.

##  IV Experiment and results

Table 1: Synchronization results under different losses. Loss is the total loss between Alice and Bob, and QBER is the quantum bit error rate. The noise filter gate width $\sigma_{g}=\sigma$. Loss(dB) | 17.6 | 20.0 | 22.7 | 26.5 | 29.7
---|---|---|---|---|---
$\tau^{B}(ns)$ |  $20.0000184$ |  $20.0000182$ |  $20.0000186$ |  $20.0000185$ |  $20.0000185$
QBER | 0.459% | 0.617% | 0.457% | 0.788% | 0.625%

We tested our qubit-based distributed frame synchronization method by
modifying a polarization-based QKD system Chen  _et al._ (2022); Huang  _et
al._ (2022). The experimental setup is illustrated in Fig. 3. Alice used
commercial lasers (LD, WT-LD200-DL, Qasky Co. LTD) to generate light pulses
with a frequency of 50 MHz. Alice’s time reference was provided by an
arbitrary waveform generator (AWG), and the polarization was encoded through a
Sagnac-based polarization modulator (Sagnac PM). ATT1 attenuated the encoded
quantum light to the single-photon level, whereas ATT2 was used to simulate
channel loss.

At the receiving station, a polarization controller (PC) was used to
compensate actively for polarization drift during transmission over the
channel. The received quantum states were decoded by the polarization analysis
module (PAM) and detected by superconducting-nanowire single-photon detectors
(SNSPD) with an average efficiency of $51.3\%$ and a dark count rate of
$2\times 10^{-6}$. The receiver measured the polarization on the Z base with a
probability of $90\%$. The detection events were recorded using a time-to-
digital converter (TDC) triggered by an internal clock. Subsequently, the
detection events were processed using synchronization algorithms implemented
on a computer (CMP).

Figure 3:  Experimental setup. AWG: arbitrary waveform generator; LD:
commercial laser diode; $\theta$-M: phase modulator; CPM: customized
polarization module; ATT: optical attenuator; PC: polarization controller;
SNSPD: superconducting-nanowire single-photon detector; PAM: polarization
analysis module; TDC: time-to-digital converter; CMP: a computer for
synchronization algorithms.

We first tested the time compensation method described in in Section II.2. We
sent polarization H/V and performed period recovery by analyzing detections.
Bob’s detections were then sorted into the corresponding time windows based on
the recovery period $\tau^{B}$. To determine the time deviation introduced by
different paths, we calculated mod ${}_{\tau^{B}}(t_{a}^{m,P})$ for the
recorded time of four polarizations. Following the method described in Section
II.2, we can estimated the time deviation caused by the path difference. The
results are presented in Fig. 4. It can be seen that the black solid, dashed,
dotted, and dash-dot lines represent obtained average time deviations
$t_{d}^{H}$, $t_{d}^{V}$, $t_{d}^{D}$, and $t_{d}^{A}$ of 0 ns, 0.65 ns, 1.48
ns, and 1.01 ns, respectively. We applied obtained time deviations to post-
processing in the following tests.

We then evaluated the frame synchronization performance under various loss
conditions using the method described in Section III. Specifically, we set the
average number of photons per pulse sent by Alice to 1 and obtained five
different losses by adjusting ATT2. The synchronization process used a
synchronization string of length $L=10^{5}$ and synchronization frame
construction factor of $M=1$, with an acquisition time of 100 s. After
performing time recovery, we estimated the quantum bit error rate (QBER), with
results as presented in Table 1. Our method demonstrated consistent recovered
time periods and time delays under different losses. This enabled us to obtain
raw key bits with a QBER below $0.457\%$, which aligns closely with the
intrinsic optical misalignment of the system.

Figure 4: Distribution of time deviation for different polarization
detections. The estimated time deviation of the four polarizations H, V, D and
A are represented by red asterisks, blue asterisks, magenta triangles and
green triangles, respectively. Solid, dashed, dotted, and dash-dot lines
represent average time deviation for H, V, P, and M, respectively.

We conducted tests on a continuously running system to demonstrate the
stability of our method. For comparison, we also implemented Qubit4Sync using
the same system. We plotted the time error, which defines as
mod$\tau^{B}(\mathcal{E}^{I}_{a}(b))$, and QBER versa the continuous
acquisition time. The experimental results are shown in Fig. 5. After the
entire system had operated for more than 80 s, Qubit4Sync was unable to
recover the clock accurately, resulting in a time error of approximately 10 ns
and a QBER of $50\%$. In contrast, our method continued to operate
effectively. Here we present experimental results for a duration of only 150
s, but our method has been applied successfully in a chip-based system and has
been running continuously, collecting data for up to 6 h Wei  _et al._ (2023).

Figure 5: Stability test of Qubit4Sync and our method. The red and black circles represent the QBER of Qubit4Sync and our method, respectively. The blue and green diamonds represent the time error of Qubit4Sync and our method, respectively. Ideally, the time error is 0 ns and QBER is 0%. Table 2: Success probability and QBER in offset recovery test. Loss is the total loss of the system, $P_{s}$ is the probability of successful execution of offset recovery, QBER is the quantum bit error rate, and $K$ is the number of synchronization frames considered by the complement operation. The length of the synchronization string is $L=10^{5}$ and the total sampling time at each loss is 20 s. Loss (dB) | $K$ | 1 | 2 | 4 | 8
---|---|---|---|---|---
26.5 | $P_{s}$ | 20.3% | 81.1% | 98.9% | 99.4%
QBER | 40.0% | 10.1% | 1.33% | 1.08%
29.7 | $P_{s}$ | 5.35% | 16.4% | 78.2% | 98.7%
QBER | 47.4% | 41.9% | 11.4% | 1.27%

Finally, we tested the performance of the distributed frame synchronization
method at resisting high loss. We set a constant length of the synchronization
string as $L=10^{5}$ and post-processed the data recorded by Bob with an
acquisition time of 20 s. The post-processing method was as described in
Section III. The results are summarized in Table 2. At a channel loss of 26.5
dB, with an increasing number of synchronization frames considered by the
complement operation described in Section III, we obtained a lower QBER
because of the high probability of the successful execution of time offset
recovery. A similar observation applies to the channel loss of 29.7 dB. In
particular, at a higher channel loss, our method can execute time offset
recovery successfully by accumulating more $K$ frames without needing to
lengthen $L$.

##  V Conclusion

We present a novel qubit-based distributed frame synchronization method
inspired by classical communication techniques. Our proposed method addresses
the limitations encountered in the Qubit4Sync approach and offers improved
performance and robustness. By incorporating periodic quantum state insertions
into the synchronization string, we enable the continuous execution of the
qubit-based synchronization algorithm and secure key bit distillation, even
when confronted with inherent system clock jitter.

The feasibility and robustness of the proposed method were demonstrated
through an experimental implementation in a polarization-based BB84 system
operating at a frequency of 50 MHz. The experimental results demonstrate the
successful establishment of synchronization using our proposed method. We
believe our synchronization algorithm is applicable to a broad range of QKD
scenarios, including drone-based QKD Isaac  _et al._ ; Tian  _et al._ (2023),
chip-based systems Paraïso  _et al._ (2021); Du  _et al._ (2023); Sax  _et
al._ (2023), and the construction of quantum networks Fröhlich  _et al._
(2013).

##  VI Acknowledgments

This study was supported by the National Natural Science Foundation of China
(Nos. 62171144 and 62031024), the Guangxi Science Foundation (No.
2021GXNSFAA220011), the Training plan for Guangxi 1000 Young and Middle-aged
Teacher and the Open Fund of IPOC (BUPT) (No. IPOC2021A02).

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