$G_{-1}^{sic}(x_{b},t_{b};x_{a},t_{a})$ may be simply calculated from (4.24a).
Of course, it also can be directly calculated in quantum mechanics. Similarly,
the $G_{+1}^{sic}(x_{b},t_{b};x_{a},t_{a})$ may be simply calculated by the
formula:

$G_{+1}^{sic}(x_{b},t_{b};x_{a},t_{a})=G_{-1}^{sic}(x_{a},t_{a};x_{b},t_{b})^{\ast},$
(4.24b)

if the $G_{-1}^{sic}(x_{b},t_{b};x_{a},t_{a})$ is already calculated in
quantum mechanics.

Below the relation is set up between the time interval $T_{m}$ and both the
initial time $t_{a}$ and final time $t_{b}$ in the $SIC$ Green function of
(4.2) or (4.22). Consider the setting that the time interval $T_{m}\geq 0$ in
(4.2) or (4.22). It is known from (4.2) and (4.23) that the
$G_{+1}^{sic}(x_{b},t_{b};x_{a},t_{a})$ is generated by the unitary propagator
$\exp\left(-iHT_{m}/\hslash\right)$ with $T_{m}\geq 0.$ Then there is
$T_{m}=t_{b}-t_{a}\geq 0.$ This corresponds to the case $a_{m}^{s}=+1$. On the
other hand, it is also known from (4.2) and (4.23) that the
$G_{-1}^{sic}(x_{b},t_{b};x_{a},t_{a})$ is obtained from
$\exp\left(+iHT_{m}/\hslash\right)$ which is the inverse of
$\exp\left(-iHT_{m}/\hslash\right).$ Therefore, there is
$T_{m}=t_{a}-t_{b}\geq 0$ or $-T_{m}=t_{b}-t_{a}\leq 0.$ This corresponds to
the case $a_{m}^{s}=-1$. These results for the two cases $a_{m}^{s}=\pm 1$
together show that in the present setting $T_{m}\geq 0$ there is the relation
$a_{m}^{s}T_{m}=t_{b}-t_{a}$ in value. Next consider another setting that the
time interval $T_{m}<0$ in (4.2) or (4.22). It can be deduced from (4.2) and
(4.23) that the $G_{-1}^{sic}(x_{b},t_{b};x_{a},t_{a})$ is generated by the
unitary propagator $\exp\left(-iH\left|T_{m}\right|/\hslash\right)$ with
$\left|T_{m}\right|>0$. Then one has $\left|T_{m}\right|=t_{b}-t_{a}>0$ or
$-T_{m}=t_{b}-t_{a}>0.$ This corresponds to the case $a_{m}^{s}=-1.$ On the
other hand, it is also known from (4.2) and (4.23) that the
$G_{+1}^{sic}(x_{b},t_{b};x_{a},t_{a})$ is generated by
$\exp\left(+iH\left|T_{m}\right|/\hslash\right)$ which is the inverse of
$\exp\left(-iH\left|T_{m}\right|/\hslash\right).$ Thus, there is
$\left|T_{m}\right|=t_{a}-t_{b}>0$ or $T_{m}=t_{b}-t_{a}<0.$ This corresponds
to the case $a_{m}^{s}=+1.$ These results for both the cases $a_{m}^{s}=\pm 1$
together show that in the present setting $T_{m}<0$ there is the relation
$a_{m}^{s}T_{m}=t_{b}-t_{a}$ in value. It therefore concludes that in either
setting (i.e., either $T_{m}\geq 0$ or $T_{m}<0)$ there is always the relation
$a_{m}^{s}T_{m}=t_{b}-t_{a}$ in value between the time interval $T_{m}$ and
both the initial time $t_{a}$ and final time $t_{b}$ in the $SIC$ Green
function of (4.2) or (4.22).

Now it is proposed to combine the energy eigenfunction expansion method with
the Feynman path integral technique [$14$] to determine a more general $SIC$
Green function. This joint method to determine exactly a $SIC$ Green function
is described below.

Since the Hamiltonian of the $U_{H}^{sic}(a_{m}^{s},T_{m})$ of (4.1) is time-
independent, both the $SIC$ unitary propagator $U_{H}^{sic}(a_{m}^{s},T_{m})$
and its $SIC$ Green function $G_{H}^{sic}(x_{b},t_{b};x_{a},t_{a})$ (or
$G_{\pm 1}^{sic}(x_{b},t_{b};x_{a},t_{a}))$ of (4.2) are not dependent upon
the initial (or final) time $t_{a}$ $\left(\text{or }t_{b}\right)$ separately
but dependent upon the time interval $t_{b}-t_{a}.$ This is the same as a $QM$
unitary propagator with time-independent Hamiltonian and its $QM$ Green
function. The $QM$ unitary propagator
$U_{H}(T_{m})=\exp\left(-iHT_{m}/\hslash\right)$ corresponds to the $SIC$
unitary propagator $U_{H}^{sic}(a_{m}^{s},T_{m})$. Its $QM$ Green function
$G_{H}(x_{b},t_{b};x_{a},t_{a})=\left\langle
x_{b},t_{b}\right|U_{H}(T_{m})\left|x_{a},t_{a}\right\rangle$ corresponds to
the $SIC$ Green function $G_{H}^{sic}(x_{b},t_{b};x_{a},t_{a})$. According to
the energy eigenfunction expansion method this $QM$ Green function may be
formally expressed as [$10,14$]

$G_{H}(x_{b},t_{b};x_{a},t_{a})=\sum_{k=0}^{\infty}\exp\left(-iE_{k}T_{m}/\hslash\right)\psi_{k}^{\ast}\left(x_{a}\right)\psi_{k}\left(x_{b}\right).$
(4.25)

One sees that this $QM$ Green function $G_{H}(x_{b},t_{b};x_{a},t_{a})$
corresponds to the $SIC$ Green function $G_{H}^{sic}(x_{b},t_{b};x_{a},t_{a})$
of (4.22). It is known from (4.22) and (4.25) that the
$G_{H}(x_{b},t_{b};x_{a},t_{a})$ is a function of the time interval
$t_{b}-t_{a}=T_{m},$ while the $G_{H}^{sic}(x_{b},t_{b};x_{a},t_{a})$ is a
function of the time interval $t_{b}-t_{a}=a_{m}^{s}T_{m}.$ By comparing
(4.25) with (4.22) it can be found that if one replaces the time interval
$T_{m}$ in the $G_{H}(x_{b},t_{b};x_{a},t_{a})$ of (4.25) with the time
interval $a_{m}^{s}T_{m},$ then this $G_{H}(x_{b},t_{b};x_{a},t_{a})$ is
exactly changed to the $SIC$ Green function
$G_{H}^{sic}(x_{b},t_{b};x_{a},t_{a})$ of (4.22). Suppose now that one can
explicitly determine the exact expression of the $QM$ Green function of (4.25)
in quantum mechanics. Then in the exact expression one replaces the time
interval $T_{m}$ with $a_{m}^{s}T_{m}.$ Such replacement, i.e.,
$T_{m}\rightarrow a_{m}^{s}T_{m},$ leads to that the exact expression of the
$QM$ Green function $G_{H}(x_{b},t_{b};x_{a},t_{a})$ of (4.25) is exactly
changed to the one of the $SIC$ Green function
$G_{H}^{sic}(x_{b},t_{b};x_{a},t_{a})$ of (4.22).

The correspondent relation between the $SIC$ Green function of (4.22) and the
$QM$ one of (4.25) may be employed to calculate a general $SIC$ Green function
of (4.22), if the $QM$ Green function of (4.25) can be exactly obtained in
quantum mechanics. Below on the basis of the correspondent relation the joint
method based on both the energy eigenfunction expansion method and the Feynman
path integration technique is proposed to determine exactly a more general
$SIC$ Green function than those of (4.9) and (4.20). The Feynman path
integration technique [$14,20$] has been well developed in quantum mechanics.
It provides a formulation of quantum mechanics that is different from the
original Heisenberg matrix mechanics and the Schrödinger wave mechanics. It is
able to treat theoretically any quantum systems in time and space in unitary
quantum dynamics. Many $QM$ Green functions for the simple and special quantum
systems can be exactly determined by the path integration technique, but most
$QM$ Green functions can not be exactly obtained for complicated quantum
systems. Now consider any quantum system whose $QM$ Green function can be
exactly determined by the path integration technique. For such quantum system
one can calculate exactly its $SIC$ Green function by the joint method. Here
for convenience the Hamiltonian of the quantum system is limited to be time-
independent. At the first step of the joint method one calculates exactly the
$QM$ Green function of the quantum system by using the path integration
technique. Denote $T_{m}$ as the time interval between the final and the
initial time in the $QM$ Green function. Evidently this $QM$ Green function is
a function of the time interval $T_{m}$. At the second step one makes the
time-interval replacement, $T_{m}\rightarrow a_{m}^{s}T_{m},$ in the exact
expression of the $QM$ Green function. Then it can be deduced from the
correspondent relation that after the time-interval replacement this $QM$
Green function is exactly changed to the $SIC$ Green function of the same
quantum system. One therefore obtains the exact expression of the $SIC$ Green
function that is correspondent to the $QM$ Green function.

In quantum mechanics there are a number of quantum systems whose $QM$ Green
functions can be calculated exactly by the path integration technique, and
there also may be the $QM$ Green functions of a quantum system with different
Hamiltonians which may be calculated exactly by the path integration
technique. In principle the $SIC$ Green functions corresponding to all these
exact $QM$ Green functions can be exactly determined by the joint method. As
typical examples, below several applications of the joint method are described
to the exact calculation of the $SIC$ Green functions in one-dimensional
coordinate space in a single-atom system. The first two applications are to
re-calculate the $SIC$ Green functions of (4.9) and (4.20) by the joint
method, respectively. As shown in (4.9) and (4.20), these two $SIC$ Green
functions are already obtained exactly by the energy eigenfunction expansion
method alone.

The $SIC$ Green function of (4.9) is generated by the quadratic Hamiltonian
$\hat{T}=\frac{1}{2m}p^{2},$ while the one of (4.20) is generated by the
quadratic Hamiltonian$\ H_{c}$ with the expression (3.7a). Here, as shown
above, for a single-atom system (e.g., a hydrogen-like atom) $p$ and $x$ may
stand for the momentum and coordinate of the COM motion of the single atom,
respectively, and they are also a pair of canonical momentum and coordinate.
For a general quantum system in time and space $p$ and $x$ may be treated as a
pair of canonical momentum and coordinate (See the paragraph with (4.36) in
detail below). In the Feynman path integration technique the $QM$ Green
function of a quantum system tends to be characterized by the Lagrangian $L$
of the quantum system and its corresponding action $S=\int Ldt.$ With the help
of the relation between the Hamiltonian $H$ and the Lagrangian $L$ in a
quantum system (See (4.36) with one degree of freedom below) it can be found
that the quadratic Lagrangians corresponding to the Hamiltonians $\hat{T}$ and
$H_{c}$ are given by
$L_{0}=\frac{1}{2}m(\frac{dx}{dt})^{2}\equiv\frac{1}{2}m\dot{x}^{2}$ and
$L_{c}=\frac{1}{2}m\dot{x}^{2}-\frac{1}{2}m\omega^{2}x^{2},$ respectively.

The $QM$ unitary quadratic propagator that is generated by the kinetic
operator $\hat{T}$ is given by
$U_{\hat{T}}(T_{m})=\exp\left(-i\hat{T}T_{m}/\hslash\right),$ while the one
that is generated by the Hamiltonian$\ H_{c}$ is given by
$U_{H_{c}}(T_{m})=\exp\left(-iH_{c}T_{m}/\hslash\right).$ The former
corresponds to the $SIC$ quadratic propagator
$U_{\hat{T}}^{sic}(a_{m}^{s},T_{m})$ in (4.4), while the latter corresponds to
the $SIC\ $quadratic propagator $U_{H_{c}}^{sic}(a_{m}^{s},T_{m})$ of (4.10).
It can prove by the path integration technique that the $QM$ Green function
that is generated by the quadratic Lagrangian $L_{0}$ or correspondent to the
$U_{\hat{T}}(T_{m})$ can be calculated exactly and may be written in the form
[$14,20$]

$G_{\hat{T}}(x_{b},t_{b};x_{a},t_{a})=\sqrt{\frac{m}{i2\pi\hslash
T_{m}}}\exp\left(i\frac{m}{2\hslash}\frac{\left(x_{b}-x_{a}\right)^{2}}{T_{m}}\right),$
(4.26)

and similarly the $QM$ Green function that is generated by the quadratic
Lagrangian $L_{c}$ or correspondent to the $U_{H_{c}}(T_{m})$ also can be
exactly calculated and may be written in the form [$14,20$]

$G_{H_{c}}(x_{b},t_{b};x_{a},t_{a})=\sqrt{\frac{m\omega}{i2\pi\hslash\sin\left(\omega
T_{m}\right)}}$ $\times\exp\left[\frac{im\omega}{2\hslash\sin\left(\omega
T_{m}\right)}\left(\left(x_{a}^{2}+x_{b}^{2}\right)\allowbreak\cos\left(\omega
T_{m}\right)-2x_{a}x_{b}\right)\right]$ (4.27)

It can be seen that both the $G_{\hat{T}}(x_{b},t_{b};x_{a},t_{a})$ and
$G_{H_{c}}(x_{b},t_{b};x_{a},t_{a})$ are the functions of the time interval
$t_{b}-t_{a}=T_{m}.$ The work in this paragraph is the first step of the joint
method.

Now the second step of the joint method can be easily carried out. In the
exact expression (4.26) of the $QM$ Green function
$G_{\hat{T}}(x_{b},t_{b};x_{a},t_{a})$ one makes the time-interval replacement
$T_{m}\rightarrow a_{m}^{s}T_{m}.$ Then it can be found that the
$G_{\hat{T}}(x_{b},t_{b};x_{a},t_{a})$ is exactly changed to the $SIC$ Green
function $G_{\hat{T}}^{sic}(x_{b},t_{b};x_{a},t_{a})$ of (4.9). One thus
obtains the exact expression of the $SIC$ Green function
$G_{\hat{T}}^{sic}(x_{b},t_{b};x_{a},t_{a})$ that is generated by the
quadratic Lagrangian $L_{0}$ and corresponds to the $QM$ Green function
$G_{\hat{T}}(x_{b},t_{b};x_{a},t_{a}).$ This example confirms that the $SIC$
Green function $G_{\hat{T}}^{sic}(x_{b},t_{b};x_{a},t_{a})$ that is exactly
determined by the joint method is completely consistent with the one of (4.9)
that is exactly obtained by the energy eigenfunction expansion method alone.
Similarly, if in the exact expression (4.27) of the
$G_{H_{c}}(x_{b},t_{b};x_{a},t_{a})$ one makes the time-interval replacement
$T_{m}\rightarrow a_{m}^{s}T_{m},$ then it can be found that the $QM$ Green
function $G_{H_{c}}(x_{b},t_{b};x_{a},t_{a})$ is exactly changed to the $SIC$
Green function $G_{H_{c}}^{sic}(x_{b},t_{b};x_{a},t_{a})$ of (4.20). The exact
expression for the $SIC$ Green function that is generated by the quadratic
Lagrangian $L_{c}$ thus is obtained by the joint method. This is the second
example that confirms that both the joint method and the energy eigenfunction
expansion method can achieve the same result. These two examples (or
applications) therefore confirm that, just like the energy eigenfunction
expansion method, the joint method can obtain the correct result in the exact
calculation of the $SIC$ unitary propagators in the coordinate representation.

As a third application, consider the $SIC$ quadratic propagator
$U_{H_{d}}^{sic}(a_{m}^{s},T_{m})$
$=\exp\left[-ia_{m}^{s}H_{d}T_{m}/\hslash\right],$ where the time-independent
$QM$ quadratic Hamiltonian $H_{d}$ is given by
$H_{d}=\frac{1}{2m}p^{2}+\frac{1}{2}m\omega^{2}x^{2}+fx.$ As shown by (4.36)
below, the quadratic Lagrangian $L_{d}$ corresponding to the Hamiltonian
$H_{d}$ is given by
$L_{d}=\frac{1}{2}m\dot{x}^{2}-\frac{1}{2}m\omega^{2}x^{2}-fx.$ It is also
time-independent. Now one wants to determine the $SIC$ quadratic propagator
$U_{H_{d}}^{sic}(a_{m}^{s},T_{m})$ in the coordinate representation, i.e.,
$G_{H_{d}}^{sic}(x_{b},t_{b};x_{a},t_{a}),$ by the joint method. Given the
$QM$ quadratic Lagrangian $L_{d}$ (or the Hamiltonian $H_{d}$) one first
determines the $QM$ quadratic propagator
$U_{H_{d}}(T_{m})=\exp\left[-iH_{d}T_{m}/\hslash\right]$ in the coordinate
representation, i.e., $G_{H_{d}}(x_{b},t_{b};x_{a},t_{a}),$ by the path
integration technique. Here the $QM$ Green function
$G_{H_{d}}(x_{b},t_{b};x_{a},t_{a})$ corresponds to the $SIC$ Green function
$G_{H_{d}}^{sic}(x_{b},t_{b};x_{a},t_{a}).$ The exact expression of the$\
G_{H_{d}}(x_{b},t_{b};x_{a},t_{a})$ is already obtained by the Feynman path
integration technique [$14,20$]. Evidently the
$G_{H_{d}}(x_{b},t_{b};x_{a},t_{a})$ is a function of the time interval
$T_{m}.$ This is the first step of the joint method. At the second step in the
exact expression of the $G_{H_{d}}(x_{b},t_{b};x_{a},t_{a})$ one makes the
time-interval replacement $T_{m}\rightarrow a_{m}^{s}T_{m}.$ This replacement
results in that the $QM$ Green function $G_{H_{d}}(x_{b},t_{b};x_{a},t_{a})$
is exactly changed to the $SIC$ Green function
$G_{H_{d}}^{sic}(x_{b},t_{b};x_{a},t_{a}).$ One thus obtains the exact
expression of the $G_{H_{d}}^{sic}(x_{b},t_{b};x_{a},t_{a})$ which may be
explicitly written as

$G_{H_{d}}^{sic}(x_{b},t_{b};x_{a},t_{a})=\sqrt{\frac{m\omega}{i2\pi\hslash\sin(\omega
a_{m}^{s}T_{m})}}$ $\times\exp\\{i\frac{m\omega}{2\hslash\sin(\omega
a_{m}^{s}T_{m})}[(x_{b}^{2}+x_{a}^{2})\cos(\omega
a_{m}^{s}T_{m})-2x_{b}x_{a}]$ $+i\frac{f}{\hslash\omega}\frac{\cos\left(\omega
a_{m}^{s}T_{m}\right)-1}{\sin\left(\omega
a_{m}^{s}T_{m}\right)}\left(x_{b}+x_{a}\right)$
$+i\allowbreak\frac{f^{2}}{2m\hslash\omega^{3}}\frac{2\cos\left(\omega
a_{m}^{s}T_{m}\right)+\left(\omega a_{m}^{s}T_{m}\right)\sin\left(\omega
a_{m}^{s}T_{m}\right)-2}{\sin\left(\omega a_{m}^{s}T_{m}\right)}\\}$ (4.28)

where the time interval $t_{b}-t_{a}=a_{m}^{s}T_{m}$. This is just the exact
expression of the $SIC$ quadratic propagator
$U_{H_{d}}^{sic}(a_{m}^{s},T_{m})$ in the coordinate representation.

As a fourth application of the joint method, here consider a general $SIC$
quadratic propagator. A general $SIC$ quadratic propagator may be written as
$U_{H_{q}}^{sic}(a_{m}^{s},T_{m})=\exp\left[-ia_{m}^{s}H_{q}T_{m}/\hslash\right],$
where the time-independent $QM$ quadratic Hamiltonian $H_{q}$ may be formally
written as

$H_{q}\equiv
H_{q}\left(p,x\right)=\frac{1}{2m}p^{2}+\frac{1}{2}b\left(xp+px\right)+cx^{2}+dp+fx+g$
(4.29)

and its corresponding time-independent quadratic Lagrangian $L_{q},$ which is
obtained from (4.36) (See below), is given by

$L_{q}\equiv
L_{q}\left(\dot{x},x\right)=\allowbreak\frac{1}{2}m\dot{x}^{2}\allowbreak-
b_{L}x\dot{x}-c_{L}x^{2}-d_{L}\dot{x}-f_{L}x-g_{L}$ (4.30)

where $b_{L}=mb,$ $c_{L}=c-\frac{1}{2}mb^{2},$ $d_{L}=md,$ $f_{L}=f-mbd,$ and
$g_{L}=g-\frac{1}{2}md^{2}.$ With the help of the relation (4.36) below the
Hamiltonian $H_{q}$ can be changed to the corresponding Lagrangian $L_{q}$ and
vice versa. Generally the action $S_{q}$ of the Lagrangian $L_{q}$ is given by
$S_{q}=\int L_{q}dt.$ The $QM$ quadratic propagator that is generated by the
quadratic Hamiltonian $H_{q}$ is given by
$U_{H_{q}}(T_{m})=\exp\left(-iH_{q}T_{m}/\hslash\right).$ It corresponds to
the $SIC$ quadratic propagator $U_{H_{q}}^{sic}(a_{m}^{s},T_{m}).$

Quantum mechanically it can prove [$14,20$] that the $QM$ quadratic Green
function that is generated by the $U_{H_{q}}(T_{m})$ or by the quadratic
Lagrangian $L_{q}$ of (4.30) can be exactly determined by the path integration
technique and may be written in a general form [$20$]

$G_{H_{q}}(x_{b},t_{b};x_{a},t_{a})=\sqrt{\frac{m}{i2\pi\hslash}(-T_{ab})}\exp\left(\frac{i}{\hslash}S_{q}(x_{b},t_{b};x_{a},t_{a})\right)$
(4.31)

where the $QM$ action $S_{q}(x_{b},t_{b};x_{a},t_{a})$ may be formally
expressed as

$S_{q}(x_{b},t_{b};x_{a},t_{a})=\frac{1}{2}m\left(T_{bb}x_{b}^{2}+T_{ab}2x_{a}x_{b}+T_{aa}x_{a}^{2}\right)+x_{a}B_{a}+x_{b}B_{b}+B_{0}.$
(4.32)

In the $QM$ Green function of (4.31) and the $QM$ action of (4.32) these
parameters $T_{aa},$ $T_{ab},$ $T_{bb},$ $B_{a},$ $B_{b},$ and $B_{0}$ each
are a real function of the time interval $T_{m},$ $e.g.,$
$T_{ab}=T_{ab}\left(T_{m}\right),$ $B_{a}=B_{a}\left(T_{m}\right),$ etc., and
they each are independent of the coordinates $x_{b}$ and $x_{a}.$ This is the
first step of the joint method that one first obtains the exact expression
(4.31) of the $QM$ Green function of the quadratic Lagrangian $L_{q}$ by the
path integration technique. The next step is that in the exact expression
(4.31) of the $QM$ Green function $G_{H_{q}}(x_{b},t_{b};x_{a},t_{a})$ one
makes the time-interval replacement, i.e., $T_{m}\rightarrow a_{m}^{s}T_{m}.$
It can be seen from (4.31) and (4.32) that this time-interval replacement is
equivalent to that all these parameters in (4.31) and (4.32) are made the
replacement of the time-interval variable:
$T_{\lambda}\left(T_{m}\right)\rightarrow
T_{\lambda}\left(a_{m}^{s}T_{m}\right)$ for $\lambda=aa,$ $ab,$ $bb$ and
$B_{\mu}\left(T_{m}\right)\rightarrow B_{\mu}\left(a_{m}^{s}T_{m}\right)$ for
$\mu=a,$ $b,$ $0.$ After the time-interval replacement the $QM$ Green function
$G_{H_{q}}(x_{b},t_{b};x_{a},t_{a})$ is exactly changed to the $SIC$ Green
function $G_{H_{q}}^{sic}(x_{b},t_{b};x_{a},t_{a})$ that is generated by the
$SIC$ quadratic propagator $U_{H_{q}}^{sic}(a_{m}^{s},t_{m})$ or by the
quadratic Lagrangian $L_{q}$. One thus obtains the exact expression of the
$SIC$ Green function $G_{H_{q}}^{sic}(x_{b},t_{b};x_{a},t_{a})$ which may be
formally written as

$G_{H_{q}}^{sic}(x_{b},t_{b};x_{a},t_{a})=\sqrt{\frac{m}{i2\pi\hslash}(-S_{ab})}\exp\left(\frac{i}{\hslash}S_{q}^{sic}(x_{b},t_{b};x_{a},t_{a})\right)$
(4.33)

and here the $SIC$ action $S_{q}^{sic}(x_{b},t_{b};x_{a},t_{a})$ may be
formally expressed as

$S_{q}^{sic}(x_{b},t_{b};x_{a},t_{a})=\frac{1}{2}m\left(S_{bb}x_{b}^{2}+S_{ab}2x_{a}x_{b}+S_{aa}x_{a}^{2}\right)+x_{a}Q_{a}+x_{b}Q_{b}+\Theta_{0},$
(4.34)

where these $SIC$ parameters $S_{aa},$ $S_{ab},$ $S_{bb},$ $Q_{a},$ $Q_{b},$
and $\Theta_{0}$ are the real functions of the time-interval variable
$t_{b}-t_{a}=a_{m}^{s}T_{m}$ and they are explicitly given by
$S_{\lambda}=T_{\lambda}\left(a_{m}^{s}T_{m}\right)$ for $\lambda=aa,$ $ab,$
$bb;$ $Q_{\mu}=B_{\mu}\left(a_{m}^{s}T_{m}\right)$ for $\mu=a,$ $b;$ and
$\Theta_{0}=B_{0}\left(a_{m}^{s}T_{m}\right).$

It is known from (4.32) and (4.34) that both the $QM$ action
$S_{q}(x_{b},t_{b};x_{a},t_{a})$ and the $SIC$ action
$S_{q}^{sic}(x_{b},t_{b};x_{a},t_{a})$ each are a quadratic function of the
coordinates $x_{a}$ and $x_{b}.$ This is the characteristic feature of a $QM\
$quadratic Green function and also a $SIC$ quadratic Green function. Due to
this characteristic feature the formal expression (4.33) is general for a
$SIC$ quadratic Green function, even if the quadratic Lagrangian $L_{q}$ of
(4.30) (or the Hamiltonian $H_{q}$ of (4.29)) carries the solution information
($a_{m}^{s}$), that is, $L_{q}=L_{q}\left(a_{m}^{s}\right)$ (or
$H_{q}=H_{q}\left(a_{m}^{s}\right)$). It can prove that the previous $SIC$
quadratic Green functions of (4.9), (4.20), and (4.28) are the special cases
of the general $SIC$ quadratic Green function of (4.33).

The final application of the joint method is to determine the $SIC$ Green
function with a time-independent $QM$ non-quadratic Lagrangian (or
Hamiltonian). This non-quadratic Hamiltonian is given by
$H_{s}=\frac{1}{2m}p^{2}+V\left(x\right)$ and the corresponding non-quadratic
Lagrangian is $L_{s}=\frac{1}{2m}p^{2}-V\left(x\right)$ (See also (4.36)
below), where the potential energy $V\left(x\right)$ is given by
$V\left(x\right)=0$ if $0<x<a$ and $V\left(x\right)=\infty$ if $0<x$ or $x>a.$
In the present example this Hamiltonian describes the COM motion of a single
atom in external infinite square potential well with walls at $x=0$ and $x=a$.
The $QM$ unitary propagator with the Hamiltonian $H_{s}$ is given by
$\exp\left(-iH_{s}T_{m}/\hslash\right),$ and here its corresponding $QM$ Green
function with the Lagrangian $L_{s}$ is denoted as
$G_{H_{s}}(x_{b},t_{b};x_{a},t_{a}).$ It can prove that by the path
integration technique the $QM$ Green function
$G_{H_{s}}(x_{b},t_{b};x_{a},t_{a})$ can be exactly determined [$20,23$], and
it is a function of the time interval $T_{m}$. Now in the exact expression
[$20,23$] of the $G_{H_{s}}(x_{b},t_{b};x_{a},t_{a})$ one makes the time-
interval replacement, i.e., $T_{m}\rightarrow a_{m}^{s}T_{m}.$ This time-
interval replacement leads to that the $QM$ Green function
$G_{H_{s}}(x_{b},t_{b};x_{a},t_{a})$ is exactly changed to the $SIC$ Green
function $G_{H_{s}}^{sic}(x_{b},t_{b};x_{a},t_{a})$ that is generated by the
$SIC$ unitary propagator $\exp\left(-ia_{m}^{s}H_{s}T_{m}/\hslash\right)$ or
by the Lagrangian $L_{s}.$ Therefore, one obtains the exact expression of the
$SIC$ Green function:

$G_{H_{s}}^{sic}(x_{b},t_{b};x_{a},t_{a})=\sqrt{\frac{m}{i2\pi\hslash\left(a_{m}^{s}T_{m}\right)}}$
$\times\sum_{n=-\infty}^{\infty}\left[\exp\left(i\frac{m}{2\hslash}\frac{\left(x_{b}-x_{a}+2na\right)^{2}}{\left(a_{m}^{s}T_{m}\right)}\right)-\exp\left(i\frac{m}{2\hslash}\frac{\left(x_{b}+x_{a}+2na\right)^{2}}{\left(a_{m}^{s}T_{m}\right)}\right)\right].$
(4.35)

This is just the $SIC$ unitary propagator
$\exp\left(-ia_{m}^{s}H_{s}T_{m}/\hslash\right)$ in the coordinate
representation. This $SIC$ Green function can not be exactly reduced to the
formal expression (4.33) of a general $SIC$ quadratic Green function. It is a
$SIC$ non-quadratic Green function.

In the above theoretical treatments based on the joint method the path
integration technique has been indirectly employed to calculate exactly a
$SIC$ unitary propagator in the coordinate representation. It is well known in
quantum mechanics that most of $QM$ Green functions can not be exactly
determined by the path integration technique. Since the joint method is
usually limited to determine an exact $SIC$ Green function which corresponds
to an exact $QM$ Green function and it is used to treat a $SIC$ unitary
propagator that is generated by a time-independent Hamiltonian, it is a
limited method. Below it is simply described how the path integration
technique is directly used to calculate (exactly or approximately) a general
$SIC$ Green function.

In classical dynamics a physical system may be described by a set of canonical
dynamical variables, i.e., the canonical momentum ($p_{i}$) and coordinate
($q_{i}$) variables $\\{q_{i},p_{i}\\}$. If the degree of freedom of the
system is $g,$ then the set contains $2g$ variables $\\{q_{i},p_{i}\\}$. Both
the Hamiltonian $H$ and the Lagrangian $L$ of the system are the functions of
the canonical momentum and coordinate variables $\\{q_{i},p_{i}\\}$,
respectively. That is, $H=H\left(\mathbf{q},\mathbf{p},t\right)$ and
$L=L\left(\mathbf{q},\mathbf{\dot{q}},t\right),$ where
$\mathbf{q=\\{}q_{i}\mathbf{\\}}$ and $\mathbf{p}=\\{p_{i}\\}$. Then according
to classical dynamics 121212See, for example, D. T. Greenwood, Classical
dynamics, Chapt. 4, Dover edition, Prentice-Hall, Inc. (1977) there is the
relation between the Hamiltonian $H$ and the Lagrangian $L:$

$H\left(\mathbf{q},\mathbf{p},t\right)=\sum_{i=1}^{g}p_{i}\dot{q}_{i}-L\left(\mathbf{q},\mathbf{\dot{q}},t\right)$
(4.36)

With the help of this relation the Hamiltonian $H$ can be changed to the
Lagrangian $L$ and vice versa. For the quantum system corresponding to the
classical physical system above the relation (4.36) still holds between the
Hamiltonian $H$ and the Lagrangian $L$ in quantum mechanics. However, in
quantum mechanics $\mathbf{q=\\{}q_{i}\mathbf{\\}}$ and
$\mathbf{p}=\\{p_{i}\\}$ in (4.36) are coordinate and momentum operators,
respectively. Of course, in the quantization transition from the classical
system to the quantum system these canonical coordinate and momentum
variables$\ \\{q_{i},p_{i}\\}$ should be expressed in Cartesian coordinates
[$10$]. In quantum mechanics a unitary propagator is usually generated with a
Hamiltonian ($H$). In the Feynman path integration technique a Green function
is usually characterized with a Lagrangian ($L$).

In the definition (4.1) the Hamiltonian used to define a $SIC$ unitary
propagator is time-independent. Does the $SIC$ unitary propagator make sense
if the Hamiltonian is time-dependent? The present theoretical work dose not
involve the concrete preparation and realization of a $SIC$ unitary
propagator. This provides convenience to define theoretically a $SIC$ unitary
propagator in a quantum system in time and space whose Hamiltonian is time-
dependent. It also could become convenient for general theoretical treatment
or calculation of a $SIC$ Green function in the framework of the path
integration technique. Theoretically a $SIC$ unitary propagator may be defined
in any quantum system in time and space whose Hamiltonian may or may not be
time-dependent. Now a $SIC$ unitary propagator, which is generated by a time-
dependent Hamiltonian $H\left(t\right),$ may be defined by

$U_{H}^{sic}(a_{m}^{s},t_{0}+a_{m}^{s}T_{m},t_{0})$
$=\lim_{N\rightarrow\infty}\exp\left[-\frac{i}{\hslash}a_{m}^{s}H\left(t_{0}+a_{m}^{s}T_{m}-\frac{1}{2}\left(a_{m}^{s}T_{m}/N\right)\right)\left(T_{m}/N\right)\right]$
$\times\exp\left[-\frac{i}{\hslash}a_{m}^{s}H\left(t_{0}+a_{m}^{s}T_{m}-\frac{3}{2}\left(a_{m}^{s}T_{m}/N\right)\right)\left(T_{m}/N\right)\right]$
$\times...\times\exp\left[-\frac{i}{\hslash}a_{m}^{s}H\left(t_{0}+\frac{3}{2}\left(a_{m}^{s}T_{m}/N\right)\right)\left(T_{m}/N\right)\right]$
$\times\exp\left[-\frac{i}{\hslash}a_{m}^{s}H\left(t_{0}+\frac{1}{2}\left(a_{m}^{s}T_{m}/N\right)\right)\left(T_{m}/N\right)\right]$
(4.37)

where $t_{0}+a_{m}^{s}T_{m}$ and $t_{0}$ are the final and the initial time of
the quantum system, respectively. It can be found that when the Hamiltonian
$H\left(t\right)$ is time-independent, the $SIC$ unitary propagator of (4.37)
is reduced to the one of (4.1). The $SIC$ unitary propagator defined by (4.37)
may correspond to the following $QM$ unitary propagator:

$U_{H}(t_{0}+T_{m},t_{0})=\lim_{N\rightarrow\infty}\exp\left[-\frac{i}{\hslash}H\left(t_{0}+T_{m}-\frac{1}{2}\left(T_{m}/N\right)\right)\left(T_{m}/N\right)\right]$
$\times\exp\left[-\frac{i}{\hslash}H\left(t_{0}+T_{m}-\frac{3}{2}\left(T_{m}/N\right)\right)\left(T_{m}/N\right)\right]$
$\times...\times\exp\left[-\frac{i}{\hslash}H\left(t_{0}+\frac{3}{2}\left(T_{m}/N\right)\right)\left(T_{m}/N\right)\right]$
$\times\exp\left[-\frac{i}{\hslash}H\left(t_{0}+\frac{1}{2}\left(T_{m}/N\right)\right)\left(T_{m}/N\right)\right]$
(4.38)

If one replaces the time interval $T_{m}$ in the $QM$ unitary propagator of
(4.38) with the time interval $a_{m}^{s}T_{m},$ then the $QM$ unitary
propagator is exactly changed to the $SIC$ unitary propagator of (4.37).

The main advantage for the definition (4.37) of a $SIC$ unitary propagator
$U_{H}^{sic}(a_{m}^{s},t_{0}+a_{m}^{s}T_{m},t_{0})$ is that the
$U_{H}^{sic}(a_{m}^{s},t_{0}+a_{m}^{s}T_{m},t_{0})$ is governed simply and
directly by a single time-dependent Hamiltonian $H(t)$ and that the
$U_{H}^{sic}(a_{m}^{s},t_{0}+a_{m}^{s}T_{m},t_{0})$ with $a_{m}^{s}=+1$ is the
inverse of the $U_{H}^{sic}(a_{m}^{s},t_{0}+a_{m}^{s}T_{m},t_{0})$ with
$a_{m}^{s}=-1$ and vice versa, that is,

$U_{H}^{sic}(+1,t_{0}+T_{m},t_{0})=U_{H}^{sic}(-1,t_{0},t_{0}+T_{m})^{+}.$
(4.39)

This relation may be called the global time-evolution-reversal symmetry.
However, such definition as (4.37) of a $SIC$ unitary propagator is usually
complicated, since it is involved in the piecewise time-independent
Hamiltonians which may be dependent upon the double-valued logical number
$a_{m}^{s}.$ Consequently it usually could be complicated to construct and
realize the $SIC$ unitary propagator in a quantum system.

A simpler definition for a $SIC$ unitary propagator generated by a time-
dependent Hamiltonian $H(t)$ may be given by

$U_{H}^{sic}(a_{m}^{s},t_{0}+a_{m}^{s}T_{m},t_{0})=\lim_{N\rightarrow\infty}\exp\left[-\frac{i}{\hslash}a_{m}^{s}H\left(t_{0}+T_{m}-\frac{1}{2}\left(T_{m}/N\right)\right)\left(T_{m}/N\right)\right]$
$\times\exp\left[-\frac{i}{\hslash}a_{m}^{s}H\left(t_{0}+T_{m}-\frac{3}{2}\left(T_{m}/N\right)\right)\left(T_{m}/N\right)\right]$
$\times...\times\exp\left[-\frac{i}{\hslash}a_{m}^{s}H\left(t_{0}+\frac{3}{2}\left(T_{m}/N\right)\right)\left(T_{m}/N\right)\right]$
$\times\exp\left[-\frac{i}{\hslash}a_{m}^{s}H\left(t_{0}+\frac{1}{2}\left(T_{m}/N\right)\right)\left(T_{m}/N\right)\right]$
(4.40)

This $SIC$ unitary propagator is obtained by combining the definition (4.1) of
the $SIC$ unitary propagator generated by a time-independent Hamiltonian with
the $QM$ unitary propagator of (4.38) that is generated by a time-dependent
Hamiltonian $H(t)$. The $SIC$ unitary propagator of (4.40) is different from
the one of (4.37) in that the piecewise time-independent Hamiltonians in
(4.40) are independent of the double-valued logical number $a_{m}^{s}$ (here
suppose that the initial time $t_{0}$ is independent of $a_{m}^{s}$).

The $SIC$ unitary propagator of (4.40) does not own the global time-evolution-
reversal symmetry of (4.39), but it owns the local time-evolution-reversal
symmetry, because in the short time interval $T_{m}/N$ every $SIC$ exponential
propagator $U_{H_{j}}^{sic}(a_{m}^{s},T_{m}/N)$ in (4.40) satisfies

$U_{H_{j}}^{sic}(+1,T_{m}/N)=U_{H_{j}}^{sic}(-1,T_{m}/N)^{+},\text{
}j=1,2,...,N$ (4.41)

Here, as defined by (4.1),
$U_{H_{j}}^{sic}(a_{m}^{s},T_{m}/N)=\exp\left[-\frac{i}{\hslash}a_{m}^{s}H_{j}T_{m}/N\right],$
where the piecewise time-independent Hamiltonian $H_{j}$ is given by

$H_{j}=H\left(t_{0}+T_{m}-\left(j-1/2\right)\left(T_{m}/N\right)\right),$

as shown in (4.40).

It could be thought that the $SIC$ unitary propagator of (4.40) is effectively
generated by the time-dependent Hamiltonian $a_{m}^{s}H(t).$ When
$a_{m}^{s}=+1,$ the Hamiltonian $a_{m}^{s}H(t)$ is the natural one (i.e.,
$H(t)$) of the quantum system. However, when $a_{m}^{s}=-1,$ the Hamiltonian
$a_{m}^{s}H(t)=-H\left(t\right)$ is not the natural one and it may be
considered as the effective time-dependent Hamiltonian of the quantum system.
Given the effective time-dependent Hamiltonian $a_{m}^{s}H(t)$ one may set up
the time-dependent $SIC$ Schrödinger equation (See Appendix A) to describe the
$SIC$ unitary propagator of (4.40).

As shown in (4.2), corresponding to the $SIC$ unitary propagator of (4.37) or
(4.40) the $SIC$ Green function may be re-expressed as
$G_{H}^{sic}(x_{b},t_{b};x_{a},t_{a})=\left\langle
x_{b},t_{b}\right|U_{H}^{sic}(a_{m}^{s},t_{0}+a_{m}^{s}T_{m},t_{0})\left|x_{a},t_{a}\right\rangle,$
where the initial time $t_{a}=t_{0}$ and in value the final time $t_{b}$ is
obtained from $a_{m}^{s}T_{m}=t_{b}-t_{0}.$

The $SIC$ unitary propagators of (4.37) and (4.40) and their corresponding
$SIC$ Green functions each still own the dual character of quantum-computing
speedup. They obey both the unitary quantum dynamics and the mathematical-
logical principle of the unstructured search problem. This means that one
needs to employ both the physical Hilbert space and the math Hilbert space to
describe completely the quantum-computing speedup process that is governed by
the $SIC$ unitary propagator of (4.37) or (4.40). This quantum-computing
speedup process is unitary and deterministic. It obeys the information
conservation principle.

The following theoretical treatment is devoted to setting up a standard path-
integration formula for a $SIC$ Green function. It is limited to one-
dimensional coordinate space. It is based on the Feynman path integration
technique [$14,20$]. Suppose that a quantum system in time and space has the
time-independent Hamiltonian $H=\hat{T}+\hat{V},$ where
$\hat{T}=\frac{1}{2m}p^{2}$ and $\hat{V}=V\left(x\right)$ are the kinetic and
potential energy operators, respectively. According to the definition (4.1)
the $SIC$ unitary propagator of the quantum system then may be written as
$U_{H}^{sic}(a_{m}^{s},T_{m})=\exp\left[-ia_{m}^{s}\left(\hat{T}+\hat{V}\right)T_{m}/\hslash\right].$
With the help of the Trotter product formula the
$U_{H}^{sic}(a_{m}^{s},T_{m})$ may be written as

$U_{H}^{sic}(a_{m}^{s},T_{m})=\lim_{N\rightarrow\infty}\left[\exp\left(-\frac{i}{\hslash}a_{m}^{s}\hat{T}T_{m}/N\right)\exp\left(-\frac{i}{\hslash}a_{m}^{s}\hat{V}T_{m}/N\right)\right]^{N}$

Then the corresponding $SIC$ Green function of (4.2) is given by

$G_{H}^{sic}(x_{b},t_{b};x_{a},t_{a})$ $=\lim_{N\rightarrow\infty}\left\langle
x_{b},t_{b}\right|\left[\exp\left(-\frac{i}{\hslash}a_{m}^{s}\hat{T}T_{m}/N\right)\exp\left(-\frac{i}{\hslash}a_{m}^{s}\hat{V}T_{m}/N\right)\right]^{N}\left|x_{a},t_{a}\right\rangle$
(4.42)

By inserting the $N-1$ coordinate identity operators $\\{\int
dx_{j}\left|x_{j}\right\rangle\left\langle x_{j}\right|\\}$ for
$j=1,2,...,N-1$ into (4.42) one obtains

$G_{H}^{sic}(x_{b},t_{b};x_{a},t_{a})=\lim_{N\rightarrow\infty}\int
dx_{1}dx_{2}...dx_{N-1}$
$\times\underset{j=0}{\overset{N-1}{\mathop{\displaystyle\prod}}}\left\langle
x_{j+1}\right|\exp\left(-\frac{i}{\hslash}a_{m}^{s}\hat{T}\varepsilon\right)\exp\left(-\frac{i}{\hslash}a_{m}^{s}\hat{V}\varepsilon\right)\left|x_{j}\right\rangle$
(4.43)

where $\varepsilon=T_{m}/N,$ $x_{0}=x_{a},$ and $x_{N}=x_{b}.$ There is the
eigenvalue equation
$\exp\left(-ia_{m}^{s}\hat{V}\varepsilon/\hslash\right)\left|x_{j}\right\rangle=$
$\exp\left(-ia_{m}^{s}V\left(x_{j}\right)\varepsilon/\hslash\right)\left|x_{j}\right\rangle$.
There is also the expression of the matrix element:

$\left\langle
x_{j+1}\right|\exp\left(-ia_{m}^{s}\hat{T}\varepsilon/\hslash\right)\left|x_{j}\right\rangle=\sqrt{\frac{m}{i2\pi\hslash\left(a_{m}^{s}\varepsilon\right)}}\exp\left[i\frac{m}{2\hslash}\left(a_{m}^{s}\varepsilon\right)\left(\frac{x_{j+1}-x_{j}}{\varepsilon}\right)^{2}\right].$

This expression can be obtained by the way that is used to determine the $SIC$
Green function of (4.9), here one needs to use the important property of the
double-valued logical number $a_{m}^{s},$ i.e.,
$\left(a_{m}^{s}\right)^{2}=1.$ Now with the help of the eigenvalue equation
and the expression one can reduce the $SIC$ Green function of (4.43) to the
form

$G_{H}^{sic}(x_{b},t_{b};x_{a},t_{a})=\lim_{N\rightarrow\infty}\int
dx_{1}dx_{2}...dx_{N-1}\left(\sqrt{\frac{m}{i2\pi\hslash\left(a_{m}^{s}\varepsilon\right)}}\right)^{N}$
$\times\exp\left[\frac{i}{\hslash}\left(a_{m}^{s}\varepsilon\right)\sum_{j=0}^{N-1}\left(\frac{1}{2}m\left(\frac{x_{j+1}-x_{j}}{\varepsilon}\right)^{2}-V\left(x_{j}\right)\right)\right]$
(4.44)

This is a standard path-integration expression of the $SIC\ $Green function.
In the limit $N\rightarrow\infty$ or $\varepsilon=T_{m}/N\rightarrow 0$ the
exponent (omitting $ia_{m}^{s}/\hslash$) in (4.44) may be written as

$\lim_{\varepsilon\rightarrow
0}\varepsilon\sum_{j=0}^{N-1}\left(\frac{1}{2}m\left(\frac{x_{j+1}-x_{j}}{\varepsilon}\right)^{2}-V\left(x_{j}\right)\right)=\int_{0}^{T_{m}}Ldt$
(4.45)

Here the $QM$ Lagrangian $L$ is given by

$L=\frac{1}{2}m\dot{x}^{2}-V\left(x\right),$ (4.46)

and the right side of (4.45) is just the $QM$ action $S:$

$S=\int_{t_{a}}^{t_{b}}Ldt.$ (4.47)

The factor
$\left(\frac{m}{i2\pi\hslash\left(a_{m}^{s}\varepsilon\right)}\right)^{N/2}$
in (4.44) is a ‘normalization constant’. It ensures that the
$G_{H}^{sic}(x_{b},t_{b};x_{a},t_{a})$ of (4.44) is unitary. Now the
$G_{H}^{sic}(x_{b},t_{b};x_{a},t_{a})$ also may be formally written as

$G_{H}^{sic}(x_{b},t_{b};x_{a},t_{a})=\int_{x\left(t_{a}\right)}^{x\left(t_{b}\right)}D\left[x\left(t\right)\right]\exp\left(\frac{i}{\hslash}a_{m}^{s}\int_{t_{a}}^{t_{b}}Ldt\right)$
(4.48)

The path integration formula (4.44) can be easily generalized for the $SIC$
unitary propagator of (4.40) that is generated by a time-dependent
Hamiltonian.

A check for correction of the path integral formula (4.44) may be carried out
by using (4.44) to calculate exactly the $SIC$ Green function
$G_{\hat{T}}^{sic}(x_{b},t_{b};x_{a},t_{a})$ that is generated by the kinetic
operator $\hat{T}=\frac{1}{2m}p^{2}.$ By using the path integral formula
(4.44) the $G_{\hat{T}}^{sic}(x_{b},t_{b};x_{a},t_{a})$ can be calculated
exactly by the way [$14,20$] that is used to calculate the corresponding $QM$
Green function $G_{\hat{T}}(x_{b},t_{b};x_{a},t_{a})$ (See also (4.26) above).
An important point is that one must repeat to use the important property
$\left(a_{m}^{s}\right)^{2}=1$ in the calculation. The detailed calculation
shows that the previous expression (4.9) of the
$G_{\hat{T}}^{sic}(x_{b},t_{b};x_{a},t_{a}),$ which is obtained by the energy
eigenfunction expansion method alone, is exactly reproduced by the path
integral formula (4.44).

In the path integration technique most $SIC$ Green functions can not be
exactly obtained. It could be feasible to employ the perturbation method to
calculate approximately a general $SIC$ Green function. Suppose that the total
Hamiltonian of a quantum system in time and space is given by $H=H_{0}+H_{1},$
where $H_{0}$ and $H_{1}$ are the main term and the perturbation term,
respectively. According to the definition (4.1) the two $SIC$ unitary
propagators $U^{sic}\left(t\right)=\exp\left(-ia_{m}^{s}Ht/\hslash\right)$ and
$U_{0}^{sic}\left(t\right)=\exp\left(-ia_{m}^{s}H_{0}t/\hslash\right)$ are
defined with the total Hamiltonian $H$ and the main Hamiltonian $H_{0},$
respectively. And the two corresponding $SIC$ Green functions
$G_{H}^{sic}(x_{b},t_{b};x_{a},t_{a})$ and
$G_{H_{0}}^{sic}(x_{b},t_{b};x_{a},t_{a})$ are obtained from (4.2),
respectively. Then the perturbation equation that these two $SIC$ Green
functions obey may be written as

$G_{H}^{sic}(x_{b},t_{b};x_{a},0)=G_{H_{0}}^{sic}(x_{b},t_{b};x_{a},0)$
$+\frac{1}{i\hslash}\int_{0}^{t}dt^{\prime}\int
dx^{\prime}G_{H_{0}}^{sic}(x_{b},t_{b};x^{\prime},t^{\prime})\left(a_{m}^{s}H_{1}\left(x^{\prime}\right)\right)G_{H}^{sic}(x^{\prime},t^{\prime};x_{a},0)$
(4.49)

where the initial time $t_{a}=0$ and the final time
$t_{b}=t_{b}-t_{a}=a_{m}^{s}t$ in value. This is the $SIC$ perturbation
equation that a $SIC$ Green function obeys. In principle it should be able to
derive this $SIC$ perturbation equation from the standard path integration
formula (4.44), but here there is not such a detailed derivation. However,
this $SIC$ perturbation equation can be directly obtained from the $SIC$
perturbation equation in operator form:

$U^{sic}\left(t\right)=U_{0}^{sic}\left(t\right)+\frac{1}{i\hslash}\int_{0}^{t}U_{0}^{sic}\left(t-t^{\prime}\right)\left(a_{m}^{s}H_{1}\right)U^{sic}\left(t^{\prime}\right)dt^{\prime}$
(4.50)

This is the $SIC$ perturbation equation that a $SIC$ unitary propagator obeys.
It is derived in detail in Appendix A on the basis of the motion equation in
the interaction picture in quantum mechanics.

If the $SIC$ Green function $G_{H_{0}}^{sic}(x_{b},t_{b};x_{a},0)$ in (4.49)
can be calculated conveniently by the path integration formula (4.44), then by
iterating the $SIC$ perturbation equation (4.49) one is able to calculate the
$SIC$ Green function $G_{H}^{sic}(x_{b},t_{b};x_{a},0)$ to any order
approximation.

So far a single $SIC$ unitary propagator and its corresponding $SIC$ Green
function have been treated explicitly. Generally a QUANSDAM (or UNIDYSLOCK)
process is a sequence of many $SIC$ unitary propagators. Therefore, it is
necessary to treat explicitly the product of two or more $SIC$ unitary
propagators and its corresponding $SIC$ Green function.

A concatenate $SIC$ unitary propagator is defined as the product of two or
more $SIC$ unitary propagators. Given two $SIC$ unitary propagators
$U_{H_{1}}^{sic}(a_{m}^{s},T_{1})$ and $U_{H_{2}}^{sic}(a_{m}^{s},T_{2})$ the
concatenate $SIC$ unitary propagator of them may be written as

$U_{H_{12}}^{sic}(a_{m}^{s},T_{12})=U_{H_{2}}^{sic}(a_{m}^{s},T_{2})U_{H_{1}}^{sic}(a_{m}^{s},T_{1})$
(4.51)

According to the definition (4.2) this concatenate $SIC$ unitary propagator is
given by
$G_{H_{12}}^{sic}\left(x_{b},t_{b};x_{a},t_{a}\right)\equiv\left\langle
x_{b},t_{b}|U_{H_{12}}^{sic}(a_{m}^{s},T_{12})|x_{a},t_{a}\right\rangle$ in
the coordinate representation. And it can prove that

$G_{H_{12}}^{sic}\left(x_{b},t_{b};x_{a},t_{a}\right)=\int_{-\infty}^{\infty}dx_{c}G_{H_{2}}^{sic}\left(x_{b},t_{b};x_{c},t_{c}\right)G_{H_{1}}^{sic}\left(x_{c},t_{c};x_{a},t_{a}\right).$
(4.52)

This integral formula of the product of a pair of $SIC$ Green functions is
obtained by first substituting (4.51) into the definition of the
$G_{H_{12}}^{sic}\left(x_{b},t_{b};x_{a},t_{a}\right)$ and then inserting the
operator identity
$\int_{-\infty}^{\infty}dx_{c}\left|x_{c},t_{c}\right\rangle\left\langle
x_{c},t_{c}\right|$ of the coordinate eigenbases
$\left\\{\left|x_{c},t_{c}\right\rangle\right\\}$ into between
$U_{H_{2}}^{sic}(a_{m}^{s},T_{2})$ and $U_{H_{1}}^{sic}(a_{m}^{s},T_{1})$. If
both the $U_{H_{1}}^{sic}(a_{m}^{s},T_{1})$ and
$U_{H_{2}}^{sic}(a_{m}^{s},T_{2})$ in (4.51) each are taken as the form of
(4.1), then in (4.52) the time interval $t_{c}-t_{a}=a_{m}^{s}T_{1}$ and
$t_{b}-t_{c}=a_{m}^{s}T_{2}.$ The integral formula (4.52) may be called the
integral product formula of the $SIC$ Green functions. By the integral formula
(4.52) one is able to determine the $SIC$ Green function that is generated by
the product of two $SIC$ unitary propagators. This integral product formula
for a pair of $SIC$ Green functions corresponds to the conventional $QM$ one
for a pair of $QM$ Green functions [$14$].

If both the $U_{H_{1}}^{sic}(a_{m}^{s},T_{1})$ and
$U_{H_{2}}^{sic}(a_{m}^{s},T_{2})$ in (4.51) are any $SIC$ quadratic
propagators, then it can prove that the concatenate $SIC$ unitary propagator
$U_{H_{12}}^{sic}(a_{m}^{s},T_{12})$ of (4.51) is a $SIC$ quadratic propagator
too. A detail proof is given below that is based on the $SIC$-Green-function
integral product formula (4.52), the $SIC$ quadratic Green functions, and the
Gaussian integration. The $SIC$ Green functions for the $SIC$ quadratic
propagators $U_{H_{1}}^{sic}(a_{m}^{s},T_{1})$ and
$U_{H_{2}}^{sic}(a_{m}^{s},T_{2})$ can be obtained from (4.33) above,
respectively. According to (4.33), formally both the $SIC$ quadratic Green
functions of the $U_{H_{1}}^{sic}(a_{m}^{s},T_{1})$ and
$U_{H_{2}}^{sic}(a_{m}^{s},T_{2})$ may be written respectively as

$G_{H_{1}}^{sic}\left(x_{c},t_{c};x_{a},t_{a}\right)=\sqrt{\frac{m}{i2\pi\hslash}(-S_{ac})}$
$\times\exp\left(\frac{i}{\hslash}\left[\frac{1}{2}m\left(S_{cc}x_{c}^{2}+S_{ac}2x_{a}x_{c}+S_{aa}x_{a}^{2}\right)+x_{a}Q_{a}+x_{c}Q_{c}+\Theta_{0}\right]\right)$
(4.53)

and

$G_{H_{2}}^{sic}(x_{b},t_{b};x_{c},t_{c})=\sqrt{\frac{m}{i2\pi\hslash}(-S_{cb}^{\prime})}$
$\times\exp\left(\frac{i}{\hslash}\left[\frac{1}{2}m\left(S_{bb}^{\prime}x_{b}^{2}+S_{cb}^{\prime}2x_{c}x_{b}+S_{cc}^{\prime}x_{c}^{2}\right)+x_{c}Q_{c}^{\prime}+x_{b}Q_{b}^{\prime}+\Theta_{0}^{\prime}\right]\right)$
(4.54)

where all these real $SIC$ parameters $S_{aa},$ $S_{ac},$ $S_{cc},$ $Q_{a},$
$Q_{c},$ and $\Theta_{0};$ $S_{cc}^{\prime},$ $S_{cb}^{\prime},$
$S_{bb}^{\prime},$ $Q_{c}^{\prime},$ $Q_{b}^{\prime},$ and
$\Theta_{0}^{\prime}$ each may depend on the double-valued logical number
$\left(a_{m}^{s}=\pm 1\right),$ e.g., $S_{aa}=S_{aa}\left(a_{m}^{s}\right),$
etc. By substituting (4.53) and (4.54) into the integral product formula
(4.52) one obtains

$G_{H_{12}}^{sic}\left(x_{b},t_{b};x_{a},t_{a}\right)=\sqrt{\frac{m}{i2\pi\hslash}(-S_{cb}^{\prime})}\sqrt{\frac{m}{i2\pi\hslash}(-S_{ac})}\exp\left(\frac{i}{\hslash}\left(\Theta_{0}+\Theta_{0}^{\prime}\right)\right)$
$\times\exp\left(\frac{i}{\hslash}\left[\frac{1}{2}m\left(S_{bb}^{\prime}x_{b}^{2}\right)+\frac{1}{2}m\left(S_{aa}x_{a}^{2}\right)+x_{b}Q_{b}^{\prime}+x_{a}Q_{a}\right]\right)$
$\times\int_{-\infty}^{\infty}dx_{c}\exp\\{\frac{i}{\hslash}[\frac{1}{2}m\left(S_{cb}^{\prime}2x_{c}x_{b}+S_{cc}^{\prime}x_{c}^{2}\right)$
$+\frac{1}{2}m\left(S_{cc}x_{c}^{2}+S_{ac}2x_{a}x_{c}\right)+x_{c}Q_{c}^{\prime}+x_{c}Q_{c}]\\}$
(4.55)

The integration on the RH side of (4.55) is a Gaussian integration about the
variable $x_{c}$. For convenience this Gaussian integration is denoted as
$G\left(x_{b},x_{a}\right).$ It can be exactly calculated. The calculated
result is given by

$G\left(x_{b},x_{a}\right)=\int_{-\infty}^{\infty}dx_{c}\exp\left(-ax_{c}^{2}+bx_{c}\right)=\sqrt{\frac{\pi}{a}}\exp\left(\frac{b^{2}}{4a}\right)$
(4.56)

where $a=-\frac{im}{2\hslash}\left(S_{cc}^{\prime}+S_{cc}\right)$ and
$b=\frac{i}{\hslash}\left[m\left(S_{cb}^{\prime}x_{b}+S_{ac}x_{a}\right)+\left(Q_{c}^{\prime}+Q_{c}\right)\right].$
Now by substituting (4.56) into (4.55) the $SIC$ Green function of (4.55) can
be reduced to the form

$G_{H_{12}}^{sic}\left(x_{b},t_{b};x_{a},t_{a}\right)=\exp\left(\frac{i}{\hslash}\left(\Theta_{0}+\Theta_{0}^{\prime}\right)\right)\exp\left(-\frac{i}{2\hslash
m}\frac{\left(Q_{c}^{\prime}+Q_{c}\right)^{2}}{\left(S_{cc}^{\prime}+S_{cc}\right)}\right)$
$\times\sqrt{\frac{m}{i2\pi\hslash}(-S_{cb}^{\prime})}\sqrt{\frac{m}{i2\pi\hslash}(-S_{ac})}\sqrt{\frac{2i\pi\hslash}{m\left(S_{cc}^{\prime}+S_{cc}\right)}}$
$\times\exp\left(\frac{i}{\hslash}\left[\frac{1}{2}m\left(S_{bb}^{\prime}x_{b}^{2}\right)+\frac{1}{2}m\left(S_{aa}x_{a}^{2}\right)+x_{b}Q_{b}^{\prime}+x_{a}Q_{a}\right]\right)$
$\times\exp\left(-\frac{i}{2\hslash
m}\frac{\left[m^{2}\left(S_{cb}^{\prime}x_{b}+S_{ac}x_{a}\right)^{2}+2m\left(Q_{c}^{\prime}+Q_{c}\right)\left(S_{cb}^{\prime}x_{b}+S_{ac}x_{a}\right)\right]}{\left(S_{cc}^{\prime}+S_{cc}\right)}\right)$
(4.57)

By this expression it can be further shown that the
$G_{H_{12}}^{sic}\left(x_{b},t_{b};x_{a},t_{a}\right)$ of (4.57) is indeed a
$SIC$ quadratic Green function. Actually, the
$G_{H_{12}}^{sic}\left(x_{b},t_{b};x_{a},t_{a}\right)$ can be formally written
as

$G_{H_{12}}^{sic}\left(x_{b},t_{b};x_{a},t_{a}\right)=\sqrt{\frac{m}{i2\pi\hslash}(-S_{ab}^{\prime\prime})}\exp\left(\frac{i}{\hslash}S_{12}^{sic}\left(x_{b},t_{b};x_{a},t_{a}\right)\right)\newline
$ (4.58)

where the $SIC$ action $S_{12}^{sic}\left(x_{b},t_{b};x_{a},t_{a}\right)$ is a
quadratic function of the coordinates $x_{b}$ and $x_{a}$ and may be formally
written as

$S_{12}^{sic}(x_{b},t_{b};x_{a},t_{a})=\frac{1}{2}m\left(S_{bb}^{\prime\prime}x_{b}^{2}+S_{ab}^{\prime\prime}2x_{a}x_{b}+S_{aa}^{\prime\prime}x_{a}^{2}\right)+x_{a}Q_{a}^{\prime\prime}+x_{b}Q_{b}^{\prime\prime}+\Theta_{0}^{\prime\prime}$
(4.59)

All these real $SIC$ parameters in (4.58) and (4.59), i.e.,
$S_{aa}^{\prime\prime},$ $S_{ab}^{\prime\prime},$ $S_{bb}^{\prime\prime},$
$Q_{a}^{\prime\prime},$ $Q_{b}^{\prime\prime},$ and
$\Theta_{0}^{\prime\prime},$ each can be explicitly obtained from (4.57). They
generally depend on the double-valued logical number $a_{m}^{s}.$ A detailed
calculation shows that they are explicitly given by

$S_{bb}^{\prime\prime}=S_{bb}^{\prime}-\frac{\left(S_{cb}^{\prime}\right)^{2}}{S_{cc}^{\prime}+S_{cc}},\text{
}S_{ab}^{\prime\prime}=-\frac{S_{cb}^{\prime}S_{ac}}{S_{cc}^{\prime}+S_{cc}},\text{
}S_{aa}^{\prime\prime}=S_{aa}-\frac{S_{ac}^{2}}{S_{cc}^{\prime}+S_{cc}},$
(4.60a)
$Q_{b}^{\prime\prime}=Q_{b}^{\prime}-\frac{S_{cb}^{\prime}\left(Q_{c}^{\prime}+Q_{c}\right)}{S_{cc}^{\prime}+S_{cc}},\text{
}Q_{a}^{\prime\prime}=Q_{a}-\frac{S_{ac}\left(Q_{c}^{\prime}+Q_{c}\right)}{S_{cc}^{\prime}+S_{cc}}\newline
,$ (4.60b)
$\Theta_{0}^{\prime\prime}=\Theta_{0}^{\prime}+\Theta_{0}-\frac{1}{2m}\frac{\left(Q_{c}^{\prime}+Q_{c}\right)^{2}}{S_{cc}^{\prime}+S_{cc}}$
(4.60c)

As shown in (4.59), the $SIC$ action
$S_{12}^{sic}\left(x_{b},t_{b};x_{a},t_{a}\right)$ in (4.58) is a quadratic
function of the coordinates $x_{b}$ and $x_{a}.$ This shows that the
$G_{H_{12}}^{sic}\left(x_{b},t_{b};x_{a},t_{a}\right)$ of (4.58) is a $SIC$
quadratic Green function. This further shows that the concatenate $SIC$
unitary propagator $U_{H_{12}}^{sic}(a_{m}^{s},T_{12})$ of (4.51), which
generates the $G_{H_{12}}^{sic}\left(x_{b},t_{b};x_{a},t_{a}\right)$ of
(4.58), is indeed a $SIC$ quadratic propagator. It therefore concludes that
the product of a pair of $SIC$ quadratic propagators is still a $SIC$
quadratic propagator. The only exception is that the
$G_{H_{12}}^{sic}\left(x_{b},t_{b};x_{a},t_{a}\right)$ of (4.58) may be
reduced to a $QM$ quadratic Green function up to a global phase factor, when
both the physical and the math Hilbert space to describe the
$G_{H_{12}}^{sic}\left(x_{b},t_{b};x_{a},t_{a}\right)$ are completely
overlapping with each other. The $SIC$ quadratic propagator
$U_{H_{12}}^{sic}(a_{m}^{s},T_{12})$ of (4.51) that generates the
$G_{H_{12}}^{sic}(x_{b},t_{b};$ $x_{a},t_{a})$ of (4.58) may not be written in
the simple form of (4.1). Thus, it is a general $SIC$ quadratic propagator.
With the help of the same method as the above one it can prove that the above
conclusion that the product of a pair of $SIC$ quadratic propagators is a
$SIC$ quadratic propagator is still available for general $SIC$ quadratic
propagators.

It can seen from the above theoretical proof that the property that the
product of a pair of $SIC$ quadratic propagators is still a $SIC$ quadratic
propagator is obtained via the Gaussian integration of the product of a pair
of $SIC$ quadratic Green functions. As shown in (4.53) and (4.54), these $SIC$
quadratic Green functions are the quadratic (or Gaussian) functions of the
coordinate variables ($x_{a},$ $x_{b},$ $x_{c}$). It is known that the product
of a pair of Gaussian functions is still a Gaussian function and the Gaussian
integration of the product is still a Gaussian function.131313Suppose that
$G_{1}\left(x,y,z\right)$ and $G_{2}\left(x,y,z\right)$ are two Gaussian
functions with variables $x,$ $y,$ and $z$. Then the product of the two
Gaussian functions,
$G_{3}\left(x,y,z\right)=G_{1}\left(x,y,z\right)G_{2}\left(x,y,z\right),$ is
still a Gaussian function. Moreover, the Gaussian integral of the product,
$G\left(x,y\right)=\int_{-\infty}^{\infty}dzG_{3}\left(x,y,z\right),$ is also
a Gaussian function of the variables $x$ and $y$. Similarly,
$G\left(x,z\right)$ and $G\left(y,z\right)$ are also Gaussian functions. Then
the Gaussian integration of the product of a pair of $SIC$ quadratic Green
functions is still a Gaussian function and hence a $SIC$ quadratic Green
function. This is the first interpretation for the property. It also may be
thought that the property is original from the symmetrical structure and
property of the quantum system with a quadratic Hamiltonian, which is the
fundamental quantum-computing-speedup resource [$1,2$]. In the quantum-
computing speedup theory the symmetrical structure and property of the quantum
system may be completely characterized by the quantum symmetry group whose
elements are the quadratic unitary operators on the physical Hilbert space and
also the one whose elements are the quadratic unitary operators on the math
Hilbert space, here both the quantum symmetry groups are the same one in the
present case. This is the second interpretation. The second interpretation is
fundamental (See Refs. [$1,2$] and also the section one above), but the first
one really plays direct role in constructing theoretically a powerful QUANSDAM
process by the Green function method.

One $SIC$ unitary propagator may be transformed to another by making a $QM$
unitary transformation. This property can be applied as well to a $SIC$
quadratic propagator. As shown in the previous section 3, it is useful for the
preparation of the $SIC$ unitary propagators by starting from the basic $SIC$
unitary operators. It is also important for those $SIC$ unitary propagators
that may not be prepared (or implemented) directly or conveniently. As a
typical example, consider the simplest$\ SIC$ quadratic propagators
$\exp\left(-ia_{m}^{s}\hat{V}_{c0}t_{m}/\hslash\right)\mathbf{\ }$with
$\hat{V}_{c0}=\frac{1}{2}m\omega_{0}^{2}x^{2}\mathbf{\ }$and
$\exp\left(-ia_{m}^{s}\hat{T}t_{m}/\hslash\right)$ with
$\hat{T}=\frac{1}{2m}p^{2}$ to be prepared in a single-atom system. Both the
$SIC$ quadratic propagators could not be prepared directly in a single-atom
system in the absence of external potential field, since such a single atom
with free COM motion has a continuous but not discrete COM-motion energy
spectrum. However, if they could be prepared conveniently in the single-atom
system with external harmonic potential field, then in principle they could be
effectively prepared in the same single-atom system in the absence of external
potential field. This may be done as follows: after (and before) they each are
prepared in the single-atom system with external harmonic potential field, one
turns off the external harmonic potential field immediately. Now according to
quantum mechanics the Heisenberg motion equation for the COM motion of a
single atom (e.g., a single ion [$17$]) in external harmonic potential field
may be written in the analytical solution form (See, for example, Page 102 in
Ref. [$26$])

$x\left(t\right)=\exp\left(iH_{c0}t/\hslash\right)x\exp\left(-iH_{c0}t/\hslash\right)=x\cos\left(\omega_{0}t\right)+\frac{p}{m\omega_{0}}\sin\left(\omega_{0}t\right)$
(4.61a)
$p\left(t\right)=\exp\left(iH_{c0}t/\hslash\right)p\exp\left(-iH_{c0}t/\hslash\right)=p\cos\left(\omega_{0}t\right)-m\omega_{0}x\sin\left(\omega_{0}t\right)$
(4.61b)

where the atomic COM Hamiltonian $H_{c0}$ is given by
$H_{c0}=\hat{T}+\hat{V}_{c0}$ with the oscillatory angular frequency
$\omega_{0}$. By setting $\omega_{0}t=\pi/2$ in the motion equation one
obtains

$\exp\left(iH_{c0}t_{c}/\hslash\right)x\exp\left(-iH_{c0}t_{c}/\hslash\right)=\frac{p}{m\omega_{0}}$
(4.62a)
$\exp\left(iH_{c0}t_{c}/\hslash\right)p\exp\left(-iH_{c0}t_{c}/\hslash\right)=-m\omega_{0}x$
(4.62b)

where $t=t_{c}=\frac{\pi}{2\omega_{0}}.$ Note that
$\exp\left(-iH_{c0}t_{c}/\hslash\right)$ is a $QM$ unitary operator. With the
help of the $QM$ unitary transformations of (4.62a) and (4.62b) one obtains
the following $QM$ unitary transformations between the two $SIC$ quadratic
propagators $\exp\left(-ia_{m}^{s}\hat{V}_{c0}t_{m}/\hslash\right)$ and
$\exp\left(-ia_{m}^{s}\hat{T}t_{m}/\hslash\right):$

$\exp\left(iH_{c0}t_{c}/\hslash\right)\exp\left(-ia_{m}^{s}\hat{V}_{c0}t_{m}/\hslash\right)\exp\left(-iH_{c0}t_{c}/\hslash\right)=\exp\left(-ia_{m}^{s}\hat{T}t_{m}/\hslash\right)$
(4.63a)
$\exp\left(iH_{c0}t_{c}/\hslash\right)\exp\left(-ia_{m}^{s}\hat{T}t_{m}/\hslash\right)\exp\left(-iH_{c0}t_{c}/\hslash\right)=\exp\left(-ia_{m}^{s}\hat{V}_{c0}t_{m}/\hslash\right)$
(4.63b)

It can be seen from (4.63a) and (4.63b) that when one of the two $SIC$
quadratic propagators (or their corresponding $SIC$ Green functions) can be
prepared or calculated conveniently, another also can be obtained conveniently
by the $QM$ unitary transformation of (4.63a) or (4.63b).

5\. Discussion

A quantum-computing speedup process owns the dual character that it obeys both
the unitary quantum dynamics and the mathematical-logical principle of a
computational problem to be solved. This is essentially different from a
conventional quantum computation (algorithm) [$32,33$] which is essentially a
purely quantum-physical process [$1$]. It takes into account the quantum
symmetry that is considered as the fundamental quantum-computing-speedup
resource. Both the dual character and the fundamental quantum-computing-
speedup resource are considered to be responsible for achieving an essential
quantum-computing speedup to solve a hard computational problem in the
quantum-computing speedup theory. Consequently from the point of view of pure
quantum mechanics both the unitary quantum dynamics and the quantum symmetry
are considered as the two pillars to build an efficient quantum algorithm to
solve a hard computational problem [$1,2$].

In contrast, it has been thought that the quantum-computational speedup for a
conventional quantum computation (algorithm) is achieved through the quantum
parallelism [$33$] based on the superposition of quantum states. Then from the
point of view of orthodox quantum mechanics it is essentially based on the
non-local effect of entanglement quantum states, and a conventional quantum
computation (algorithm) is indeterministic.

The largest difficulty facing the conventional quantum computation [$32,33$]
is perhaps that it is subjected to the square speedup limit on solving an
unstructured search problem, although most hard computational problems can not
be solved in an exponential quantum-computational speedup in the conventional
quantum computation.

In quantum mechanics there are two fundamentally different laws that govern
the time evolution process of a quantum system: the quantum measurement and
the unitary quantum dynamics [$13$]. The unitary time evolution process is
deterministic in the sense that in isolated quantum system state changes
deterministically in the process. In contrast, the quantum measurement is
indeterministic because in quantum system state changes in probability in the
quantum measurement. In orthodox quantum mechanics it is more reasonable to
say that the unitary quantum dynamics is computationally theoretical tool than
that it is fundamental physical law. The existing research works on the non-
local effect of entanglement quantum states in the past decades further
strengthen orthodox quantum mechanics. The conventional quantum computation
[$32,33$] is based on orthodox quantum mechanics.

Is independent mathematical-logical principle of a computational problem
necessary in the quantum-computing speedup theory? It is essential and
necessary. If it was not independent of any physical laws, then ultimately it
would be contained by orthodox quantum mechanics and the quantum-computing
speedup theory would lose its own independence and existence. Therefore, only
when the mathematical-logical principle is taken into account, can the
determinacy of the unitary quantum dynamics become substantial in a quantum-
computing speedup process [$1$]. The mathematical-logical principle of an
unstructured search problem to be solved was not considered as an independent
principle, which is independent of the unitary quantum dynamics and the
quantum symmetry, to make contribution to the quantum-searching speedup until
the early 2001 [$3$]. According to the mathematical-logical principle of the
unstructured search problem the real solution to the search problem is
completely determined in the $HSSS$ quantum search process. The real solution
state cannot be made a random transition between the physical Hilbert space
and the corresponding math Hilbert space and it is always in the physical
Hilbert space during the $HSSS$ quantum search process [$1$]. Suppose here
that existence of the real solution state reflects the reality in physics.
Then it may be deduced from these that if the reality is fundamental and
universal, then so is the determinism in physics.

The conventional quantum computation itself is not able to make certain for
whether the mathematical-logical principle of a computational problem is
obeyed or not in a conventional quantum computation (algorithm) to solve the
computational problem, because the conventional quantum computation
(algorithm) obeys the indeterministic (or probabilistic) quantum-physical law,
and indeterminism (or probability) is fundamental in orthodox quantum
mechanics and also in the conventional quantum computation. The conventional
quantum computation which tends to be unitary [$32$] contains the Bennett’s
reversible computation [$11,12$] of classical physics.141414This is due to
that quantum physics contains classical physics. Apparently, from the
viewpoint of the quantum-computing speedup theory [$1,2$] it seems that the
mathematical-logical principle of a computational problem should be obeyed,
when a conventional quantum computation (algorithm) solves the problem.
However, that is not so from the point of view of the quantum-mechanical basis
of the conventional quantum computation (i.e., orthodox quantum mechanics). A
conventional quantum computation (algorithm) is essentially a purely quantum-
physical process [$1$]. Note that indeterminism (or probability) is
fundamental in orthodox quantum mechanics and hence any quantum-physical
process is fundamentally indeterministic. The conventional quantum computation
(algorithm) to solve the computational problem therefore is also
indeterministic. Consequently, whether the mathematical-logical principle of
the computational problem is obeyed is indeterministic in the conventional
quantum computation (algorithm).

A quantum-computing speedup process to solve a computational problem still
employs the quantum measurement to output the final computational result. Then
the mathematical-logical principle of the computational problem is still
obeyed during the quantum measurement. This leads to that there are two
fundamentally different ways (deterministic vs. indeterministic) to describing
the quantum measurement outputting the final computational result, when a
quantum-computing speedup process and a conventional quantum computation
(algorithm) solve the same computational problem, respectively.

As far as the $HSSS$ quantum search process is concerned, the real solution to
the unstructured search problem exists and can never be changed in the $HSSS$
quantum search process and especially during the quantum measurement, because
it is determined uniquely by the mathematical-logical principle of the
unstructured search problem. In order to show clearly the essential difference
between the $HSSS$ quantum search process and a conventional quantum search
algorithm in the quantum measurement to output the final computational result,
here consider the special case that the $HSSS$ quantum search process simply
consists of the duality-character oracle operations (See the section 2 above)
and the relevant $QM$ unitary operators, it does not employ the search-space
dynamical reduction, and it is simply performed in an $n-$qubit quantum system
only. On the one hand, suppose that after this special $HSSS$ quantum search
process is applied to the initial state which is an expansion series of the
usual computational bases of the quantum system, just before the quantum
measurement the quantum state in the physical Hilbert space of the $n-$qubit
quantum system is written as

$\left|\Psi_{x_{0}}\right\rangle=B_{x_{0}}\left|x_{0}\right\rangle+\sum_{x=0,x\neq
x_{0}}^{N-1}B_{x}\left|x\right\rangle$

where, as shown in the section 2, $\left|x_{0}\right\rangle$ is the real
solution state, and $\\{\left|x\right\rangle\\}$ which includes
$\left|x_{0}\right\rangle$ is the usual computational basis set with base
number $N=2^{n}$. The coefficients in the final state
$\left|\Psi_{x_{0}}\right\rangle$ satisfy the normalization condition:
$\sum_{x=0}^{N-1}\left|B_{x}\right|^{2}=1.$ Particularly the coefficient
$B_{x_{0}}$ of the real solution state $\left|x_{0}\right\rangle$ generally
satisfies $\left|B_{x_{0}}\right|^{2}\leq 1.$ On the other hand, consider that
the conventional quantum search algorithm is applied to the same initial
state. Assume that after the conventional quantum search algorithm is applied,
just before the quantum measurement the quantum state of the $n-$qubit quantum
system is written as

$\left|\Psi_{x_{0}^{\prime}}^{\prime}\right\rangle=B_{x_{0}^{\prime}}^{\prime}\left|x_{0}^{\prime}\right\rangle+\sum_{x=0,x\neq
x_{0}^{\prime}}^{N-1}B_{x}^{\prime}\left|x\right\rangle$

where $\left|x_{0}^{\prime}\right\rangle$ is the marked state of the
conventional quantum search algorithm with $x_{0}^{\prime}=x_{0}$ to be found,
and $\\{\left|x\right\rangle\\}$ which includes
$\left|x_{0}^{\prime}\right\rangle$ is still the usual computational basis
set. The final state $\left|\Psi_{x_{0}^{\prime}}^{\prime}\right\rangle$
satisfies the normalization condition:
$\sum_{x=0}^{N-1}\left|B_{x}^{\prime}\right|^{2}=1,$ and the coefficient
$B_{x_{0}^{\prime}}^{\prime}$ of the marked state
$\left|x_{0}^{\prime}\right\rangle$ generally satisfies
$\left|B_{x_{0}^{\prime}}^{\prime}\right|^{2}\leq 1.$ Now by quantum measuring
the final state $\left|\Psi_{x_{0}}\right\rangle$ one is able to output the
final computational result for the $HSSS$ quantum search process. Likewise, by
quantum measuring the final state
$\left|\Psi_{x_{0}^{\prime}}^{\prime}\right\rangle$ one may obtain the final
computational result of the conventional quantum search algorithm. The quantum
measurement is the final step for any quantum search process (or algorithm)
above. At this final step there is still the essential difference between the
$HSSS$ quantum search process and the conventional quantum search algorithm.
This can be shown below.

First of all, though both the final states $\left|\Psi_{x_{0}}\right\rangle$
and $\left|\Psi_{x_{0}^{\prime}}^{\prime}\right\rangle$ each are a finite
series of the same $N$ usual computational bases of the $n-$qubit quantum
system, there is still their essential difference. According to the
theoretical analysis for the duality-character oracle operations of the $HSSS$
quantum search process, as shown in the section 2, it can prove theoretically
that in the final state $\left|\Psi_{x_{0}}\right\rangle$ only the real
solution state $\left|x_{0}\right\rangle$ is a member of the unstructured
search space, while any other computational base $\left|x\right\rangle$ than
$\left|x_{0}\right\rangle$ is not. In contrast, in the final state
$\left|\Psi_{x_{0}^{\prime}}^{\prime}\right\rangle$ of the conventional
quantum search algorithm the marked state $\left|x_{0}^{\prime}\right\rangle$
is naturally a member of the unstructured search space, but any other
computational base $\left|x\right\rangle$ than
$\left|x_{0}^{\prime}\right\rangle$ is also a member of the unstructured
search space. How can this essential difference between the final states
$\left|\Psi_{x_{0}}\right\rangle$ and
$\left|\Psi_{x_{0}^{\prime}}^{\prime}\right\rangle$ cause the deterministic
vs. indeterministic difference between the $HSSS$ quantum search process and
the conventional quantum search algorithm?

It seems that there is apparently no difference between the $HSSS$ quantum
search process and the conventional quantum search algorithm at the final step
if both the amplitudes $B_{x_{0}}$ in the final state
$\left|\Psi_{x_{0}}\right\rangle$ and $B_{x_{0}^{\prime}}^{\prime}$ in the
final state $\left|\Psi_{x_{0}^{\prime}}^{\prime}\right\rangle$ satisfy
$\left|B_{x_{0}^{\prime}}^{\prime}\right|^{2}=\left|B_{x_{0}}\right|^{2}=1,$
because for any one of both the quantum search processes (or algorithms) the
output state is the state $\left|x_{0}\right\rangle$ ($x_{0}^{\prime}=x_{0}$)
only and the final computational result is the real solution ($x_{0}$) to the
unstructured search problem. While for the $HSSS$ quantum search process this
final step is evidently deterministic, apparently it seems to be also
deterministic for the conventional quantum search algorithm. However, once
$\left|B_{x_{0}^{\prime}}^{\prime}\right|^{2},$
$\left|B_{x_{0}}\right|^{2}<1,$ the essential difference is exposed between
the $HSSS$ quantum search process and the conventional quantum search
algorithm. The essential difference is analyzed in detail in the following two
parts $(1)$ and $(2)$ for the conventional quantum search algorithm and the
$HSSS$ quantum search process, respectively.

$\left(1\right)$ This part is devoted to the conventional quantum search
algorithm. Suppose that the probability
$\left|B_{x_{0}^{\prime}}^{\prime}\right|^{2}$ of the marked state
$\left|x_{0}^{\prime}\right\rangle$ is the maximum one in the final state
$\left|\Psi_{x_{0}^{\prime}}^{\prime}\right\rangle$ and it is still less than
one, i.e., $\left|B_{x_{0}^{\prime}}^{\prime}\right|^{2}<1.$ According to
orthodox quantum mechanics each time after the quantum measurement is
performed the final state $\left|\Psi_{x_{0}^{\prime}}^{\prime}\right\rangle$
is changed in probability to some computational basis state (e.g.,
$\left|x_{0}^{\prime}\right\rangle$ or $\left|x\right\rangle$). As shown
above, this basis state of the final state
$\left|\Psi_{x_{0}^{\prime}}^{\prime}\right\rangle$ is always a member (or
candidate solution state) of the unstructured search space, regardless of the
marked state $\left|x_{0}^{\prime}\right\rangle$ or any other basis state
$\left|x\right\rangle$. It is just the output state of the conventional
quantum search algorithm after performing the quantum measurement one time,
and hence it is considered as a measurement-outputting real solution state to
the unstructured search problem. This is consistent with the ‘inverse’
statement that the real solution state may be possibly considered as any
candidate solution state in the final state
$\left|\Psi_{x_{0}^{\prime}}^{\prime}\right\rangle$ before the real solution
is found, and as stated in the section 2, this ‘inverse’ statement means the
indeterministic description. Of course, due to
$\left|B_{x_{0}^{\prime}}^{\prime}\right|^{2}<1$ this measurement-outputting
real solution state may or may not be finally assigned as the real solution
state to the unstructured search problem. However, by repeating many times to
run the conventional quantum search algorithm including the quantum
measurement of the final state
$\left|\Psi_{x_{0}^{\prime}}^{\prime}\right\rangle$ one can obtain the
probability distribution for all the measurement-outputting real solution
states. Then with the help of this probability distribution one can find the
measurement-outputting real solution state with maximum probability, i.e., the
marked state $\left|x_{0}^{\prime}\right\rangle$ with $x_{0}^{\prime}=x_{0}$.
Finally this marked state, which is found by the conventional quantum search
algorithm, is assigned as the real solution state to the unstructured search
problem.

The detailed analysis above reveals the indeterministic property for the
conventional quantum search algorithm that the quantum measurement, which is
indeterministic and fundamental in orthodox quantum mechanics, determines the
real solution state to the unstructured search problem instead of the
mathematical-logical principle of the unstructured search problem. This really
means that while it is able to find the real solution to the unstructured
search problem, the conventional quantum search algorithm itself can not make
certain for whether the mathematical-logical principle of the unstructured
search problem is obeyed or not. This is consistent with the previous general
conclusion that the conventional quantum computation itself is not able to
make certain for whether the mathematical-logical principle of a computational
problem is obeyed or not in a conventional quantum computation (algorithm) to
solve the computational problem. The reason for these comes from orthodox
quantum mechanics. According to orthodox quantum mechanics, not only whether
the mathematical-logical principle of the unstructured search problem is
obeyed is uncertain in the conventional quantum search algorithm, but also
whether the state $\left|x_{0}\right\rangle$ can be assigned as the real
solution state to the unstructured search problem or not is also uncertain
before the quantum measurement.

$\left(2\right)$ This part is devoted to the $HSSS$ quantum search process.
Consider first the basic case that the coefficient $B_{x_{0}}$ of the real
solution state $\left|x_{0}\right\rangle$ is given by
$\left|B_{x_{0}}\right|^{2}=1$ in the final state
$\left|\Psi_{x_{0}}\right\rangle.$ Now by quantum measurement the final state
$\left|\Psi_{x_{0}}\right\rangle$ is changed deterministically to the real
solution state $\left|x_{0}\right\rangle.$ The latter state is just the output
state of the $HSSS$ quantum search process. Next consider that the absolute
coefficient $B_{x_{0}}$ is smaller than one or $\left|B_{x_{0}}\right|^{2}<1.$
For convenience here suppose that the term $B_{x_{0}}\left|x_{0}\right\rangle$
is the dominating term, while the term $\sum_{x=0,x\neq
x_{0}}^{N-1}B_{x}\left|x\right\rangle$ is small term in the final state
$\left|\Psi_{x_{0}}\right\rangle,$ that is,
$\left|B_{x_{0}}\right|^{2}>\sum_{x=0,x\neq
x_{0}}^{N-1}\left|B_{x}\right|^{2}.$ Then by quantum measurement the final
state $\left|\Psi_{x_{0}}\right\rangle$ is changed to the real solution state
$\left|x_{0}\right\rangle$ or any other basis state $\left|x\right\rangle$
than $\left|x_{0}\right\rangle$ in probability. As shown above, in this case
only the real solution state $\left|x_{0}\right\rangle$ is a member of the
unstructured search space, while any other basis state $\left|x\right\rangle$
than $\left|x_{0}\right\rangle$ is not. Thus, in the final state
$\left|\Psi_{x_{0}}\right\rangle$ any basis state $\left|x\right\rangle$ other
than $\left|x_{0}\right\rangle$ is not a candidate solution state to the
unstructured search problem. As shown in the section 2, this is original from
the duality-character oracle operation of the $HSSS$ quantum search process.
This is the substantial difference for the $HSSS$ quantum search process from
the conventional quantum search algorithm. Consequently, if by the quantum
measurement the final state $\left|\Psi_{x_{0}}\right\rangle$ is changed to
the basis state $\left|x\right\rangle,$ one cannot say that the $HSSS$ quantum
search process obtains a measurement-outputting real solution state. This also
means that assigning the basis state $\left|x\right\rangle$ to the real
solution state is not consistent with the mathematical-logical principle of
the unstructured search problem. The real solution to the unstructured search
problem has nothing to do with quantum measurement. It is uniquely determined
by the mathematical-logical principle of the unstructured search problem. The
mathematical-logical principle therefore must be obeyed in the quantum
measurement. Once this is taken into account, it is only reasonable that the
final state $\left|\Psi_{x_{0}}\right\rangle$ is changed to the real solution
state $\left|x_{0}\right\rangle$ by the quantum measurement, although
$\left|B_{x_{0}}\right|^{2}<1.$ Therefore, in theory the $HSSS$ quantum search
process is still deterministic even if $\left|B_{x_{0}}\right|^{2}<1.$ Now one
needs to further consider reasonably the case that the final state
$\left|\Psi_{x_{0}}\right\rangle$ is changed to the basis state
$\left|x\right\rangle$ by the quantum measurement. This is related to the
small term $\sum_{x=0,x\neq x_{0}}^{N-1}B_{x}\left|x\right\rangle$ of the
final state $\left|\Psi_{x_{0}}\right\rangle,$ which is a finite series of all
the basis states $\\{\left|x\right\rangle\\}$ except the real solution state
$\left|x_{0}\right\rangle.$ Here it is not discussed whether the term could
make a contribution to the overlapping integrations between the final states
of the physical and math Hilbert spaces. Equivalently speaking, the small term
is always changed to a basis state $\left|x\right\rangle$
($\neq\left|x_{0}\right\rangle$) by the quantum measurement. Thus, it does not
make a positive contribution to the determination of the real solution state
by the quantum measurement. It disturbs the determination of the real solution
state. Then it is an error source to determining the real solution state by
the quantum measurement for the $HSSS$ quantum search process. Theoretically,
it vanishes for the $HSSS$ quantum search process whose final state is only
the real solution state.

The quantum-computing speedup theory mainly deals with the quantum-computing
speedup and its mechanism. It involves the fundamental aspects of quantum
physics, i.e., unitarity, symmetry, reality, and determinism.

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Appendix A. The perturbation equations for a $SIC$ unitary propagator and a
$SIC$ Green function

The $SIC$ perturbation equations are derived below. Suppose that the time-
independent Hamiltonian of a quantum system in time and space is given by

$H=H_{0}+H_{1}\newline $ (A1)

where $H_{0}$ is the main term and $H_{1}$ is the perturbation term. With the
total Hamiltonian $H$ and the main Hamiltonian $H_{0}$ two $SIC$ unitary
propagators are defined by
$U^{sic}\left(t\right)=\exp\left(-ia_{m}^{s}Ht/\hslash\right)$ and
$U_{0}^{sic}\left(t\right)=\exp\left(-ia_{m}^{s}H_{0}t/\hslash\right),$
respectively. In the Schrödinger picture the time evolution process that is
governed by the $SIC$ unitary propagator $U^{sic}\left(t\right)$ is written as

$\left|\Psi_{s}^{sic}\left(t\right)\right\rangle=U^{sic}\left(t\right)\left|\Psi_{s}^{sic}\left(0\right)\right\rangle$
(A2)

where $\left|\Psi_{s}^{sic}\left(t\right)\right\rangle$ is the $SIC$ state at
the time $t$ in the Schrödinger picture and
$\left|\Psi_{s}^{sic}\left(0\right)\right\rangle$ is the initial state, which
may or may not carry the solution information, at the initial time $t=0.$ The
Schrödinger equation that describes the time evolution process (A2) is given
by

$i\hslash\frac{d}{dt}\left|\Psi_{s}^{sic}\left(t\right)\right\rangle=a_{m}^{s}H\left|\Psi_{s}^{sic}\left(t\right)\right\rangle$
(A3)

The $SIC$ Schrödinger equation (A3) and the two $SIC$ unitary propagators
$U^{sic}\left(t\right)$ and $U_{0}^{sic}\left(t\right)$ are theoretical basis
to set up the perturbation equations that a $SIC$ unitary propagator and a
$SIC$ Green function obey, respectively.

Generally, if the Hamiltonian $H$ is time-dependent, i.e.,
$H=H\left(t\right),$ then the $SIC$ Schrödinger equation (A3) is rewritten as

$i\hslash\frac{d}{dt}\left|\Psi_{s}^{sic}\left(t\right)\right\rangle=a_{m}^{s}H\left(t\right)\left|\Psi_{s}^{sic}\left(t\right)\right\rangle$

This time-dependent $SIC$ Schrödinger equation also may be used to define the
$SIC$ unitary propagator (4.40) that is generated by a time-dependent
Hamiltonian $H(t)$ in a quantum system in time and space.

Now the interaction picture is defined by

$\left|\Psi_{I}^{sic}\left(t\right)\right\rangle=U_{0}^{sic}\left(t\right)^{+}\left|\Psi_{s}^{sic}\left(t\right)\right\rangle$
(A4)
$H_{I}^{sic}\left(t\right)=U_{0}^{sic}\left(t\right)^{+}H_{1}U_{0}^{sic}\left(t\right)\newline
$ (A5)

where $\left|\Psi_{I}^{sic}\left(t\right)\right\rangle$ is the $SIC$ state in
the time $t$ in the interaction picture. It is clear that at the initial time
$t=0$ one has the initial state
$\left|\Psi_{I}^{sic}\left(0\right)\right\rangle=\left|\Psi_{s}^{sic}\left(0\right)\right\rangle.$
Here the initial state $\left|\Psi_{I}^{sic}\left(0\right)\right\rangle$ or
$\left|\Psi_{s}^{sic}\left(0\right)\right\rangle$ can be arbitrary. The
interaction Hamiltonian $H_{I}^{sic}\left(t\right)$ carries the solution
information, although the perturbation term $H_{1}$ does not.

The motion equation for the state
$\left|\Psi_{I}^{sic}\left(t\right)\right\rangle$ in the interaction picture
can be derived from (A4) and (A5). It is given by

$i\hslash\frac{d}{dt}\left|\Psi_{I}^{sic}\left(t\right)\right\rangle=a_{m}^{s}H_{I}^{sic}\left(t\right)\left|\Psi_{I}^{sic}\left(t\right)\right\rangle$
(A6)

This equation has the formal solution:

$\left|\Psi_{I}^{sic}\left(t\right)\right\rangle=U_{I}^{sic}\left(t\right)\left|\Psi_{I}^{sic}\left(0\right)\right\rangle$
(A7)

where the $SIC$ unitary propagator $U_{I}^{sic}\left(t\right)$ in the
interaction picture can be formally written as, as shown from the motion
equation (A10) below,

$U_{I}^{sic}\left(t\right)=\hat{T}\exp\left(-\frac{i}{\hslash}a_{m}^{s}\int_{0}^{t}H_{I}^{sic}\left(t^{\prime}\right)dt^{\prime}\right)$
(A8)

with the time-ordering operator $\hat{T}$. At the initial time $t=0$ one has
$U_{I}^{sic}\left(0\right)=E$ (the unity operator).

Now by using the equations (A2) and (A7) it can be obtained from the
definition (A4) of the interaction picture that
$U_{I}^{sic}\left(t\right)\left|\Psi_{I}^{sic}\left(0\right)\right\rangle=U_{0}^{sic}\left(t\right)^{+}U^{sic}\left(t\right)\times$
$\left|\Psi_{s}^{sic}\left(0\right)\right\rangle.$ Note that the initial state
$\left|\Psi_{I}^{sic}\left(0\right)\right\rangle=\left|\Psi_{s}^{sic}\left(0\right)\right\rangle$
is arbitrary. Then one has

$U^{sic}\left(t\right)=U_{0}^{sic}\left(t\right)U_{I}^{sic}\left(t\right)$
(A9)

On the other hand, with the help of the equation (A7) and the fact that the
initial state $\left|\Psi_{I}^{sic}\left(0\right)\right\rangle$ is arbitrary
the motion equation (A6) of the state
$\left|\Psi_{I}^{sic}\left(t\right)\right\rangle$ can be reduced to the
following motion equation of the $SIC$ unitary propagator
$U_{I}^{sic}\left(t\right)$ in the interaction picture:

$i\hslash\frac{d}{dt}U_{I}^{sic}\left(t\right)=a_{m}^{s}H_{I}^{sic}\left(t\right)U_{I}^{sic}\left(t\right)$
(A10)

with $U_{I}^{sic}\left(0\right)=E$ at the initial time $t=0$. The formal
solution of the motion equation (A10) is just given by (A8).

By integrating the motion equation (A10) one obtains

$U_{I}^{sic}\left(t\right)=E+\frac{1}{i\hslash}a_{m}^{s}\int_{0}^{t}H_{I}^{sic}\left(t^{\prime}\right)U_{I}^{sic}\left(t^{\prime}\right)dt^{\prime}$
(A11)

Then by using this equation and the equation (A9) one further obtains

$U^{sic}\left(t\right)=U_{0}^{sic}\left(t\right)+\frac{1}{i\hslash}a_{m}^{s}U_{0}^{sic}\left(t\right)\int_{0}^{t}H_{I}^{sic}\left(t^{\prime}\right)U_{I}^{sic}\left(t^{\prime}\right)dt^{\prime}$
(A12)

It can prove by (A5) and (A9) that
$H_{I}^{sic}\left(t^{\prime}\right)U_{I}^{sic}\left(t^{\prime}\right)=U_{0}^{sic}\left(t^{\prime}\right)^{+}H_{1}U^{sic}\left(t^{\prime}\right).$
By using this relation the equation (A12) can be rewritten in the form

$U^{sic}\left(t\right)=U_{0}^{sic}\left(t\right)+\frac{1}{i\hslash}\int_{0}^{t}U_{0}^{sic}\left(t-t^{\prime}\right)\left(a_{m}^{s}H_{1}\right)U^{sic}\left(t^{\prime}\right)dt^{\prime}$
(A13)

This is the $SIC$ perturbation equation that the $SIC$ unitary propagators
$U^{sic}\left(t\right)$ and $U_{0}^{sic}\left(t\right)$ obey. This operator
perturbation equation directly leads to that the $SIC$ Green functions
$G_{H}^{sic}(x_{b},t_{b};x_{a},t_{a})$ (generated by $U^{sic}\left(t\right)$)
and $G_{H_{0}}^{sic}(x_{b},t_{b};$ $x_{a},t_{a})$ (generated by
$U_{0}^{sic}\left(t\right)$) obey the following equation:

$G_{H}^{sic}(x_{b},t_{b};x_{a},0)=G_{H_{0}}^{sic}(x_{b},t_{b};x_{a},0)$
$+\frac{1}{i\hslash}\int_{0}^{t}dt^{\prime}\int
dx^{\prime}G_{H_{0}}^{sic}(x_{b},t_{b};x^{\prime},t^{\prime})\left(a_{m}^{s}H_{1}\left(x^{\prime}\right)\right)G_{H}^{sic}(x^{\prime},t^{\prime};x_{a},0)$
(A14)

where the initial time $t_{a}=0$ and the final time
$t_{b}=t_{b}-t_{a}=a_{m}^{s}t$ in value. This is the $SIC$ perturbation
equation that a $SIC$ Green function obeys.

It can prove that if in (A13) one makes the replacements
$U^{sic}\left(\tau\right)\rightarrow U\left(\tau\right),$
$U_{0}^{sic}\left(\tau\right)\rightarrow U_{0}\left(\tau\right),$ and
$\left(a_{m}^{s}H_{1}\right)\rightarrow H_{1},$ where $\tau=t,$
$t-t^{\prime},$ or $t^{\prime}$, then the $SIC$ perturbation equation (A13) is
exactly changed to the $QM$ perturbation equation that a $QM$ unitary
propagator obeys. Here the $QM$ unitary propagators $U\left(\tau\right)$ and
$U_{0}\left(\tau\right)$ correspond to the $SIC$ unitary propagators
$U^{sic}\left(\tau\right)$ and $U_{0}^{sic}\left(\tau\right),$ respectively.
Similarly it can prove that if in (A14) one makes the replacements
$G_{H}^{sic}(x_{b},t_{b};x_{a},t_{a})\rightarrow
G_{H}(x_{b},t_{b};x_{a},t_{a}),$
$G_{H_{0}}^{sic}(x_{b},t_{b};x_{a},t_{a})\rightarrow
G_{H_{0}}(x_{b},t_{b};x_{a},t_{a}),$ and
$a_{m}^{s}H_{1}\left(x^{\prime}\right)\rightarrow
H_{1}\left(x^{\prime}\right),$ then the $SIC$ perturbation equation (A14) is
exactly changed to the $QM$ perturbation equation that a $QM$ Green function
obeys [$10,14$].