st}(\eta;k)}\|_{HS(L^{2}_{Q};H^{k})}^{2}\leq 2M^{2}K_{B}^{2}(1+\eta),$ (7.23)

for any $0\leq t\leq T.$ In conclusion, condition (HB) is satisfied with
$\Theta_{*}^{2}=2M^{2}K_{B}^{2}(\varepsilon^{-1}+\eta)$. ∎

###### Proof of Proposition 7.1.

Collecting the results in Lemmas 7.3–7.5, the estimates
(LABEL:eq:st:a:priori:bnd:e) and (7.13) can be combined to yield

$\displaystyle\mathbb{E}\left[\sup_{0\leq t\leq t_{\rm st}(\eta;k)\wedge
T}N_{\varepsilon;k}(t)^{p}\right]$ (7.24) $\displaystyle\qquad\quad\leq
K^{p}\Bigg{(}\|v(0)\|_{H^{k}}^{2p}+\sigma^{2p}(p^{p}+\log(T)^{p})+(\sigma^{4}+\eta)^{p}\mathbb{E}\left[\sup_{0\leq
t\leq t_{\rm st}(\eta;k)\wedge T}N_{\varepsilon;k}(t)^{p}\right]\Bigg{)}.$

The result hence readily follows by restricting the size of
$\sigma^{4}+{\eta}$, . ∎

###### Proof of Proposition 2.5.

The bound can immediately be deduced from Proposition 7.1 by following the
proof of [27, Thm. 2.4] to undo the time transformation, which only affects
the value of the constants. In more detail, recall the perturbation $v(t)$
defined in (6.1) together with the stochastic time transformed perturbation
$\bar{v}(\tau)$ defined in (6.18). We now write

$\bar{N}_{\varepsilon,k}(\tau)=\|\bar{v}(\tau)\|_{H^{k}}^{2}+\int_{0}^{\tau}e^{-\varepsilon(\tau-\sigma)}\|\bar{v}(\sigma)\|_{H^{k+1}}^{2}\mathrm{d}\sigma,$
(7.25)

define the associated stopping time $\bar{t}_{\rm st}(\eta;k),$ and recall the
similar expressions (2.53) and (2.54) for $v(t)$. Observe the inequality

$\sup_{0\leq t\leq t_{\rm st}(\eta;k)\wedge T}N_{\varepsilon;k}(t)\leq
K_{\kappa}\sup_{0\leq\tau\leq\bar{t}_{\rm st}(K_{\kappa}^{-1}\eta;k)\wedge
K_{\kappa}T}\bar{N}_{K_{\kappa}^{-1}\varepsilon;k}(\tau),$ (7.26)

where the constant $K_{\kappa}$ is as in (6.17). This inequality can be
obtained by tracing through the proof of [27, Prop. 6.4] and utilising the
implication

$\sup_{0\leq t\leq T}N_{\varepsilon;k}(t)>\eta\implies\sup_{0\leq\tau\leq
K_{\kappa}T}\bar{N}_{K_{\kappa}^{-1}\varepsilon;k}(\tau)>K_{\kappa}^{-1}\eta,$
(7.27)

which is equivalent to

$t_{\rm st}(\eta)<T\implies\bar{t}_{\rm st}(K_{\kappa}^{-1}\eta)<K_{\kappa}T.$
(7.28)

This completes the proof. ∎

##  Appendix A List of main functions

In this appendix we provide an overview of the main functions that are used in
this paper. Throughout this section, we take $k=0$ if (Hf-Lip) is satisfied or
$k=1$ if (Hf-Cub) is satisfied. We assume furthermore that (HSt), (Hq), and
(HCor) hold with this choice of $k$. In addition, we take
$c,\gamma\in\mathbb{R}$ and $\xi\in L^{2}_{Q}$. We emphasise that we have kept
our naming conventions as close as possible to those considered in [27, 30] to
prevent confusion.

We start by choosing a smooth non-decreasing cut-off function

$\chi_{\rm low}:\mathbb{R}\to[\tfrac{1}{4},\infty)$ (A.1)

that satisfies the properties

$\chi_{\rm
low}(\vartheta)=\frac{1}{4}|\mathbb{T}|^{d-1},\quad\vartheta\leq\frac{1}{4}|\mathbb{T}|^{d-1},\quad\chi_{\rm
low}(\vartheta)=\vartheta,\quad\vartheta\geq\frac{1}{2}|\mathbb{T}|^{d-1},$
(A.2)

together with a smooth non-increasing cut-off function

$\chi_{\rm high}:\mathbb{R}_{+}\to[0,1]$ (A.3)

that satisfies the properties

$\chi_{\rm high}(\vartheta)=1,\quad\vartheta\leq 2+\|\Phi_{0}-\Phi_{\rm
ref}\|_{L^{2}(\mathcal{D};\mathbb{R}^{n})},\quad\chi_{\rm
high}(\vartheta)=0,\quad\vartheta\geq 3+\|\Phi_{0}-\Phi_{\rm
ref}\|_{L^{2}(\mathcal{D};\mathbb{R}^{n})}.$ (A.4)

For any $u\in\mathcal{U}_{L^{2}(\mathcal{D};\mathbb{R}^{n})}$, these cut-offs
can be used to define

$\chi_{h}(u,\gamma)=\chi_{\rm high}(\|u-T_{\gamma}\Phi_{\rm
ref}\|_{L^{2}(\mathcal{D};\mathbb{R}^{n})})\quad\text{and}\quad\chi_{l}(u,\gamma)=\big{[}\chi_{\rm
low}\big{(}-\langle u,T_{\gamma}\psi_{\rm
tw}^{\prime}\rangle_{L^{2}(\mathcal{D};\mathbb{R}^{n})}\big{)}\big{]}^{-1}.$
(A.5)

We note that when (HPar) is satisfied and we take $u=T_{\gamma}[\Phi+v]$ with

$\|v\|_{L^{2}(\mathcal{D};\mathbb{R}^{n})}\leq\min\\{1,|\mathbb{T}|^{\frac{d-1}{2}}[4\|\psi_{\rm
tw}\|_{H^{1}(\mathbb{R};\mathbb{R}^{n})}]^{-1}\\},$ (A.6)

then we have

$\chi_{h}(u,\gamma)=1\quad\text{and}\quad\chi_{l}(u,\gamma)=-\big{[}\langle
u,T_{\gamma}\psi_{\rm
tw}^{\prime}\rangle_{L^{2}(\mathcal{D};\mathbb{R}^{n})}\big{]}^{-1}.$ (A.7)

Indeed, we may compute

$\begin{array}[]{lcl}\|u-T_{\gamma}\Phi_{\rm
ref}\|_{L^{2}(\mathcal{D};\mathbb{R}^{n})}&\leq&\|\Phi-\Phi_{0}\|_{L^{2}(\mathcal{D};\mathbb{R}^{n})}+\|\Phi_{0}-\Phi_{\rm
ref}\|_{L^{2}(\mathcal{D};\mathbb{R}^{n})}+\|v\|_{L^{2}(\mathcal{D};\mathbb{R}^{n})}\\\\[5.69046pt]
&\leq&2+\|\Phi_{0}-\Phi_{\rm
ref}\|_{L^{2}(\mathcal{D};\mathbb{R}^{n})},\end{array}$ (A.8)

together with

$\begin{array}[]{lcl}|\langle\Phi_{0}-\Phi-v,\psi_{\rm
tw}^{\prime}\rangle_{L^{2}(\mathcal{D};\mathbb{R}^{n})}|&\leq&\big{[}\|\Phi_{0}-\Phi\|_{L^{2}(\mathcal{D};\mathbb{R}^{n})}+\|v\|_{L^{2}(\mathcal{D};\mathbb{R}^{n})}\big{]}|\mathbb{T}|^{\frac{d-1}{2}}\|\psi^{\prime}_{\rm
tw}\|_{L^{2}(\mathbb{R};\mathbb{R}^{n})}\\\\[5.69046pt]
&\leq&\frac{1}{2}|\mathbb{T}|^{d-1},\end{array}$ (A.9)

to conclude

$\begin{array}[]{lcl}-\langle\Phi+v,\psi_{\rm
tw}^{\prime}\rangle_{L^{2}(\mathcal{D};\mathbb{R}^{n})}&=&-\langle\Phi_{0},\psi_{\rm
tw}^{\prime}\rangle_{L^{2}(\mathcal{D};\mathbb{R}^{n})}+\langle\Phi_{0}-\Phi-v,\psi_{\rm
tw}^{\prime}\rangle_{L^{2}(\mathcal{D};\mathbb{R}^{n})}\\\\[5.69046pt]
&\geq&|\mathbb{T}|^{d-1}-\frac{1}{2}|\mathbb{T}|^{d-1}.\end{array}$ (A.10)

Again taking $u\in\mathcal{U}_{L^{2}(\mathcal{D};\mathbb{R}^{n})}$, we
introduce the scalar function

$b(u,\gamma)[\xi]=-\chi_{h}(u,\gamma)^{2}\chi_{l}(u,\gamma)\langle
g(u)[\xi],T_{\gamma}\psi_{\rm tw}\rangle_{L^{2}(\mathcal{D};\mathbb{R}^{n})},$
(A.11)

together with

$\kappa_{\sigma}(u,\gamma)=1+\frac{\sigma^{2}}{2}\|b(u,\gamma)\|^{2}_{HS(L^{2}_{Q};\mathbb{R})},$
(A.12)

and the associated quantities

$\displaystyle\nu_{\sigma}^{(1)}(u,\gamma)$
$\displaystyle=\kappa_{\sigma}(u,\gamma)-1,$ (A.13)
$\displaystyle\nu_{\sigma}^{(-1)}(u,\gamma)$
$\displaystyle=\kappa_{\sigma}(u,\gamma)^{-1}-1,$ (A.14)
$\displaystyle\nu_{\sigma}^{(-1/2)}(u,\gamma)$
$\displaystyle=\kappa_{\sigma}(u,\gamma)^{-1/2}-1.$ (A.15)

These expressions are all well-defined by Corollary 4.9 and Lemma 4.10.

For $u\in\mathcal{U}_{L^{2}(\mathcal{D};\mathbb{R}^{n})}$, the map
$g(u):L^{2}_{Q}\to L^{2}(\mathcal{D};\mathbb{R}^{n})$ has a formal adjoint
$g^{\rm adj}(u):L^{2}(\mathcal{D};\mathbb{R}^{n})\to L^{2}_{Q},$ that acts as

$g^{\rm adj}(u)[\zeta]=Qg(u)^{\top}[\zeta],$ (A.16)

where the matrix transpose is taken in a pointwise fashion. Indeed, for
$\xi\in L^{2}_{Q}$ and $\zeta\in L^{2}(\mathcal{D};\mathbb{R}^{n})$ we compute

$\langle g(u)[\xi],\zeta\rangle_{L^{2}(\mathcal{D};\mathbb{R}^{n})}=\langle
Q^{-1/2}\xi,Q^{-1/2}g^{\rm
adj}(u)[\zeta]\rangle_{L^{2}(\mathcal{D};\mathbb{R}^{m})}=\langle\xi,g^{\rm
adj}(u)[\zeta]\rangle_{L^{2}_{Q}}.$ (A.17)

The fact that both maps $g(u)$ and $g^{\rm adj}(u)$ are well-defined follows
from Lemmas 4.5 and 4.6, together with Lemma 4.4 and the computation

$\begin{array}[]{lcl}||Qg(u)^{\top}[\zeta]||_{L^{2}_{Q}}&=&||Q^{1/2}g(u)^{\top}[\zeta]||_{L^{2}(\mathcal{D};\mathbb{R}^{m})}\\\\[5.69046pt]
&\leq&||p||_{L^{2}(\mathcal{D};\mathbb{R}^{m\times
m})}||g(u)^{\top}[\zeta]||_{L^{1}(\mathcal{D};\mathbb{R}^{m})}\\\\[5.69046pt]
&\leq&||p||_{L^{2}(\mathcal{D};\mathbb{R}^{m\times
m})}||g(u)^{\top}||_{L^{2}(\mathcal{D};\mathbb{R}^{m\times
n})}||\zeta||_{L^{2}(\mathcal{D};\mathbb{R}^{n})}.\end{array}$ (A.18)

This allows us to introduce the $L^{2}_{Q}$-valued function

$\widetilde{\mathcal{K}}_{C}(u,\gamma)=\chi_{l}(u,\gamma)\chi_{h}(u,\gamma)Qg(u)^{\top}T_{\gamma}\psi_{\rm
tw},$ (A.19)

together with the $L^{2}(\mathcal{D};\mathbb{R}^{n})$-valued function

$\mathcal{K}_{C}(u,\gamma)=-\chi_{h}(u,\gamma)g(u)\widetilde{\mathcal{K}}_{C}(u,\gamma)=-\chi_{h}(u,\gamma)^{2}\chi_{l}(u,\gamma)g(u)[g^{\rm
adj}(u)T_{\gamma}\psi_{\rm tw}].$ (A.20)

Note that a short computation shows

$\|b(u,\gamma)\|_{HS(L^{2}_{Q};\mathbb{R})}^{2}=\chi_{h}(u,\gamma)^{4}\chi_{l}(u,\gamma)^{2}\langle
g(u)g^{\rm adj}(u)T_{\gamma}\psi_{\rm tw},T_{\gamma}\psi_{\rm
tw}\rangle_{L^{2}(\mathcal{D};\mathbb{R}^{n})},$ (A.21)

which provides a more explicit representation for (A.12).

Now taking131313If (Hf-Lip) holds, it suffices to take
$u\in\mathcal{U}_{L^{2}(\mathcal{D};\mathbb{R}^{n})}$. However (A.24) is not
necessarily well-defined in this case.
$u\in\mathcal{U}_{H^{k+1}(\mathcal{D};\mathbb{R}^{n})}$, we are ready to
define the scalar function

$\begin{array}[]{lcl}a_{\sigma}(u,\gamma;c)&=&-\chi_{l}(u,\gamma)\Big{[}\langle
f(u)+\sigma^{2}h(u),T_{\gamma}\psi_{\rm
tw}\rangle_{L^{2}(\mathcal{D};\mathbb{R}^{n})}-\langle
cu+\sigma^{2}\mathcal{K}_{C}(u,\gamma),\partial_{x}\psi_{\rm
tw}\rangle_{L^{2}(\mathcal{D};\mathbb{R}^{n})}\\\\[5.69046pt]
&&\qquad\qquad\qquad+\kappa_{\sigma}(u,\gamma)\langle
u,T_{\gamma}\psi^{\prime\prime}_{\rm
tw}\rangle_{L^{2}(\mathcal{D};\mathbb{R}^{n})}\Big{]};\end{array}$ (A.22)

see Lemmas 4.11 and 4.24. In addition, we define the
$H^{k}(\mathcal{D};\mathbb{R}^{n})$-valued function

$\displaystyle\mathcal{J}_{\sigma}(u,\gamma;c)$
$\displaystyle=\kappa_{\sigma}(u,\gamma)^{-1}\left(f(u)+c\partial_{x}u+\sigma^{2}h(u)+\sigma^{2}[\partial_{x}\mathcal{K}_{C}(u,\gamma)]\right),$
(A.23)

where the well-posedness follows from the bounds (4.62) and (4.75), together
with Lemma 4.6 and Corollary 4.29. Note that

$\begin{array}[]{lcl}a_{\sigma}(u,\gamma;c)&=&-\chi_{l}(u,\gamma)\kappa_{\sigma}(u,\gamma)\big{(}\langle\mathcal{J}_{\sigma}(u,\gamma;c),T_{\gamma}\psi_{\rm
tw}\rangle_{L^{2}(\mathcal{D};\mathbb{R}^{n})}+\langle u,T_{\gamma}\psi_{\rm
tw}^{\prime\prime}\rangle_{L^{2}(\mathcal{D};\mathbb{R}^{n})}\big{)}.\end{array}$
(A.24)

Exploiting the translational invariance of our nonlinearities and the noise,
we obtain the commutation relations

$T_{\gamma}f(u)=f(T_{\gamma}u),\qquad
T_{\gamma}g(u)[\xi]=g(T_{\gamma}u)[T_{\gamma}\xi],\qquad T_{\gamma}g^{\rm
adj}(u)[\zeta]=g^{\rm adj}(T_{\gamma}u)[T_{\gamma}\zeta].$ (A.25)

In particular, we see that

$a_{\sigma}(u,\gamma;c)=a_{\sigma}(T_{-\gamma}u,0;c),\qquad
b(u,\gamma)[\xi]=b(T_{-\gamma}u,0)[T_{-\gamma}\xi],$ (A.26)

and similar identities hold for $\kappa_{\sigma}$, $\mathcal{J}_{\sigma}$ and
the cut-off functions (A.5). This subsequently allows us to eliminate the
dependence on $\gamma$ in the sequel.

Assuming (HPar) and taking $v\in H^{k+1}(\mathcal{D};\mathbb{R}^{n})$, we
introduce the expressions

$\begin{array}[]{lcl}\mathcal{R}_{\sigma}(v;c,\Phi)&=&\Delta_{y}v+\kappa_{\sigma}(\Phi+v,0)\big{[}\partial_{x}^{2}v+\Phi^{\prime\prime}+\mathcal{J}_{\sigma}(\Phi+v,0;c)\big{]}+a_{\sigma}(\Phi+v,0,c)\partial_{x}(\Phi+v),\\\\[5.69046pt]
\mathcal{S}(v;\Phi)[\xi]&=&g(\Phi+v)[\xi]+\partial_{x}(\Phi+v)b(\Phi+v,0)[\xi].\end{array}$
(A.27)

We remark that $\mathcal{S}(v;\Phi)[\xi]\in H^{k}(\mathcal{D};\mathbb{R}^{n})$
and $\mathcal{R}_{\sigma}(v;c,\Phi)\in H^{k-1}(\mathcal{D};\mathbb{R}^{n})$,
and refer to §5 for the subtle interpretation of the latter. By construction,
we have $\langle\mathcal{S}(v;\Phi)[\xi],\psi_{\rm
tw}\rangle_{L^{2}(\mathcal{D};\mathbb{R}^{n})}=0$ whenever (A.6) is satisfied.

Finally, recalling the family $(\Phi_{\sigma},c_{\sigma})$ constructed in
Proposition 2.3, we define the $H^{k}(\mathcal{D};\mathbb{R}^{n})$-valued
expressions

$\begin{array}[]{lcl}\mathcal{N}_{\sigma}(v)&=&\Phi_{\sigma}^{\prime\prime}+\mathcal{J}_{\sigma}(\Phi_{\sigma}+v,0;c_{\sigma})-c_{0}\partial_{x}v-Df(\Phi_{0})v\\\\[5.69046pt]
&&\qquad\qquad+\kappa_{\sigma}(\Phi_{\sigma}+v;0)^{-1}a_{\sigma}(\Phi_{\sigma}+v,0,c_{\sigma})\partial_{x}(\Phi_{\sigma}+v),\\\\[5.69046pt]
\mathcal{M}_{\sigma}(v)[\xi]&=&\kappa_{\sigma}(\Phi_{\sigma}+v,0)^{-1/2}\mathcal{S}(v;\Phi_{\sigma})[\xi],\end{array}$
(A.28)

for any $v\in H^{k+1}(\mathcal{D};\mathbb{R}^{n})$. Indeed, note that these
expressions no longer involve second derivatives of $v$. In fact, upon
introducing the intermediate function

$\begin{array}[]{lcl}\mathcal{N}_{I;\sigma}(v)&=&\Phi_{\sigma}^{\prime\prime}+\mathcal{J}_{\sigma}(\Phi_{\sigma}+v,0;c_{\sigma})-c_{0}\partial_{x}v-Df(\Phi_{0})v\\\\[5.69046pt]
&=&\Phi_{\sigma}^{\prime\prime}+\mathcal{J}_{\sigma}(\Phi_{\sigma}+v,0;c_{\sigma})-[\mathcal{L}_{\rm
tw}-\partial_{xx}]v,\end{array}$ (A.29)

we may use the identity $\mathcal{L}_{\rm tw}^{\rm adj}\psi_{\rm tw}=0$ to
arrive at the convenient representation

$\mathcal{N}_{\sigma}(v)=\mathcal{N}_{I;\sigma}(v)-\chi_{l}(\Phi_{\sigma}+v,0)\langle\mathcal{N}_{I;\sigma}(v),\psi_{\rm
tw}\rangle_{L^{2}(\mathcal{D};\mathbb{R}^{n})}[\partial_{x}\Phi+\partial_{x}v].$
(A.30)

One now readily verifies that $\langle\mathcal{N}_{\sigma}(v),\psi_{\rm
tw}\rangle_{L^{2}(\mathcal{D};\mathbb{R}^{n})}=0$ holds whenever (A.6) is
satisfied.

##  Appendix B Moment bounds and tail probabilities

We briefly review here the technique of passing back and forth between moment
estimates and tail probabilities. Similar results can be found in [29, Sec.
2]; see also [91, 99].

######  Lemma B.1.

Consider a nonnegative random variable $X$. Suppose that there exists two
constants $\Theta_{1}>0$ and $\Theta_{2}>0$ so that the moment bound

$\mathbb{E}\left[X^{p}\right]\leq p^{p}\Theta_{1}^{p}+\Theta_{2}^{p}$ (B.1)

holds for all integers $p\geq 1$. Then for every $\vartheta>0$ we have the
estimate

$\mathbb{P}(X\geq\vartheta)\leq
3\exp\left(\frac{\Theta_{2}}{2e\Theta_{1}}\right)\exp\left(-\frac{\vartheta}{2e\Theta_{1}}\right).$
(B.2)

###### Proof.

The proof is similar to that of [29, Lem 2.2]. Let $\lambda>0$ be arbitrary.
Then by an exponential Markov inequality, we obtain

$\displaystyle\mathbb{P}(X\geq\vartheta)$ $\displaystyle\leq
e^{-\lambda\vartheta}\mathbb{E}[e^{\lambda X}]$ (B.3)
$\displaystyle=e^{-\lambda\vartheta}\sum_{p=0}^{\infty}\frac{\lambda^{p}}{p!}\mathbb{E}[X^{p}]$
$\displaystyle\leq
e^{-\lambda\vartheta}\left[\sum_{p=0}^{\infty}\lambda^{p}e^{p}\Theta_{1}^{p}+\sum_{p=0}^{\infty}\frac{\lambda^{p}}{p!}\Theta_{2}^{2p}\right],$

exploiting the identity $p!\geq p^{p}e^{-p}$ in the second inequality above.
Upon choosing $\lambda=(2e\Theta_{1})^{-1}$, we obtain the tail probability
$\mathbb{P}(X\geq\vartheta)\leq
e^{-\lambda\vartheta}[2+e^{\lambda\Theta_{2}}]\leq
3e^{\lambda\Theta_{2}}e^{-\lambda\vartheta}$, which proves the assertion. ∎

######  Lemma B.2.

Fix two constants $A\geq 2$ and $\Theta_{1}>0$ and consider a nonnegative
random variable $X$ that satisfies the estimate

$\mathbb{P}(X\geq\vartheta)\leq
2A\exp\left(-\frac{\vartheta}{2e\Theta_{1}}\right)$ (B.4)

for all $\vartheta>0$. Then for any $p\geq 1$ we have the moment bound

$\mathbb{E}\left[X^{p}\right]\leq\big{(}p^{p}+\log(A)^{p}\big{)}(8e\Theta_{1})^{p}.$
(B.5)

###### Proof.

This follows directly from [29, Lem. 2.3]. ∎

######  Corollary B.3.

Consider $N\geq 2$ nonnegative random variables $X_{1},X_{2},\ldots,X_{N}$ and
suppose that there exists two constants $\Theta_{1}>0$ and $\Theta_{2}>0$ so
that the moment bound

$\mathbb{E}[X_{i}^{p}]\leq p^{p}\Theta_{1}^{p}+\Theta_{2}^{p}$ (B.6)

holds for all integers $p\geq 1$ and each $i\in\\{1,\ldots,N\\}$. Then for any
$p\geq 1$ we have the maximal bound

$\mathbb{E}\max_{i\in\\{1,\ldots,N\\}}X_{i}^{p}\leq\big{(}p^{p}+\log(N)^{p}+\Theta_{2}^{p}\Theta_{1}^{-p}\big{)}\left(24e\Theta_{1}\right)^{p}.$
(B.7)

###### Proof.

For any $\vartheta>0$, we invoke Lemma B.1 and the observation $(2e)^{-1}<1$
to obtain the estimate

$\mathbb{P}\left(\max_{i\in\\{1,\ldots,N\\}}X_{i}\geq\vartheta\right)\leq\sum_{i=1}^{N}\mathbb{P}(X_{i}\geq\vartheta)\leq
3N\exp(\Theta_{2}\Theta_{1}^{-1})\exp\left(-\frac{\vartheta}{2e\Theta_{1}}\right).$
(B.8)

The assertion follows by appealing to Lemma B.2. In particular, we take
$A=\frac{3}{2}N\exp(\Theta_{2}\Theta_{1}^{-1})$, use the inequality
$(a+b+c)^{p}\leq 3^{p-1}(a^{b}+b^{p}+c^{p})$ and note that
$\log(\frac{3}{2})<1$. ∎

##  Appendix C Fourier analysis

The Fourier transform can be defined for any locally compact Abelian group [6,
12, 18, 78, 83], and thus in particular for
$S\in\mathcal{\\{}\mathbb{T}^{d},\mathcal{D}\\}$ with
$\mathcal{D}=\mathbb{R}\times\mathbb{T}^{d-1}$. The Pontryagin dual of $S$,
denoted by $\widehat{S}$, for the spaces $S=\mathbb{T}^{d}$ and
$S=\mathcal{D}$ are $\widehat{S}=\mathbb{Z}^{n}$ and
$\widehat{S}=\mathbb{R}\times\mathbb{Z}^{d-1},$ respectively.

Let $V$ be a seperable Banach space. For any function $u\in
L^{1}(\mathbb{T}^{d};V)\cap L^{2}(\mathbb{T}^{d};V)$, we define the Fourier
transform to be

$\hat{u}(\xi)=\frac{1}{|\mathbb{T}|^{d}}\int_{\mathbb{T}^{d}}u(y)e^{-\frac{2\pi
i}{|\mathbb{T}|}\langle\xi,y\rangle}\mathrm{d}y,\quad\xi\in\mathbb{Z}^{d},$
(C.1)

while for any function $u\in L^{1}(\mathcal{D};V)\cap L^{2}(\mathcal{D};V)$,
we have as Fourier transform

$\hat{u}(\omega,\xi)=\frac{1}{|\mathbb{T}|^{d-1}}\int_{\mathbb{R}\times\mathbb{T}^{d-1}}u(x,y)e^{-2\pi
i\omega x}e^{-\frac{2\pi
i}{|\mathbb{T}|}\langle\xi,y\rangle}\mathrm{d}x\,\mathrm{d}y,\quad(\omega,\xi)\in\mathbb{R}\times\mathbb{Z}^{d-1}.$
(C.2)

The mapping $\mathcal{F}:f\to\hat{f}$ extends to an isometric isomorphism from
$L^{2}(S;V)$ to $L^{2}(\widehat{S};V)$ for any choice of $S$. In particular,
the inversion formula for $S=\mathcal{D}$ is given by [45, 72]

$u(x,y)=\sum_{\xi\in\mathbb{Z}^{d-1}}\int_{\mathbb{R}}\hat{u}(\omega,\xi)e^{2\pi
i\omega x}e^{\frac{2\pi i}{|\mathbb{T}|}\langle\xi,y\rangle}\mathrm{d}x$ (C.3)

and Plancherel’s identity holds, i.e.,

$\displaystyle\|f\|_{L^{2}(\mathcal{D};V)}^{2}$
$\displaystyle=\int_{\mathcal{D}}\|f(\mathbf{x})\|_{V}^{2}\mathrm{d}\mathbf{x}=\int_{\mathbb{R}\times\mathbb{T}^{d-1}}\|f(x,y)\|_{V}^{2}\mathrm{d}x\,\mathrm{d}y$
(C.4)
$\displaystyle=\frac{1}{|\mathbb{T}|^{d-1}}\sum_{\xi\in\mathbb{Z}^{d-1}}\int_{\mathbb{R}}\|\hat{f}(\omega,\xi)\|_{V}^{2}\mathrm{d}\omega=\frac{1}{|\mathbb{T}|^{d-1}}\int_{\widehat{\mathcal{D}}}\|\hat{f}(\boldsymbol{\xi})\|_{V}^{2}\mathrm{d}\boldsymbol{\xi}=\frac{1}{|\mathbb{T}|^{d-1}}\|\hat{f}\|_{L^{2}(\widehat{\mathcal{D}};V)}.$

Here we have introduced the notation
$\mathrm{d}\boldsymbol{\xi}=\mathrm{d}\omega\mathrm{d}\xi,$ where
$\mathrm{d}\xi$ is the counting measure on $\mathbb{Z}^{d-1}.$ The factor
$1/{|\mathbb{T}|^{d-1}}$ is a consequence of not having normalised the
Lebesgue induced measure on $\mathbb{T}^{d-1}$.

Recall that the Sobolev spaces $H^{k}(S;V)$ can be characterised by means of
the Fourier transform [68]. Indeed, an equivalent norm is given by

$\vvvert
u\vvvert_{H^{k}(S;V)}=\int_{\widehat{S}}(1+|\boldsymbol{\xi}|^{2})^{k}\|\hat{f}(\boldsymbol{\xi})\|_{V}^{2}\mathrm{d}\boldsymbol{\xi},$
(C.5)

where $\boldsymbol{\xi}\in\widehat{S}$. This equivalence follows readily from
the fact that $\widehat{\partial^{\alpha}f}=\xi^{\alpha}\hat{f}$ holds and by
exploiting Plancherel’s identity. In a similar fashion, Parseval’s identity
holds, which yields an inner product on $H^{k}(S;V)$ in terms of the Fourier
transform. The norm also gives the inner product by polarisation.

## References

  * [1] P. Acquistapace  “Evolution operators and strong solutions of abstract linear parabolic equations”  In _Differential and Integral Equations_ 1.4, 1988, pp. 433–457
  * [2] P. Acquistapace and B. Terreni  “On fundamental solutions for abstract parabolic equations”  In _Differential Equations in Banach Spaces_ Springer Berlin Heidelberg, 1986, pp. 1–11
  * [3] Z.. Adams and J. MacLaurin  “The Isochronal Phase of Stochastic PDE and Integral Equations: Metastability and Other Properties”  In _arXiv preprint arXiv:2210.10681_ , 2022
  * [4] A. Agresti and M.. Veraar  “The critical variational setting for stochastic evolution equations”  In _arXiv preprint arXiv:2206.00230_ , 2022
  * [5] H. Berestycki, F. Hamel and H. Matano  “Bistable traveling waves around an obstacle”  In _Comm. Pure Appl. Math._ 62.6, 2009, pp. 729–788
  * [6] V.. Bogachev and M… Ruas  “Measure theory”  Springer, 2007
  * [7] H. Brézis  “Functional analysis, Sobolev spaces and partial differential equations”  Springer, 2011
  * [8] P.-L. Chow  “Stochastic partial differential equations”  ChapmanHall/CRC, 2007
  * [9] G. Da Prato, A. Jentzen and M. Röckner  “A mild Itô formula for SPDEs”  In _Transactions of the American Mathematical Society_ 372.6, 2019, pp. 3755–3807
  * [10] G. Da Prato and J. Zabczyk  “Ergodicity for infinite dimensional systems”  Cambridge university press, 1996
  * [11] G. Da Prato and J. Zabczyk  “Stochastic equations in infinite dimensions”  Cambridge university press, 2014
  * [12] A. Deitmar and S. Echterhoff  “Principles of harmonic analysis”  Springer, 2014
  * [13] S. Dirksen  “Tail bounds via generic chaining”  In _Electron. J. Probab_ 20.53, 2015, pp. 1–29
  * [14] K. Eichinger, M.. Gnann and C. Kuehn  “Multiscale analysis for traveling-pulse solutions to the stochastic FitzHugh–Nagumo equations”  In _The Annals of Applied Probability_ 32.5 Institute of Mathematical Statistics, 2022, pp. 3229–3282
  * [15] K.J. Engel and R. Nagel  “One-parameter semigroups for linear evolution equations”  Springer, 2000
  * [16] L.. Evans  “An introduction to stochastic differential equations”  American Mathematical Soc., 2012
  * [17] L.. Evans  “Partial differential equations”  American Mathematical Society, 2022
  * [18] A. Friedman  “Foundations of modern analysis”  Courier Corporation, 1982
  * [19] J. García-Ojalvo and J. Sancho  “Noise in spatially extended systems”  Springer Science & Business Media, 2012
  * [20] R.. Gardner  “Existence and stability of travelling wave solutions of competition models: a degree theoretic approach”  In _Journal of Differential equations_ 44.3 Academic Press, 1982, pp. 343–364
  * [21] L. Gawarecki and V. Mandrekar  “Stochastic differential equations in infinite dimensions: with applications to stochastic partial differential equations”  Springer Science & Business Media, 2010
  * [22] M.. Gnann, R… Westdorp and J. Winden  “Solitary waves in a stochastic parametrically forced nonlinear Schrödinger equation”  In _arXiv preprint arXiv:2403.04625_ , 2024
  * [23] M. Gómez-Gesteria et al.  “Vulnerability in excitable Belousov-Zhabotinsky medium: from 1D to 2D”  In _Physica D: Nonlinear Phenomena_ 76.4 Elsevier, 1994, pp. 359–368
  * [24] M. Hairer  “An introduction to stochastic PDEs”  In _arXiv preprint arXiv:0907.4178_ , 2009
  * [25] M. Hairer  “A theory of regularity structures”  In _Inventiones mathematicae_ 198.2 Springer, 2014, pp. 269–504
  * [26] J.. Hale and L.. Peletier  “Exact homoclinic and heteroclinic solutions of the Gray–Scott model for autocatalysis”  In _SIAM Journal on Applied Mathematics_ 61.1 SIAM, 2000, pp. 102–130
  * [27] C… Hamster and H.. Hupkes  “Stability of traveling waves for reaction-diffusion equations with multiplicative noise”  In _SIAM Journal on Applied Dynamical Systems_ 18.1 SIAM, 2019, pp. 205–278
  * [28] C… Hamster and H.. Hupkes  “Stability of traveling waves for systems of reaction-diffusion equations with multiplicative noise”  In _SIAM Journal on Mathematical Analysis_ 52.2 SIAM, 2020, pp. 1386–1426
  * [29] C… Hamster and H.. Hupkes  “Stability of traveling waves on exponentially long timescales in stochastic reaction-diffusion equations”  In _SIAM Journal on Applied Dynamical Systems_ 19.4 SIAM, 2020, pp. 2469–2499
  * [30] C… Hamster and H.. Hupkes  “Travelling waves for reaction-diffusion equations forced by translation invariant noise”  In _Physica D: Nonlinear Phenomena_ 401, 2020, pp. 132233
  * [31] P. Heijster, A. Doelman and T.. Kaper  “Pulse dynamics in a three-component system: stability and bifurcations”  In _Physica D: Nonlinear Phenomena_ 237.24 Elsevier, 2008, pp. 3335–3368
  * [32] T. Hytönen, J. Van Neerven, M. Veraar and L. Weis  “Analysis in Banach Spaces: Volume II: Probabilistic Methods and Operator Theory”  Springer, 2018
  * [33] T. Hytönen, J. Neerven, M. Veraar and L. Weis  “Analysis in Banach Spaces: Volume III: Harmonic Analysis and Spectral Theory”  Springer Nature, 2023
  * [34] J. Inglis and J. MacLaurin  “A general framework for stochastic traveling waves and patterns, with application to neural field equations”  In _SIAM Journal on Applied Dynamical Systems_ 15.1 SIAM, 2016, pp. 195–234
  * [35] A. Jentzen and P.. Kloeden  “Taylor approximations for stochastic partial differential equations”  SIAM, 2011
  * [36] S. Kadar, J. Wang and K. Showalter  “Noise-supported travelling waves in sub-excitable media”  In _Nature_ 391.6669 Nature Publishing Group UK London, 1998, pp. 770–772
  * [37] N. Kalton and L. Weis  “The $H^{\infty}$-Functional Calculus and Square Function Estimates”, 2015  arXiv:1411.0472
  * [38] T. Kapitula  “On the stability of traveling waves in weighted $L^{\infty}$ spaces”  In _Journal of Differential equations_ 112.1 Elsevier, 1994, pp. 179–215
  * [39] T. Kapitula  “Multidimensional Stability of Planar Travelling Waves”  In _Trans. Amer. Math. Soc._ 249, 1997, pp. 257–269
  * [40] T. Kapitula and K. Promislow  “Spectral and dynamical stability of nonlinear waves”  Springer, 2013
  * [41] I. Karatzas and S. Shreve  “Brownian motion and stochastic calculus”  Springer Science & Business Media, 2012
  * [42] A. Karczewska  “Stochastic integral with respect to cylindrical Wiener process”  In _arXiv preprint math/0511512_ , 2005
  * [43] D.. Kessler and H. Levine  “Stability of traveling waves in the Belousov-Zhabotinskii reaction”  In _Physical Review A_ 41.10 APS, 1990, pp. 5418
  * [44] P.. Kloeden and E. Platen  “Stochastic differential equations”  Springer, 1992
  * [45] I. Kluvánek  “Fourier transforms of vector-valued functions and measures”  In _Studia math_ 37.1970, 1970, pp. 1–12
  * [46] H.. Krug, L. Pohlmann and L. Kuhnert  “Analysis of the modified complete Oregonator accounting for oxygen sensitivity and photosensitivity of Belousov-Zhabotinskii systems”  In _Journal of Physical Chemistry_ 94.12 ACS Publications, 1990, pp. 4862–4866
  * [47] J. Kruger and W. Stannat  “Front propagation in stochastic neural fields: a rigorous mathematical framework”  In _SIAM Journal on Applied Dynamical Systems_ 13.3 SIAM, 2014, pp. 1293–1310
  * [48] N.. Krylov and B.. Rozovskii  “Stochastic evolution equations”  In _Journal of Soviet mathematics_ 16 Springer, 1981, pp. 1233–1277
  * [49] C. Kuehn  “Travelling waves in monostable and bistable stochastic partial differential equations”  In _Jahresbericht der Deutschen Mathematiker-Vereinigung_ 122 Springer, 2020, pp. 73–107
  * [50] C. Kuehn, J. MacLaurin and G. Zucal  “Stochastic rotating waves”  In _Stochastics and Dynamics_ 22.07 World Scientific, 2022, pp. 2240029
  * [51] C. Kuehn and A. Neamţu  “Pathwise mild solutions for quasilinear stochastic partial differential equations”  In _Journal of Differential Equations_ 269.3, 2020, pp. 2185–2227
  * [52] E. Lang  “A multiscale analysis of traveling waves in stochastic neural fields”  In _SIAM Journal on Applied Dynamical Systems_ 15.3 SIAM, 2016, pp. 1581–1614
  * [53] E. Lang and W. Stannat  “$L^{2}$-stability of traveling wave solutions to nonlocal evolution equations”  In _Journal of Differential Equations_ 261.8 Elsevier, 2016, pp. 4275–4297
  * [54] J.. Leach, J.. Merkin and S.. Scott  “The initiation and propagation of travelling waves on membrane interfaces in the Belousov-Zhabotinskii reaction”  In _Philosophical Transactions of the Royal Society of London. Series A: Physical and Engineering Sciences_ 345.1675 The Royal Society London, 1993, pp. 229–258
  * [55] J.. León and D. Nualart  “Stochastic evolution equations with random generators”  In _Annals of probability_ JSTOR, 1998, pp. 149–186
  * [56] C.. Levermore and J.. Xin  “Multidimensional stability of traveling waves in a bistable reaction–diffusion equation, II”  In _Communications in partial differential equations_ 17.11-12 Taylor & Francis, 1992, pp. 1901–1924
  * [57] J.-G. Liu and J. Wang  “The best constant for $L^{\infty}$-type Gagliardo-Nirenberg inequalities”  In _Quarterly of Applied Mathematics_ , 2023
  * [58] W. Liu and M. Röckner  “SPDE in Hilbert space with locally monotone coefficients”  In _Journal of Functional Analysis_ 259.11 Elsevier, 2010, pp. 2902–2922
  * [59] W. Liu and M. Röckner  “Stochastic partial differential equations: an introduction”  Springer, 2015
  * [60] G.. Lord and V. Thümmler  “Computing stochastic traveling waves”  In _SIAM Journal on Scientific Computing_ 34.1 SIAM, 2012, pp. B24–B43
  * [61] A. Lunardi  “Linear and nonlinear diffusion problems”  In _Lecture Notes_ , 2004
  * [62] A. Lunardi  “Interpolation theory”  Springer, 2018
  * [63] J. MacLaurin  “Phase reduction of waves, patterns, and oscillations subject to spatially extended noise”  In _SIAM Journal on Applied Mathematics_ 83.3 SIAM, 2023, pp. 1215–1244
  * [64] J.. Merkin  “Travelling waves in the Oregonator model for the BZ reaction”  In _IMA journal of applied mathematics_ 74.4 OUP, 2009, pp. 622–643
  * [65] P.. Meyer and G. Denzel  “Probability and Potentials”  Blaisdell Publishing Company, 1966
  * [66] A.. Mikhailov, L. Schimansky-Geier and W. Ebeling  “Stochastic motion of the propagating front in bistable media”  In _Physics Letters A_ 96.9 Elsevier, 1983, pp. 453–456
  * [67] B. Sz Nagy  “Über Integralungleichungen zwischen einer Funktion und ihrer Ableitung”  In _Acta Univ. Szeged. Sect. Sci. Math_ 10, 1941, pp. 64–74
  * [68] T. Nau  “Vector-valued Fourier transform and Fourier series”  In _Lp-Theory of Cylindrical Boundary Value Problems: An Operator-Valued Fourier Multiplier and Functional Calculus Approach_ Springer, 2012, pp. 9–23
  * [69] J… Neerven and M.. Veraar  “Maximal estimates for stochastic convolutions in 2-smooth Banach spaces and applications to stochastic evolution equations”  In _arXiv: Probability_ , 2020
  * [70] J…. Neerven and M.. Veraar  “Maximal inequalities for stochastic convolutions and pathwise uniform convergence of time discretisation schemes”  In _Stochastics and Partial Differential Equations: Analysis and Computations_ Springer, 2021, pp. 1–66
  * [71] J…. Neerven, M.. Veraar and L. Weis  “Stochastic integration in UMD Banach spaces”, 2007
  * [72] S. Okada  “The Fourier transform of vector-valued functions”  In _Glasgow mathematical journal_ 26.2 Cambridge University Press, 1985, pp. 181–186
  * [73] C. Prévôt and M. Röckner  “A concise course on stochastic partial differential equations”  Springer, 2007
  * [74] M. Pronk and M.. Veraar  “A new approach to stochastic evolution equations with adapted drift”  In _Journal of Differential Equations_ 256.11, 2014, pp. 3634–3683
  * [75] M. Pronk and M.. Veraar  “Forward integration, convergence and non-adapted pointwise multipliers”  In _Infinite Dimensional Analysis, Quantum Probability and Related Topics_ 18.01 World Scientific, 2015, pp. 1550005
  * [76] P.. Protter  “Stochastic Integration and Differential Equations”, Stochastic Modelling and Applied Probability  Springer-Verlag Berlin Heidelberg, 2005
  * [77] M. Röckner, S. Shang and T. Zhang  “Well-posedness of stochastic partial differential equations with fully local monotone coefficients”  In _arXiv preprint arXiv:2206.01107_ , 2022
  * [78] W. Rudin  “Fourier analysis on groups”  Courier Dover Publications, 2017
  * [79] F. Russo and P. Vallois  “Intégrales progressive, rétrograde et symétrique de processus non adaptés”  In _Comptes rendus de l’Académie des sciences. Série 1, Mathématique_ 312.8, 1991, pp. 615–618
  * [80] F. Russo and P. Vallois  “Forward, backward and symmetric stochastic integration”  In _Probability theory and related fields_ 97 Springer, 1993, pp. 403–421
  * [81] F. Russo and P. Vallois  “The generalized covariation process and Itô formula”  In _Stochastic Processes and their applications_ 59.1 Elsevier, 1995, pp. 81–104
  * [82] B. Sandstede  “Stability of travelling waves”  In _Handbook of dynamical systems_ 2 Elsevier, 2002, pp. 983–1055
  * [83] M. Scalamandre  “Harmonic Analysis on lca Groups”, 2017
  * [84] I. Sendiña-Nadal et al.  “Wave propagation in a medium with disordered excitability”  In _Physical review letters_ 80.24 APS, 1998, pp. 5437
  * [85] A.. Skorokhod  “On a generalization of a stochastic integral”  In _Theory of Probability & Its Applications_ 20.2 SIAM, 1976, pp. 219–233
  * [86] R. Sowers  “Large deviations for the invariant measure of a reaction-diffusion equation with non-Gaussian perturbations”  In _Probability Theory and Related Fields_ 92 Springer, 1992, pp. 393–421
  * [87] W. Stannat  “Stability of travelling waves in stochastic Nagumo equations”  In _arXiv preprint arXiv:1301.6378_ , 2013
  * [88] W. Stannat  “Stability of travelling waves in stochastic bistable reaction-diffusion equations”  In _arXiv preprint arXiv:1404.3853_ , 2014
  * [89] O. Steinbock, V.. Zykov and S.. Müller  “Wave propagation in an excitable medium along a line of a velocity jump”  In _Physical Review E_ 48.5 APS, 1993, pp. 3295
  * [90] A. Świech  “A PDE approach to large deviations in Hilbert spaces”  In _Stochastic processes and their applications_ 119.4 Elsevier, 2009, pp. 1081–1123
  * [91] M. Talagrand  “The generic chaining: upper and lower bounds of stochastic processes”  Springer Science & Business Media, 2005
  * [92] G. Tessitore and J. Zabczyk  “Wong-Zakai approximations of stochastic evolution equations”  In _Journal of Evolution Equations_ 6.4 Birkhaüser, 2006, pp. 621–655
  * [93] H. Triebel  “Interpolation Theory, Function Spaces, Differential Operators”, North-Holland Mathematical Library 18  Elsevier Science, 1978
  * [94] K. Twardowska and A. Nowak  “On the relation between the Itô and Stratonovich integrals in Hilbert spaces”  In _Annales Mathematicae Silesianae_ 18, 2004, pp. 49–63
  * [95] N.. Van Kampen  “Itô versus stratonovich”  In _Journal of Statistical Physics_ 24 Springer, 1981, pp. 175–187
  * [96] S… Varadhan  “Large deviations”  In _The Annals of Probability_ 36.2 Institute of Mathematical Statistics, 2008, pp. 397–419
  * [97] V.. Vavilin, A.M. Zhabotinsky and A.N. Zaikin  “Effect of ultraviolet radiation on the oscillating oxidation reaction of malonic acid derivatives”  In _Russ. J. Phys. Chem_ 42, 1968, pp. 1649
  * [98] M.. Veraar  “Non-autonomous stochastic evolution equations and applications to stochastic partial differential equations”  In _Journal of Evolution Equations_ 10.1 Springer, 2010, pp. 85–127
  * [99] M.. Veraar and L. Weis  “A note on maximal estimates for stochastic convolutions”  In _Czechoslovak mathematical journal_ 61.3 Springer, 2011, pp. 743–758
  * [100] J.. Walsh  “An introduction to stochastic partial differential equations”  In _École d’Été de Probabilités de Saint Flour XIV - 1984_ Springer Berlin Heidelberg, 1986, pp. 265–439
  * [101] L. Weis  “The $H^{\infty}$ holomorphic functional calculus for sectorial operators—a survey”  In _Partial Differential Equations and Functional Analysis: The Philippe Clément Festschrift_ Springer, 2006, pp. 263–294


ukdate []

# Coherence and entanglement in Grover and Harrow-Hassidim-Lloyd algorithm

Changchun Feng  LMIB(Beihang University), Ministry of Education  School of
Mathematical Sciences, Beihang University, Beijing 100191, China  Lin Chen
<EMAIL_ADDRESS>(corresponding author) LMIB(Beihang University), Ministry
of Education  School of Mathematical Sciences, Beihang University, Beijing
100191, China  International Research Institute for Multidisciplinary Science,
Beihang University, Beijing 100191, China  Li-Jun Zhao<EMAIL_ADDRESS>(corresponding author) LMIB(Beihang University), Ministry of Education  School
of Mathematical Sciences, Beihang University, Beijing 100191, China

()

###### Abstract

Coherence, discord and geometric measure (GM) of entanglement are important
tools for measuring physical resources. We compute them at every steps of the
Grover’s algorithm. We summarize these resources’s patterns of change. These
resources are getting smaller at the step oracle and are getting bigger or
invariant at the step diffuser. This result is similar to the entanglement’s
pattern of change in Grover’s algorithm. Furthermore, we compute GM at every
steps of the Harrow-Hassidim-Lloyd algorithm.

###### pacs:

03.65.Ud, 03.67.Mn

##  I Introduction

Quantum entanglement plays an important role as a physical resource in quantum
information processing 1935ES ; 2000MA ; 2007Horo . It is widely used in
various quantum information processing tasks such as quantum computing
2005Experimental , teleportation 2004Deterministic , dense coding 2002Quantum
, cryptography 2020Entanglement and quantum key distribution Xu2020 . Quantum
coherence constitutes a powerful resource for quantum metrology 2004VG ;
2014RD and entanglement creation 2005JK ; arXivAS and is at the root of a
number of intriguing phenomena of wide-ranging impact in quantum optics 1963RJ
; 1991MO ; 1994AA ; 1995DF , quantum information 2000MA , solid state physics
2012CM , and thermodynamics 1978LH ; 2014LA . The representatives of the
quantum algorithm are Shor’ factoring 1994PW and Grover’s search 1996LK
algorithms. A few years ago another algorithm called Harrow Hassidim-Lloyd
(HHL) algorithm was developed. It can compute the inverse of sparse matrix.
The HHL algorithm is known to be optimal in the matrix inversion task. Grover
algorithm is an unstructured search algorithm running on a quantum computer,
and is one of the typical algorithms of quantum computing.

Quantum entanglement is investigated in Grover alogorithm or HHL algorithm
2022MR . In this paper we examine a question: ‘how the coherence, discord and
GM change in Grover alogorithm or HHL algorithm?’. In order to explore this
issue we firstly concentrate on the Grover algorithm. We compute the coherence
in Subsec. III.1. We compute discord in every steps in Lemmas 4, 5, 6 and 7,
respectively. We compute GM in every steps in Lemmas 8, 9, 10 and 11,
respectively. Then we show the tables of coherence, discord and GM in Tables
1, 2 and 3, respectively. We obtain that the variation trends of these
physical quantities are getting smaller in the step oracle and getting bigger
or invariant in the step diffuser. Furthermore, we concentrate on the HHL
algorithm. We compute the GM in every steps of the HHL algorithm in Lemmas 12,
13 and 14, respectively.

In addition, some geometrically motivated entanglement measures have been
providing us with new insights into quantum entanglement, e.g. entanglement of
formation PhysRevA.54.3824 , relative entropy of entanglement
PhysRevLett.78.2275 ; PhysRevA.57.1619 , global robustness PhysRevA.68.012308
; PhysRevA.59.141 and squashed entanglement 2004CM . Besides providing a
simple geometric picture, they are closely related to some operationally
motivated entanglement measures, e.g. entanglement of distillation
PhysRevA.54.3824 and entanglement cost PatrickMHayden_2001 . In the future, we
will investigate how these entanglement measures change in quantum algorithms.

The rest of this paper is organized as follows. In Sec. II we introduce the
preliminary facts, such as the definitions about coherence, discord, geometric
measure of entanglement and lemmas about geometric measure. In Sec. III we
investigate the coherence, discord, geometric measure of the quantum states in
the steps of Grover’s Algorithm respectively. In Sec. IV we investigate the
geometric measure of the quantum states in three steps of HHL Algorithm
respectively. Finally we conclude in Sec. V.

##  II Preliminaries

Quantum coherence is conventionally associated with the capability of a
quantum state to exhibit quantum interference phenomena 1995DF . Frozen
coherence is the distance between the quantum state $\rho$ and the incoherent
state $\delta_{\rho}$, which is the closest irrelevant state of $\rho$. Then
we have

$\displaystyle\mathcal{C}(\rho)=\mathcal{D}(\rho,\delta_{\rho})=\mathop{\rm
min}_{\rho^{\prime}}\mathcal{D}(\rho,\rho^{\prime}).$ (1)

Then we introduce the distance norm. Firstly we introduce the definition of
the Frobenius norm BOTTCHER20081864 ,

$\displaystyle||A||_{F}=\sqrt{\sum_{i=1}^{m}\sum_{j=1}^{n}|a_{i,j}|^{2}}=\sqrt{\mathop{\rm
Tr}(A^{\dagger}A)}.$ (2)

We use the Frobenius norm as the distance norm as follows,

$\displaystyle\mathcal{D}(A,B)=||A-B||_{F}=\sqrt{\mathop{\rm
Tr}((A-B)^{\dagger}(A-B))},$ (3)

where $A$ and $B$ are two matrices. Frobenius norm is used in the detection of
glottal closure instants 1994MA .

Classical quantum discord revolves around information theory Bera_2018 . If we
measure the lack of information by entropy, this definition of correlations is
captured by the mutual information

$\displaystyle I(A:B)=S(A)+S(B)-S(AB),$ (4)

where $S(X)$ is the von Neumann entropy $S(X)=-\mathop{\rm
Tr}(\rho_{X}\log\rho_{X})$ and $\rho_{X}$ is a quantum state of system $X$.
For classical variables, Bayes’ rule defines a conditional probability as
$p_{x|y}=p_{xy}/p_{y}$. This implies an equivalent form for the classical
mutual information

$\displaystyle J_{cl}(B|A)=S(B)-S(B|A),$ (5)

where the conditional entropy $S(B|A)=\sum_{a}p_{a}S(B|a)$ is the average of
entropies $S(B|a)=-\sum_{b}p_{b|a}\log p_{b|a}$. The notion of classicality
related to quantum discord revolves around information theory 2000Zurek ;
2001Hend ; oz01 .

We introduce positive-operator-valued measure (POVM) on subsystem A. The
measurement is described by a POVM with elements $E_{a}=M_{a}^{\dagger}M_{a}$,
where $M_{a}$ is the measurement operator and $a$ is the classical outcome.
Moreover we have $\sum_{a}E_{a}=I$. The initial state $\rho_{AB}$ is
transformed under the measurement to

$\displaystyle\rho_{AB}\rightarrow\rho_{AB}^{\prime}=\sum_{a}(M_{a}\otimes
I_{B})\rho_{AB}(M_{a}\otimes I_{B})^{\dagger},$ (6)

where party $A$ observes outcome a with probability

$\displaystyle p_{a}=\mathop{\rm Tr}((E_{a}\otimes I)\rho_{AB}),$ (7)

and B has the conditional state

$\displaystyle\rho_{B|a}=\mathop{\rm Tr}_{A}((E_{a}\otimes
I)\rho_{AB})/p_{a}.$ (8)

Then we define the conditional entropy
$S(B|{E_{a}})=\sum_{a}p_{a}S(\rho_{B|a})$. Using Equations (7),(8) we have

$\displaystyle J(B|{E_{a}})$
$\displaystyle=S(B)-S(B|{E_{a}})=S(B)-\sum_{a}p_{a}S(\rho_{B|a})$ (9)
$\displaystyle=S(B)-\sum_{a}\mathop{\rm Tr}(E_{a}\rho_{AB})S({\mathop{\rm
Tr}}_{A}(E_{a}\rho_{AB})/\mathop{\rm Tr}(E_{a}\rho_{AB})).$

We will quantify the classical correlations of the state. Then independently
of a measurement $J(B|{E_{a}})$ is maximized over all measurements,

$\displaystyle J(B|A)=\mathop{\rm max}_{E_{a}}J(B|{E_{a}}).$ (10)

The quantum discord of the state $\rho_{AB}$ under the measurement
$\\{E_{a}\\}$ is defined as a difference between total correlations.

$\displaystyle D(B|A)=I(A:B)-J(B|A)=\mathop{\rm
min}_{\\{E_{a}\\}}\sum_{a}p_{a}S(\rho_{B|a})+S(A)-S(AB).$ (11)

GM is closely related to the construction of optimal entanglement witnesses
PhysRevA.68.042307 and discrimination of quantum states under LOCC
PhysRevLett.96.040501 ; PhysRevA.77.012104 ; Markham_2007 . In condensed
matter physics, GM is useful for studying quantum many-body systems, such as
characterizing ground state properties and detecting phase transitions odv2008
; 2008Orus2 . We give the definition as follows 2010Additivity .

###### Definition 1

Suppose $\rho$ is an $N$-qubit state. GM measures the closest distance in
terms of overlap between the state $\rho$ and the set of separable states, or,
equivalently, the set of pure product states. Formally, GM is defined as

$\displaystyle\Lambda^{2}(\rho):=\mathop{\rm max}_{\sigma\in SEP}\mathop{\rm
Tr}(\rho\sigma)=\mathop{\rm max}_{|\phi\rangle\in
PRO}\langle\phi|\rho|\phi\rangle,$ (12) $\displaystyle
G(\rho)=-2\log\Lambda(\rho).$ (13)

Here, PRO denotes the set of fully pure product states in the Hilbert space
$\otimes^{N}_{j=1}\mathcal{H}_{j}$. Any pure product state maximizing (12) is
the closest product state of $\rho$.

Then we introduce two lemmas about GM2010Additivity .

###### Lemma 2

In GM, the closest product state to any $N$-partite pure or mixed symmetric
state with $N>3$ is necessarily symmetric.

A density matrix is called non-negative if all its entries in the
computational basis are non-negative.

###### Lemma 3

In GM, the closest product state to a non-negative state can be chosen to be
non-negative.

These two lemmas are useful for finding the closest product state and
computing GM in every steps of the Grover algorithm or HHL algorithm.

##  III Grover’s Algorithm

Let us consider a set $\mathcal{S}=\\{|j\rangle|j=0,1,...,N-1\\}$, $\langle
j_{1}|j_{2}\rangle=\delta_{j_{1}j_{2}}$. Grover’s algorithm 1996LK ; 1997LK
tries to find a particular quantum state $|\psi_{G}\rangle\in\mathcal{S}$.
Grover’s algorithm is made up of three unitary transformations called
superposition, oracle and diffuser, respectively. The superposition transforms
the initial state $|0\rangle^{\otimes n}$ to a superposed state, where all
states in $\mathcal{S}$ are superposed with equal probability amplitude. This
can be achieved by making use of the Hadamard gate,

$H=\frac{1}{\sqrt{2}}\begin{bmatrix}1&1\\\ 1&-1\end{bmatrix}.$ (14)

Therefore, after superposition transformation the initial state is changed
into

$|s\rangle=H^{\otimes n}|0\rangle^{\otimes
n}=\frac{1}{\sqrt{N}}\sum_{j=0}^{N-1}|j\rangle,$ (15)

where $N=2^{n}$. The oracle and diffuser are described by the unitary
operators $U_{oracle}=I-2|\psi_{G}\rangle\langle\psi_{G}|$ and
$U_{diffuser}=2|s\rangle\langle s|-I$. The oracle changes a sign of
$|\psi_{G}\rangle$ in $|s\rangle$. The diffuser increases the probability
amplitude of $|\psi_{G}\rangle$ from $U_{oracle}|s\rangle$.

Even though Grover’s algorithm is optimal as a quantum searching algorithm
1999CZ , such maximal creation and complete annihilation of entanglement do
not occur for large $N$. For example, let us consider the case $N=8$. If
$\psi_{G}=|7\rangle=|111\rangle$, Grover’s algorithm changes the quantum state
as

$\displaystyle|\psi_{1}\rangle=U_{oracle}|s\rangle=\frac{1}{2\sqrt{2}}(|0\rangle+|1\rangle+|2\rangle+|3\rangle+|4\rangle+|5\rangle+|6\rangle-|7\rangle),$
(16)
$\displaystyle|\psi_{2}\rangle=U_{diffuser}|\psi_{1}\rangle=\frac{1}{4\sqrt{2}}(|0\rangle+|1\rangle+|2\rangle+|3\rangle+|4\rangle+|5\rangle+|6\rangle+5|7\rangle),$
(17)
$\displaystyle|\psi_{3}\rangle=U_{oracle}|\psi_{2}\rangle=\frac{1}{4\sqrt{2}}(|0\rangle+|1\rangle+|2\rangle+|3\rangle+|4\rangle+|5\rangle+|6\rangle-5|7\rangle),$
(18)
$\displaystyle|\psi_{4}\rangle=U_{diffuser}|\psi_{3}\rangle=\frac{1}{-8\sqrt{2}}(|0\rangle+|1\rangle+|2\rangle+|3\rangle+|4\rangle+|5\rangle+|6\rangle-11|7\rangle).$
(19)

###  III.1 Coherence by the Frobenius norm

In this subsection, we will use the Frobenius norm to calculate the coherence.
Suppose $\delta_{\rho_{1}}$ is the closest irrelevant state of $\rho_{1}$.
Then we have $\delta_{\psi_{1}}=\mathop{\rm
diag}(x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},x_{7},x_{8})$ and
$\sum_{i=1}^{8}x_{i}=1$, $x_{i}\geq 0$.

$\displaystyle\mathcal{C}_{F}(\rho_{1})$
$\displaystyle=\mathcal{D}_{F}(\rho_{1},\delta_{\rho_{1}})=||\rho_{1}-\delta_{\rho_{1}}||_{F},$
(20)

where

$\displaystyle\rho_{1}-\delta_{\rho_{1}}$
$\displaystyle=\frac{1}{8}\begin{bmatrix}1-8x_{1}&1&1&1&1&1&1&-1\\\
1&1-8x_{2}&1&1&1&1&1&-1\\\ 1&1&1-8x_{3}&1&1&1&1&-1\\\
1&1&1&1-8x_{4}&1&1&1&-1\\\ 1&1&1&1&1-8x_{5}&1&1&-1\\\
1&1&1&1&1&1-8x_{6}&1&-1\\\ 1&1&1&1&1&1&1-8x_{7}&-1\\\
-1&-1&-1&-1&-1&-1&-1&1-8x_{8}\end{bmatrix}.$ (21)

Then we obtain that $\mathcal{C}_{F}(\rho_{1})=\mathop{\rm
min}_{x_{i}}\sqrt{\frac{1}{64}\sum_{i=1}^{8}(1-8x_{i})^{2}+\frac{7}{8}}$. We
have
$\sum_{i=1}^{8}(1-x_{i})^{2}\geq\frac{(8-8\sum_{i=1}^{8}x_{i})^{2}}{8}=0$. The
equation is equal when $1-8x_{1}=1-8x_{2}=\cdots=1-8x_{8}=0$. So we get that
$\mathcal{C}_{F}(\rho_{1})=\sqrt{\frac{7}{8}}=\frac{\sqrt{14}}{4}$.

Similarly, we have that

$\displaystyle\mathcal{C}_{F}(\rho_{2})=\mathcal{D}_{F}(\rho_{2},\delta_{\rho_{2}})=||\rho_{2}-\delta_{\rho_{2}}||_{F},$
(22)

where

$\displaystyle\rho_{2}-\delta_{\rho_{2}}$ (23) $\displaystyle=$
$\displaystyle\frac{1}{32}\begin{bmatrix}1-32x_{1}&1&1&1&1&1&1&5\\\
1&1-32x_{2}&1&1&1&1&1&5\\\ 1&1&1-32x_{3}&1&1&1&1&5\\\
1&1&1&1-32x_{4}&1&1&1&5\\\ 1&1&1&1&1-32x_{5}&1&1&5\\\
1&1&1&1&1&1-32x_{6}&1&5\\\ 1&1&1&1&1&1&1-32x_{7}&5\\\
5&5&5&5&5&5&5&25-32x_{8}\end{bmatrix}.$

$\mathcal{C}_{F}(\rho_{2})=\mathop{\rm
min}_{x_{i}}\sqrt{\frac{1}{1024}(\sum_{i=1}^{7}(1-32x_{i})^{2}+(25-32x_{8})^{2})+\frac{49}{128}}=\frac{7\sqrt{2}}{16}$.
When $x_{i}=\frac{1}{32}$ for $i\in[1,7]$, $x_{8}=\frac{25}{32}$, the
expression gets the minimum value.

For $\rho_{3}$, we have that

$\mathcal{C}_{F}(\rho_{3})=\mathcal{D}_{F}(\rho_{3},\delta_{\rho_{3}})=||\rho_{3}-\delta_{\rho_{3}}||_{F},$
(24)

where

$\displaystyle\rho_{3}-\delta_{\rho_{3}}$ (25) $\displaystyle=$
$\displaystyle\frac{1}{128}\begin{bmatrix}1-32x_{1}&1&1&1&1&1&1&-5\\\
1&1-32x_{2}&1&1&1&1&1&-5\\\ 1&1&1-32x_{3}&1&1&1&1&-5\\\
1&1&1&1-32x_{4}&1&1&1&-5\\\ 1&1&1&1&1-32x_{5}&1&1&-5\\\
1&1&1&1&1&1-32x_{6}&1&-5\\\ 1&1&1&1&1&1&1-32x_{7}&-5\\\
-5&-5&-5&-5&-5&-5&-5&25-32x_{8}\end{bmatrix}.$

$\mathcal{C}_{F}(\rho_{3})=\mathop{\rm
min}_{x_{i}}\sqrt{\frac{1}{1024}(\sum_{i=1}^{7}(1-32x_{i})^{2}+(25-32x_{8})^{2})+\frac{49}{128}}=\frac{7\sqrt{2}}{16}$.
When $x_{i}=\frac{1}{32}$ for $i\in[1,7]$, $x_{8}=\frac{25}{32}$, the
expression gets the minimum value.

For $\rho_{4}$, we have that

$\mathcal{C}_{F}(\rho_{4})=\mathcal{D}_{F}(\rho_{4},\delta_{\rho_{4}})=||\rho_{4}-\delta_{\rho_{4}}||_{F},$
(26)

where

$\displaystyle\rho_{4}-\delta_{\rho_{4}}$ (27) $\displaystyle=$
$\displaystyle\frac{1}{128}\begin{bmatrix}1-128x_{1}&1&1&1&1&1&1&-11\\\
1&1-128x_{2}&1&1&1&1&1&-11\\\ 1&1&1-128x_{3}&1&1&1&1&-11\\\
1&1&1&1-128x_{4}&1&1&1&-11\\\ 1&1&1&1&1-128x_{5}&1&1&-11\\\
1&1&1&1&1&1-128x_{6}&1&-11\\\ 1&1&1&1&1&1&1-128x_{7}&-11\\\
-11&-11&-11&-11&-11&-11&-11&121-128x_{8}\end{bmatrix}.$

$\mathcal{C}_{F}(\rho_{3})=\mathop{\rm
min}_{x_{i}}\sqrt{\frac{1}{128^{2}}(\sum_{i=1}^{7}(1-128x_{i})^{2}+(121-128x_{8})^{2})+\frac{1736}{128^{2}}}=\frac{\sqrt{434}}{64}$.
When $x_{i}=\frac{1}{128}$ for $i\in[1,7]$, $x_{8}=\frac{121}{128}$, the
expression gets the minimum value.

We make a table to show the change of the coherence in every step in Grover’s
algorithm.

Table 1: The coherence of every step in Grover’s algorithm Quantum state | Coherence |  Value (keep two demical places)
---|---|---
$|\psi_{1}\rangle$ | $\frac{\sqrt{14}}{4}$ | $0.95$
$|\psi_{2}\rangle$ | $\frac{7\sqrt{2}}{16}$ | $0.62$
$|\psi_{3}\rangle$ | $\frac{7\sqrt{2}}{16}$ | $0.62$
$|\psi_{4}\rangle$ | $\frac{\sqrt{434}}{64}$ | $0.33$

In this table, wo conclude that the coherence of state in Grover’s algorithm
is getting smaller in the step oracle.

We have known that in Grover’s algorithm, the three-tangle and concurrences of
the mixed states is also reduced in the step oracle.

###  III.2 Discord

In this subsection, we will investigate the quantum discord in the process of
Grover’s Algorithm. Firstly, we investigate the discord of the quantum state
$\rho_{1}=|\psi_{1}\rangle\langle\psi_{1}|$ in (16). We have the following
observation.

###### Lemma 4

Suppose
$|\psi_{1}\rangle=U_{oracle}|s\rangle=\frac{1}{2\sqrt{2}}(|0\rangle+|1\rangle+|2\rangle+|3\rangle+|4\rangle+|5\rangle+|6\rangle-|7\rangle)$.
For $\rho_{1}=|\psi_{1}\rangle\langle\psi_{1}|$, we denote $A_{1},B_{1}$ and
$C_{1}$ as three subsystems of $\rho_{1}$. Then the discord

$\displaystyle D(B_{1}C_{1}|A_{1})=D(A_{1}C_{1}|B_{1})=D(A_{1}B_{1}|C_{1})$
(28)
$\displaystyle=D(C_{1}|A_{1}B_{1})=D(B_{1}|A_{1}C_{1})=D(A_{1}|B_{1}C_{1})$
(29)
$\displaystyle=-\frac{1}{4}\log{\frac{1}{4}}-\frac{3}{4}\log{\frac{3}{4}}$
(30) $\displaystyle\approx 0.81.$ (31)

###### Proof.

For $\rho_{1}=|\psi_{1}\rangle\langle\psi_{1}|$. Then we get the following
expression by equation (11)

$\displaystyle D(B_{1}C_{1}|A_{1})=\mathop{\rm
min}_{\\{E_{a}\\}}\sum_{a}p_{a}S(B_{1}C_{1}|a)+S(A_{1})-S(\rho_{1}).$ (32)
$\displaystyle D(C_{1}|A_{1}B_{1})=\mathop{\rm
min}_{\\{E_{a}\\}}\sum_{a}p_{a}S(C_{1}|a)+S(A_{1}B_{1})-S(\rho_{1}).$ (33)

We obtain that $\rho_{1}=\frac{1}{8}\begin{bmatrix}1&1&1&1&1&1&1&-1\\\
1&1&1&1&1&1&1&-1\\\ 1&1&1&1&1&1&1&-1\\\ 1&1&1&1&1&1&1&-1\\\
1&1&1&1&1&1&1&-1\\\ 1&1&1&1&1&1&1&-1\\\ 1&1&1&1&1&1&1&-1\\\
-1&-1&-1&-1&-1&-1&-1&1\end{bmatrix}\\\ $,

$\rho_{A_{1}}=\rho_{B_{1}}=\rho_{C_{1}}=\frac{1}{8}(3(|0\rangle+|1\rangle)(\langle
0|+\langle 1|)+(|0\rangle-|1\rangle)(\langle 0|-\langle
1|))=\frac{1}{8}\begin{bmatrix}4&2\\\ 2&4\end{bmatrix}\\\ $.

$\rho_{B_{1}C_{1}}=\rho_{A_{1}B_{1}}=\rho_{A_{1}C_{1}}=\frac{1}{8}((|00\rangle+|01\rangle+|10\rangle-|11\rangle)(\langle
00|+\langle 01|+\langle 10|-\langle
11|)+(|00\rangle+|01\rangle+|10\rangle+|11\rangle)(\langle 00|+\langle
01|+\langle 10|+\langle 11|))=\frac{1}{8}\begin{bmatrix}2&2&2&0\\\ 2&2&2&0\\\
2&2&2&0\\\ 0&0&0&2\end{bmatrix}$.

We get that two eigenvalues of $\rho_{A_{1}}$ are $\frac{1}{4},\frac{3}{4}$.
Four eigenvalues of $\rho_{B_{1}C_{1}}$ are $\frac{1}{4},\frac{3}{4},0,0$. So
the von Neumann entropy
$S(\rho_{A_{1}})=-\frac{1}{4}\log{\frac{1}{4}}-\frac{3}{4}\log{\frac{3}{4}}=0.81$.
The von Neumann entropy
$S(\rho_{B_{1}C_{1}})=-\frac{1}{4}\log{\frac{1}{4}}-\frac{3}{4}\log{\frac{3}{4}}$.

For $\mathop{\rm min}_{\\{E_{a}\\}}\sum_{a}p_{a}S(B_{1}C_{1}|a)$ in (32), we
suppose $\\{E_{a}\\}$ is made of $E_{a_{1}}$, $E_{a_{2}}$ and $E_{a_{3}}$.
$E_{a_{1}}$, $E_{a_{2}}$ and $E_{a_{3}}$ have the following expression,

$E_{a_{1}}=\mathop{\rm diag}(1,1,1,1,0,0,0,0),$ (34) $E_{a_{2}}=\mathop{\rm
diag}(0,0,0,0,1,1,1,0),$ (35) $E_{a_{3}}=\mathop{\rm diag}(0,0,0,0,0,0,0,1).$
(36)

In this case, we can get the following result by (9).

$\displaystyle\sum_{a}p_{a}S(B_{1}C_{1}|a)=\sum_{a_{i}}\mathop{\rm
Tr}(E_{a_{i}}\rho_{1})S(\mathop{\rm Tr}_{A}(E_{a_{i}}\rho_{1})/\mathop{\rm
Tr}(E_{a_{i}}\rho_{1}))=0$ (37)

By (32) we get that

$\begin{split}D(B_{1}C_{1}|A_{1})&=\mathop{\rm
min}_{\\{E_{a}\\}}\sum_{a}p_{a}S(B_{1}C_{1}|a)+S(A_{1})-S(\rho_{1})\\\
&=S(A_{1})=-\frac{1}{4}\log{\frac{1}{4}}-\frac{3}{4}\log{\frac{3}{4}}\\\
&\approx 0.81.\end{split}$ (38)

For $\mathop{\rm min}_{\\{E_{a}\\}}\sum_{a}p_{a}S(C_{1}|a)$ in (32), we
suppose $\\{E_{a}\\}$ is made of $E_{a_{1}^{\prime}}$ and
$E_{a_{2}^{\prime}}$. $E_{a_{1}^{\prime}}$, $E_{a_{2}^{\prime}}$ and
$E_{a_{3}^{\prime}}$ have the following expression,

$E_{a_{1}^{\prime}}=\mathop{\rm diag}(1,1,1,1,1,1,0,0),$ (39)
$E_{a_{2}^{\prime}}=\mathop{\rm diag}(0,0,0,0,0,0,1,0),$ (40)
$E_{a_{3}^{\prime}}=\mathop{\rm diag}(0,0,0,0,0,0,0,1).$ (41)

In this case, we can get the following result by (9).

$\displaystyle\sum_{a}p_{a}S(C_{1}|a)=\sum_{a_{i}^{\prime}}\mathop{\rm
Tr}(E_{a_{i}^{\prime}}\rho_{1})S(\mathop{\rm
Tr}_{AB}(E_{a_{i}^{\prime}}\rho_{1})/\mathop{\rm
Tr}(E_{a_{i}^{\prime}}\rho_{1}))=0$ (42)

By (32) we get that

$\begin{split}D(C_{1}|A_{1}B_{1})&=\mathop{\rm
min}_{\\{E_{a}\\}}\sum_{a}p_{a}S(C_{1}|a)+S(A_{1}B_{1})-S(\rho_{1})\\\
&=S(A_{1}B_{1})=-\frac{1}{4}\log{\frac{1}{4}}-\frac{3}{4}\log{\frac{3}{4}}\\\
&\approx 0.81.\end{split}$ (43)

Then we calculate
$D(A_{1}C_{1}|B_{1})=D(A_{1}B_{1}|C_{1})=-\frac{1}{4}\log{\frac{1}{4}}-\frac{3}{4}\log{\frac{3}{4}}$
in the same way as calculating $D(A_{1}B_{1}|C_{1})$, and calculate
$D(B_{1}|A_{1}C_{1})=D(A_{1}|B_{1}C_{1})=-\frac{1}{4}\log{\frac{1}{4}}-\frac{3}{4}\log{\frac{3}{4}}$
in the same way as calculating $D(C_{1}|A_{1}B_{1})$.  $\sqcap$$\sqcup$

Then we consider the discords of three quantum states
$\rho_{2}=|\psi_{2}\rangle\langle\psi_{2}|$,
$\rho_{3}=|\psi_{3}\rangle\langle\psi_{3}|$ and
$\rho_{4}=|\psi_{4}\rangle\langle\psi_{4}|$ in the Grover’s algorithm’s
unitary transforms.

###### Lemma 5

Suppose
$|\psi_{2}\rangle=U_{diffuser}|\psi_{1}\rangle=\frac{1}{4\sqrt{2}}(|0\rangle+|1\rangle+|2\rangle+|3\rangle+|4\rangle+|5\rangle+|6\rangle+5|7\rangle)$.
For $\rho_{2}=|\psi_{2}\rangle\langle\psi_{2}|$, we denote $A_{2},B_{2}$ and
$C_{2}$ as three subsystems of $\rho_{2}$. Then the discord

$\displaystyle D(B_{2}C_{2}|A_{2})=D(A_{2}C_{2}|B_{2})=D(A_{2}B_{2}|C_{2})$
(44)
$\displaystyle=D(C_{2}|A_{2}B_{2})=D(B_{2}|A_{2}C_{2})=D(A_{2}|B_{2}C_{2})$
$\displaystyle=-\frac{1}{8}(4+\sqrt{13}))\log{\frac{1}{8}(4+\sqrt{13})}-\frac{1}{8}(4-\sqrt{13})\log{\frac{1}{8}(4-\sqrt{13})}$
$\displaystyle\approx 0.28.$

###### Lemma 6

Suppose
$|\psi_{3}\rangle=U_{oracle}|\psi_{2}\rangle=\frac{1}{4\sqrt{2}}(|0\rangle+|1\rangle+|2\rangle+|3\rangle+|4\rangle+|5\rangle+|6\rangle-5|7\rangle)$.
For $\rho_{3}=|\psi_{3}\rangle\langle\psi_{3}|$, we denote $A_{3},B_{3}$ and
$C_{3}$ as three subsystems of $\rho_{3}$. Then the discord

$\displaystyle D(B_{3}C_{3}|A_{3})=D(A_{3}C_{3}|B_{3})=D(A_{3}B_{3}|C_{3})$
(45)
$\displaystyle=D(C_{3}|A_{3}B_{3})=D(B_{3}|A_{3}C_{3})=D(A_{3}|B_{3}C_{3})$
$\displaystyle=-\frac{1}{16}(8+\sqrt{37}))\log{\frac{1}{16}(8+\sqrt{37})}-\frac{1}{16}(8-\sqrt{37})\log{\frac{1}{16}(8-\sqrt{37})}$
$\displaystyle\approx 0.52$

$D(B_{3}C_{3}|A_{3})=S(A_{3})=0.52,D(C_{3}|A_{3}B_{3})=S(A_{3}B_{3})\approx
0.52.$ (46)

###### Lemma 7

Suppose
$|\psi_{4}\rangle=U_{diffuser}|\psi_{3}\rangle=\frac{1}{-8\sqrt{2}}(|0\rangle+|1\rangle+|2\rangle+|3\rangle+|4\rangle+|5\rangle+|6\rangle-11|7\rangle)$.
For $\rho_{4}=|\psi_{4}\rangle\langle\psi_{4}|$, we denote $A_{4},B_{4}$ and
$C_{4}$ as three subsystems of $\rho_{4}$. Then the discord

$\displaystyle D(B_{4}C_{4}|A_{4})=D(A_{4}C_{4}|B_{4})=D(A_{1}B_{1}|C_{4})$
(47)
$\displaystyle=D(C_{4}|A_{4}B_{4})=D(B_{4}|A_{4}C_{4})=D(A_{4}|B_{4}C_{4})$
$\displaystyle=-\frac{1}{32}(16+\sqrt{229}))\log{\frac{1}{32}(16+\sqrt{229})}-\frac{1}{32}(16-\sqrt{229})\log{\frac{1}{32}(16-\sqrt{229})}$
$\displaystyle\approx 0.17$
$D(B_{4}C_{4}|A_{4})=S(A_{4})=0.17,D(C_{4}|A_{4}B_{4})=S(A_{4}B_{4})\approx
0.17.$ (48)

The proofs of these three lemmas are similar to proof of Lemma 4. Then we make
a table to show the change of the discord in every step in Grover’s algorithm.

Table 2: The discord of every step in Grover’s algorithm Quantum state | Discord |  Value (keep two demical places)
---|---|---
$|\psi_{1}\rangle$ | $-\frac{1}{4}\log{\frac{1}{4}}-\frac{3}{4}\log{\frac{3}{4}}$ | $0.81$
$|\psi_{2}\rangle$ | $-\frac{1}{8}(4+\sqrt{13}))\log{\frac{1}{8}(4+\sqrt{13})}-\frac{1}{8}(4-\sqrt{13})\log{\frac{1}{8}(4-\sqrt{13})}$ | $0.28$
$|\psi_{3}\rangle$ | $-\frac{1}{16}(8+\sqrt{37}))\log{\frac{1}{16}(8+\sqrt{37})}-\frac{1}{16}(8-\sqrt{37})\log{\frac{1}{16}(8-\sqrt{37})}$ | $0.52$
$|\psi_{4}\rangle$ | $-\frac{1}{32}(16+\sqrt{229}))\log{\frac{1}{32}(16+\sqrt{229})}-\frac{1}{32}(16-\sqrt{229})\log{\frac{1}{32}(16-\sqrt{229})}$ | $0.17$

In this table, we conclude that the discord of state in Grover’s algorithm is
getting smaller in the step oracle. This trend is similar to the the change of
the coherence in every step in Grover’s algorithm.

###  III.3 Geometric measure of entanglement

In this subsection, we will investigate GM in the process of Grover’s
Algorithm. Firstly, we investigate GM of the quantum state
$\rho_{1}=|\psi_{1}\rangle\langle\psi_{1}|$ in (16). We have the following
observation.

###### Lemma 8

Suppose
$|\psi_{1}\rangle=U_{oracle}|s\rangle=\frac{1}{2\sqrt{2}}(|0\rangle+|1\rangle+|2\rangle+|3\rangle+|4\rangle+|5\rangle+|6\rangle-|7\rangle)$.
For $\rho_{1}=|\psi_{1}\rangle\langle\psi_{1}|$, we get that
$G(\rho_{1})\approx 0.56$.

###### Proof.

We recall the definition of Geometric measure of entanglement in Definition 1.
Then we have

$\displaystyle\Lambda^{2}(\rho_{1}):=\mathop{\rm max}_{\sigma\in
SEP}\mathop{\rm Tr}(\rho_{1}\sigma_{1})=\mathop{\rm max}_{|\phi_{1}\rangle\in
PRO}\langle\phi_{1}|\rho_{1}|\phi_{1}\rangle.$ (49)

We have known that $|\phi_{1}\rangle$ is fully pure product states in the
Hilbert space. By Lemma 2, we get that $|\phi_{1}\rangle$ is a symmetric
state. Then we suppose
$|\phi_{1}\rangle=(\cos\alpha|0\rangle+e^{i\beta}\sin\alpha|1\rangle)^{\otimes
3}$ is a closest product state. Then we have that

$\displaystyle|\phi_{1}\rangle=$
$\displaystyle\cos^{3}\alpha|000\rangle+\cos^{2}\alpha\sin\alpha
e^{i\beta}(|001\rangle+|010\rangle+|100\rangle)$ (50)
$\displaystyle+\cos\alpha\sin^{2}\alpha
e^{2i\beta}(|011\rangle+|101\rangle+|110\rangle)+\sin^{3}\alpha
e^{3i\beta}|111\rangle,$ $\displaystyle\Lambda^{2}(\rho_{1})$
$\displaystyle=\mathop{\rm max}_{|\phi_{1}\rangle\in
PRO}\langle\phi_{1}|\rho_{1}|\phi_{1}\rangle=\mathop{\rm
max}_{x_{i}}\frac{1}{8}|\sum_{i=1}^{7}x_{i}-x_{8}|^{2},$ (51)

where $x_{1}=\cos^{3}\alpha$, $x_{2}=x_{3}=x_{5}=\cos^{2}\alpha\sin\alpha
e^{i\beta}$, $x_{4}=x_{6}=x_{7}=\cos\alpha\sin^{2}\alpha e^{2i\beta}$, and
$x_{8}=\sin^{3}\alpha e^{3i\beta}$.

Furthermore, we have

$\displaystyle\Lambda^{2}(\rho_{1}):=\mathop{\rm max}_{|\phi_{1}\rangle\in
PRO}\langle\phi_{1}|\rho_{1}|\phi_{1}\rangle=\mathop{\rm
max}_{a,b}\frac{1}{8}|a+ib|^{2}=\mathop{\rm
max}_{a,b}\frac{1}{8}(a^{2}+b^{2}),$ (52)

where
$a=\cos^{3}\alpha+3\cos^{2}\alpha\sin\alpha\cos\beta+3\cos\alpha\sin^{2}\alpha\cos
2\beta-\sin^{3}\alpha\cos 3\beta$ and
$b=3\cos^{2}\alpha\sin\alpha\sin\beta+3\cos\alpha\sin^{2}\alpha\sin
2\beta-\sin^{3}\alpha\sin 3\beta$. Then we calculate the maximum value of the
function

$\displaystyle f_{1}=\frac{1}{8}(a^{2}+b^{2}).$ (53)

Using Mathematica we get the maximum value of $f_{1}$ is approximately $0.67$
, when $\alpha\approx 0.59$ and $\beta\approx 0$. So the GM of $\rho_{1}$,

$\displaystyle G(\rho_{1})=-2\log\Lambda(\rho_{1})\approx 0.56.$ (54)

$\sqcap$$\sqcup$

Then we investigate GM of three quantum states
$\rho_{2}=|\psi_{2}\rangle\langle\psi_{2}|$,
$\rho_{3}=|\psi_{3}\rangle\langle\psi_{3}|$ and
$\rho_{4}=|\psi_{4}\rangle\langle\psi_{4}|$ in the Grover’s algorithm. We have
the following observations.

###### Lemma 9

Suppose
$|\psi_{2}\rangle=U_{diffuser}|\psi_{1}\rangle=\frac{1}{4\sqrt{2}}(|0\rangle+|1\rangle+|2\rangle+|3\rangle+|4\rangle+|5\rangle+|6\rangle+5|7\rangle)$.
For $\rho_{2}=|\psi_{2}\rangle\langle\psi_{2}|$, we get that
$G(\rho_{2})\approx 0.11$.

###### Proof.

We recall the definition of Geometric measure of entanglement in Definition 1.
Then we have

$\displaystyle\Lambda^{2}(\rho_{2}):=\mathop{\rm max}_{\sigma\in
SEP}\mathop{\rm Tr}(\rho_{2}\sigma_{2})=\mathop{\rm max}_{|\phi_{2}\rangle\in
PRO}\langle\phi_{2}|\rho_{2}|\phi_{2}\rangle.$ (55)

We have known that $|\phi_{2}\rangle$ is fully pure product states in the
Hilbert space. Using Lemma 2 and Lemma 3, we get that $|\phi_{2}\rangle$ is a
symmetric and non-negative state. Then we suppose
$|\phi_{2}\rangle=(\cos\alpha|0\rangle+\sin\alpha|1\rangle)^{\otimes 3}$ is a
closest product state, where $0\leq\alpha\leq\frac{\pi}{2}$.

$\displaystyle\Lambda^{2}(\rho_{2})$
$\displaystyle=\langle\phi_{2}|\rho_{2}|\phi_{2}\rangle=\frac{1}{32}(\sum_{i=1}^{7}x_{i}+5x_{8})^{2},$
(56)

where $x_{1}=\cos^{3}\alpha$, $x_{2}=x_{3}=x_{5}=\cos^{2}\alpha\sin\alpha$,
$x_{4}=x_{6}=x_{7}=\cos\alpha\sin^{2}\alpha$, and $x_{8}=\sin^{3}\alpha$.

Using Mathematica we get the maximum of $\Lambda^{2}(\rho_{2})$ is $0.92$ when
$\alpha\approx 1.28$. So the GM of $\rho_{2}$,

$\displaystyle G(\rho_{2})=-2\log\Lambda(\rho_{2})\approx 0.11.$ (57)

$\sqcap$$\sqcup$

###### Lemma 10

Suppose
$|\psi_{3}\rangle=U_{oracle}|\psi_{2}\rangle=\frac{1}{4\sqrt{2}}(|0\rangle+|1\rangle+|2\rangle+|3\rangle+|4\rangle+|5\rangle+|6\rangle-5|7\rangle)$.
For $\rho_{3}=|\psi_{3}\rangle\langle\psi_{3}|$, we get that
$G(\rho_{3})\approx 0.24$.

###### Proof.

We recall the definition of Geometric measure of entanglement in Definition 1.
Then we have

$\displaystyle\Lambda^{2}(\rho_{3}):=\mathop{\rm max}_{\sigma\in
SEP}\mathop{\rm Tr}(\rho_{3}\sigma_{3})=\mathop{\rm max}_{|\phi_{3}\rangle\in
PRO}\langle\phi_{3}|\rho_{3}|\phi_{3}\rangle.$ (58)

We have known that $|\phi_{3}\rangle$ is fully pure product states in the
Hilbert space. By Lemma 2, we get that $|\phi_{1}\rangle$ is a symmetric
state. Then we suppose
$|\phi_{3}\rangle=(\cos\alpha|0\rangle+e^{i\beta}\sin\alpha|1\rangle)^{\otimes
3}$ is a closest product state.

$\displaystyle\Lambda^{2}(\rho_{3})$ $\displaystyle=\mathop{\rm
max}_{|\phi_{3}\rangle\in
PRO}\langle\phi_{3}|\rho_{3}|\phi_{3}\rangle=\mathop{\rm
max}_{x_{i}}\frac{1}{32}|\sum_{i=1}^{7}x_{i}-5x_{8}|^{2},$ (59)

where $x_{1}=\cos^{3}\alpha$, $x_{2}=x_{3}=x_{5}=\cos^{2}\alpha\sin\alpha
e^{i\beta}$, $x_{4}=x_{6}=x_{7}=\cos\alpha\sin^{2}\alpha e^{2i\beta}$, and
$x_{8}=\sin^{3}\alpha e^{3i\beta}$.

Furthermore, we have

$\displaystyle\Lambda^{2}(\rho_{3})=\mathop{\rm max}_{|\phi_{3}\rangle\in
PRO}\langle\phi_{3}|\rho_{3}|\phi_{3}\rangle=\mathop{\rm
max}_{a,b}\frac{1}{32}|a+ib|^{2}=\mathop{\rm
max}_{a,b}\frac{1}{32}(a^{2}+b^{2}),$ (60)

where
$a=\cos^{3}\alpha+3\cos^{2}\alpha\sin\alpha\cos\beta+3\cos\alpha\sin^{2}\alpha\cos
2\beta-5\sin^{3}\alpha\cos 3\beta$ and
$b=3\cos^{2}\alpha\sin\alpha\sin\beta+3\cos\alpha\sin^{2}\alpha\sin
2\beta-5\sin^{3}\alpha\sin 3\beta$. Then we calculate the maximum value of the
function

$\displaystyle f_{3}=\frac{1}{32}(a^{2}+b^{2}).$ (61)

Using Mathematica we get the maximum value of $f_{3}$ is approximately $0.85$
, when $\alpha\approx 1.43$ and $\beta=\pi$. So the GM of $\rho_{3}$,

$\displaystyle G(\rho_{3})=-2\log\Lambda(\rho_{3})\approx 0.24.$ (62)

$\sqcap$$\sqcup$

###### Lemma 11

Suppose
$|\psi_{4}\rangle=U_{diffuser}|\psi_{3}\rangle=\frac{1}{-8\sqrt{2}}(|0\rangle+|1\rangle+|2\rangle+|3\rangle+|4\rangle+|5\rangle+|6\rangle-11|7\rangle)$.
For $\rho_{4}=|\psi_{4}\rangle\langle\psi_{4}|$, we get that
$G(\rho_{4})=0.05$.

###### Proof.

We recall the definition of geometric measure of entanglement in Definition 1.
Then we have

$\displaystyle\Lambda^{2}(\rho_{4}):=\mathop{\rm max}_{\sigma\in
SEP}\mathop{\rm Tr}(\rho_{4}\sigma_{4})=\mathop{\rm max}_{|\phi_{4}\rangle\in
PRO}\langle\phi_{4}|\rho_{4}|\phi_{4}\rangle.$ (63)

We have known that $|\phi_{4}\rangle$ is fully pure product states in the
Hilbert space. By Lemma 2, we suppose
$|\phi_{4}\rangle=(\cos\alpha|0\rangle+e^{i\beta}\sin\alpha|1\rangle)^{\otimes
3}$ is a closest product state.

$\displaystyle\Lambda^{2}(\rho_{4})$ $\displaystyle=\mathop{\rm
max}_{|\phi_{4}\rangle\in
PRO}\langle\phi_{4}|\rho_{4}|\phi_{4}\rangle=\mathop{\rm
max}_{x_{i}}\frac{1}{128}|\sum_{i=1}^{7}x_{i}-11x_{8}|^{2},$ (64)

where $x_{1}=\cos^{3}\alpha$, $x_{2}=x_{3}=x_{5}=\cos^{2}\alpha\sin\alpha
e^{i\beta}$, $x_{4}=x_{6}=x_{7}=\cos\alpha\sin^{2}\alpha e^{2i\beta}$, and
$x_{8}=\sin^{3}\alpha e^{3i\beta}$. Furthermore, we have

$\displaystyle\begin{split}\Lambda^{2}(\rho_{4})&=\mathop{\rm
max}_{|\phi_{4}\rangle\in PRO}\langle\phi_{4}|\rho_{4}|\phi_{4}\rangle\\\
&=\mathop{\rm max}_{x_{i}}\frac{1}{128}(\sum_{i=1}^{7}x_{i}-11x_{8})^{2}\\\
&=\mathop{\rm max}_{a,b}\frac{1}{128}|a+ib|^{2}\\\ &=\mathop{\rm
max}_{a,b}\frac{1}{128}(a^{2}+b^{2}),\end{split}$ (65)

where
$a=\cos^{3}\alpha+3\cos^{2}\alpha\sin\alpha\cos\beta+3\cos\alpha\sin^{2}\alpha\cos
2\beta-11\sin^{3}\alpha\cos 3\beta$ and
$b=3\cos^{2}\alpha\sin\alpha\sin\beta+3\cos\alpha\sin^{2}\alpha\sin
2\beta-11\sin^{3}\alpha\sin 3\beta$. Then we calculate the maximum value of
the function

$\displaystyle f_{4}=\frac{1}{128}(a^{2}+b^{2}).$ (66)

Using Mathematica we get the maximum value of $f_{4}$ is approximately $0.96$
, when $\alpha\approx 1.64$ and $\beta\approx 0$. So the GM of $\rho_{4}$,

$\displaystyle G(\rho_{4})=-2\log\Lambda(\rho_{4})\approx 0.05.$ (67)

$\sqcap$$\sqcup$

Then we make a table to show the change of GM in every step in Grover’s
algorithm.

Table 3: The GM of state in every step in Grover’s algorithm Quantum state | GM
---|---
$|\psi_{1}\rangle$ | $0.56$
$|\psi_{2}\rangle$ | $0.11$
$|\psi_{3}\rangle$ | $0.24$
$|\psi_{4}\rangle$ | $0.05$

In this table, wo conclude that GM of state of the step oracle in Grover’s
algorithm is getting bigger and that of the step diffuser in Grover’s
algorithm is getting smaller. The trend of change of GM is similar to discord.

##  IV HHL Algorithm

The HHL algorithm 2009AW consists of three steps, which are quantum phase
estimation (QPE), R($\lambda^{-1}$)-rotation, and inverse QPE. These three
steps were experimentally and explicitly realized by selecting a linear
equation $Ax=b$, where

$\displaystyle A=\frac{1}{2}\begin{bmatrix}3&1\\\ 1&3\end{bmatrix},$ (68)
$\displaystyle b=\begin{bmatrix}b_{0}\\\ b_{1}\end{bmatrix},$ (69)

where $b_{0}^{2}+b_{1}^{2}=1$.

We discuss how efficiently the HHL algorithm utilizes the entanglement as we
discussed previously in the Grover’s algorithm. By this reason we will compute
the entanglement at each stage of the HHL algorithm. The first three-qubit
state after the QPE stage is simply

$\displaystyle|\psi_{1}\rangle=\frac{1}{2}[(b_{0}-b_{1})|01\rangle\otimes(|0\rangle-|1\rangle)+(b_{0}+b_{1})|10\rangle\otimes(|0\rangle+|1\rangle)].$
(70)

Then we have

$\displaystyle\rho_{1}=|\psi_{1}\rangle\\!\langle\psi_{1}|=\frac{1}{4}\begin{bmatrix}0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&0\\\
0&0&(b_{0}-b_{1})^{2}&-(b_{0}-b_{1})^{2}&b_{0}^{2}-b_{1}^{2}&b_{0}^{2}-b_{1}^{2}&0&0\\\
0&0&-(b_{0}-b_{1})^{2}&(b_{0}-b_{1})^{2}&b_{1}^{2}-b_{0}^{2}&b_{1}^{2}-b_{0}^{2}&0&0\\\
0&0&b_{0}^{2}-b_{1}^{2}&b_{1}^{2}-b_{0}^{2}&(b_{0}+b_{1})^{2}&(b_{0}+b_{1})^{2}&0&0\\\
0&0&b_{0}^{2}-b_{1}^{2}&b_{1}^{2}-b_{0}^{2}&(b_{0}+b_{1})^{2}&(b_{0}+b_{1})^{2}&0&0\\\
0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0\end{bmatrix}.$ (71)

The second three-qubit state’s spectral decomposition is

$\displaystyle\rho_{2}=p|\phi_{1}\rangle\\!\langle\phi_{1}|+(1-p)|\phi_{2}\rangle\\!\langle\phi_{2}|,$
(72)

where

$\displaystyle|\phi_{1}\rangle=\frac{x_{1}}{\sqrt{2}}(|010\rangle-|011\rangle)+\frac{x_{2}}{\sqrt{2}}(|100\rangle+|101\rangle),$
(73)
$\displaystyle|\phi_{2}\rangle=-\frac{x_{2}}{\sqrt{2}}(|010\rangle-|011\rangle)+\frac{x_{1}}{\sqrt{2}}(|100\rangle+|101\rangle),$
(74) $\displaystyle
p=\frac{1}{2}[1+\sqrt{1-4\beta_{1}^{2}\beta_{2}^{2}(1-\gamma^{2})}],\gamma=\sqrt{(1-C^{2})(1-\frac{C^{2}}{4})}+\frac{C^{2}}{2},$
(75) $\displaystyle C=(\sin\frac{\pi}{4}+2\sin\frac{\pi}{8})\approx 0.736,$
(76)
$\displaystyle\beta_{1}=\frac{1}{\sqrt{2}}(b_{0}-b_{1}),\beta_{2}=\frac{1}{\sqrt{2}}(b_{0}+b_{1}),$
(77)

and we have

$\displaystyle x_{1}=\frac{a_{1}}{\sqrt{a_{1}^{2}+a_{2}^{2}}},\qquad
x_{2}=\frac{a_{2}}{\sqrt{a_{1}^{2}+a_{2}^{2}}},$ (78)

with

$\displaystyle
a_{1}=\beta_{1}[1+\sqrt{1-4\beta_{1}^{2}\beta_{2}^{2}(1-\gamma^{2})}-2\beta_{2}^{2}(1-\gamma^{2})],$
(79) $\displaystyle
a_{2}=\beta_{2}\gamma[1+\sqrt{1-4\beta_{1}^{2}\beta_{2}^{2}(1-\gamma^{2})}].$
(80)

Then we have

$\displaystyle{\large\rho_{2}=\frac{1}{2}\begin{bmatrix}0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&0\\\
0&0&px_{1}^{2}+(1-p)x_{2}^{2}&-px_{1}^{2}-(1-p)x_{2}^{2}&(2p-1)x_{1}x_{2}&(2p-1)x_{1}x_{2}&0&0\\\
0&0&-px_{1}^{2}-(1-p)x_{2}^{2}&px_{1}^{2}+(1-p)x_{2}^{2}&(1-2p)x_{1}x_{2}&(1-2p)x_{1}x_{2}&0&0\\\
0&0&(2p-1)x_{1}x_{2}&(1-2p)x_{1}x_{2}&(1-p)x_{1}^{2}+px_{2}^{2}&(1-p)x_{1}^{2}+px_{2}^{2}&0&0\\\
0&0&(2p-1)x_{1}x_{2}&(1-2p)x_{1}x_{2}&(1-p)x_{1}^{2}+px_{2}^{2}&(1-p)x_{1}^{2}+px_{2}^{2}&0&0\\\
0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0\end{bmatrix}.}$ (81)

The third three-qubit state’s spectral decomposition is

$\displaystyle\rho_{3}=q|\varphi_{1}\rangle\\!\langle\varphi_{1}|+(1-q)|\varphi_{2}\rangle\\!\langle\varphi_{2}|,$
(82)

where

$\displaystyle q=\frac{1}{2}[1+\sqrt{1-4(AC_{2}-BC_{1})^{2}}],$ (83)

with

$\displaystyle
A=\frac{1}{2}[(b_{0}-b_{1})\sqrt{1-C^{2}}+(b_{0}+b_{1})\sqrt{1-\frac{C^{2}}{4}}],$
(84) $\displaystyle
B=\frac{1}{2}[-(b_{0}-b_{1})\sqrt{1-C^{2}}+(b_{0}+b_{1})\sqrt{1-\frac{C^{2}}{4}}],$
(85) $\displaystyle C_{1}=C\frac{3b_{0}-b_{1}}{4}\qquad
C_{2}=C\frac{-b_{0}+3b_{1}}{4}.$ (86)

One can show $A^{2}+B^{2}+C_{1}^{2}+C_{2}^{2}=b_{0}^{2}+b_{1}^{2}=1$
explicitly. $|\varphi_{1}\rangle$ and $|\varphi_{2}\rangle$ are

$\displaystyle|\varphi_{1}\rangle=|00\rangle\otimes(y_{1}|0\rangle+y_{2}|1\rangle),|\varphi_{2}\rangle=|00\rangle\otimes(-y_{2}|0\rangle+y_{1}|1\rangle),$
(87)

where

$\displaystyle y_{1}=\frac{f_{1}}{\sqrt{f_{1}^{2}+f_{2}^{2}}},$ (88)
$\displaystyle y_{2}=\frac{f_{2}}{\sqrt{f_{1}^{2}+f_{2}^{2}}},$ (89)

with

$\displaystyle
f_{1}=A^{2}-B^{2}+C_{1}^{2}+C_{2}^{2}+\sqrt{1-4(AC_{2}-BC_{1})^{2}},\qquad
f_{2}=2(AB+C_{1}C_{2}).$ (90)

Then we have

$\displaystyle\rho_{3}=\begin{bmatrix}qy_{1}^{2}+(1-q)y_{2}^{2}&(2q-1)y_{1}y_{2}&0&0&0&0&0&0\\\
(2q-1)y_{1}y_{2}&(1-q)y_{1}^{2}+qy_{2}^{2}&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&0\end{bmatrix}.$ (91)

###  IV.1 Geometric measure of three states in HHL algorithm

Then we investigate the GM of three states in HHL algorithm.

####  IV.1.1 First state in HHL algorithm

Firstly, we investigate the first state
$\rho_{1}=|\psi_{1}\rangle\\!\langle\psi_{1}|$ in HHL algorithm.

###### Lemma 12

Suppose
$|\psi_{1}\rangle=\frac{1}{2}[(b_{0}-b_{1})|01\rangle\otimes(|0\rangle-|1\rangle)+(b_{0}+b_{1})|10\rangle\otimes(|0\rangle+|1\rangle)]$.
For $\rho_{1}=|\psi_{1}\rangle\langle\psi_{1}|$, we have
$G(\rho_{1})=-2\log(\mathop{\rm
max}\\{\frac{|b_{0}-b_{1}|^{2}}{2},\frac{|b_{0}+b_{1}|^{2}}{2}\\})$.

###### Proof.

We know that $U^{\dagger}\rho_{1}U$ has the same GM with $\rho_{1}$, where $U$
is a local unitary matrix. Then we choose the $U$ such that

$\displaystyle|\psi_{1}^{\prime}\rangle=U|\psi_{1}\rangle=\frac{1}{\sqrt{2}}[(b_{0}-b_{1})|000\rangle+(b_{0}+b_{1})|111\rangle].$
(92)

Then we have

$\displaystyle\rho_{1}^{\prime}=|\psi_{1}^{\prime}\rangle\\!\langle\psi_{1}^{\prime}|=\frac{1}{2}\begin{bmatrix}(b_{0}-b_{1})^{2}&0&0&0&0&0&0&b_{0}^{2}-b_{1}^{2}\\\
0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0\\\
b_{0}^{2}-b_{1}^{2}&0&0&0&0&0&0&(b_{0}+b_{1})^{2}\end{bmatrix}.$ (93)

We recall geometric measure of entanglement in Definition 1. Then we have

$\displaystyle\Lambda^{2}(\rho_{1}^{\prime}):=\mathop{\rm max}_{\sigma\in
SEP}\mathop{\rm Tr}(\rho_{1}^{\prime}\sigma_{1})=\mathop{\rm
max}_{|\phi_{1}\rangle\in
PRO}\langle\phi_{1}|\rho_{1}^{\prime}|\phi_{1}\rangle=\mathop{\rm
max}\\{\frac{|b_{0}-b_{1}|^{2}}{2},\frac{|b_{0}+b_{1}|^{2}}{2}\\}.$ (94)

Then

$\displaystyle G(\rho_{1})=G(\rho_{1}^{\prime})=-2\log(\mathop{\rm
max}\\{\frac{|b_{0}-b_{1}|^{2}}{2},\frac{|b_{0}+b_{1}|^{2}}{2}\\}).$ (95)

$\sqcap$$\sqcup$

####  IV.1.2 Second state in HHL algorithm

Then we investigate the second state $\rho_{2}$ in HHL algorithm.

###### Lemma 13

We recall
$\rho_{2}=p|\phi_{1}\rangle\\!\langle\phi_{1}|+(1-p)|\phi_{2}\rangle\\!\langle\phi_{2}|$
in Eq. (72), we have $G(\rho_{2})=-2\log(\mathop{\rm
max}\\{f(0),f(x^{\prime})\\})$, where $x^{\prime}$ is the solution of

$\displaystyle f^{\prime}=$
$\displaystyle\cos^{5}\alpha(a\cos\alpha-6a\sin\alpha+3b(\sec^{2}\alpha-1)^{\frac{1}{2}}\sec^{2}\alpha\tan\alpha$
(96)
$\displaystyle-6b(\sec^{2}\alpha-1)^{\frac{3}{2}}\sin\alpha+6c(\sec^{2}\alpha-1)^{2}\sec^{2}\alpha\tan\alpha-6c(\sec^{2}\alpha-1)^{3}\sin\alpha).$

###### Proof.

For $|\phi_{1}\rangle$ and $|\phi_{2}\rangle$ in Eq. (73). Let unitary
matrices $U_{1}=I\otimes I\otimes V_{1}$ and $U_{2}=I\otimes I\otimes V_{2}$
such that

$\displaystyle|\phi_{1}^{\prime}\rangle=U_{1}|\phi_{1}\rangle=\frac{x_{1}}{\sqrt{2}}|000\rangle+\frac{x_{2}}{\sqrt{2}}|111\rangle,$
(97)
$\displaystyle|\phi_{2}^{\prime}\rangle=U_{2}|\phi_{2}\rangle=-\frac{x_{2}}{\sqrt{2}}|000\rangle+\frac{x_{1}}{\sqrt{2}}|111\rangle.$
(98)

Then we have

$\displaystyle\rho_{2}^{\prime}=\frac{1}{2}\begin{bmatrix}px_{1}^{2}+(1-p)x_{2}^{2}&0&0&0&0&0&0&(2p-1)x_{1}x_{2}\\\
0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0\\\
(2p-1)x_{1}x_{2}&0&0&0&0&0&0&(1-p)x_{1}^{2}+px_{2}^{2}\end{bmatrix}.$ (99)

We recall geometric measure of entanglement in Definition 1. Then we have

$\displaystyle\Lambda^{2}(\rho_{2})=\Lambda^{2}(\rho_{2}^{\prime}):=\mathop{\rm
max}_{\sigma\in SEP}\mathop{\rm Tr}(\rho_{2}^{\prime}\sigma_{2})=\mathop{\rm
max}_{|\phi_{2}\rangle\in
PRO}\langle\phi_{2}|\rho_{2}^{\prime}|\phi_{2}\rangle.$ (100)

We know that $|\psi_{2}\rangle$ is a fully pure product state in the Hilbert
space. By Lemma 2, we obtain that $|\psi_{2}\rangle$ is a symmetric state.
Suppose
$|\psi_{2}\rangle=(\cos\alpha|0\rangle+e^{i\beta}\sin\alpha|1\rangle)^{\otimes
3}$ is a closest product state. Then we have that

$\displaystyle|\psi_{2}\rangle=$
$\displaystyle\cos^{3}\alpha|000\rangle+\cos^{2}\alpha\sin\alpha
e^{i\beta}(|001\rangle+|010\rangle+|100\rangle)$ (101)
$\displaystyle+\cos\alpha\sin^{2}\alpha
e^{2i\beta}(|011\rangle+|101\rangle+|110\rangle)+\sin^{3}\alpha
e^{3i\beta}|111\rangle,$ $\displaystyle\Lambda^{2}(\rho_{2}^{\prime})=$
$\displaystyle\mathop{\rm max}_{|\psi_{2}\rangle\in
PRO}\langle\psi_{2}|\rho_{2}^{\prime}|\psi_{2}\rangle$ (102) $\displaystyle=$
$\displaystyle\mathop{\rm
max}\\{\frac{1}{2}((px_{1}^{2}+(1-p)x_{2}^{2})\cos^{6}\alpha+((1-p)x_{1}^{2}+px_{2}^{2})\sin^{6}\alpha$
$\displaystyle+(4p-2)x_{1}x_{2}\cos^{3}\alpha\sin^{3}\alpha(e^{3i\beta}+e^{-3i\beta}))\\}$
$\displaystyle\leq$ $\displaystyle\mathop{\rm
max}\\{\frac{1}{2}((px_{1}^{2}+(1-p)x_{2}^{2})\cos^{6}\alpha+((1-p)x_{1}^{2}+px_{2}^{2})\sin^{6}\alpha$
$\displaystyle+\lvert(8p-4)x_{1}x_{2}\cos^{3}\alpha\sin^{3}\alpha\rvert\\},$

Then we will transform this problem to find the maximum value of $f$, i.e.,

$\displaystyle
f=a\cos^{6}\alpha+b\cos^{3}\alpha\sin^{3}\alpha+c\sin^{6}\alpha,$ (103)

where

$\displaystyle a=\frac{1}{2}(px_{1}^{2}+(1-p)x_{2}^{2}),$ (104) $\displaystyle
b=\lvert(4p-2)x_{1}x_{2}\rvert,$ (105) $\displaystyle
c=\frac{1}{2}((1-p)x_{1}^{2}+px_{2}^{2}).$ (106)

Then we have

$\displaystyle
f=\cos^{6}\alpha(a+b(\sec^{2}\alpha-1)^{\frac{3}{2}}+c(\sec^{2}\alpha-1)^{3}).$
(107)

Then

$\displaystyle f^{\prime}=$
$\displaystyle\cos^{5}\alpha(a\cos\alpha-6a\sin\alpha+3b(\sec^{2}\alpha-1)^{\frac{1}{2}}\sec^{2}\alpha\tan\alpha$
(108)
$\displaystyle-6b(\sec^{2}\alpha-1)^{\frac{3}{2}}\sin\alpha+6c(\sec^{2}\alpha-1)^{2}\sec^{2}\alpha\tan\alpha-6c(\sec^{2}\alpha-1)^{3}\sin\alpha).$

Suppose $x^{\prime}$ is the solution of $f^{\prime}=0$. Then we have

$\displaystyle G(\rho_{2})=G(\rho_{2}^{\prime})=-2\log(\mathop{\rm
max}\\{f(0),f(x^{\prime})\\}).$ (109)

$\sqcap$$\sqcup$

####  IV.1.3 Third state in HHL algorithm

Then we investigate the third state $\rho_{3}$ in HHL algorithm.

###### Lemma 14

For
$\rho_{3}=q|\varphi_{1}\rangle\\!\langle\varphi_{1}|+(1-q)|\varphi_{2}\rangle\\!\langle\varphi_{2}|$
in Eq. (82), we obtain that $G(\rho_{3})=-2log(\mathop{\rm max}\\{q,1-q\\})$.

###### Proof.

We know that
$\rho_{3}=q|\varphi_{1}\rangle\\!\langle\varphi_{1}|+(1-q)|\varphi_{2}\rangle\\!\langle\varphi_{2}|$.
We recall geometric measure of entanglement in Definition 1. Then we have

$\displaystyle\Lambda^{2}(\rho_{3}):=\mathop{\rm max}_{\sigma\in
SEP}\mathop{\rm Tr}(\rho_{3}\sigma_{2})=\mathop{\rm max}\\{q,1-q\\}.$ (110)

Then

$\displaystyle G(\rho_{3})=-2\log(\mathop{\rm max}\\{q,1-q\\}).$ (111)

$\sqcap$$\sqcup$

##  V conclusion

We have investigated the coherence, discord and GM of quantum states in the
steps of Grover’s Algorithm respectively. Then we show the tables of
coherence, discord and GM in Tables 1, 2 and 3 respectively. We also conclude
that the variation trends of these physical quantitie are getting smaller in
the step oracle and are getting bigger or invariant at the step diffuser. Then
we investigate GM of quantum states in three steps of HHL Algorithm in Lemmas
12, 13 and 14, respectively.

These results help investigate the Grover’s Algorithm and HHL Algorithm. In
the future we plan to investigate more physical quantities about quantum
states in Grover’s Algorithm and HHL Algorithm.

## Acknowledgements

LC was supported by the NNSF of China (Grant No. 11871089). LJZ was supported
by the NNSF of China (Grant No. 12101031), and the Fundamental Research Funds
for the Central Universities (Grant Nos. KG12080401 and ZG216S1902).

## References

  * [1] E. Schrödinger.  Die gegenwärtige situation in der quantenmechanik.  Naturwissenschaften, 23(50):844–849, 1935.
  * [2] M. A. Nielsen and I. L. Chuang.  Quantum computation and quantum information.  Cambridge University Press.
  * [3] Ryszard Horodecki, Pawel Horodecki, Michal Horodecki, and Karol Horodecki.  Quantum entanglement.  Rev. Mod. Phys., 81:865–942, Jun 2009.
  * [4] P. Walther, K. J. Resch, T. Rudolph, E. Schenck, H. Weinfurter, V. Vedral, M. Aspelmeyer, and A. Zeilinger.  Experimental one-way quantum computing.  Nature, 434:169–176, 2005.
  * [5] M. Riebe, H Haffner, C. F. Roos, W Hansel, J. Benhelm, and Lancaster Gpt.  Deterministic quantum teleportation with atoms.  Nature, 429(6993):734–7, 2004.
  * [6] X. Li, Q. Pan, J. Jing, J. Zhang, and K. Peng.  Quantum dense coding exploiting a bright einstein-podolsky-rosen beam.  Physical Review Letters, 88(4):047904, 2002.
  * [7] J. Yin, Y. H. Li, S. K. Liao, M. Yang, and J. W. Pan.  Entanglement-based secure quantum cryptography over 1,120 kilometres.  Nature, 582(7813):1–5, 2020.
  * [8] Feihu Xu, Xiongfeng Ma, Qiang Zhang, Hoi-Kwong Lo, and Jian-Wei Pan.  Secure quantum key distribution with realistic devices.  Rev. Mod. Phys., 92:025002, May 2020.
  * [9] S. Lloyd V. Giovannetti and L. Maccone.  Quantum-enhanced measurements: Beating the standard quantum limit.  Science, 306:1330–1336.
  * [10] Rafal Demkowicz-Dobrzański and Lorenzo Maccone.  Using entanglement against noise in quantum metrology.  Phys. Rev. Lett., 113:250801, Dec 2014.
  * [11] János K. Asbóth, John Calsamiglia, and Helmut Ritsch.  Computable measure of nonclassicality for light.  Phys. Rev. Lett., 94:173602, May 2005.
  * [12] Alexander Streltsov, Uttam Singh, Himadri Shekhar Dhar, Manabendra Nath Bera, and Gerardo Adesso.  Measuring quantum coherence with entanglement.  Phys. Rev. Lett., 115:020403, Jul 2015.
  * [13] Roy J. Glauber.  Coherent and incoherent states of the radiation field.  Phys. Rev., 131(6):2766–2788.
  * [14] Marlan O. Scully.  Enhancement of the index of refraction via quantum coherence.  Phys. Rev. Lett., 67:1855–1858, Sep 1991.
  * [15] Andreas Albrecht.  Some remarks on quantum coherence.  Journal of Modern Optics, 41(12):2467–2482, 1994.
  * [16] D. F. Walls and G. J. Milburn.  Quantum optics.  Springer-Verlag.
  * [17] Y.-N. Chen G.-Y. Chen C.-M. Li, N. Lambert and F. Nori.  Witnessing quantum coherence: from solid-state to biological systems.  Scientific Reports, 2(1):885.
  * [18] Ford and H. L.  Quantum coherence effects and the second law of thermodynamics.  Proceedings of the Royal Society A Mathematical Physical & Engineering Sciences, 364(1717):227–236, 1978.
  * [19] L. A. Correa, J. P. Palao, D. Alonso, and G. Adesso.  Quantum-enhanced absorption refrigerators.  Scientific Reports, 4:3949, 2014.
  * [20] P. W. Shor.  Algorithms for quantum computation: Discrete logarithms and factoring.  Proc. 35th Annual Symposium on Foundations of Computer Science.
  * [21] L. K. Grover.  A fast quantum mechanical algorithm for database search.  Proc. 28th Annual ACM Symposium on the Theory of Computing.
  * [22] Eylee Jung Chang-Yong Woo Mi-Ra Hwang, MuSeong Kim1 and DaeKil Park.  Tripartite entanglement and matrix inversion quantum algorithm.
  * [23] Charles H. Bennett, David P. DiVincenzo, John A. Smolin, and William K. Wootters.  Mixed-state entanglement and quantum error correction.  Phys. Rev. A, 54:3824–3851, Nov 1996.
  * [24] V. Vedral, M. B. Plenio, M. A. Rippin, and P. L. Knight.  Quantifying entanglement.  Phys. Rev. Lett., 78:2275–2279, Mar 1997.
  * [25] V. Vedral and M. B. Plenio.  Entanglement measures and purification procedures.  Phys. Rev. A, 57:1619–1633, Mar 1998.
  * [26] Aram W. Harrow and Michael A. Nielsen.  Robustness of quantum gates in the presence of noise.  Phys. Rev. A, 68:012308, Jul 2003.
  * [27] Guifré Vidal and Rolf Tarrach.  Robustness of entanglement.  Phys. Rev. A, 59:141–155, Jan 1999.
  * [28] Matthias Christandl and Andreas Winter.  ”squashed entanglement”: An additive entanglement measure.  Journal of Mathematical Physics, 45(3):829, 2004.
  * [29] Patrick M Hayden, Michal Horodecki, and Barbara M Terhal.  The asymptotic entanglement cost of preparing a quantum state.  Journal of Physics A: Mathematical and General, 34(35):6891, aug 2001.
  * [30] Albrecht Böttcher and David Wenzel.  The frobenius norm and the commutator.  Linear Algebra and its Applications, 429(8):1864–1885, 2008.
  * [31] Changxue Ma, Y. Kamp, and L.F. Willems.  A frobenius norm approach to glottal closure detection from the speech signal.  IEEE Transactions on Speech and Audio Processing, 2(2):258–265, 1994.
  * [32] Anindita Bera, Tamoghna Das, Debasis Sadhukhan, Sudipto Singha Roy, Aditi Sen(De), and Ujjwal Sen.  Quantum discord and its allies: a review of recent progress.  Reports on Progress in Physics, 81(2):024001, dec 2017.
  * [33] W. H. Zurek.  Einselection and decoherence from an information theory perspective.  Annalen Der Physik, 9(11-12):855–864, 2010.
  * [34] L. Henderson and V. Vedral.  Classical, quantum and total correlations.  Journal of Physics A General Physics, 34(35):6899–6905, 2001.
  * [35] Harold Ollivier and Wojciech H. Zurek.  Quantum discord: A measure of the quantumness of correlations.  Phys. Rev. Lett., 88:017901, Dec 2001.
  * [36] Tzu-Chieh Wei and Paul M. Goldbart.  Geometric measure of entanglement and applications to bipartite and multipartite quantum states.  Phys. Rev. A, 68:042307, Oct 2003.
  * [37] M. Hayashi, D. Markham, M. Murao, M. Owari, and S. Virmani.  Bounds on multipartite entangled orthogonal state discrimination using local operations and classical communication.  Phys. Rev. Lett., 96:040501, Feb 2006.
  * [38] Masahito Hayashi, Damian Markham, Mio Murao, Masaki Owari, and Shashank Virmani.  Entanglement of multiparty-stabilizer, symmetric, and antisymmetric states.  Phys. Rev. A, 77:012104, Jan 2008.
  * [39] Damian Markham, Akimasa Miyake, and Shashank Virmani.  Entanglement and local information access for graph states.  New Journal of Physics, 9(6):194, jun 2007.
  * [40] R. Orús, S. Dusuel, and J. Vidal.  Equivalence of critical scaling laws for many-body entanglement in the lipkin-meshkov-glick model.  Phys. Rev. Lett., 101:025701, 2008.
  * [41] Román Orús.  Geometric entanglement in a one-dimensional valence-bond solid state.  Phys. Rev. A, 78:062332, Dec 2008.
  * [42] H. Zhu, L. Chen, and M. Hayashi.  Additivity and non-additivity of multipartite entanglement measures.  New Journal of Physics, 12(8):2099–2154, 2010.
  * [43] Lov K. Grover.  Quantum mechanics helps in searching for a needle in a haystack.  Phys. Rev. Lett., 79:325–328, Jul 1997.
  * [44] Christof Zalka.  Grover’s quantum searching algorithm is optimal.  Phys. Rev. A, 60:2746–2751, Oct 1999.
  * [45] Aram W. Harrow, Avinatan Hassidim, and Seth Lloyd.  Quantum algorithm for linear systems of equations.  Phys. Rev. Lett., 103:150502, Oct 2009.