# Control-enhanced multiparameter quantum estimation

Jing Liu  Department of Mechanical and Automation Engineering, The Chinese
University of Hong Kong, Shatin, Hong Kong  Haidong Yuan  Department of
Mechanical and Automation Engineering, The Chinese University of Hong Kong,
Shatin, Hong Kong

###### Abstract

Most studies in multiparameter estimation assume the dynamics is fixed and
focus on identifying the optimal probe state and the optimal measurements. In
practice, however, controls are usually available to alter the dynamics, which
provides another degree of freedom. In this paper we employ optimal control
methods, particularly the gradient ascent pulse engineering (GRAPE), to design
optimal controls for the improvement of the precision limit in multiparameter
estimation. We show that the controlled schemes not only capable to provide a
higher precision limit, but also have a higher stability to the inaccuracy of
the time point performing the measurements. This high time stability will
benefit the practical metrology where it is hard to perform the measurement at
a very accurate time point due to the response time of the measurement
apparatus.

##  I Introduction

Quantum metrology has seen a rapid development recently Giovannetti2011 ;
GIOV04 ; cavesprd ; rosetta ; Fujiwara2008 ; Paris09 ; Escher2011 ; Demkowicz1
; Tsang2013 ; Rafal2012 ; durkin2011 ; Alipour2014 ; Chaves2013 ; Correa2015 ;
Toth ; Berni ; Xing2016 ; Tan2013 ; Tsang2016 ; LiuJPA2016 ; LiuNJP2016 , due
to its wide applications in quantum imaging Shihreview ; Boydreview , quantum
sensing Degenreview  and quantum Hamiltonian identification Shabani2011 ;
Cole2006 ; Zhang2014 ; Wang2016 . In most of these applications there are
usually multiple unknown parameters, which falls in the subject of
multiparameter quantum estimation. Due to its generality, multiparameter
quantum estimation is gaining increasing attention. The recent discovery that
simultaneous estimation of multiple parameters can provide a higher precision
limit than estimating each parameter individually Humphreys2013 ; Dattareview
; Dattamegfield ; Liu2016 ; Dattagauss ; Yue2014 ; Zhang2017  further prompts
the development of this field Dattareview .

Most existing schemes in multiparameter quantum estimation assume the dynamics
is fixed, focus on the identification of the optimal probe states and optimal
measurements Humphreys2013 ; Dattareview ; Dattamegfield ; Liu2016 ;
Dattagauss ; Yue2014 ; Zhang2017 ; Yao2014 ; Yao2014a ; Cheng2014 ; Knott2016
; Fan2014 ; BerryPRX ; Ciampini2016 . In practice, however, additional
controls usually can be employed to alter the dynamics for further improvement
of the precision limit. Systematic methods to obtain optimal controls for the
improvement of the precision limit in multiparameter quantum estimation are
highly desired.

Controls have been recently employed to improve the precision limit for
single-parameter quantum estimation YuanPRL ; Liuarxiv ; Pang2017 . For
multiparameter quantum estimation, systematic methods to obtain the optimal
controls are still lacking, so far optimal controls have only been obtained
under unitary dynamics Yuan2016 . In this paper we employ the gradient ascent
pulse engineering (GRAPE) Khaneja05  to obtain optimal controls for
multiparameter quantum estimation under general Markovian dynamics, this
provides a systematic method on the design of optimally controlled schemes for
multiparameter quantum estimation.

GRAPE has been widely applied as the pulse-engineering technique in quantum
information, including the implementation of logic gates for spin systems
Rowland2012 , Bose-Einstein condensates Jager2014  and nitrogen-vacancy
centers Wrachtrup2014 ; Du2015 . GRAPE allows a high flexibility in the
designed pulse shapes and no pulse families need to be assumed in advance
Khaneja05 . However, if practical constraints do exist, they can also be
easily incorporated into GRAPE Dridi2015 . Thus, GRAPE provides a versatile
tool for designing controlled schemes for quantum parameter estimation
Liuarxiv .

Controls can not only improve the sensitivity, but also improve the stability,
which is another important factor to consider in practical quantum metrology
Degenreview ; Durkin2010 . In practice the measurement apparatus usually has a
response time, so the actual time point performing the measurements may be
different from the theoretical value. Thus, the stability of the precision
limit to the inaccuracy in the measurement time are important and needs to be
considered. We provide several examples to show the advantages of the
controlled schemes and demonstrate that controls obtained from GRAPE can
improve both the precision limit and the stability. Specifically we studied
the single- and two-spin systems with dephasing noises and demonstrated the
improvement of the sensitivity and the stability provided by the controls.

##  II multiparameter estimation

Typically to estimate multiple parameters encoded in some quantum state
$\rho_{\vec{x}}$, where $\vec{x}=(x_{1},x_{2},...,x_{d})$, one needs to
perform a set of positive-operator valued measurements (POVM) $\\{E(y)\\}$
($\sum_{y}E(y)=\openone$ with $\openone$ the identity) on the state, which
will give the result $y$ with probability
$p_{y|\vec{x}}=\mathrm{Tr}(\rho_{\vec{x}}E(y))$. From the measurement result
one can then construct an estimator
$\vec{\hat{x}}=(\hat{x}_{1},\hat{x}_{2},...,\hat{x}_{d})$. According to the
Cramér-Rao bound, the covariance of any unbiased estimator is bounded below by
the (classical) Fisher information matrix as Helstrom ; Holevo

$C\geq\mathcal{F}_{\mathrm{cl}}^{-1}.$ (1)

Here $C$ denotes the covariance matrix with the entries
$C_{\alpha\beta}:=\sum_{y}(\hat{x}_{\alpha}-x_{\alpha})(\hat{x}_{\beta}-x_{\beta})p_{y|\vec{x}}$,
$\alpha,\beta\in\\{1,2,...,d\\}$, and $\mathcal{F}_{\mathrm{cl}}$ denotes the
classical Fisher information matrix (CFIM) with the $\alpha\beta$th entry
given by

$\mathcal{F}_{\mathrm{cl},\alpha\beta}:=\sum_{y}\frac{(\partial_{x_{\alpha}}p_{y|\vec{x}})(\partial_{x_{\beta}}p_{y|\vec{x}})}{p_{y|\vec{x}}}.$
(2)

The covariance matrix can be further bounded by the quantum Fisher information
matrix

$C\geq\mathcal{F}_{\mathrm{cl}}^{-1}\geq\mathcal{F}_{\mathrm{q}}^{-1},$ (3)

where $\mathcal{F}_{\mathrm{q}}$ is the quantum Fisher information matrix
(QFIM) with the $\alpha\beta$th entry given by
$\mathcal{F}_{\mathrm{q},\alpha\beta}=\frac{1}{2}\mathrm{Tr}(\rho\\{L_{\alpha},L_{\beta}\\})$.
$L_{\alpha}$ is the symmetric logarithmic derivative for $x_{\alpha}$ which is
the solution to the equation $\partial_{x_{\alpha}}\rho=(\rho
L_{\alpha}+L_{\alpha}\rho)/2$. In multiparameter quantum estimation the bound
given by the QFIM is usually not saturable Dattareview ; Matsumoto2002 ;
Ragy2016 ; Pezze2017 , therefore we will focus on the classical Cramér-Rao
bound. In this paper we will take
$|\delta\vec{\hat{x}}|^{2}:=\sum^{d}_{i=1}\delta^{2}\hat{x}_{i}$ as the figure
of merit, from the Cramér-Rao bound we have

$|\delta\vec{\hat{x}}|^{2}\geq\mathrm{Tr}\mathcal{F}_{\mathrm{cl}}^{-1}.$ (4)

Thus the controls should be designed to minimize
$\mathrm{Tr}\mathcal{F}_{\mathrm{cl}}^{-1}$.

For the case with only two parameters, Eq. (4) reduces to
$|\delta\vec{\hat{x}}|^{2}\geq F^{-1}_{\mathrm{cl,e}}$ Liu2016 , where
$F_{\mathrm{cl,e}}=\det\mathcal{F}_{\mathrm{cl}}/\mathrm{Tr}\mathcal{F}_{\mathrm{cl}}$,
here $\det(\cdot)$ denotes the determinant. For the cases with more than two
parameters, it is in general difficult to obtain the analytical expression for
$\mathcal{F}^{-1}_{\mathrm{cl}}$ directly, which means this function is not
very friendly to a gradient-based algorithm in general as in each iteration
the gradient needs to be evaluated numerically which is very demanding
computationally. Usually the bound will be further relaxed to ease the
computation. Since $[\mathcal{F}^{-1}_{\mathrm{cl}}]_{\alpha\alpha}\geq
1/\mathcal{F}_{\mathrm{cl},\alpha\alpha}$, we then have

$\mathrm{Tr}\mathcal{F}^{-1}_{\mathrm{cl}}\geq\sum_{\alpha}\frac{1}{\mathcal{F}_{\mathrm{cl},\alpha\alpha}}.$
(5)

We will then also take the following quantity

$f_{0}(T)=\left(\sum_{\alpha}\frac{1}{\mathcal{F}_{\mathrm{cl},\alpha\alpha}(T)}\right)^{-1}$
(6)

as an objective function to be maximized. However, the obtained controls will
be eventually evaluated by their effects on the figure of merit
$\mathrm{Tr}\mathcal{F}^{-1}_{\mathrm{cl}}$, which characterizes the precision
limit of $\delta\vec{\hat{x}}$. In many cases this provides efficient
algorithms lead to near optimal controls.

##  III Algorithm

Figure 1: (Color online) The flow chart of GRAPE algorithm for multiparameter
estimation.

We consider the systems whose dynamics can be described by the master equation

$\partial_{t}{\rho}(t)=\mathcal{L}[\rho(t)],$ (7)

where $\mathcal{L}(\rho)=-i[H,\rho]+\Gamma(\rho)$, here $\Gamma(\rho)$
describes the effect of the noises and $H$ denotes the Hamiltonian. The
Hamiltonian can be decomposed as Khaneja05 ; Ernst1987

$H=H_{0}(\vec{x})+\sum_{k=1}^{p}V_{k}(t)H_{k},$ (8)

where $H_{0}(\vec{x})$ is the Hamiltonian for the free evolution which
contains the unknown parameters $\vec{x}$, $\sum_{k=1}^{p}V_{k}(t)H_{k}$ is
the control Hamiltonian with $V_{k}(t)$ denotes the amplitude of $k$th control
field.

The solution of Eq. (7) at the target time $T$ can be approximated as
$\rho(T)=\Pi_{i=1}^{m}\exp(\Delta t\mathcal{L}_{i})\rho(0)$, with
$\mathcal{L}_{i}$ the super-operator for the $i$th time step and $m=T/\Delta
t$ sufficiently large, here the multiplication in $\rho(T)$ is taken from
right to left in the increasing order of $i$.

We then employ gradient ascent pulse engineering (GRAPE) Khaneja05  to obtain
optimal controls that can maximize $f_{0}(T)$ ($F_{\mathrm{cl,e}}$ for two-
parameter estimation). The flow of the algorithm, shown in Fig. 1, is as
following:

  1. 1.

guess a set of initial values for $V_{k}(j)$ ($V_{k}(j)$ is the $k$th control
at the $j$th time step);

  2. 2.

evolve the dynamics with the controls;

  3. 3.

calculate the objective function $f_{0}(T)$ ($F_{\mathrm{cl,e}}$ for two-
parameter estimation);

  4. 4.

calculate the gradient $\frac{\delta f_{0}(T)}{\delta V_{k}(j)}$
($\frac{\delta F_{\mathrm{cl,e}}}{\delta V_{k}(j)}$ for two-parameter
estimation);

  5. 5.

update $V_{k}(j)$ to $V_{k}(j)+\epsilon\frac{\delta f_{0}(T)}{\delta
V_{k}(j)}$;

  6. 6.

go to step 2 until the objective function converges.

The detailed calculation of the gradient for $f_{0}(T)$, which is based on the
gradient of the entries of CFIM and an essential step for GRAPE, can be found
in appendix A.

GRAPE can be further improved via the quasi-Newton optimization Fouquieres2011
, which utilize the gradient history to construct an approximation to the
Hessian matrix. Davidon-Fletcher-Powell formula DFP  and Broyden-Fletcher-
Goldfarb-Shanno algorithm BFGS  are two well applied methods for this
construction. This quasi-Newton optimization can also be utilized here in
quantum parameter estimation for the improvement of GRAPE performance.

Apart from GRAPE, other optimization methods, such as Krotov’s method
krotovbook ; Machnes2011 ; Reich2012 ; Goerz2015 ; Kochreview , hybrid update
method Machnes2011  and hybrid optimization method Goerz2015  can all be
employed for controlled quantum parameter estimation. The comparison of these
methods for quantum parameter estimation will be studied in future works.

##  IV Application

We demonstrate the effect of controls with several examples in Hamiltonian
parameter estimation.

###  IV.1 Noiseless scenario

_Example 1_ : Consider a two-qubit system where one qubit is in a magnetic
field and the other qubit acts as an ancillary qubit. The Hamiltonian of this
system is $H=H_{0}+H_{\mathrm{c}}(t)$ where

$H_{0}=\vec{B}\cdot\vec{\sigma}^{(1)},$ (9)

$\vec{\sigma}^{(1)}=(\sigma^{(1)}_{1},\sigma^{(1)}_{2},\sigma^{(1)}_{3})$ with
$\sigma^{(1)}_{i}=\sigma_{i}\otimes\openone_{2}$, here
$\sigma_{1},\sigma_{2},\sigma_{3}$ are Pauli matrices, $\openone_{2}$ denotes
the 2 by 2 identity matrix. Similarly we will use
$\vec{\sigma}^{(2)}=(\sigma^{(2)}_{1},\sigma^{(2)}_{2},\sigma^{(2)}_{3})$ to
denote $\sigma^{(2)}_{i}=\openone_{2}\otimes\sigma_{i}$. With the absence of
control, the QFIM can be attained via Bell measurement Yuan2016 . For
controlled scheme, the local controls on the qubits

$H_{\mathrm{c}}(t)=\sum_{i=1,2}\vec{V}_{i}(t)\cdot\vec{\sigma}^{(i)}$ (10)

are employed. This problem has been studied previously and analytical
solutions for optimal controls have been obtained Yuan2016 . It has been shown
that the optimal controls is to reverse the free evolution and the optimal
initial state is any maximally entangled state, such as
$\frac{1}{\sqrt{2}}\left(|00\rangle\pm|11\rangle\right)$, and the optimal
measurement is the projective measurement on the Bell basis:
$|\Phi^{\pm}\rangle=\frac{1}{\sqrt{2}}\left(|00\rangle\pm|11\rangle\right)$
and
$|\Psi^{\pm}\rangle=\frac{1}{\sqrt{2}}\left(|01\rangle\pm|10\rangle\right)$.
The precision limit for the three parameters $(B,\theta,\phi)$ characterizing
the magnetic field
$\vec{B}=(B\sin\theta\cos\phi,B\sin\theta\sin\phi,B\cos\theta)$ under optimal
control is Yuan2016

$\mathrm{Tr}\mathcal{F}^{-1}_{\mathrm{cl}}=\frac{3}{4T^{2}}.$ (11)

As a confirmation of validity of GRAPE, we compare the performance of the
controls obtained from GRAPE (the objective function is $f_{0}$). It can be
seen from Fig. 2(a) that the performance of the controls (red dots) obtained
from GRAPE coincides with the analytical solutions, which shows that GRAPE is
capable to identify the optimal controls in this case.

Figure 2: (Color online) $\mathrm{Tr}\mathcal{F}^{-1}_{\mathrm{cl}}$ as a
function of target time $T$. The solid black and dashed blue line represent
the noiseless and noisy performance of non-controlled schemes for (a) example
1; (b) example 2 and (c) example 3.The red dots and cyan stars represent
noiseless and noisy performance of controlled schemes in three examples. The
controls are obtained via GRAPE with the objective function $f_{0}(T)$ for (a)
and (b) and $F_{\mathrm{cl,e}}$ for (c). The dotted green line represent the
optimal total precision limits. The dephasing rate in (a) is $\gamma=0.2$ and
in (b) and (c) is $\gamma_{1}=\gamma_{2}=0.1$. The true values in (a) are
$1,\pi/4,\pi/4$ for $B,\theta,\phi$; in (b) are $1,1.2,0.1$ for
$\omega_{1},\omega_{2},g$; in (c) are $1,1.2$ for $x_{1},x_{2}$.

_Example 2_ : Next we consider the Hamiltonian estimation for a two-qubit
system with ZZ coupling, which is a widely used model under strong field. The
Hamiltonian in this case is

$H_{0}=\omega_{1}\sigma^{(1)}_{3}+\omega_{2}\sigma^{(2)}_{3}+g\sigma^{(1)}_{3}\sigma^{(2)}_{3},$
(12)

where $\omega_{1}$, $\omega_{2}$ and $g$ are the parameters to be estimated.

We first consider the case without controls. The optimal probe state for this
system is $|++\rangle$ where
$|+\rangle=\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)$ (see detailed derivations
in appendix C) and the corresponding QFIM is
$\mathcal{F}_{\mathrm{q}}(T)=4T^{2}\openone_{3}$ ($\openone_{3}$ denotes the 3
by 3 identity matrix). This QFIM is attainable, which is due to the fact that
$\sigma^{(1)}_{3}$, $\sigma^{(2)}_{3}$ and $\sigma^{(1)}_{3}\sigma^{(2)}_{3}$
commute with each other. The optimal measurement exists Humphreys2013 ;
Dattareview  but practically challenging to implement. In practice, usually
the local measurement
$E_{\pm}=\\{|++\rangle\langle++|,|+-\rangle\langle+-|,|-+\rangle\langle-+|,|--\rangle\langle--|\\}$
is used instead, under this local measurement, the precision (solid black line
in Fig. 2(b)) is typically much worse than the optimal measurement, and it
oscillates with time, which indicates that the time stability of the local
measurement is quite poor, i.e., a small error in the time point chosen to
perform the measurement can result very different precisions.

Now if controls are employed, both the precision and the time stability can be
improved. This can be seen from Fig. 2(b), the precision of controlled scheme
under the local measurement $E_{\pm}$ (red dots in Fig. 2(b)) essentially
coincides with the precision under the optimal measurement (dotted green line
in Fig. 2(b)), it also does not oscillate as in the non-controlled case. This
shows that controls can improve both the precision limit and the time
stability.

_Example 3_ : In this example we consider the Hamiltonian estimation with XXZ
coupling. The Hamiltonian is

$H_{0}=-x_{1}\left(\sigma^{(1)}_{1}\sigma^{(2)}_{1}+\sigma^{(1)}_{2}\sigma^{(2)}_{2}\right)-x_{2}\sigma^{(1)}_{3}\sigma^{(2)}_{3},$
(13)

where $x_{1}$ and $x_{2}$ are coupling parameters to be estimated. First
without controls, the optimal QFIM is

$\mathcal{F}_{\mathrm{q},\mathrm{opt}}=\left(\begin{array}[]{cc}8T^{2}&0\\\
0&4T^{2}\end{array}\right),$ (14)

which can be attained with the optimal probe state
$|\psi_{0,\mathrm{opt}}\rangle=\frac{1}{\sqrt{2}}|0\rangle(|0\rangle+i|1\rangle)$
(see details in appendix D). The optimal total precision limit for the non-
controlled scheme then reads

$\mathrm{Tr}\mathcal{F}^{-1}_{\mathrm{q,opt}}=\frac{3}{8T^{2}}.$ (15)

In this case the QFIM is attainable as the generators for $x_{1}$ and $x_{2}$
commute. However, if we use the more practical local measurement $E_{\pm}$,
the CFIM for non-controlled scheme is
$\mathcal{F}_{\mathrm{cl}}=4T^{2}\openone_{2}$, this indicates that the local
measurement, $E_{\pm}$, is only optimal for $x_{2}$. But if we add controls
that obtained from GRAPE with $F_{\mathrm{cl,e}}$ as the objective function,
then the precision limit(red dots in Fig. 2(c)) can reach the optimal values
even under the local measurement $E_{\pm}$.

###  IV.2 Noisy scenario

We now consider the dynamics with dephasing noises in all examples discussed
above.

_Example 1_ : Again we consider a two-qubit system where the magnetic field
only acts on the first qubit and the second qubit acts as an ancillary system.
For example, in Nitrogen vacancy center, the electron spin can act as the
sensing qubit while a neighboring nuclear spin can act as an ancillary system.
Recall that the Hamiltonian is given by $H=H_{0}+H_{\mathrm{c}}(t)$ where
$H_{0}=\vec{B}\cdot\vec{\sigma}^{(1)}$ and
$H_{\mathrm{c}}(t)=\sum_{i=1,2}\vec{V}_{i}(t)\cdot\vec{\sigma}^{(i)}$.

Assume that the sensing qubit (first qubit) suffers from the dephasing noise,
the dynamics is then described by the master equation

$\partial_{t}\rho=-i[H,\rho]+\frac{\gamma}{2}\left(\sigma^{(1)}_{3}\rho\sigma^{(1)}_{3}-\rho\right).$
(16)

Figure 2(a) (cyan stars) shows the advantages of controlled scheme in
improving the precision limit. From the figure it can also be seen that the
controls also suppress the oscillations, thus improve the time stability.

_Example 2_ : Consider the Hamiltonian with ZZ coupling
$H=H_{0}+H_{\mathrm{c}}(t)$ where
$H_{0}=\omega_{1}\sigma^{(1)}_{3}+\omega_{2}\sigma^{(2)}_{3}+g\sigma^{(1)}_{3}\sigma^{(2)}_{3}$
and $H_{\mathrm{c}}(t)=\sum_{i=1,2}\vec{V}_{i}(t)\cdot\vec{\sigma}^{(i)}$.
With the presence of dephasing, the dynamics is described by

$\partial_{t}\rho=-i[H,\rho]+\sum_{i=1,2}\frac{\gamma_{i}}{2}\left(\sigma_{3}^{(i)}\rho\sigma_{3}^{(i)}-\rho\right),$
(17)

where $\gamma_{i}$ is the dephasing rate for $i$th qubit. Fig. 2(b) shows that
the controlled scheme improves the precision limit under the local measurement
$E_{\pm}$. It can also be seen that the controls suppressed the oscillations
in the precision limit, thus improved the time stability.

_Example 3_ : Consider the Hamiltonian estimation with XXZ coupling where
$H_{0}=-x_{1}(\sigma^{(1)}_{1}\sigma^{(2)}_{1}+\sigma^{(1)}_{2}\sigma^{(2)}_{2})-x_{2}\sigma^{(1)}_{3}\sigma^{(2)}_{3}$
and the control Hamiltonian is
$H_{\mathrm{c}}(t)=\sum_{i=1,2}\vec{V}_{i}(t)\cdot\vec{\sigma}^{(i)}$, both
qubits have the dephasing. The master equation for this system is in the form
of Eq. (17), and we will assume the dephasing rates are the same for both
qubits, i.e., $\gamma_{1}=\gamma_{2}=\gamma$. The CFIM for non-controlled
scheme under the practical local measurement $E_{\pm}$ is given by (see
detailed derivation in appendix D)

$\mathcal{F}_{\mathrm{cl}}=2T^{2}\left(\begin{array}[]{cc}\delta_{+}+\delta_{-}&\delta_{+}-\delta_{-}\\\
\delta_{+}-\delta_{-}&\delta_{+}+\delta_{-}\end{array}\right),$ (18)

where

$\delta_{\pm}:=\frac{\cos^{2}\left[2T(x_{1}\pm x_{2})\right]}{e^{2\gamma
T}-\sin^{2}\left[2T(x_{1}\pm x_{2})\right]}.$ (19)

The precision limit for non-controlled scheme is then

$\mathrm{Tr}\mathcal{F}_{\mathrm{cl}}^{-1}=\frac{1}{4T^{2}}\left(\frac{1}{\delta_{+}}+\frac{1}{\delta_{-}}\right).$
(20)

It can be seen that when $2T(x_{1}\pm x_{2})=\pi/2+n\pi$ ($n=0,1,2...$) this
precision blows up, which corresponds to the peaks in Fig. 2(c).

After adding controls, which are obtained from GRAPE with the objective
function $F_{\mathrm{cl,e}}$, the precision limit (cyan stars in Fig. 2(c)) is
improved compared the uncontrolled case (dashed blue line in Fig. 2(c)). Also,
the oscillations is suppressed, indicating that the controlled scheme provides
a higher time stability.

##  V Summary

In this paper, we employed the optimal control method, GRAPE in particular, in
designing optimal controls for the improvement of the precision limit in
multiparameter quantum estimation. Advantages on the precision limit of
controlled schemes are demonstrated through three examples, including the
estimation of a magnetic field, Hamiltonian estimation with ZZ and XXZ
couplings. The controls not only improve the precision limit, but also the
time stability, especially for noisy systems.

###### Acknowledgements.

H.Y. acknowledges partial financial support from RGC of Hong Kong with Grant
No. 538213. J.L. acknowledges partial financial support from the Natural
Science Foundation of Zhejiang Province under Grant No. LY18A050003. We would
like to thank the anonymous referees for the helpful suggestions.

##  Appendix A Calculation of gradient for the entries of CFIM

It is known that the entry of CFIM is

$\mathcal{F}_{\mathrm{cl},\alpha\beta}=\sum_{y}\frac{(\partial_{x_{\alpha}}p_{y|\vec{x}})(\partial_{x_{\beta}}p_{y|\vec{x}})}{p_{y|\vec{x}}}.$
(21)

where $p_{y|\vec{x}}=\mathrm{Tr}(\rho(T)E(y))$. Here $E(y)$ is a POVM
measurement which satisfies $\sum_{y}E(y)=\openone$. To calculate the
gradient, we first need to know

$\displaystyle\frac{\delta p_{y|\vec{x}}}{\delta V_{k}(j)}$ $\displaystyle=$
$\displaystyle\mathrm{Tr}\left[\frac{\delta\rho(T)}{\delta
V_{k}(j)}E(y)\right]$ (22) $\displaystyle=$
$\displaystyle\mathrm{Tr}\left[D_{j+1}^{m}\frac{\delta\rho_{j}}{\delta
V_{k}(j)}E(y)\right]$ $\displaystyle=$ $\displaystyle-\Delta
t\mathrm{Tr}\left[E(y)\mathcal{M}_{j}^{(1)}\right],$

where $\mathcal{M}_{j}^{(1)}=iD_{j+1}^{m}H_{k}^{\times}\rho_{j}$. One then can
obtain

$\displaystyle\frac{\delta\left(\partial_{x_{\alpha}}p_{y|\vec{x}}\right)}{\delta
V_{k}(j)}$ $\displaystyle=$ $\displaystyle-i\Delta
t\mathrm{Tr}\left[E(y)\partial_{x_{\alpha}}\\!\\!\left(D_{j+1}^{m}H_{k}^{\times}\rho_{j}\right)\right]$
(23) $\displaystyle=$ $\displaystyle-i\Delta
t\mathrm{Tr}\Big{\\{}E(y)\Big{[}\left(\partial_{x_{\alpha}}D_{j+1}^{m}\right)H_{k}^{\times}\rho_{j}$
$\displaystyle+D_{j+1}^{m}H_{k}^{\times}\partial_{x}\rho_{j}\Big{]}\Big{\\}}.$

Based on Ref. Liuarxiv , we know

$\displaystyle\partial_{x_{\alpha}}D_{j+1}^{m}$ $\displaystyle=$
$\displaystyle\Delta
t\sum_{i=j+1}^{m}D_{i+1}^{m}\left(\partial_{x_{\alpha}}\mathcal{L}_{i}\right)D_{j+1}^{i},$
(24) $\displaystyle\partial_{x_{\alpha}}\rho_{j}$ $\displaystyle=$
$\displaystyle\Delta
t\sum_{i=1}^{j}D_{i+1}^{j}\left(\partial_{x_{\alpha}}\mathcal{L}_{i}\right)\rho_{i},$
(25)

where $\dot{H}^{\times}_{0}=[\partial_{x_{\alpha}}H_{0},\cdot]$. For $j\neq
m$,

$\displaystyle\frac{\delta\left(\partial_{x_{\alpha}}p_{y|\vec{x}}\right)}{\delta
V_{k}(j)}$ $\displaystyle=$
$\displaystyle-\Delta^{2}t\mathrm{Tr}\Big{[}\\!\Big{(}\\!E(y)\\!\\!\sum_{i=j+1}^{m}\\!\\!D_{i+1}^{m}\dot{H}_{0}^{\times}D_{j+1}^{i}H_{k}^{\times}\rho_{j}$
(26)
$\displaystyle+\sum_{i=1}^{j}D_{j+1}^{m}H_{k}^{\times}D_{i+1}^{j}\dot{H}_{0}^{\times}\rho_{i}\Big{)}\Big{]}.$

for $j=m$, there is

$\displaystyle\frac{\delta\left(\partial_{x_{\alpha}}p_{y|\vec{x}}\right)}{\delta
V_{k}(m)}$ $\displaystyle=$ $\displaystyle-i\Delta
t\mathrm{Tr}\left[E(y)H_{k}^{\times}\partial_{x}\rho_{m}\right]$
$\displaystyle=$
$\displaystyle-\Delta^{2}t\mathrm{Tr}\left[E(y)H_{k}^{\times}\sum_{i=1}^{m}D_{i+1}^{j}\dot{H}_{0}^{\times}\rho_{i}\right].$

Thus, combined above equations, we have

$\frac{\delta\left(\partial_{x_{\alpha}}p_{y|\vec{x}}\right)}{\delta
V_{k}(j)}=-\Delta^{2}t\mathrm{Tr}\left\\{E(y)\left[\mathcal{M}_{j,\alpha}^{(2)}+\mathcal{M}_{j,\alpha}^{(3)}\right]\right\\},$

where $\mathcal{M}_{j,\alpha}^{(2)}$ and $\mathcal{M}_{j,\alpha}^{(3)}$ are

$\displaystyle\mathcal{M}_{j,\alpha}^{(2)}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{j}\mathcal{D}_{j+1}^{m}H_{k}^{\times}\mathcal{D}_{i+1}^{j}\dot{H}_{0}^{\times}(\rho_{i}),$
(27) $\displaystyle\mathcal{M}_{j,\alpha}^{(3)}$ $\displaystyle=$
$\displaystyle(1-\delta_{jm})\\!\\!\\!\sum_{i=j+1}^{m}\\!\\!\mathcal{D}_{i+1}^{m}\dot{H}_{0}^{\times}\mathcal{D}_{j+1}^{i}H_{k}^{\times}(\rho_{j}).$

Utilizing above expressions, the gradient for
$\mathcal{F}_{\mathrm{cl},\alpha\beta}$ is

$\displaystyle\frac{\delta\mathcal{F}_{\mathrm{cl},\alpha\beta}(T)}{\delta
V_{k}(j)}$ $\displaystyle=$ $\displaystyle\Delta
t\mathrm{Tr}\left(\tilde{L}_{2,\alpha\beta}\mathcal{M}_{j}^{(1)}\right)$ (28)
$\displaystyle-\Delta^{2}t\mathrm{Tr}\left[\tilde{L}_{1,\beta}\left(\\!\mathcal{M}_{j,\alpha}^{(2)}\\!+\\!\mathcal{M}_{j,\alpha}^{(3)}\\!\right)\right]$
$\displaystyle-\Delta^{2}t\mathrm{Tr}\left[\tilde{L}_{1,\alpha}\left(\\!\mathcal{M}_{j,\beta}^{(2)}\\!+\\!\mathcal{M}_{j,\beta}^{(3)}\\!\right)\right],$

where $\tilde{L}_{1,\alpha(\beta)}$ and $\tilde{L}_{2,\alpha\beta}$ are

$\displaystyle\tilde{L}_{1,\alpha(\beta)}$ $\displaystyle=$
$\displaystyle\sum_{y}\left(\partial_{x_{\alpha(\beta)}}\ln
p_{y|\vec{x}}\right)E(y),$ (29) $\displaystyle\tilde{L}_{2,\alpha\beta}$
$\displaystyle=$ $\displaystyle\sum_{y}\left(\partial_{x_{\alpha}}\ln
p_{y|\vec{x}}\right)\left(\partial_{x_{\beta}}\ln p_{y|\vec{x}}\right)E(y).$
(30)

For two parameter estimation, the objective function is

$F_{\mathrm{cl,e}}=\frac{\det\mathcal{F}_{\mathrm{cl}}}{\mathrm{Tr}\mathcal{F}_{\mathrm{cl}}}=\frac{\mathcal{F}_{\mathrm{cl},\alpha\alpha}\mathcal{F}_{\mathrm{cl},\beta\beta}-\mathcal{F}^{2}_{\mathrm{cl},\alpha\beta}}{\mathcal{F}_{\mathrm{cl},\alpha\alpha}+\mathcal{F}_{\mathrm{cl},\beta\beta}}.$
(31)

The corresponding gradient is

$\frac{\delta F_{\mathrm{cl,e}}}{\delta
V_{k}(j)}=\\!\\!\sum_{\alpha\neq\beta}\\!\\!\frac{\mathcal{F}^{2}_{\mathrm{cl},\beta\beta}+\mathcal{F}^{2}_{\mathrm{cl},\alpha\beta}}{\mathrm{Tr}^{2}\mathcal{F}_{\mathrm{cl}}}\frac{\delta\mathcal{F}_{\mathrm{cl},\alpha\alpha}}{\delta
V_{k}(j)}-\frac{2\mathcal{F}_{\mathrm{cl},\alpha\beta}}{\mathrm{Tr}\mathcal{F}_{\mathrm{cl}}}\frac{\delta\mathcal{F}_{\mathrm{cl},\alpha\beta}}{\delta
V_{k}(j)}$ (32)

For parameter estimation with more than two parameters, recall the objective
function is

$f_{0}(T)=\left(\sum_{\alpha}\frac{1}{\mathcal{F}_{\mathrm{cl},\alpha\alpha}(T)}\right)^{-1}.$
(33)

The gradient of $f_{0}(T)$ then reads

$\frac{\delta f_{0}(T)}{\delta
V_{k}(j)}=\sum_{\alpha}\frac{f^{2}_{0}(T)}{\mathcal{F}^{2}_{\mathrm{cl},\alpha\alpha}}\frac{\delta\mathcal{F}_{\mathrm{cl},\alpha\alpha}}{\delta
V_{k}(j)}.$ (34)

##  Appendix B Calculation of noisy CFIM and QFIM in the estimation of
magnetic field

In this appendix we show the detailed calculation in the estimation of
magnetic field. Recall that the scenario is a two-qubit system where the
magnetic field only acts on the first qubit and the second qubit acts as an
ancillary system, the Hamiltonian is given by $H=H_{0}+H_{\mathrm{c}}(t)$
where

$H_{0}=\vec{B}\cdot\vec{\sigma}^{(1)},$ (35)

$\vec{\sigma}^{(1)}=(\sigma^{(1)}_{1},\sigma^{(1)}_{2},\sigma^{(1)}_{3})$ with
$\sigma^{(1)}_{i}=\sigma_{i}\otimes\openone_{2}$, where
$\sigma_{1},\sigma_{2},\sigma_{3}$ are Pauli matrices, $\openone_{2}$ denotes
the 2 by 2 identity matrix. Similarly we denote
$\vec{\sigma}^{(2)}=(\sigma^{(2)}_{1},\sigma^{(2)}_{2},\sigma^{(2)}_{3})$ with
$\sigma^{(2)}_{i}=\openone_{2}\otimes\sigma_{i}$.

With the initial state $\frac{1}{\sqrt{2}}(|00\rangle+|11\rangle)$, the
evolved state without noise can be straightforwardly obtained in the basis
$\\{|00\rangle,|01\rangle,|10\rangle,|11\rangle\\}$ as below

$|\psi(T)\rangle=\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}\cos(BT)-i\sin(BT)\cos\theta\\\
-i\sin(BT)\sin\theta e^{-i\phi}\\\ -i\sin(BT)\sin\theta e^{i\phi}\\\
\cos(BT)+i\sin(BT)\cos\theta\\\ \end{array}\right).$ (36)

For the further calculation, we rewrite the corresponding density matrix in
block form
$|\psi(T)\rangle\langle\psi(T)|=\frac{1}{2}\left(\begin{array}[]{cc}X&Y\\\
Y^{\dagger}&Q\end{array}\right),$ where $X$, $Y$ and $Q$ are 2 by 2 matrices.
Recall the master equation involving the noise is

$\partial_{t}\rho=-i[H,\rho]+\frac{\gamma}{2}\left(\sigma^{(1)}_{3}\rho\sigma^{(1)}_{3}-\rho\right).$
(37)

The solution for above equation can be expressed in the following form

$\rho(T)=\frac{1}{2}\left(\begin{array}[]{cc}X&e^{-\gamma T}Y\\\ e^{-\gamma
T}Y^{\dagger}&Q\end{array}\right).$ (38)

Performing the Bell measurements, the probability distribution is

$\displaystyle p_{\Phi^{+}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}[(1+e^{-\gamma T})\\!\cos^{2}(BT)\\!+\\!(1-e^{-\gamma
T})\\!\sin^{2}(BT)\\!\cos^{2}\theta],$ $\displaystyle p_{\Phi^{-}}$
$\displaystyle=$ $\displaystyle\frac{1}{2}[(1-e^{-\gamma
T})\\!\cos^{2}(BT)\\!+\\!(1+e^{-\gamma T})\\!\sin^{2}(BT)\\!\cos^{2}\theta],$
$\displaystyle p_{\Psi^{+}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sin^{2}(BT)\sin^{2}\theta[1+e^{-\gamma
T}\cos(2\phi)],$ $\displaystyle p_{\Psi^{-}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sin^{2}(BT)\sin^{2}\theta[1-e^{-\gamma
T}\cos(2\phi)].$

According to the probabilities, the entries of CFIM can then be calculated as
below

$\mathcal{F}_{\mathrm{cl},\phi\phi}=\frac{4e^{-2\gamma
T}\sin^{2}(BT)\sin^{2}\theta\sin^{2}(2\phi)}{1-e^{-2\gamma
T}\cos^{2}(2\phi)},$ (39)

$\mathcal{F}_{\mathrm{cl},\theta\phi}=\mathcal{F}_{\mathrm{cl},B\phi}=0$ and

$\displaystyle\mathcal{F}_{\mathrm{cl},\theta\theta}$ $\displaystyle=$
$\displaystyle
4\sin^{2}(BT)\left\\{\cos^{2}\theta+\frac{\sin^{2}(BT)\sin^{2}\theta\cos^{2}\theta[(1+3e^{-2\gamma
T})\cos^{2}(BT)+(1-e^{-2\gamma
T})\sin^{2}(BT)\cos^{2}\theta]}{[\cos^{2}(BT)+\sin^{2}(BT)\cos^{2}\theta]^{2}-e^{-2\gamma
T}[\cos^{2}(BT)-\sin^{2}(BT)\cos^{2}\theta]^{2}}\right\\},$
$\displaystyle\mathcal{F}_{\mathrm{cl},BB}$ $\displaystyle=$ $\displaystyle
4T^{2}\cos^{2}(BT)\Big{\\{}\sin^{2}\theta+\sin^{2}(BT)\times$
$\displaystyle\frac{[\sin^{4}\theta+e^{-2\gamma
T}(1+\cos^{2}\theta)^{2}][1-\sin^{2}(BT)\sin^{2}\theta]+2e^{-2\gamma
T}\sin^{2}\theta(1+\cos^{2}\theta)[-\cos^{2}(BT)+\sin^{2}(BT)\cos^{2}\theta]}{[\cos^{2}(BT)+\sin^{2}(BT)\cos^{2}\theta]^{2}-e^{-2\gamma
T}[\cos^{2}(BT)-\sin^{2}(BT)\cos^{2}\theta]^{2}}\Big{\\}},$
$\displaystyle\mathcal{F}_{\mathrm{cl},B\theta}$ $\displaystyle=$
$\displaystyle T\sin(2BT)\sin(2\theta)\Big{\\{}1+\sin^{2}(BT)\times$
$\displaystyle\frac{(1+e^{-2\gamma
T})\cos^{2}(BT)\sin^{2}\theta+(1-e^{-2\gamma
T})\sin^{2}(BT)\cos^{2}\theta\sin^{2}\theta-2e^{-2\gamma
T}\cos^{2}(BT)(1+\cos^{2}\theta)}{[\cos^{2}(BT)+\sin^{2}(BT)\cos^{2}\theta]^{2}-e^{-2\gamma
T}[\cos^{2}(BT)-\sin^{2}(BT)\cos^{2}\theta]^{2}}\Big{\\}}.$

Although the CFIM has analytical solution, it is still difficult to obtain the
general solution of $\mathrm{Tr}\mathcal{F}^{-1}_{\mathrm{cl}}$ analytically.
Here we consider the case that $\theta$ is small. The expression of
$\mathrm{Tr}\mathcal{F}^{-1}_{\mathrm{cl}}$ approximates to

$\displaystyle\mathrm{Tr}\mathcal{F}^{-1}_{\mathrm{cl}}$
$\displaystyle\approx$ $\displaystyle\frac{e^{2\gamma
T}-\cos^{2}(2\phi)}{2[1-\cos(2BT)]\sin^{2}(2\phi)}\theta^{-2}-\frac{1+\cos(2BT)}{1-\cos(4BT)}$
(40) $\displaystyle-\frac{1-2e^{2\gamma T}+\cos(4BT)}{4T^{2}[1-\cos(4BT)]}.$

The non-zero eigenvalues of the noisy density matrix are
$\lambda_{\pm}=\frac{1}{2}(1\pm e^{-\gamma T})$, which are independent of the
magnetic field. The eigenstates for $\lambda_{\pm}$ are
$|\lambda_{+}\rangle=|\psi(T)\rangle$ and

$|\lambda_{-}\rangle=\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}\cos(BT)-i\sin(BT)\cos\theta\\\
-i\sin(BT)\sin\theta e^{-i\phi}\\\ i\sin(BT)\sin\theta e^{i\phi}\\\
\cos(BT)+i\sin(BT)\cos\theta\end{array}\right).$ (41)

It is known that the QFIM can be calculated via the density matrix’s non-zero
eigenvalues and corresponding eigenstates Liu2014 ; Liu2014a . Utilizing above
eigenstates, The entries of QFIM can be calculated as below

$\mathcal{F}_{\mathrm{q},BB}=4T^{2}\left(\cos^{2}\theta e^{-2\gamma
T}+\sin^{2}\theta\right),$ (42)

and

$\displaystyle\mathcal{F}_{\mathrm{q},\theta\theta}$ $\displaystyle=$
$\displaystyle 4\sin^{2}(BT)\Big{\\{}\cos^{2}\theta$ (43)
$\displaystyle+\sin^{2}\theta\left[e^{-2\gamma
T}\cos^{2}(BT)+\sin^{2}(BT)\right]\\!\\!\Big{\\}}\\!,$

and

$\mathcal{F}_{\mathrm{q},\phi\phi}\\!=\\!4\sin^{2}\\!\theta\sin^{2}(BT)\\!\left[1\\!-\\!(1-e^{-2\gamma
T})\sin^{2}\\!\theta\sin^{2}(BT)\right]\\!.$ (44)

The off-diagonal entries are

$\mathcal{F}_{\mathrm{q},B\theta}=(1-e^{-2\gamma T})T\sin(2BT)\sin(2\theta),$
(45)

and

$\mathcal{F}_{\mathrm{q},B\phi}=-2(1-e^{-2\gamma
T})T\sin(2\theta)\sin\theta\sin^{2}(BT),$ (46)

and

$\mathcal{F}_{\mathrm{q},\theta\phi}=2(1-e^{-2\gamma
T})\sin^{3}\theta\sin(2BT)\sin^{2}(BT).$ (47)

it can be seen that the Bell measurement fails to be optimal when noise
exists.

##  Appendix C Detailed calculation in Hamiltonian estimation with ZZ coupling

In this appendix we show the detailed calculation in the hamiltonian
estimation with ZZ coupling. The corresponding Hamiltonian is in the form

$H_{0}=\omega_{1}\sigma^{(1)}_{3}+\omega_{2}\sigma^{(2)}_{3}+g\sigma^{(1)}_{3}\sigma^{(2)}_{3},$
(48)

where $\omega_{1}$, $\omega_{2}$ and $g$ are the parameters to be estimated.
It is known that for pure states, the element of QFIM is

$\displaystyle\mathcal{F}_{\mathrm{q},\alpha\beta}$ $\displaystyle=$
$\displaystyle
4\mathrm{Re}\big{(}\langle\partial_{\alpha}\psi(T)|\partial_{\beta}\psi(T)\rangle$
(49)
$\displaystyle-\langle\partial_{\alpha}\psi(T)|\psi(T)\rangle\langle\psi(T)|\partial_{\beta}\psi(T)\rangle\big{)}.$

In this case, the evolved state $|\psi(T)\rangle=e^{-iHT}|\psi_{0}\rangle$.
Utilizing this form, one can have the diagonal entries of the QFIM as below,

$\displaystyle\mathcal{F}_{\mathrm{q},\omega_{1}\omega_{2}}$ $\displaystyle=$
$\displaystyle
4T^{2}\left(1-\langle\psi_{0}|\sigma^{(1)}_{3}|\psi_{0}\rangle^{2}\right),$
(50) $\displaystyle F_{\mathrm{q},\omega_{2}\omega_{2}}$ $\displaystyle=$
$\displaystyle
4T^{2}\left(1-\langle\psi_{0}|\sigma^{(2)}_{3}|\psi_{0}\rangle^{2}\right),$
(51) $\displaystyle F_{\mathrm{q},gg}$ $\displaystyle=$ $\displaystyle
4T^{2}\left(1-\langle\psi_{0}|\sigma^{(1)}_{3}\sigma^{(2)}_{3}|\psi_{0}\rangle^{2}\right).$
(52)

The off-diagonal entries are

$\displaystyle\mathcal{F}_{\mathrm{q},\omega_{1}\omega_{2}}\\!$
$\displaystyle=$
$\displaystyle\\!4T^{2}\\!\\!\left(\\!\langle\psi_{0}|\sigma^{(1)}_{3}\sigma^{(2)}_{3}|\psi_{0}\rangle-\langle\psi_{0}|\sigma^{(1)}_{3}|\psi_{0}\rangle\langle\psi_{0}|\sigma^{(2)}_{3}|\psi_{0}\rangle\\!\right)\\!,$
$\displaystyle\mathcal{F}_{\mathrm{q},\omega_{1}g}\\!$ $\displaystyle=$
$\displaystyle\\!4T^{2}\\!\\!\left(\\!\langle\psi_{0}|\sigma^{(2)}_{3}|\psi_{0}\rangle-\langle\psi_{0}|\sigma^{(1)}_{3}|\psi_{0}\rangle\langle\psi_{0}|\sigma^{(1)}_{3}\sigma^{(2)}_{3}|\psi_{0}\rangle\\!\right)\\!,$
$\displaystyle\mathcal{F}_{\mathrm{q},\omega_{2}g}\\!$ $\displaystyle=$
$\displaystyle\\!4T^{2}\\!\\!\left(\\!\langle\psi_{0}|\sigma^{(1)}_{3}|\psi_{0}\rangle-\langle\psi_{0}|\sigma^{(2)}_{3}|\psi_{0}\rangle\langle\psi_{0}|\sigma^{(1)}_{3}\sigma^{(2)}_{3}|\psi_{0}\rangle\\!\right)\\!.$

From the expressions of QFIM, one can see that the optimal QFIM is in the form

$\mathcal{F}_{\mathrm{q,max}}=4T^{2}\openone_{3},$ (53)

where $\openone_{3}$ is a 3 by 3 identity matrix. This optimal QFIM can be
obtained if the probe state satisfies the following equations

$\displaystyle\langle\psi_{0}|\sigma^{(1)}_{3}|\psi_{0}\rangle$
$\displaystyle=$ $\displaystyle 0,$ (54)
$\displaystyle\langle\psi_{0}|\sigma^{(2)}_{3}|\psi_{0}\rangle$
$\displaystyle=$ $\displaystyle 0,$ (55)
$\displaystyle\langle\psi_{0}|\sigma^{(1)}_{3}\sigma^{(2)}_{3}|\psi_{0}\rangle$
$\displaystyle=$ $\displaystyle 0.$ (56)

Denote the probe state as
$|\psi_{0}\rangle=a|00\rangle+b|01\rangle+c|10\rangle+d|11\rangle$, above
equations are equivalent to

$\displaystyle|a|^{2}+|b|^{2}-|c|^{2}-|d|^{2}$ $\displaystyle=$ $\displaystyle
0,$ (57) $\displaystyle|a|^{2}+|c|^{2}-|b|^{2}-|d|^{2}$ $\displaystyle=$
$\displaystyle 0,$ (58) $\displaystyle|a|^{2}+|d|^{2}-|b|^{2}-|d|^{2}$
$\displaystyle=$ $\displaystyle 0.$ (59)

Taking into account the normalization $|a|^{2}+|b|^{2}+|c|^{2}+|d|^{2}=1$, the
only solution is $|a|^{2}=|b|^{2}=|c|^{2}=|d|^{2}=1/4.$ Thus, the optimal
probe state is in the form

$|\psi_{0,\mathrm{opt}}\rangle=\frac{1}{2}\left(|00\rangle+e^{i\phi_{1}}|01\rangle+e^{i\phi_{2}}|10\rangle+e^{i\phi_{3}}|11\rangle\right),$

where $\phi_{1}$, $\phi_{2}$ and $\phi_{3}$ are relative phases. The simplest
one is $\phi_{1}=\phi_{2}=\phi_{3}=0$, i.e.,
$|\psi_{0,\mathrm{opt}}\rangle=|++\rangle$, where
$|+\rangle=(|0\rangle+|1\rangle)/\sqrt{2}$.

Now we discuss the noisy case. Consider the dephasing noise for both qubits,
the master equation for the system is

$\partial_{t}\rho=-i[H,\rho]+\sum_{i=1,2}\frac{\gamma_{i}}{2}\left(\sigma_{3}^{(i)}\rho\sigma_{3}^{(i)}-\rho\right).$
(60)

The specific solution of $\rho(T)$ is

$\left(\begin{array}[]{cccc}\rho_{00}(0)&\rho_{01}(0)e^{-i2(g+\omega_{2})T-\gamma_{2}T}&\rho_{02}(0)e^{-i2(g+\omega_{1})T-\gamma_{1}T}&\rho_{03}(0)e^{-i2(\omega_{1}+\omega_{2})T-(\gamma_{1}+\gamma_{2})T}\\\
\rho_{10}(0)e^{i2(g+\omega_{2})T-\gamma_{2}T}&\rho_{11}(0)&\rho_{12}(0)e^{-i2(\omega_{1}-\omega_{2})T-(\gamma_{1}+\gamma_{2})T}&\rho_{13}(0)e^{i2(g-\omega_{1})T-\gamma_{1}T}\\\
\rho_{20}(0)e^{i2(g+\omega_{1})T-\gamma_{1}T}&\rho_{21}(0)e^{i2(\omega_{1}-\omega_{2})T-(\gamma_{1}+\gamma_{2})T}&\rho_{22}(0)&\rho_{23}(0)e^{i2(g-\omega_{2})T-\gamma_{2}T}\\\
\rho_{30}(0)e^{i2(\omega_{1}+\omega_{2})T-(\gamma_{1}+\gamma_{2})T}&\rho_{31}(0)e^{-i2(g-\omega_{1})T-\gamma_{1}T}&\rho_{32}(0)e^{-i2(g-\omega_{2})T-\gamma_{2}T}&\rho_{33}(0)\end{array}\right),$

where $\rho_{ij}(0)$ is the initial values of the entries for density matrix.
For the probe state $|++\rangle$, $\rho_{ij}(0)=1/4$ for any $i$ and $j$. The
probability distribution for the measurement
$\\{|++\rangle\langle++|,|+-\rangle\langle+-|,|-+\rangle\langle-+|,|--\rangle\langle--|\\}$
are

$\displaystyle p_{++}(T)\\!$ $\displaystyle=$
$\displaystyle\\!\frac{1}{4}\\!\left\\{1+e^{-\gamma_{1}T}\\!\cos(2gT)\cos(2\omega_{1}T)+e^{-\gamma_{2}T}\\!\cos(2gT)\cos(2\omega_{2}T)+e^{-(\gamma_{1}+\gamma_{2})T}\\!\cos(2\omega_{1}T)\cos(2\omega_{2}T)\right\\}\\!,$
(61) $\displaystyle p_{+-}(T)\\!$ $\displaystyle=$
$\displaystyle\\!\frac{1}{4}\\!\left\\{1+e^{-\gamma_{1}T}\\!\cos(2gT)\cos(2\omega_{1}T)-e^{-\gamma_{2}T}\\!\cos(2gT)\cos(2\omega_{2}T)-e^{-(\gamma_{1}+\gamma_{2})T}\\!\cos(2\omega_{1}T)\cos(2\omega_{2}T)\right\\}\\!,$
(62) $\displaystyle p_{-+}(T)\\!$ $\displaystyle=$
$\displaystyle\\!\frac{1}{4}\\!\left\\{1-e^{-\gamma_{1}T}\\!\cos(2gT)\cos(2\omega_{1}T)+e^{-\gamma_{2}T}\\!\cos(2gT)\cos(2\omega_{2}T)-e^{-(\gamma_{1}+\gamma_{2})T}\\!\cos(2\omega_{1}T)\cos(2\omega_{2}T)\right\\}\\!,$
(63) $\displaystyle p_{--}(T)\\!$ $\displaystyle=$
$\displaystyle\\!\frac{1}{4}\\!\left\\{1-e^{-\gamma_{1}T}\\!\cos(2gT)\cos(2\omega_{1}T)-e^{-\gamma_{2}T}\\!\cos(2gT)\cos(2\omega_{2}T)+e^{-(\gamma_{1}+\gamma_{2})T}\\!\cos(2\omega_{1}T)\cos(2\omega_{2}T)\right\\}\\!.$
(64)

##  Appendix D Detailed calculation in Hamiltonian estimation with XXZ
coupling

In this appendix we show the detailed calculation in the Hamiltonian
estimation with XXZ coupling. Recall the Hamiltonian for the XXZ model is

$H_{0}=-x_{1}\left(\sigma_{1}^{(1)}\sigma_{1}^{(2)}+\sigma_{2}^{(1)}\sigma_{2}^{(2)}\right)-x_{2}\sigma_{3}^{(1)}\sigma_{3}^{(2)}.$
(65)

Because
$[\sigma_{3}^{(1)}\sigma_{3}^{(2)},\sigma_{1}^{(1)}\sigma_{1}^{(2)}]=[\sigma_{3}^{(1)}\sigma_{3}^{(2)},\sigma_{2}^{(1)}\sigma_{2}^{(2)}]=0$,
the diagonal entries for QFIM can then be calculated as

$\displaystyle\mathcal{F}_{\mathrm{q},x_{1}x_{1}}$ $\displaystyle=$
$\displaystyle
4T^{2}\Big{(}2-2\langle\psi_{0}|\sigma_{3}^{(1)}\sigma_{3}^{(2)}|\psi_{0}\rangle$
(66)
$\displaystyle-\langle\psi_{0}|\sigma_{1}^{(1)}\sigma_{1}^{(2)}+\sigma_{2}^{(1)}\sigma_{2}^{(2)}|\psi_{0}\rangle^{2}\Big{)},$

and
$\mathcal{F}_{\mathrm{q},x_{2}x_{2}}=4T^{2}\left(1-\langle\psi_{0}|\sigma_{3}^{(1)}\sigma_{3}^{(2)}|\psi_{0}\rangle^{2}\right)$.
The off-diagonal entry is

$\displaystyle\mathcal{F}_{\mathrm{q},x_{1}x_{2}}$ $\displaystyle=$
$\displaystyle-4T^{2}\langle\psi_{0}|\sigma_{1}^{(1)}\sigma_{1}^{(2)}+\sigma_{2}^{(1)}\sigma_{2}^{(2)}|\psi_{0}\rangle$
(67)
$\displaystyle\times\left(1+\langle\psi_{0}|\sigma_{3}^{(1)}\sigma_{3}^{(2)}|\psi_{0}\rangle\right).$

From these expressions, one can find the optimal QFIM is (in the basis
$\\{x_{1},x_{2}\\}$)

$\mathcal{F}_{\mathrm{q},\mathrm{opt}}=4\left(\begin{array}[]{cc}2T^{2}&0\\\
0&T^{2}\end{array}\right),$ (68)

which can be obtained when the probe state $|\psi_{0}\rangle$ satisfies

$\displaystyle\langle\psi_{0}|\sigma_{3}^{(1)}\sigma_{3}^{(2)}|\psi_{0}\rangle$
$\displaystyle=$ $\displaystyle 0,$ (69)
$\displaystyle\langle\psi_{0}|\sigma_{1}^{(1)}\sigma_{1}^{(2)}+\sigma_{2}^{(1)}\sigma_{2}^{(2)}|\psi_{0}\rangle$
$\displaystyle=$ $\displaystyle 0.$ (70)

Expand the probe state in the computational basis
$\\{|00\rangle,|01\rangle,|10\rangle,|11\rangle\\}$, i.e.,
$|\psi_{0}\rangle=a|00\rangle+b|01\rangle+c|10\rangle+d|11\rangle$, above
equations reduce to

$\displaystyle|a|^{2}-|b|^{2}-|c|^{2}+|d|^{2}$ $\displaystyle=$ $\displaystyle
0,$ (71) $\displaystyle b^{*}c+bc^{*}$ $\displaystyle=$ $\displaystyle 0.$
(72)

Taking into account the normalization relation, above equations can be
reorganized into

$\displaystyle|a|^{2}+|d|^{2}$ $\displaystyle=$ $\displaystyle\frac{1}{2},$
(73) $\displaystyle|b|^{2}+|c|^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2},$ (74)
$\displaystyle\cos\left(\phi_{c}-\phi_{b}\right)$ $\displaystyle=$
$\displaystyle 0,$ (75)

where $\phi_{b}$ and $\phi_{c}$ are the arguments of $b$ and $c$,
respectively. Any probe state satisfies these conditions is the optimal probe
state. One of them is

$|\psi_{0,\mathrm{opt}}\rangle=\frac{1}{\sqrt{2}}|0\rangle\left(|0\rangle+i|1\rangle\right),$
(76)

which will be used as the probe state in the following calculation. The
evolved state can then be further calculated as

$\displaystyle|\psi_{\mathrm{T},\mathrm{opt}}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left[2\cos(x_{2}T)-e^{-ix_{2}T}\right]|00\rangle$
(77)
$\displaystyle+\frac{i}{\sqrt{2}}e^{-ix_{2}T}\cos\left(2x_{1}T\right)|01\rangle$
$\displaystyle-\frac{1}{\sqrt{2}}\sin\left(2x_{1}T\right)e^{-ix_{2}T}|10\rangle.$

Neglecting the global phase $e^{-ix_{2}T}$, this evolved state is equivalent
to

$|\psi_{\mathrm{T},\mathrm{opt}}\rangle\\!=\\!\frac{1}{\sqrt{2}}\\!\left[e^{i2x_{2}T}|00\rangle+\\!i\\!\cos(2x_{1}T)|01\rangle\\!-\\!\sin(2x_{1}T)\\!|10\rangle\right]\\!.$
(78)

Now we perform a practical measurement
$\\{|++\rangle\langle++|,|+-\rangle\langle+-|,|-+\rangle\langle-+|,|--\rangle\langle--|\\}$
and calculate the corresponding CFIM. The probabilities are

$\displaystyle p_{++}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\left\\{1-\sin\left[2(x_{1}-x_{2})T\right]\right\\},$
(79) $\displaystyle p_{+-}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\left\\{1-\sin\left[2(x_{1}+x_{2})T\right]\right\\},$
(80) $\displaystyle p_{-+}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\left\\{1+\sin\left[2(x_{1}+x_{2})T\right]\right\\},$
(81) $\displaystyle p_{--}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\left\\{1+\sin\left[2(x_{1}-x_{2})T\right]\right\\}.$
(82)

The corresponding CFIM is (in the basis $\\{x_{1},x_{2}\\}$)

$\mathcal{F}_{\mathrm{cl}}=\left(\begin{array}[]{cc}4T^{2}&0\\\ 0&4T^{2}\\\
\end{array}\right).$ (83)

For $x_{2}$, this measurement is optimal, but for $x_{1}$, it is not. Thus, it
is not optimal for joint measurement.

Now we involve the dephasing noise in both qubits. The master equation for the
system is

$\partial_{t}\rho=-i[H,\rho]+\sum_{i=1,2}\frac{\gamma_{i}}{2}\left(\sigma_{3}^{(i)}\rho\sigma_{3}^{(i)}-\rho\right).$
(84)

Taking into account the probe state $|\psi_{0,\mathrm{opt}}\rangle$, the
solution for this equation is

$\displaystyle\rho_{00}(T)$ $\displaystyle=$
$\displaystyle\rho_{00}(0)=\frac{1}{2},$ (85) $\displaystyle\rho_{33}(T)$
$\displaystyle=$ $\displaystyle\rho_{33}(0)=0,$ (86)
$\displaystyle\rho_{03}(T)$ $\displaystyle=$
$\displaystyle\rho_{03}(0)e^{-(\gamma_{1}+\gamma_{2})T}=0.$ (87)

For $\rho_{01}$ and $\rho_{02}$, the differential equations are

$\displaystyle\partial_{t}\rho_{01}$ $\displaystyle=$
$\displaystyle\left(i2x_{2}-\gamma_{2}\right)\rho_{01}-i2x_{1}\rho_{02},$ (88)
$\displaystyle\partial_{t}\rho_{02}$ $\displaystyle=$
$\displaystyle\left(i2x_{2}-\gamma_{1}\right)\rho_{02}-i2x_{1}\rho_{01}.$ (89)

The solutions are

$\displaystyle\rho_{01}(T)$ $\displaystyle=$
$\displaystyle-\frac{i}{2}e^{\left(i2x_{2}-\gamma_{2}\right)T}\cos\left(2x_{1}T\right),$
(90) $\displaystyle\rho_{02}(T)$ $\displaystyle=$
$\displaystyle-\frac{1}{2}e^{\left(i2x_{2}-\gamma_{1}\right)T}\sin\left(2x_{1}T\right).$
(91)

For $\rho_{13}$ and $\rho_{23}$, the differential equations are

$\displaystyle\partial_{t}\rho_{13}$ $\displaystyle=$
$\displaystyle-i2x_{2}\rho_{13}-\gamma_{1}\rho_{13}+i2x_{2}\rho_{23},$ (92)
$\displaystyle\partial_{t}\rho_{23}$ $\displaystyle=$ $\displaystyle
i2x_{1}\rho_{13}-i2x_{2}\rho_{23}-\gamma_{2}\rho_{23}.$ (93)

Taking into account the initial condition, the solutions are $\rho_{13}(T)=0$
and $\rho_{23}(T)=0$. For $\rho_{11}$, $\rho_{12}$ and $\rho_{22}$, the
differential equations are

$\displaystyle\partial_{t}\rho_{12}$ $\displaystyle=$
$\displaystyle-\left(\gamma_{1}+\gamma_{2}\right)\rho_{12}-i2x_{1}\left(\rho_{11}-\rho_{22}\right),$
(94) $\displaystyle\partial_{t}\rho_{11}$ $\displaystyle=$
$\displaystyle-i2x_{1}\left(\rho_{12}-\rho_{21}\right),$ (95)
$\displaystyle\partial_{t}\rho_{22}$ $\displaystyle=$ $\displaystyle
i2x_{1}\left(\rho_{12}-\rho_{21}\right).$ (96)

Taking into account the initial condition the solution for these differential
equations are

$\rho_{12}(T)=-\frac{i}{4}\sin\left(4x_{1}T\right)e^{-(\gamma_{1}+\gamma_{2})T},$
(97)

and

$\displaystyle\rho_{11}(T)$ $\displaystyle=$
$\displaystyle\frac{1}{2}-4x_{1}^{2}\frac{1-e^{-(\gamma_{1}+\gamma_{2})T}\cos\left(4x_{1}T\right)}{16x_{1}^{2}+\left(\gamma_{1}+\gamma_{2}\right)^{2}}$
(98)
$\displaystyle+x_{1}\frac{e^{-(\gamma_{1}+\gamma_{2})T}\left(\gamma_{1}+\gamma_{2}\right)\sin\left(4x_{1}T\right)}{16x_{1}^{2}+\left(\gamma_{1}+\gamma_{2}\right)^{2}}\\!,$

and $\rho_{22}(T)=1/2-\rho_{11}(T)$.

Next we perform the measurement
$\\{|++\rangle\langle++|,|+-\rangle\langle+-|,|-+\rangle\langle-+|,|--\rangle\langle--|\\}$.
The corresponding probabilities are

$\displaystyle p_{++}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\left[1-\sin\left(2x_{1}T\right)\cos\left(2x_{2}T\right)e^{-\gamma_{1}T}+\cos\left(2x_{1}T\right)\sin\left(2x_{2}T\right)e^{-\gamma_{2}T}\right],$
(99) $\displaystyle p_{+-}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\left[1-\sin\left(2x_{1}T\right)\cos\left(2x_{2}T\right)e^{-\gamma_{1}T}-\cos\left(2x_{1}T\right)\sin\left(2x_{2}T\right)e^{-\gamma_{2}T}\right],$
(100) $\displaystyle p_{-+}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\left[1+\sin\left(2x_{1}T\right)\cos\left(2x_{2}T\right)e^{-\gamma_{1}T}+\cos\left(2x_{1}T\right)\sin\left(2x_{2}T\right)e^{-\gamma_{2}T}\right],$
(101) $\displaystyle p_{--}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\left[1+\sin\left(2x_{1}T\right)\cos\left(2x_{2}T\right)e^{-\gamma_{1}T}-\cos\left(2x_{1}T\right)\sin\left(2x_{2}T\right)e^{-\gamma_{2}T}\right].$
(102)

The CFIM for this probability distribution is complicated. Here we consider a
simple case that $\gamma_{1}=\gamma_{2}=\gamma$. With this condition,

$\displaystyle\mathcal{F}_{\mathrm{cl},x_{1}x_{1}}$ $\displaystyle=$
$\displaystyle\mathcal{F}_{\mathrm{cl},x_{2}x_{2}}=2T^{2}\left(\delta_{+}+\delta_{-}\right),$
$\displaystyle\mathcal{F}_{\mathrm{cl},x_{1}x_{2}}$ $\displaystyle=$
$\displaystyle 2T^{2}\left(\delta_{+}-\delta_{-}\right),$

where the coefficients

$\displaystyle\delta_{+}$ $\displaystyle=$
$\displaystyle\frac{\cos^{2}\left[2T(x_{1}+x_{2})\right]}{e^{2\gamma
T}-\sin^{2}\left[2T(x_{1}+x_{2})\right]},$ (103) $\displaystyle\delta_{-}$
$\displaystyle=$
$\displaystyle\frac{\cos^{2}\left[2T(x_{1}-x_{2})\right]}{e^{2\gamma
T}-\sin^{2}\left[2T(x_{1}-x_{2})\right]}.$ (104)

Thus, the total variance is in the form

$\mathrm{Tr}\mathcal{F}_{\mathrm{cl}}^{-1}=\frac{1}{4T^{2}}\left(\frac{1}{\delta_{+}}+\frac{1}{\delta_{-}}\right).$
(105)

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