# Supplemental Material for: High-Fidelity CZ Gates in Double Quantum Dot –
Circuit QED Systems Beyond the Rotating-Wave Approximation

Guangzhao Yang  School of Physical and Mathematical Sciences, Nanyang
Technological University, 21 Nanyang Link, Singapore 637371, Singapore  Marek
Gluza  School of Physical and Mathematical Sciences, Nanyang Technological
University, 21 Nanyang Link, Singapore 637371, Singapore  Si Yan Koh  School
of Physical and Mathematical Sciences, Nanyang Technological University, 21
Nanyang Link, Singapore 637371, Singapore  Calvin Pei Yu Wong  Institute of
Materials Research and Engineering (IMRE), Agency for Science, Technology and
Research (A*STAR), Singapore 138634, Singapore  Kuan Eng Johnson Goh  School
of Physical and Mathematical Sciences, Nanyang Technological University, 21
Nanyang Link, Singapore 637371, Singapore  Institute of Materials Research and
Engineering (IMRE), Agency for Science, Technology and Research (A*STAR),
Singapore 138634, Singapore  Department of Physics, Faculty of Science,
National University of Singapore, Singapore 117551, Singapore  Bent Weber
School of Physical and Mathematical Sciences, Nanyang Technological
University, 21 Nanyang Link, Singapore 637371, Singapore  Hui Khoon Ng  Yale-
NUS College, Singapore 138527, Singapore  Centre for Quantum Technologies,
National University of Singapore, Singapore 117543, Singapore  Teck Seng Koh
School of Physical and Mathematical Sciences, Nanyang Technological
University, 21 Nanyang Link, Singapore 637371, Singapore

###### Contents

  1. I Charge qubit parametric drive
  2. II Derivation of exact Hamiltonians
  3. III Hamiltonian vectorization procedure
  4. IV  Fourier decomposition of exact Hamiltonians
  5. V Analytical approximation for fidelity and high-fidelity conditions
  6. VI Quantum gate with decoherence: superoperator calculations
  7. VII Numerical method
  8. VIII Strong coupling gate time correction

##  I Charge qubit parametric drive

The charge qubit Hamiltonian is
$\hat{H}_{\text{DQD}}=\frac{1}{2}(\epsilon\hat{\tau}_{z}+t_{c}\hat{\tau}_{x})$.
In the eigenbasis,
$\hat{H}^{\prime}_{\text{DQD}}=\hat{U}^{\dagger}\hat{H}_{\text{DQD}}\hat{U}=\frac{\omega_{c}}{2}\hat{\sigma}_{z}$,
where $\omega_{c}=\sqrt{\epsilon^{2}+t_{c}^{2}}$ and
$\hat{U}=\cos\frac{\theta}{2}\hat{I}-i\sin\frac{\theta}{2}\hat{\tau}_{y}$ and
$\tan\theta=t_{c}/\epsilon$. The dipole-field interaction is
$\hat{H}_{\text{int}}=g_{c}\hat{\tau}_{z}\left(\hat{a}+\hat{a}^{\dagger}\right)$.
In the eigenbasis of $\hat{H}_{\text{DQD}}$, it becomes

$\displaystyle\hat{H}_{\text{int}}^{\prime}=\left(\hat{a}+\hat{a}^{\dagger}\right)\left(g_{z}\hat{\sigma}_{z}-g_{x}\hat{\sigma}_{x}\right),$
(S.1)

where $g_{x}=g_{c}\sin\theta=g_{c}\frac{t_{c}}{\omega_{c}}$, and
$g_{z}=g_{c}\cos\theta=g_{c}\frac{\epsilon}{\omega_{c}}$. Since tunnel
coupling is non-negative, a drive applied to $t_{c}$ can be written as,

$\displaystyle t_{c}(t)=t_{c0}+\Omega\sin(\omega t+\phi^{\prime})\geq 0,$
(S.2)

where $t_{c0}$ is a positive constant and $\Omega$ is the amplitude. The
condition $t_{c}\geq 0$ leads to $t_{c0}\geq|2\Omega|$. Orbital detuning is
then

$\displaystyle\epsilon(t)=\sqrt{\omega_{c}^{2}-(t_{c0}+\Omega\sin(\omega
t+\phi^{\prime}))^{2}}.$ (S.3)

Since $\epsilon(t)$ has to be real, this implies
$t_{c0}+|\Omega|\leq\omega_{c}$.

The red-sideband transition terms arise from a unitary transformation of the
$g_{x}\hat{\sigma}_{x}$ term. To maximize coupling strength, and satisfy the
preceding conditions simultaneously, we choose $\Omega=t_{c0}=\omega_{c}/2$.
Therefore,

$\displaystyle g_{x}(t)=$ $\displaystyle
g_{c}\frac{t_{c}(t)}{\omega_{c}}=g_{c}\frac{\Omega}{\omega_{c}}(1+\sin(\omega
t+\phi^{\prime}))=\frac{g_{c}}{2}(1-\cos(\omega t+\phi^{\prime}+\pi/2))$ (S.4)
$\displaystyle=$ $\displaystyle g_{c}\sin(\frac{\omega t+\phi}{2})^{2},$ (S.5)

where we have made the substitution $\phi=\phi^{\prime}+\pi/2$ in the last
line. Since $g_{z}(t)^{2}+g_{x}(t)^{2}=g_{c}^{2}$, we have the qubit-field
interaction,

$\displaystyle\hat{H}_{\text{int}}^{\prime}=g_{c}\left(\hat{a}+\hat{a}^{\dagger}\right)\left(\sqrt{1-\sin(\frac{\omega
t+\phi}{2})^{4}}\hat{\sigma}_{z}-\sin(\frac{\omega
t+\phi}{2})^{2}\hat{\sigma}_{x}\right),$ (S.6)

from the simultaneous parametric drives

$\displaystyle t_{c}(t)=$ $\displaystyle\Omega(1+\sin(\omega
t+\phi^{\prime}))=2\Omega\sin(\frac{\omega t+\phi}{2})^{2},$ (S.7)
$\displaystyle\epsilon(t)=$
$\displaystyle\sqrt{\omega_{c}^{2}-\Omega^{2}(1+\sin(\omega
t+\phi^{\prime}))^{2}}=2\Omega\sqrt{1-\sin(\frac{\omega t+\phi}{2})^{4}}.$
(S.8)

##  II Derivation of exact Hamiltonians

To derive the exact and red-sideband Hamiltonians from the spin and charge
qubit Hamiltonians $\hat{H}_{s}^{\prime},\hat{H}_{c}^{\prime}$ in the main
text, we use the unitary transformations

$\displaystyle\hat{U}_{s}(t)$ $\displaystyle=e^{-i\omega
t\left(\hat{a}^{\dagger}\hat{a}+\sum_{q}\frac{1}{2}\hat{\sigma}_{z}^{(q)}\right)}e^{-i\left(\phi+\frac{\pi}{2}\right)\frac{1}{2}\sum_{q}\hat{\sigma}_{z}^{(q)}}e^{-it\left(\Delta_{0}\hat{a}^{\dagger}\hat{a}+\Omega^{(j)}\hat{\sigma}_{y}^{(j)}\right)}e^{i\frac{\pi}{4}\sum_{q}\hat{\sigma}_{x}^{(q)}}\quad\text{(spin)},$
(S.9) $\displaystyle\hat{U}_{c}(t)$
$\displaystyle=e^{-\sum_{q}\frac{i\pi}{4}\hat{\sigma}_{z}^{(q)}}e^{-\sum_{q}\frac{i\pi}{4}\hat{\sigma}_{y}^{(q)}}e^{-\sum_{q}\frac{i\pi}{4}\hat{\sigma}_{z}^{(q)}}e^{-i\omega_{r}t\hat{a}^{\dagger}\hat{a}-i\sum_{q}\left(\Omega
t-\frac{\pi}{4}\right)\hat{\sigma}_{y}^{(q)}}e^{i\frac{\pi}{4}\sum_{q}\hat{\sigma}_{x}^{(q)}}\quad\text{(charge)},$
(S.10)

where the index $q=1,2$, labels the qubits. The unitary transformations allow
us to obtain the exact Hamiltonians, which are functions of the coupling
strength, drive parameters $(\Omega,\omega,\phi)$ and time:

$\displaystyle\hat{H}_{E,s}^{(j)}(g_{s},\Omega,\omega,\phi,t)$
$\displaystyle=\hat{H}_{R}^{(j)}+\hat{V}_{\text{red}}^{(j)}(t)+\hat{V}_{\text{blue}}^{(j)}(t)+\hat{V}_{\text{long}}^{(j)}(t)+\hat{V}_{\text{qubit}}^{(j)}(t),$
(S.11) $\displaystyle\hat{H}_{E,c}^{(j)}(g_{c}/2,\Omega,\omega,\phi,t)$
$\displaystyle=\hat{H}_{R}^{(j)}+\hat{V}_{\text{red},t_{c}}^{(j)}(t)+\hat{V}_{\text{blue},t_{c}}^{(j)}(t)+\hat{V}_{\text{long},\epsilon}^{(j)}(t).$
(S.12)

Here, we explicitly indicate with a superscript index $j$ to label the qubit
that is coupled to the photon. This was omitted for notational simplicity in
the main text.

After applying RWA, both the exact Hamiltonians simplify to the red-sideband
Hamiltonian, which is now only a function of effective qubit-photon coupling
strength $g$ and driving phase $\phi$,

$\displaystyle\hat{H}_{R}^{(j)}(g\in\\{g_{s},g_{c}/2\\},\phi)=\frac{ig}{2}(e^{i\phi}\hat{a}\hat{\sigma}_{+}^{(j)}-e^{-i\phi}\hat{a}^{\dagger}\hat{\sigma}_{-}^{(j)}),$
(S.13)

We note that in the red-sideband Hamiltonian, the effective charge qubit-
photon interaction strength is $g_{c}/2$ compared to the effective spin qubit-
photon interaction strength $g_{s}$.

The time-dependent terms in spin qubit Hamiltonian are

$\displaystyle\hat{V}_{\text{qubit}}^{(j)}(t)$ $\displaystyle=\Omega
e^{2i\Omega t}\sin(2\omega t+2\phi)\hat{\sigma}_{+}^{(j)}+\Omega e^{-2i\Omega
t}\sin(2\omega t+2\phi)\hat{\sigma}_{-}^{(j)}+\Omega\cos(2\omega
t+2\phi)\hat{\sigma}^{(j)}_{z}$ (S.14)
$\displaystyle\hat{V}_{\text{red}}^{(j)}(t)$
$\displaystyle=\frac{ig_{s}}{2}\left(-e^{-i2\omega
t-i\phi}\hat{a}\hat{\sigma}_{+}^{(j)}+e^{i2\omega
t+i\phi}\hat{a}^{\dagger}\hat{\sigma}_{-}^{(j)}\right)$ (S.15)
$\displaystyle\hat{V}_{\text{blue}}^{(j)}(t)$ $\displaystyle=-g_{s}\sin(\omega
t+\phi)\left(e^{-i(4\Omega+\omega)t}\hat{a}\hat{\sigma}_{-}^{(j)}+e^{i(4\Omega+\omega)t}\hat{a}^{\dagger}\hat{\sigma}_{+}^{(j)}\right)$
(S.16) $\displaystyle\hat{V}_{\text{long}}^{(j)}(t)$
$\displaystyle=-g_{s}\cos(\omega
t+\phi)\left(e^{-i(2\Omega+\omega)t}\hat{a}+e^{i(2\Omega+\omega)t}\hat{a}^{\dagger}\right)\hat{\sigma}^{(j)}_{z},$
(S.17)

and those in the charge qubit Hamiltonian are

$\displaystyle\hat{V}_{\text{red},t_{c}}^{(j)}(t)$
$\displaystyle=\frac{ig_{c}}{4}\left((e^{-2i\omega t-i\phi}-2e^{-i\omega
t})\hat{a}\hat{\sigma}_{+}^{(j)}-(e^{2i\omega t+i\phi}-2e^{i\omega
t})\hat{a}^{\dagger}\hat{\sigma}_{-}^{(j)}\right)$ (S.18)
$\displaystyle\hat{V}_{\text{blue},t_{c}}^{(j)}(t)$
$\displaystyle=ig_{c}\sin(\frac{\omega
t+\phi}{2})^{2}\left(e^{-i(4\Omega+\omega)t}\hat{a}\hat{\sigma}_{-}^{(j)}-e^{i(4\Omega+\omega)t}\hat{a}^{\dagger}\hat{\sigma}_{+}^{(j)}\right)$
(S.19) $\displaystyle\hat{V}_{\text{long},\epsilon}^{(j)}(t)$
$\displaystyle=g_{c}\sqrt{1-\sin(\frac{\omega
t+\phi}{2})^{4}}\left(e^{-i(2\Omega+\omega)t}\hat{a}+e^{i(2\Omega+\omega)t}\hat{a}^{\dagger}\right)\hat{\sigma}^{(j)}_{z}$
(S.20)

##  III Hamiltonian vectorization procedure

In this section we will describe ideas for efficient bookkeeping in both
analytical and numerical computations, which enable us to derive the exact
Hamiltonian without RWA in the interaction picture, which is defined by
transformation (S.9) and (S.10) from the system Hamiltonian. This will make
use of representation theory, namely we will keep track of interaction picture
transformations needed for the derivation of the CZ gate which are exactly
solvable, i.e. either Gaussian or single-qubit unitaries.

First, we will denote by

$\displaystyle\hat{\Theta}^{(q)}=\frac{1}{\sqrt{2}}\begin{pmatrix}\hat{\sigma}_{0}^{(q)},&\hat{\sigma}_{x}^{(q)},&\hat{\sigma}_{y}^{(q)},&\hat{\sigma}_{z}^{(q)}\end{pmatrix}$
(S.21)

the vector collecting the 4 operator basis (where $\hat{\sigma}_{0}^{(q)}$ is
the identity operator in qubit Hilbert space) generators of qubit $q$.
Secondly, we will consider

$\displaystyle\hat{\Theta}^{(0)}=\begin{pmatrix}\mathbbm{\hat{1}},&\hat{a},&\hat{a}^{\dagger},&\hat{a}^{\dagger}\hat{a}\end{pmatrix}$
(S.22)

which will allow us to keep track of transformations on the photon mode. We
will use $\hat{\Theta}^{(q)}$ and $\hat{\Theta}^{(0)}$ to express the
Hamiltonian of the full system. To this end, it will be useful to introduce a
single vectorization index $\lambda$ which will allow us to keep track of
products of operators $\hat{\Theta}^{(q)}$ on qubit $q$ with operators on the
photon. For any $\mu_{0},\mu_{1},\mu_{2}\in\\{0,1,2,3\\}$ which label the
operator of qubit $q$ and the oscillator, respectively, we define
$\lambda(\mu_{0},\mu_{1},\mu_{2})$ in a one-to-one correspondence such that
$\hat{\Theta}_{\lambda}=\hat{\Theta}^{(0)}_{\mu_{0}}\hat{\Theta}^{(1)}_{\mu_{1}}\hat{\Theta}^{(2)}_{\mu_{2}}$.
For example
$\hat{\Theta}_{\lambda(2,2,0)}=\hat{a}^{\dagger}\hat{\sigma}_{y}^{(1)}$. In
the following, it will be possible to express any operator of interest
$\hat{O}$ as

$\displaystyle\hat{O}=\sum_{\lambda}O^{\lambda}\hat{\Theta}_{\lambda}.$ (S.23)

In particular, this will be true for the Hamiltonians and for the generators
of the interaction picture transformations involved in the derivations.

Next, we will discuss how the relevant transformations can be expressed in the
formalism using $\hat{\Theta}^{(j)}$ and $\hat{\Theta}^{(0)}$. Firstly let us
consider a rotation
$\hat{R}^{(j)}_{\eta}(\theta)=\exp\left[-i\theta\hat{\sigma}^{(j)}_{\eta}\right]$
for qubit $j$ generated by $\sigma_{\mu}$ by the angle $\theta$

$\displaystyle\hat{R}^{(j)}_{\eta}(\theta)^{\dagger}\sigma^{(j)}_{\eta^{\prime}}\hat{R}^{(j)}_{\eta}(\theta)=\exp\left[i\theta\hat{\sigma}^{(j)}_{\eta}\right]\hat{\sigma}^{(j)}_{\eta^{\prime}}\exp\left[-i\theta\hat{\sigma}^{(j)}_{\eta}\right]$
$\displaystyle=\cos(2\theta)\sigma^{(j)}_{\eta^{\prime}}-\epsilon_{\eta,\eta^{\prime},\eta^{\prime\prime}}\sin(2\theta)\hat{\sigma}^{(j)}_{\eta^{\prime\prime}}+2(\sin\theta)^{2}\delta_{\eta\eta^{\prime}}\hat{\sigma}^{(j)}_{\eta}$
(S.24)

When $j=1$, we assign $\eta=\mu$, and we assign $\eta=\mu^{\prime}$ for $j=2$.
Equivalently, we can write this using the $\hat{\Theta}$ notation as

$\displaystyle\hat{O}=\langle O,\hat{\Theta}\rangle$ (S.25)

where $O$ is the vector of coefficients as in Eq. (S.23) and
$\langle\cdot,\cdot\rangle$ implements the summation of the coefficients with
the operator entries in the $\hat{\Theta}^{(j)}$, similarly to the ordinary
Euclidean scalar product or Einstein summation convention. By linearity we
have the transformation we have

$\displaystyle\hat{R}^{(j)}_{\eta}(\theta)^{\dagger}\hat{O}\hat{R}^{(j)}_{\eta}(\theta)=$
$\displaystyle\langle\hat{O},\hat{\Theta}^{(o)}\hat{R}^{(j)}_{\eta}(\theta)^{\dagger}\hat{\Theta}^{(j)}\hat{R}^{(j)}_{\eta}(\theta)\hat{\Theta}^{(j^{\prime})}\rangle$
(S.26) $\displaystyle=$
$\displaystyle\langle\hat{O},\mathscr{R}^{(j)}_{\eta}(\theta)\hat{\Theta}\rangle$
(S.27) $\displaystyle=$
$\displaystyle\langle\mathscr{R}^{(j)}_{\eta}(\theta)^{T}\hat{O},\hat{\Theta}\rangle$
(S.28)

where

$\displaystyle\mathscr{R}_{0}=\mathbbm{\hat{1}}_{4},\quad\mathscr{R}_{1}(\theta)$
$\displaystyle=\begin{pmatrix}1&0&0&0\\\ 0&1&0&0\\\
0&0&\cos{2\theta}&-\sin{2\theta}\\\
0&0&\sin{2\theta}&\cos{2\theta}\end{pmatrix},\quad\mathscr{R}_{2}(\theta)$
$\displaystyle=\begin{pmatrix}1&0&0&0\\\ 0&\cos{2\theta}&0&\sin{2\theta}\\\
0&0&1&0\\\
0&-\sin{2\theta}&0&\cos{2\theta}\end{pmatrix},\quad\mathscr{R}_{3}(\theta)$
$\displaystyle=\begin{pmatrix}1&0&0&0\\\ 0&\cos{2\theta}&-\sin{2\theta}&0\\\
0&\sin{2\theta}&\cos{2\theta}&0\\\ 0&0&0&1\end{pmatrix}$ (S.29)

and

$\displaystyle\mathscr{R}^{(1)}_{\mu}=\mathbbm{\hat{1}}_{o}\otimes\mathscr{R}_{\mu}\otimes\mathscr{R}_{0},\quad\mathscr{R}^{(2)}_{\mu^{\prime}}$
$\displaystyle=\mathbbm{\hat{1}}_{o}\otimes\mathscr{R}_{0}\otimes\mathscr{R}_{\mu^{\prime}}$
(S.30)

This switch from the adjoint action by $\mathscr{R}^{(j)}_{\eta}$ on the
operators to linear action on the coefficients is equivalent to the exact
solution (S.24). It is useful, however, for bookkeping: If we compose
rotations then we can simply multiply the corresponding matrices.

Next we consider operations on operators acting on the photon mode using the
formalism with $\hat{\Theta}^{(0)}$. The transformation
$\hat{U}^{(0)}(\theta)=\exp\left[-i\theta\hat{a}^{\dagger}\hat{a}\right]$ acts
as

$\displaystyle\exp\left[i\theta\hat{a}^{\dagger}\hat{a}\right]\hat{a}\exp\left[-i\theta\hat{a}^{\dagger}\hat{a}\right]=\hat{a}e^{-i\theta}=\left(\exp\left[i\theta\hat{a}^{\dagger}\hat{a}\right]\hat{a}^{\dagger}\exp\left[-i\theta\hat{a}^{\dagger}\hat{a}\right]\right)^{\dagger}$
$\displaystyle=\left(\hat{a}^{\dagger}e^{i\theta}\right)^{\dagger}$ (S.31)

and trivially on the occupation number operator

$\displaystyle\exp\left[i\theta\hat{a}^{\dagger}\hat{a}\right]\hat{a}^{\dagger}\hat{a}\exp\left[-i\theta\hat{a}^{\dagger}\hat{a}\right]$
$\displaystyle=\hat{a}^{\dagger}\hat{a}$ (S.32)

This means that given a vector of coefficients $O$ defining the operator
$\hat{O}=\langle O,\hat{\Theta}^{(0)}\rangle$ acting on the photon mode, we
can write its rotation as

$\displaystyle\hat{U}^{(0)}(\theta)^{\dagger}\hat{O}\hat{U}^{(0)}(\theta)$
$\displaystyle=\langle\hat{O},\hat{U}^{(0)}(\theta)^{\dagger}\hat{\Theta}^{(0)}\hat{U}^{(0)}(\theta)\hat{\Theta}^{(1)}\hat{\Theta}^{(2)}\rangle$
(S.33)
$\displaystyle=\langle\hat{O},\mathscr{U}^{(0)}(\theta)\hat{\Theta}\rangle$
(S.34)
$\displaystyle=\langle\mathscr{U}^{(0)}(\theta)\hat{O},\hat{\Theta}\rangle$
(S.35)

where

$\displaystyle\mathscr{U}^{(0)}(\theta)$
$\displaystyle=\text{diag}\left(1,e^{-i\theta},e^{i\theta},1\right)\otimes\mathbbm{\hat{1}}_{4}\otimes\mathbbm{\hat{1}}_{4}\
.$ (S.36)

We then conclude that there are three $4\times 4$ matrices for the qubit
rotation and one $4\times 4$ for the photon rotation, where the identity
generators never changes due to the rotation unitary operators. The photon
space generators (identity, number, annihilation, and creation operator)
transformation can be described by a diagonal matrix. We are able to conclude
the matrices since there are restricted set of generators.

According to the transformation of Hamiltonian:

$\displaystyle\hat{H}=\hat{U}^{\dagger}\left(\hat{H}_{0}-i\frac{\partial}{\partial
t}\right)\hat{U}$ (S.37)

we are able to write the superoperator form of the time-dependent unitary
transformation:

$\displaystyle\mathscr{U}(t;\omega_{r},\omega_{j},\omega_{k})\hat{H}=\mathscr{U}^{(0)}(\omega_{r}t)\mathscr{R}_{\mu}^{(j)}(\omega_{j}t)\mathscr{R}_{\mu^{\prime}}^{(k)}(\omega_{k}t)\hat{H}-\omega_{r}\hat{\Theta}^{(0)}_{3}-\omega_{j}\hat{\Theta}^{(j)}_{\mu}-\omega_{k}\hat{\Theta}^{(k)}_{\mu^{\prime}},$
(S.38)

whereas the time-independent transformations take a simpler form:

$\displaystyle\mathscr{U}(\theta_{0},\theta_{j},\theta_{k})\hat{H}=\mathscr{U}^{(0)}(\theta_{0})\mathscr{R}_{\mu}^{(j)}(\theta_{j})\mathscr{R}_{\mu^{\prime}}^{(k)}(\theta_{k})\hat{H}$
(S.39)

It can be shown that we can focus on the coupled-qubit subspace of the
$\hat{H}_{0}$ since after the first transformation $\hat{U}_{1}(t)=e^{-i\omega
t\left(\hat{a}^{\dagger}\hat{a}+\sum_{q}\frac{1}{2}\hat{\sigma}_{z}^{(q)}\right)}$,
the idle qubit Hamiltonian is rotated away.

We can therefore write the transformation superoperator acting on the
vectorized elements of $\hat{H}_{0}^{(j)}$ in

$\displaystyle\hat{H}=\mathscr{U}\left(0,-\frac{\pi}{4},-\frac{\pi}{4}\right)\mathscr{U}\left(t;2\Omega,\Omega,0\right)\mathscr{U}\left(0,\frac{\phi}{2}+\frac{\pi}{4},\frac{\phi}{2}+\frac{\pi}{4}\right)\mathscr{U}\left(t;\omega,\omega,\omega\right)\hat{H}_{0}^{(j)}$
(S.40)

After all four unitary transformations, we want to represent the vectorized
operator in the new qubit generators defined by:

$\displaystyle\hat{\Theta}^{(q)}_{+}+\hat{\Theta}^{(q)}_{-}$
$\displaystyle=\hat{\Theta}^{(q)}_{1}$
$\displaystyle-i\hat{\Theta}^{(q)}_{+}+i\hat{\Theta}^{(q)}_{-}$
$\displaystyle=\hat{\Theta}^{(q)}_{2}$

in which
$\hat{\Theta}^{(q)}_{\\{+,-\\}}=\frac{1}{\sqrt{2}}\sigma^{(q)}_{\\{+,-\\}}$,
and the zeroth and third generators are unchanged. It can be easily deduced
that the qubit-subspace operator under the new generators representation can
be calculated by:

$\displaystyle\begin{pmatrix}O^{\prime 0}\\\ O^{\prime 1}\\\ O^{\prime 2}\\\
O^{\prime 3}\end{pmatrix}=\begin{pmatrix}1&0&0&0\\\ 0&1&-i&0\\\ 0&1&i&0\\\
0&0&0&1\end{pmatrix}\begin{pmatrix}O^{0}\\\ O^{1}\\\ O^{2}\\\
O^{3}\end{pmatrix}$ (S.41)

where elements $O^{\prime\eta}$ in the $\hat{\Theta}^{(q)}_{\\{0,+,-,3\\}}$
generator can be expressed by $O^{\eta}$, elements of
$\hat{\Theta}^{(q)}_{\\{0,1,2,3\\}}$ generator.

$\displaystyle\hat{H}_{R}(\zeta)$
$\displaystyle=\sum_{\lambda}h_{R}^{\lambda}(\zeta)\,\hat{\Theta}_{\lambda}$
(S.42) $\displaystyle\hat{H}_{E}(\zeta;t)$
$\displaystyle=\sum_{\lambda}h_{E}^{\lambda}(\zeta;t)\,\hat{\Theta}_{\lambda}$
(S.43)

##  IV  Fourier decomposition of exact Hamiltonians

The exact Hamiltonian (S.11) and (S.12) can be decomposed by complex periodic
function $e^{i\varpi t}$ at angular frequency $\varpi$. Using the vectorized
formulation (S.42), we are able to write
$h^{\lambda}_{E}(\zeta;\tau)=h^{\lambda}(\zeta;0)+\sum_{\tilde{\varpi}}h^{\lambda}_{E}(\zeta;i\tilde{\varpi}\tau)$,
where $\tilde{\varpi}$ denote all nonzero frequencies. $h^{\lambda}(\zeta;0)$
denotes the constant terms in the Hamiltonian.

In the following discussion, the vectorization index
$\lambda(\mu_{0},\mu_{1},\mu_{2})$ are adjusted to $\lambda(\mu_{0},\mu_{j})$
where $j$ labels the coupled qubit, since the uncoupled qubit does not
contribute terms in the interaction picture.

In our case, they are $h_{E}^{\lambda(1,+)}(\zeta;0)=\frac{ie^{i\phi}}{2}$ and
$h_{E}^{\lambda(2,-)}(\zeta;0)=-\frac{ie^{-i\phi}}{2}$ for both spin and
charge qubit exact Hamiltonians.

For the spin qubit Hamiltonian, all Fourier components can be found by
expanding the sine and cosine functions, e.g.,

$\displaystyle h_{E}^{\lambda(1,-)}(\zeta;\tau)$
$\displaystyle=-\sin(\tilde{\omega}\tau+\phi)e^{-i(4\tilde{\Omega}+\tilde{\omega})\tau}=\frac{ie^{i\phi}}{2}e^{-i4\tilde{\Omega}\tau}-\frac{ie^{-i\phi}}{2}e^{-i(4\tilde{\Omega}+2\tilde{\omega})\tau}$
$\displaystyle=h_{E}^{\lambda(1,-)}(\zeta;-i4\tilde{\Omega}\tau)+h_{E}^{\lambda(1,-)}(\zeta;-i(4\tilde{\Omega}+2\tilde{\omega})\tau)$

The terms in $\hat{V}_{\text{qubit}}(t)=\sum_{\mu_{j}\neq
0}h_{E}^{\lambda(0,\mu_{j})}(\zeta;\tau)\hat{\Theta}_{\lambda(0,\mu_{j})}$ are
proportional to $\frac{\tilde{\Omega}}{2}$ for
$\tilde{\varpi}\in\pm\\{2\tilde{\omega},2(\tilde{\omega}+\tilde{\Omega}),2(\tilde{\omega}-\tilde{\Omega})\\}$.
The rest of the terms have non-zero frequencies
$\tilde{\varpi}\in\pm\\{2\tilde{\Omega},4\tilde{\Omega},2(\tilde{\omega}+\tilde{\Omega}),2(\tilde{\omega}+2\tilde{\Omega}),2\tilde{\omega}\\}$.

The Fourier components of the charge qubit Hamiltonian can be found in the
same way except $h_{E}^{\lambda(1,3)}$ and $h_{E}^{\lambda(2,3)}$
corresponding to the longitudinal interaction
$\hat{V}_{\text{long},\epsilon}(t)$. The $\epsilon(t)$ drive can be decomposed
by

$\displaystyle\sqrt{1-\sin(\frac{\omega t+\phi}{2})^{4}}$
$\displaystyle=\sum_{u=0}^{\infty}\frac{(-1)^{u}\sqrt{\pi}}{2u!\Gamma\left(\frac{3}{2}-u\right)}\left(\frac{1}{2i}\right)^{4u}\left(e^{\frac{i\omega
t+i\phi}{2}}-e^{-\frac{i\omega t+i\phi}{2}}\right)^{4u}$
$\displaystyle=\sum_{u=0}^{\infty}\sum_{v=0}^{4u}\frac{(-1)^{u+v}\sqrt{\pi}C^{4u}_{v}}{2u!\Gamma\left(\frac{3}{2}-u\right)(2i)^{4u}}\,e^{i(2u-v)(\omega
t+\phi)}$ (S.44)
$\displaystyle=\sum_{u=0}^{\infty}\sum_{v=0}^{4u}E_{uv}\,e^{i(2u-v)(\omega
t+\phi)}$

wherein the relationship between the generalized binomial coefficient and the
gamma function
$C^{\frac{1}{2}}_{n}=\frac{\Gamma\left(\frac{3}{2}\right)}{\Gamma\left(n+1\right)\Gamma\left(\frac{3}{2}-n\right)}$
has been used.

The longitudinal terms contain constant when
$\tilde{\Omega}=\frac{\tilde{\omega}_{r}}{2}\left(1-\frac{1}{p}\right)$, where
$p=2u-v$ and $-2u\leq p\leq 2u,p\neq 0$. Therefore, we denote the longitudinal
interaction Hamiltonian
$\hat{V}_{\text{long},\epsilon}=(D(\tau)\hat{a}+D(\tau)^{*}\hat{a}^{\dagger})\hat{\sigma}^{(j)}_{z}$,
where
$D(\tau)=2\sqrt{1-\sin(\frac{\tilde{\omega}\tau+\phi}{2})^{4}}e^{-i\tilde{\omega}_{r}\tau}$.
The longitudinal interaction with frequency $\varpi$ is then

$\displaystyle h_{E}^{\lambda(1,3)}(\zeta;i\tilde{\varpi}\tau)$
$\displaystyle=\left(2e^{ip\phi}\sum_{p\tilde{\omega}-\tilde{\omega}_{r}=\tilde{\varpi}}E_{uv}\right)e^{i\tilde{\varpi}\tau}$
$\displaystyle h_{E}^{\lambda(2,3)}(\zeta;i\tilde{\varpi}\tau)$
$\displaystyle=\left(2e^{ip\phi}\sum_{p\tilde{\omega}+\tilde{\omega}_{r}=\tilde{\varpi}}E_{uv}\right)e^{i\tilde{\varpi}\tau}$

It is noted that the magnitude of the large $\tilde{\varpi}$ ($p$ is large) is
negligible compared to that of the small $\tilde{\varpi}$ ($p$ is small),
since $C^{4u}_{2u-p_{1}}\ll C^{4u}_{2u-p_{2}}$ if $p_{1}>p_{2}$ when $u$ is
large. The terms with $u_{1}$ is much smaller than that with $u_{2}$ when
$u_{1}>u_{2}$ since $2^{-4u_{1}}\ll 2^{-4u_{2}}$ when $u_{1},u_{2}$ is large.

##  V Analytical approximation for fidelity and high-fidelity conditions

The fidelity of the exact unitary evolution can be approximated by truncating
the photon Fock space at $N$,

$\displaystyle
F=\frac{1}{4N}\Bigg{|}\Tr[\Bigg{(}\mathcal{T}_{\rightarrow}\prod_{\zeta}\exp[i\int_{\tau_{\text{i}\zeta}}^{\tau_{\text{f}\zeta}}d\tau\hat{H}_{R}(\zeta;\tau)\Bigg{]}\hat{Z}^{\dagger}\hat{M}_{0}\Bigg{)}\Bigg{(}\hat{M}_{0}\hat{Z}\mathcal{T}_{\leftarrow}\prod_{\zeta}\exp[-i\int_{\tau_{\text{i}\zeta}}^{\tau_{\text{f}\zeta}}d\tau\hat{H}_{E}(\zeta;\tau)\Bigg{]}\Bigg{)}\Bigg{]}\Bigg{|}$
(S.45)

Calculating
$\hat{Z}^{\dagger}\hat{M}_{0}\hat{Z}=\ket{0}\bra{0}\otimes\hat{\sigma}_{0}\otimes\hat{\sigma}_{0}$,
we can write

$\displaystyle
F=\frac{1}{4}\Bigg{|}\Tr\bra{0}\hat{S}_{E}(5)\hat{S}_{E}(4)\hat{S}_{E}(3)\hat{S}_{E}(2)\hat{S}_{E}(1)\hat{S}_{R}(1)^{\dagger}\hat{S}_{R}(2)^{\dagger}\hat{S}_{R}(3)^{\dagger}\hat{S}_{R}(4)^{\dagger}\hat{S}_{R}(5)^{\dagger}\ket{0}\Bigg{|}.$
(S.46)

Expanding each sideband transition according to Neumann series until the
second order,

$\displaystyle\hat{S}_{R}(\zeta;\Delta\tau_{\zeta},\phi_{\zeta})^{\dagger}$
$\displaystyle\approx\mathbbm{\hat{1}}+i\int_{\tau_{\text{i}\zeta}}^{\tau_{\text{f}\zeta}}d\tau\hat{H}_{R}(\zeta)-\frac{1}{2}\left(\int_{\tau_{\text{i}\zeta}}^{\tau_{\text{f}\zeta}}d\tau\hat{H}_{R}(\zeta)\right)^{2}$
$\displaystyle=\mathbbm{\hat{1}}+i\hat{I}_{R1}(\zeta)-\hat{I}_{R2}(\zeta)$
(S.47) $\displaystyle\hat{S}_{E}(\zeta;\Delta\tau_{\zeta},\phi_{\zeta})$
$\displaystyle\approx\mathbbm{\hat{1}}-i\int_{\tau_{\text{i}\zeta}}^{\tau_{\text{f}\zeta}}d\tau\hat{H}_{E}(\zeta;\tau)-\int_{\tau_{\text{i}\zeta}}^{\tau_{\text{f}\zeta}}d\tau_{1}\int_{\tau_{\text{i}\zeta}}^{\tau_{1}}d\tau_{2}\,\hat{H}_{E}(\zeta;\tau_{1})\hat{H}_{E}(\zeta;\tau_{2})$
$\displaystyle=\mathbbm{\hat{1}}-i\hat{I}_{E1}(\zeta)-\hat{I}_{E2}(\zeta)$
(S.48)

the analytical approximation to fidelity up to the second order is

$\displaystyle
F_{2}=\Bigg{|}1+\frac{1}{4}\Tr\bra{0}\Bigg{(}-\sum_{\zeta}\hat{I}_{R2}(\zeta)-\sum_{\zeta}\hat{I}_{E2}(\zeta)-\sum_{\zeta<\zeta^{\prime}}\hat{I}_{R1}(\zeta)\hat{I}_{R1}(\zeta^{\prime})-\sum_{\zeta>\zeta^{\prime}}\hat{I}_{E1}(\zeta)\hat{I}_{E1}(\zeta^{\prime})+\sum_{\zeta\zeta^{\prime}}\hat{I}_{E1}(\zeta)\hat{I}_{R1}(\zeta^{\prime})\Bigg{)}\ket{0}\Bigg{|}.$
(S.49)

We gain insight from calculating the first order and second exact Hamiltonian
integrals,

$\displaystyle\hat{I}_{E1}(\zeta)$
$\displaystyle=\sum_{\lambda}\hat{\Theta}_{\lambda}\int_{\tau_{\text{i}\zeta}}^{\tau_{\text{f}\zeta}}d\tau\,h^{\lambda}_{E}(\zeta;\tau)=\sum_{\lambda}\hat{\Theta}_{\lambda}I^{\lambda}_{E1}(\zeta),$
(S.50) $\displaystyle\hat{I}_{E2}(\zeta)$
$\displaystyle=\sum_{\lambda\lambda^{\prime}}\hat{\Theta}_{\lambda}\hat{\Theta}_{\lambda^{\prime}}\int_{\tau_{\text{i}\zeta}}^{\tau_{\text{f}\zeta}}d\tau_{1}\int_{\tau_{\text{i}\zeta}}^{\tau_{1}}d\tau_{2}\,h^{\lambda}_{E}(\zeta;\tau_{1})h^{\lambda^{\prime}}_{E}(\zeta;\tau_{2})=\sum_{\lambda\lambda^{\prime}}\hat{\Theta}_{\lambda}\hat{\Theta}_{\lambda^{\prime}}I^{\lambda\lambda^{\prime}}_{E2}(\zeta).$
(S.51)

Note that for the Hamiltonians, $h^{\lambda}(\zeta;0)=h^{\lambda}_{R}(\zeta)$
when $\lambda=5,10$, and $h^{\lambda}(\zeta;0)=0$ otherwise. Therefore, the
integrals have the form

$\displaystyle I^{\lambda}_{R1}(\zeta)=$ $\displaystyle
h^{\lambda}(\zeta;0)\Delta\tau_{\zeta}$ (S.52) $\displaystyle
I^{\lambda\lambda^{\prime}}_{R2}(\zeta)=$
$\displaystyle\frac{\Delta\tau_{\zeta}^{2}}{2}h^{\lambda}(\zeta;0)h^{\lambda^{\prime}}(\zeta;0)=\frac{1}{2}I^{\lambda}_{R1}(\zeta)I^{\lambda^{\prime}}_{R1}(\zeta)$
(S.53) $\displaystyle I^{\lambda}_{E1}(\zeta)=$ $\displaystyle
h^{\lambda}(\zeta;0)\Delta\tau_{\zeta}+\sum_{\tilde{\varpi}}\frac{h^{\lambda}_{E}(\zeta;i\tilde{\varpi}\tau)}{i\tilde{\varpi}}\Big{|}_{\tau_{\text{i}\zeta}}^{\tau_{\text{f}\zeta}}$
(S.54) $\displaystyle I^{\lambda\lambda^{\prime}}_{E2}(\zeta)=$
$\displaystyle\frac{\Delta\tau_{\zeta}^{2}}{2}h^{\lambda}(\zeta;0)h^{\lambda^{\prime}}(\zeta;0)+h^{\lambda}(\zeta;0)\sum_{\tilde{\varpi}^{\prime}}\left(\frac{h^{\lambda^{\prime}}_{E}(\zeta;i\tilde{\varpi}^{\prime}\tau)}{(i\tilde{\varpi}^{\prime})^{2}}\Big{|}_{\tau_{\text{i}\zeta}}^{\tau_{\text{f}\zeta}}-\frac{h^{\lambda^{\prime}}_{E}(\zeta;i\tilde{\varpi}^{\prime}\tau_{\text{i}\zeta})\Delta\tau_{\zeta}}{i\tilde{\varpi}^{\prime}}\right)$
$\displaystyle+\sum_{\tilde{\varpi}}\left(\frac{h^{\lambda}_{E}(\zeta;i\tilde{\varpi}\tau)}{\tilde{\varpi}^{2}}\Big{|}_{\tau_{\text{i}\zeta}}^{\tau_{\text{f}\zeta}}+\frac{h^{\lambda}_{E}(\zeta;i\tilde{\varpi}\tau_{\text{f}\zeta})\Delta\tau_{\zeta}}{i\tilde{\varpi}}\right)h^{\lambda^{\prime}}(\zeta;0)$
$\displaystyle+\sum_{\tilde{\varpi}\tilde{\varpi}^{\prime}}\frac{1}{i\tilde{\varpi}^{\prime}}\left(\frac{h^{\lambda}_{E}(\zeta;i\tilde{\varpi}\tau)h^{\lambda^{\prime}}_{E}(\zeta;i\tilde{\varpi}^{\prime}\tau)}{i(\tilde{\varpi}+\tilde{\varpi}^{\prime})}\Big{|}_{\tau_{\text{i}\zeta}}^{\tau_{\text{f}\zeta}}-\frac{h^{\lambda}_{E}(\zeta;i\tilde{\varpi}\tau)h^{\lambda^{\prime}}_{E}(\zeta;i\tilde{\varpi}^{\prime}\tau_{\text{i}\zeta})}{i\tilde{\varpi}}\Big{|}_{\tau_{\text{i}\zeta}}^{\tau_{\text{f}\zeta}}\right)$
(S.55)

For high fidelity, we observe from the integrals that all the frequency
$\tilde{\varpi}$-dependent terms should be vanishingly small, so the high
fidelity condition from the first order integral is
$\absolutevalue{\tilde{\varpi}}\gg\absolutevalue{h^{\lambda}_{E}(\zeta;i\tilde{\varpi}\tau)}$.
From the second order integral, the disappearance of the
$\tilde{\varpi},\tilde{\varpi}^{\prime}$-dependent terms demonstrate more
implicit conditions for high fidelity, which means the fidelity varies swiftly
when we change the frequency parameters $\Omega$ and $\omega_{r}$. However, a
straightforward way to make them disappear is
$\absolutevalue{\tilde{\varpi}},\absolutevalue{\tilde{\varpi}^{\prime}}\gg\absolutevalue{h^{\lambda}_{E}(\zeta;i\tilde{\varpi}\tau)},\absolutevalue{h^{\lambda^{\prime}}_{E}(\zeta;i\tilde{\varpi}^{\prime}\tau)},\Delta\tau_{s}$.

In the exact Hamiltonians for both systems, the smallest non-zero frequency is
$2\tilde{\Omega}$ in the resonant regime. To make the term
$\absolutevalue{\frac{h^{\lambda}_{E}(\zeta;i2\tilde{\Omega}\tau_{\text{f}\zeta})\Delta\tau_{s}}{i2\tilde{\Omega}}}$
smaller than some small threshold, say $1/8$, we demand
$\tilde{\Omega}>\max\Delta\tau_{s}=\sqrt{2}\pi\approx 4.44$. This defines a
lower boundary of the high fidelity region across all $\tilde{\Omega}$.

The RWA Hamiltonian shares the same first order integrals

$\displaystyle I^{0}_{R}(\zeta)$ $\displaystyle=\frac{i\tau
e^{i\phi}}{2}\Bigg{|}^{\tau_{\text{f}\zeta}}_{\tau_{\text{i}\zeta}}$
$\displaystyle I^{1}_{R}(\zeta)$ $\displaystyle=-\frac{i\tau
e^{-i\phi}}{2}\Bigg{|}^{\tau_{\text{f}\zeta}}_{\tau_{\text{i}\zeta}}$

For the spin qubit Hamiltonian, the first order integrals are

$\displaystyle I^{1}_{E1}(\zeta)$
$\displaystyle=\frac{\tilde{\Omega}\left(-i\tilde{\Omega}\sin{\left(2\tilde{\omega}\tau+2\phi\right)}+2\omega\cos{\left(2\tilde{\omega}\tau+2\phi\right)}\right)e^{i\tilde{\Omega}\tau}}{\tilde{\Omega}^{2}-4\tilde{\omega}^{2}}\Bigg{|}^{\tau_{\text{f}\zeta}}_{\tau_{\text{i}\zeta}}$
(S.56) $\displaystyle I^{2}_{E1}(\zeta)$
$\displaystyle=\frac{\tilde{\Omega}\left(i\tilde{\Omega}\sin{\left(2\tilde{\omega}\tau+2\phi\right)}+2\tilde{\omega}\cos{\left(2\tilde{\omega}\tau+2\phi\right)}\right)e^{-i\tilde{\Omega}\tau}}{\tilde{\Omega}^{2}-4\tilde{\omega}^{2}}\Bigg{|}^{\tau_{\text{f}\zeta}}_{\tau_{\text{i}\zeta}}$
(S.57) $\displaystyle I^{3}_{E1}(\zeta)$
$\displaystyle=\frac{\tilde{\Omega}\sin{\left(2\tilde{\omega}\tau+2\phi\right)}}{2\tilde{\omega}}\Bigg{|}^{\tau_{\text{f}\zeta}}_{\tau_{\text{i}\zeta}}$
(S.58) $\displaystyle I^{5}_{E1}(\zeta)$ $\displaystyle=\frac{i\tau
e^{i\phi}}{2}+\frac{e^{-2i\tilde{\omega}\tau-i\phi}}{4\tilde{\omega}}\Bigg{|}^{\tau_{\text{f}\zeta}}_{\tau_{\text{i}\zeta}}$
(S.59) $\displaystyle I^{6}_{E1}(\zeta)$
$\displaystyle=-\frac{\left(4i\tilde{\Omega}\sin{\left(\tilde{\omega}\tau+\phi\right)}+i\tilde{\omega}\sin{\left(\tilde{\omega}\tau+\phi\right)}+\tilde{\omega}\cos{\left(\tilde{\omega}\tau+\phi\right)}\right)e^{-i\tau\left(4\tilde{\Omega}+\tilde{\omega}\right)}}{8\tilde{\Omega}\left(2\tilde{\Omega}+\tilde{\omega}\right)}\Bigg{|}^{\tau_{\text{f}\zeta}}_{\tau_{\text{i}\zeta}}$
(S.60) $\displaystyle I^{7}_{E1}(\zeta)$
$\displaystyle=\frac{\left(-2i\tilde{\Omega}\cos{\left(\tilde{\omega}\tau+\phi\right)}+\tilde{\omega}\sin{\left(\tilde{\omega}\tau+\phi\right)}-i\tilde{\omega}\cos{\left(\tilde{\omega}\tau+\phi\right)}\right)e^{-i\tau\left(2\tilde{\Omega}+\tilde{\omega}\right)}}{4\tilde{\Omega}\left(\tilde{\Omega}+\tilde{\omega}\right)}\Bigg{|}^{\tau_{\text{f}\zeta}}_{\tau_{\text{i}\zeta}}$
(S.61) $\displaystyle I^{9}_{E1}(\zeta)$
$\displaystyle=\frac{\left(4i\tilde{\Omega}\sin{\left(\tilde{\omega}\tau+\phi\right)}+i\tilde{\omega}\sin{\left(\tilde{\omega}\tau+\phi\right)}-\tilde{\omega}\cos{\left(\tilde{\omega}\tau+\phi\right)}\right)e^{i\tau\left(4\tilde{\Omega}+\tilde{\omega}\right)}}{8\tilde{\Omega}\left(2\tilde{\Omega}+\tilde{\omega}\right)}\Bigg{|}^{\tau_{\text{f}\zeta}}_{\tau_{\text{i}\zeta}}$
(S.62) $\displaystyle I^{10}_{E1}(\zeta)$ $\displaystyle=-\frac{i\tau
e^{-i\phi}}{2}+\frac{e^{2i\tilde{\omega}\tau+i\phi}}{4\tilde{\omega}}\Bigg{|}^{\tau_{\text{f}\zeta}}_{\tau_{\text{i}\zeta}}$
(S.63) $\displaystyle I^{11}_{E1}(\zeta)$
$\displaystyle=\frac{\left(2i\tilde{\Omega}\cos{\left(\tilde{\omega}\tau+\phi\right)}+\tilde{\omega}\sin{\left(\tilde{\omega}\tau+\phi\right)}+i\tilde{\omega}\cos{\left(\tilde{\omega}\tau+\phi\right)}\right)e^{i\tau\left(2\tilde{\Omega}+\tilde{\omega}\right)}}{4\tilde{\Omega}\left(\tilde{\Omega}+\tilde{\omega}\right)}\Bigg{|}^{\tau_{\text{f}\zeta}}_{\tau_{\text{i}\zeta}}$
(S.64)

with $\omega=\omega_{r}-2\Omega$, and the evaluation of upper and lower limit
act on the expression to its left side.

To roughly estimate the upper boundary of the high-fidelity region, we first
observe the first two exact integrals and try to set the their product smaller
than a threshold, which we take as $1/2^{9}$:

$\displaystyle\frac{\tilde{\Omega}^{4}+4\tilde{\omega}^{2}\tilde{\Omega}^{2}}{(\tilde{\Omega}^{2}-4\tilde{\omega}^{2})^{2}}<\frac{1}{2^{9}},$
(S.65)

since the other ratios are definitely smaller than this one for positive
$\tilde{\omega}_{r}$ and $\tilde{\Omega}$. We solve for
$\tilde{\omega}_{r}/\tilde{\Omega}$, and obtain
$\tilde{\omega}_{r}/\tilde{\Omega}>\frac{1}{2}(4+\sqrt{257+32\sqrt{65}})\approx
13.347$. By observing the numerical evidence, we conclude the high-fidelity
condition: a lower and upper boundary define a high-fidelity plateau.

$\displaystyle\tilde{\Omega}\in\left(\sqrt{2}\pi,\frac{\tilde{\omega}_{r}}{13.347}\right)$
(S.66)

For the charge qubit, the longitudinal terms contain a constant at
$\tilde{\Omega}=\frac{\tilde{\omega}_{r}}{2}\left(1-\frac{1}{p}\right)$ as
discussed before. The unwanted constant terms diminishes the fidelity at these
values. Additionally, when $\tilde{\Omega}=\frac{\tilde{\omega}_{r}}{2}$, the
driving frequency $\tilde{\omega}=0$, so that the red sideband transition
terms vanish in stage $\zeta=1,2,4,5$. The evaluation of the second-order
integral gives the same lower limit of $\tilde{\Omega}$ as it is for the spin
qubit. Therefore, the high-fidelity condition of the charge qubit is

$\displaystyle\tilde{\Omega}\in\mathbbm{R}^{+}\setminus\Biggl{\\{}[0,\frac{\sqrt{2}\pi}{2}),\frac{\tilde{\omega}_{r}}{2},\frac{\tilde{\omega}_{r}}{2}\left(1-\frac{1}{p}\right)\Biggr{\\}}$
(S.67)

##  VI Quantum gate with decoherence: superoperator calculations

In this section we will discuss superoperators relevant for open system
dynamics. Again, we will use similar bookkeeping techniques as previously
discussed.

In the protocol for the CZ gate, there are 7 stages ordered by: five of them
denoted by $\zeta$ is the sideband transition evolution; one of two-qubit
rotation and the final projection to 0-photon subspace. For example, the
unitary evolution $\hat{U}_{\text{CZ}}$ of the density matrix
$\hat{\rho}=\hat{U}_{\text{CZ}}\hat{\rho}(0)\hat{U}_{\text{CZ}}^{{\dagger}}$
of the ideal CZ $=\hat{U}_{cz}$ gate will be denoted as

$\displaystyle\hat{\rho}$ $\displaystyle=\mathscr{U}_{\text{CZ}}\hat{\rho}(0)$
$\displaystyle=\mathscr{M}_{0}\mathscr{Z}\mathcal{T}_{\leftarrow}\prod_{\zeta=1}^{5}\mathscr{U}(\zeta;\tau_{\text{f}\zeta},\tau_{\text{i}\zeta})$
$\displaystyle=\mathscr{M}_{0}\mathscr{Z}\mathscr{U}(5;\tau_{\text{f}5},\tau_{\text{i}5})\mathscr{U}(4;\tau_{\text{f}4},\tau_{\text{i}4})\mathscr{U}(3;\tau_{\text{f}3},\tau_{\text{i}3})\mathscr{U}(2;\tau_{\text{f}2},\tau_{\text{i}2})\mathscr{U}(1;\tau_{\text{f}1},\tau_{\text{i}1})$
(S.68)

wherein

$\displaystyle\mathscr{M}_{0}\hat{\rho}$
$\displaystyle=\hat{M}_{0}\rho\hat{M}_{0}$ (S.69)
$\displaystyle\mathscr{Z}\hat{\rho}$
$\displaystyle=\hat{Z}^{(1)}\left(\frac{\pi}{\sqrt{2}}\right)\hat{Z}^{(2)}\left(-\frac{\pi}{\sqrt{2}}\right)\hat{\rho}\hat{Z}^{(2)}\left(-\frac{\pi}{\sqrt{2}}\right)^{\dagger}\hat{Z}^{(1)}\left(\frac{\pi}{\sqrt{2}}\right)^{\dagger}$
(S.70)
$\displaystyle\mathscr{U}(\zeta;\tau_{\text{f}\zeta},\tau_{\text{i}\zeta})\hat{\rho}$
$\displaystyle=\hat{S}_{R}(\zeta;\Delta\tau,\phi)\hat{\rho}\,\hat{S}_{R}(\zeta;\Delta\tau,\phi)^{\dagger}$
(S.71)

We begin the discussion of open system dynamics in Schrödinger picture where
in general the Lindblad master equation reads [1]

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\hat{\rho}(t)=\mathscr{L}\hat{\rho}(t)=-i\commutator{\hat{H}}{\hat{\rho}(t)}+\sum_{\ell}\gamma_{\ell}\left(\hat{L}_{\ell}\hat{\rho}(t)\hat{L}^{{\dagger}}_{\ell}-\frac{1}{2}\anticommutator{\hat{L}^{{\dagger}}_{\ell}\hat{L}_{\ell}}{\hat{\rho}(t)}\right)$
(S.72)

For the system at hand we model qubit dephasing using the Lindblad operator
$\hat{L}_{1,2}=\hat{\sigma}_{z}^{(1,2)}$ and take rates
$\gamma_{1}=\gamma_{2}=\gamma$ to be the same for both qubits. In turn, for
the photon mode, we will take the rate $\gamma_{3}=\kappa$ and Lindblad
operator $\hat{L}_{3}=\hat{a}$. Thus, explicitly the Lindblad master equation
for the system will be

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\hat{\rho}(t)=-i\commutator{\hat{H}_{E}}{\hat{\rho}(t)}+\sum_{q=1,2}\gamma\left(\hat{\sigma}_{z}^{(q)}\hat{\rho}\hat{\sigma}_{z}^{(q)}-\hat{\rho}\right)+\kappa\left(\hat{a}\hat{\rho}\hat{a}^{\dagger}-\frac{1}{2}\hat{a}^{\dagger}\hat{a}\hat{\rho}-\frac{1}{2}\hat{\rho}\hat{a}^{\dagger}\hat{a}\right).$
(S.73)

Transforming the master equation into the corresponding interaction picture
with stage index $\zeta$:

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}\tau}\hat{\rho}(\tau)=\mathscr{L}(\zeta;\tau)\hat{\rho}(\tau)=-i\commutator{\hat{H}_{E}(\zeta;\tau)}{\hat{\rho}(\tau)}+\sum_{\ell}\tilde{\gamma}_{\ell}\left(\hat{L}_{\ell}(\zeta;\tau)\hat{\rho}(\tau)\hat{L}_{\ell}(\zeta;\tau)^{{\dagger}}-\frac{1}{2}\anticommutator{\hat{L}_{\ell}(\zeta;\tau)^{{\dagger}}\hat{L}_{\ell}(\zeta;\tau)}{\hat{\rho}(\tau)}\right)$
(S.74)

where for spin qubit system, using indices $j,k$ to indicate coupled and
uncoupled qubits,

$\displaystyle\hat{L}_{j}(\zeta;\tau)$
$\displaystyle=-\sin(2\tilde{\Omega}\tau)\hat{\sigma}_{x}^{(j_{\zeta})}-\cos(2\tilde{\Omega}\tau)\hat{\sigma}_{y}^{(j_{\zeta})}$
(S.75) $\displaystyle\hat{L}_{k}(\zeta;\tau)$
$\displaystyle=-\hat{\sigma}_{y}^{(k_{\zeta})}$ (S.76)
$\displaystyle\hat{L}_{3}(\tau)$
$\displaystyle=e^{-i(\tilde{\omega}+2\tilde{\Omega})\tau}\hat{a}$ (S.77)

and charge qubit system:

$\displaystyle\hat{L}_{j}(\zeta;\tau)$ $\displaystyle=\hat{\sigma}_{z}^{(j)}$
(S.78) $\displaystyle\hat{L}_{k}(\zeta;\tau)$
$\displaystyle=\hat{\sigma}_{z}^{(k)}$ (S.79) $\displaystyle\hat{L}_{3}(\tau)$
$\displaystyle=e^{-i(\tilde{\omega}+\tilde{\Omega})\tau}\hat{a}$ (S.80)

The dissipation rates in the Lindblad master equation are rescaled against the
coupling constant: $\gamma/g\rightarrow\tilde{\gamma}$, and
$\kappa/g\rightarrow\tilde{\kappa}$. Next, we formulate the notation for a
general solution [1] of a time-dependent Lindblad master equation (S.74) in
stage $\zeta$:

$\displaystyle\hat{\rho}(\tau_{\text{f}\zeta})=\mathscr{S}(\zeta;\tau_{\text{f}\zeta},\tau_{\text{i}\zeta})\hat{\rho}(\tau_{\text{i}\zeta})$
$\displaystyle=\mathcal{T}_{\leftarrow}\exp\left[\int_{\tau_{\text{i}\zeta}}^{\tau_{\text{f}\zeta}}d\tau\mathscr{L}(\zeta;\tau)\right]\hat{\rho}(\tau_{\text{i}\zeta})$
$\displaystyle\approx\mathcal{T}_{\leftarrow}\prod_{\xi=0}^{T-1}\exp\left[d\tau\mathscr{L}(\zeta;\tau_{\xi})\right]$
(S.81)

wherein $T-1$ is the total number of discretized time steps, $\xi$ is the time
increment index, and
$d\tau=\frac{\tau_{\text{f}\zeta}-\tau_{\text{i}\zeta}}{T-1}$ is the increment
magnitude. The approximate equal becomes exact when $T\rightarrow\infty$.
$\mathscr{S}(s;\tau_{\text{f}},\tau_{\text{i}})$, which is independent of
initial states, and defines a smooth semigroup evolution from an arbitrary
density matrix for stage $\zeta$, from which we can find the Markovian,
dissipative quantum gate.

Now we connect the CZ gate protocol to the open system discussion by replacing
the unitary evolution superoperator
$\mathscr{U}(s;\tau_{\text{f}\zeta},\tau_{\text{i}\zeta})$ by semigroup
evolution $\mathscr{S}(\zeta;\tau_{\text{f}\zeta},\tau_{\text{i}\zeta})$:

$\displaystyle\mathscr{S}_{cz}=\mathscr{M}_{0}\mathscr{Z}\left(\mathcal{T}_{\leftarrow}\prod_{\zeta=0}^{4}\mathscr{S}(\zeta;\tau_{\text{f}\zeta},\tau_{\text{i}\zeta})\right).$
(S.82)

The vectorized superoperators can be written in matrix form [2], and the
elements are found under set of generators
$\hat{\Theta}_{\lambda}=\hat{\Theta}^{(o)}_{\mu_{o}}\hat{\Theta}^{(1)}_{\mu_{1}}\hat{\Theta}^{(2)}_{\mu_{2}}$,
where
$\hat{\Theta}^{(o)}_{\mu_{o}}=\ket{m}\bra{n}\otimes\hat{\sigma}_{0}\otimes\hat{\sigma}_{0}$,
and
$\hat{\Theta}^{(q=1,2)}=\frac{1}{\sqrt{2}}\begin{pmatrix}\hat{\sigma}_{0}^{(q)},&\hat{\sigma}_{x}^{(q)},&\hat{\sigma}_{y}^{(q)},&\hat{\sigma}_{z}^{(q)}\end{pmatrix}$.

In general, a superoperator $\mathscr{O}$ acting on the density matrix can be
represented by a matrix with elements:

$\displaystyle\tensor{\mathscr{O}}{{}^{\mu}_{\nu}}=\text{Tr}\left[\hat{\Theta}_{\mu}^{\dagger}\mathscr{O}\hat{\Theta}_{\nu}\right].$
(S.83)

For a continuous time-dependent (described by time $t$) and composite
superoperator that can be decomposed into summation of terms that consist a
time-dependent function and time-independent joint superoperators (but may
depend on stages $\zeta$) act on each subspace labeled by $m\in\\{0,1,2\\}$
for oscillator and two qubits

$\displaystyle\mathscr{O}(\zeta;t)$
$\displaystyle=\sum_{u}f_{u}(\zeta;t)\mathscr{O}_{u}(\zeta)=\sum_{u=0}f_{u}(\zeta;t)\prod_{m}\mathscr{O}^{(m)}_{u}(\zeta)$
(S.84)
$\displaystyle=\sum_{u}f_{u}(\zeta;t)\mathscr{O}_{u}^{(0)}(\zeta)\mathscr{O}_{u}^{(1)}(\zeta)\mathscr{O}_{u}^{(2)}(\zeta).$
(S.85)

The decomposed superoperator $\mathscr{O}_{u}(\zeta)$ has the tensor product
factor $\mathscr{O}^{(\zeta)}_{u}(\zeta)$, which satisfy
$\commutator{\mathscr{O}^{(m)}_{u}(\zeta)}{\mathscr{O}^{(m^{\prime})}_{u^{\prime}}(\zeta^{\prime})}\Big{|}_{m\neq
m^{\prime}}=0$ since they act on different subspace. Therefore, we can write
the superoperator tensor product factors $\mathscr{O}^{(m=0)}_{u}(\zeta)$,
$\mathscr{O}^{(m=1)}_{u}(\zeta)$, and $\mathscr{O}^{(m=2)}_{u}(\zeta)$ in any
order without lossing generality.

In the photon-qubits system, we have the generator set

$\displaystyle\hat{\Theta}_{\lambda}=\hat{\Theta}^{(0)}_{\mu_{0}}\hat{\Theta}^{(1)}_{\mu_{1}}\hat{\Theta}^{(2)}_{\mu_{2}}$
(S.86)

where the index in each subspace $\mu_{m}\in\\{0,1,\ldots,d_{m}^{2}\\}$.

The elements of a superoperator can then be found:

$\displaystyle\tensor{\mathscr{O}(\zeta;t)}{{}^{\mu_{o}\mu_{1}\mu_{2}}_{\nu_{o}\nu_{1}\nu_{2}}}$
$\displaystyle=\sum_{u}\Tr\left[f_{u}(\zeta;t)\,\hat{\Theta}^{(0)\dagger}_{\mu_{o}}\mathscr{O}_{u}^{(0)}(\zeta)\hat{\Theta}^{(0)}_{\nu_{o}}\,\hat{\Theta}^{(1)\dagger}_{\mu_{1}}\mathscr{O}_{u}^{(1)}(\zeta)\hat{\Theta}^{(1)}_{\nu_{1}}\,\hat{\Theta}^{(2)\dagger}_{\mu_{2}}\mathscr{O}_{u}^{(2)}(\zeta)\hat{\Theta}^{(2)}_{\nu_{2}}\right]$
$\displaystyle\tensor{\mathscr{O}(\zeta;t)}{{}^{\lambda}_{\lambda^{\prime}}}$
$\displaystyle=\sum_{u}f_{u}(\zeta;t)\,\tensor{\mathscr{O}_{u}^{(0)}(\zeta)}{{}^{\mu_{0}}_{\nu_{0}}}\,\tensor{\mathscr{O}_{u}^{(1)}(\zeta)}{{}^{\mu_{1}}_{\nu_{1}}}\,\tensor{\mathscr{O}_{u}^{(2)}(\zeta)}{{}^{\mu_{2}}_{\nu_{2}}}.$
(S.87)

By using Eq. (VI), we can find the matrix representation of Lindblad
superoperator $\mathscr{L}(\zeta;t)$ and therefore obtain the semigroup
evolution superoperator by Eq. (S.82).

Finally we check the fidelity [3] of the protocol using:

$\displaystyle\mathcal{F}(\tilde{\Omega},\tilde{\omega}_{r},\tilde{\gamma},\tilde{\kappa})$
$\displaystyle=\frac{1}{d^{2}}\absolutevalue{\Tr\left[\mathscr{U}_{cz}^{{\dagger}}\mathscr{S}_{cz}\right]}$
(S.88) $\displaystyle F(\tilde{\Omega},\tilde{\omega}_{r})$
$\displaystyle=\frac{1}{d}\absolutevalue{\Tr\left[U_{cz}^{{\dagger}}U_{E}\right]}$
(S.89)

where $d=4$ is the dimension of the two-qubit Hilbert space. When
$\mathscr{S}_{cz}=\mathscr{U}_{cz}$, i.e., both of them are unitary, the
fidelity is $1$. When the $\mathscr{S}_{cz}$ is completely orthogonal to
$\mathscr{U}_{cz}$, the fidelity is $0$, which meets the fundamental
requirements to define a measure of quantum operator fidelity. The same is
true for operator fidelity $F$.

##  VII Numerical method

In the numerical evaluation of semigroup evolution in Eq. (S.82), to ensure
convergence, the total number of time steps assigned is based on the maximum
frequency of oscillation in the evolution. We assign 64 time instances for the
shortest period of oscillation so that for each stage $\zeta$, the total
number of discretized time instances is:

$\displaystyle
T=\left\lceil\frac{\max(\Delta\tau_{\zeta})\tilde{\omega}_{\text{max}}}{2\pi}\times
64+1\right\rceil,$ (S.90)

where the ceiling notation $\lceil~{}\rceil$ rounds up to the nearest integer.

To illustrate, for spin qubit, stage $\zeta=3$ has the longest duration
$\sqrt{2}\pi$ when with highest frequency $2\tilde{\omega}+4\tilde{\Omega}$ is
contained in two blue-sideband terms in the exact Hamiltonian, Eq. (S.11). For
the other stages, the assigned number of time steps is larger than 64 per
shortest period of oscillation, thereby enhancing numerical convergence.The
results of the numerical convergence at the optimal point are shown in Figure
1, which justifies the choice of $2^{6}=64$ in Eq. (S.90) that ensures
numerical accuracy while rendering the computation time manageable.

Figure 1: Fidelity at the optimal point of the spin qubit is plotted against
different $2^{n}$ samples per shortest oscillation period. In our numerical
calculations, the sampling number of $2^{6}=64$ is chosen to ensure numerical
accuracy while rendering the computation time manageable. This gives an error
in fidelity of less than $5\times 10^{-6}$. Therefore, in the main text, we
report fidelities $\mathcal{F}$ to 5 decimal places.

For charge qubit, although the highest frequency in Hamiltonian (S.12) is
infinity, the amplitude of high frequency terms is negligible, as discussed in
Section IV. Therefore, we take the same sampling rate as for the spin qubit,
without losing numerical convergence.

We utilize experimental parameters from Ref. [4]: photon frequency
$\omega_{r}/2\pi=5.846~{}\text{GHz}$, the coupling strength
$(g_{s},g_{c})/2\pi=(11~{}\text{MHz},40~{}\text{MHz})$, the qubit dephasing
rate $(\gamma_{s},\gamma_{c})/2\pi=(5~{}\text{MHz},35~{}\text{MHz})$, and the
photon loss rate $\kappa/2\pi=1.3~{}\text{MHz}$. The detuning $\Delta=2\Omega$
is adjustable when the photon energy is fixed.

The normalized parameters are shown in Table 1.

Normalized Parameters | Spin | Charge
---|---|---
$\tilde{\omega}_{r}$ | 531.455 | 292.30
$\tilde{\gamma}$ | 0.455 | 1.750
$\tilde{\kappa}$ | 0.118 | 0.065
Table 1: The normalized experimental parameters for both qubit systems from
Ref. [4].

##  VIII Strong coupling gate time correction

We eliminate the $\hat{\sigma}^{(j)}_{z}$ terms in Eq. (S.11) by with an
additional transformation [5, 6]
$\hat{U}_{5}(t)=\exp[-\frac{i\Omega}{2\omega}\sin(2\omega
t+2\phi)\hat{\sigma}^{(j)}_{z}]$, so the transformed spin qubit Hamiltonian
$\hat{H}_{E,s}^{\prime}=\hat{U}_{5}(t)^{\dagger}\left(\hat{H}_{E,s}-i\frac{\partial}{\partial
t}\right)\hat{U}_{5}(t)$ is

$\displaystyle\hat{H}_{E,s}^{\prime}=$ $\displaystyle\Omega e^{2i\Omega
t+iy\sin(2\omega t+2\phi)}\sin(2\omega t+2\phi)\hat{\sigma}_{+}^{(j)}+\Omega
e^{-2i\Omega t-iy\sin(2\omega t+2\phi)}\sin(2\omega
t+2\phi)\hat{\sigma}_{-}^{(j)}$ $\displaystyle-g\hat{a}\Bigg{(}e^{-i\omega
t+iy\sin(2\omega t+2\phi)}\sin(\omega
t+\phi)\hat{\sigma}_{+}^{(j)}+e^{-i(4\Omega+\omega)t-iy\sin(2\omega
t+2\phi)}\sin(\omega t+\phi)\hat{\sigma}_{-}^{(j)}$
$\displaystyle+e^{-i(2\Omega+\omega)t}\cos(\omega
t+\phi)\hat{\sigma}^{(j)}_{z}\Bigg{)}$ (S.91)
$\displaystyle-g\hat{a}^{\dagger}\Bigg{(}e^{i(4\Omega+\omega)t+iy\sin(2\omega
t+2\phi)}\sin(\omega t+\phi)\hat{\sigma}_{+}^{(j)}+e^{i\omega t-iy\sin(2\omega
t+2\phi)}\sin(\omega t+\phi)\hat{\sigma}_{-}^{(j)}$
$\displaystyle+e^{i(2\Omega+\omega)t}\cos(\omega
t+\phi)\hat{\sigma}^{(j)}_{z}\Bigg{)}$

where $y=\Omega/2\omega>0$. Factorizing out the common function
$g(t)=g\cos(y\sin(2\omega t+2\phi))$ from sideband transition terms
$\hat{a}\hat{\sigma}_{+}^{(j)}$ and $\hat{a}^{\dagger}\hat{\sigma}_{-}^{(j)}$
and then dropping the rest of time-dependent terms, we have a new RWA sideband
transition Hamiltonian

$\displaystyle\hat{H}_{R}^{(j)}(g(t),\phi)=\frac{ig(t)}{2}(e^{i\phi}\hat{a}\hat{\sigma}_{+}^{(j)}-e^{-i\phi}\hat{a}^{\dagger}\hat{\sigma}_{-}^{(j)}),$
(S.92)

where now, the effective coupling $g(t)$ is time-dependent. To achieve the
same CZ gate, equating

$\displaystyle\int_{\tau_{\text{i}\zeta}^{\prime}}^{\tau_{\text{f}\zeta}^{\prime}}ds\cos(y\sin(2\omega
s+2\phi_{s}))=\Delta\tau_{s}$ (S.93)

we have the new gate time $\Delta\tau^{\prime}_{s}$ for each step that can be
solved using

$\displaystyle
J_{0}(y)\Delta\tau^{\prime}_{s}+\sum_{n=1}^{\infty}\frac{J_{2n}(y)}{2\omega
n}\sin(2n(2\omega
t+2\phi_{s}))\Big{|}^{\tau_{\text{i}\zeta}^{\prime}+\Delta\tau^{\prime}}_{\tau_{\text{i}\zeta}^{\prime}}=\Delta\tau_{s}$
(S.94)

where $J_{m}(z)$ are $m$-th order Bessel function. We have absorbed the real
part of the evolution caused by $\hat{\sigma}^{(j)}_{z}$ into the gate time.

However, numerical result shows that there are no improvement of fidelity with
this strong-coupling correction of the gate time. The left qubit terms still
act as an limit of high-fidelity region. One possible way of completely
eliminate the qubit terms is to transform numerically according to

$\displaystyle\hat{U}_{5}^{\prime}(t)=\mathcal{T}_{\leftarrow}\exp[-i\Omega\int_{0}^{t}dt\left(e^{2i\Omega
t}\sin(2\omega t+2\phi)\hat{\sigma}_{+}^{(j)}+e^{-2i\Omega t}\sin(2\omega
t+2\phi)\hat{\sigma}_{-}^{(j)}+\cos(2\omega
t+2\phi)\hat{\sigma}^{(j)}_{z}\right)].$ (S.95)

However, the cost of performing the transformation numerically is the loss of
analytical insight.

## References

  * Breuer and Petruccione [2002] H.-P. Breuer and F. Petruccione,  _The theory of open quantum systems_ (Oxford University Press, USA, 2002).
  * Greenbaum [2015] D. Greenbaum, Introduction to quantum gate set tomography, arXiv preprint arXiv:1509.02921 (2015).
  * Wang  _et al._ [2008] X. Wang, C.-S. Yu, and X. Yi, An alternative quantum fidelity for mixed states of qudits, Physics Letters A 373, 58 (2008).
  * Mi  _et al._ [2018] X. Mi, M. Benito, S. Putz, D. M. Zajac, J. M. Taylor, G. Burkard, and J. R. Petta, A coherent spin–photon interface in silicon, Nature 555, 599 (2018).
  * Ashhab  _et al._ [2007] S. Ashhab, J. R. Johansson, A. M. Zagoskin, and F. Nori, Two-level systems driven by large-amplitude fields, Physical Review A 75, 063414 (2007).
  * Lü and Zheng [2012] Z. Lü and H. Zheng, Effects of counter-rotating interaction on driven tunneling dynamics: Coherent destruction of tunneling and Bloch-Siegert shift, Physical Review A 86, 023831 (2012).


# RE-AdaptIR: Improving Information Retrieval
through Reverse Engineered Adaptation

William Fleshman  Benjamin Van Durme
Johns Hopkins University
<EMAIL_ADDRESS>
###### Abstract

Large language models (LLMs) fine-tuned for text-retrieval have demonstrated
state-of-the-art results across several information retrieval (IR) benchmarks.
However, supervised training for improving these models requires numerous
labeled examples, which are generally unavailable or expensive to acquire. In
this work, we explore the effectiveness of extending reverse engineered
adaptation to the context of information retrieval (RE-AdaptIR). We use RE-
AdaptIR to improve LLM-based IR models using only unlabeled data. We
demonstrate improved performance both in training domains as well as zero-shot
in domains where the models have seen no queries. We analyze performance
changes in various fine-tuning scenarios and offer findings of immediate use
to practitioners.

RE-AdaptIR: Improving Information Retrieval
through Reverse Engineered Adaptation

William Fleshman and Benjamin Van Durme Johns Hopkins University
<EMAIL_ADDRESS>

##  1 Introduction

Information retrieval (IR) is a fundamental component of various modern
applications, powering search engines, recommender systems, and various data
analytics pipelines. Recently, large language models (LLMs) have achieved
state-of-the art results on dense text retrieval, identifying and ranking the
most relevant text for a given query by comparing learned vector
representations of the text (Reimers and Gurevych, 2019; Khattab and Zaharia,
2020; Karpukhin et al., 2020; Izacard et al., 2022; Ma et al., 2023; Jiang et
al., 2023; Weller et al., 2024). The effectiveness of text retrieval systems
have a direct impact on numerous domains, including healthcare, finance, and
social media, where accurate and timely access to information is critical.
Retrieval is also critical in the context of retrieval augmented generation
(RAG), enabling LLMs access to external resources when constructing a response
(Lewis et al., 2020). For these reasons, we seek a practical and efficient
approach for improving existing text retrieval models.

Figure 1: In RE-AdaptIR, RE-Adapt is extended to an existing retrieval model
to isolate what was learned during labeled contrastive training. The
pretrained model is fine-tuned on unlabeled in-domain documents and readapted
for text retrieval. The new retriever outperforms the original on both in-
domain and zero-shot information retrieval tasks.

Supervised fine-tuning of LLMs for text retrieval tasks has become a widely
adopted approach, leveraging their pretrained language understanding
capabilities to achieve state-of-the-art results on various benchmarks (Ma et
al., 2023; Jiang et al., 2023; Weller et al., 2024). However, adapting an LLM
for text retrieval requires labeled datasets, with numerous example queries
and documents both related and unrelated to forming a helpful response. This
poses a significant challenge to improving these systems, as data annotation
or synthetic generation can be too expensive, difficult, and error-prone
(Fredriksson et al., 2020; Desmond et al., 2021). Making matters worse, fine-
tuning an existing LLM on new domains can cause forgetting, a decreased
performance on previously capable tasks (McCloskey and Cohen, 1989; Kotha et
al., 2024). Fleshman and Van Durme (2024) recently proposed reverse engineered
adaptation (RE-Adapt), an approach for solving similar dilemmas faced when
fine-tuning existing instruction-tuned models. Here we introduce RE-AdaptIR,
an extension of RE-Adapt to IR models, leveraging unlabeled data to improve
existing text-retrieval LLMs (Figure 1). Specifically we:

  * •

Extend RE-Adapt to the information retrieval setting and apply RE-AdaptIR to
two state-of-the-art text retrieval models: RepLLaMA and e5-Mistral;

  * •

Demonstrate improved performance both in-domain and zero-shot across 14
datasets; and

  * •

Explore the importance of fine-tuning on data relevant to test-time queries
and the impact different scenarios have on performance.

##  2 Background

###  2.1 Retrieval Models

The transformer architecture is a natural choice for text retrieval models, as
it embeds text into dense vector representations (Vaswani et al., 2023). LLMs
pretrained on massive amounts of text have demonstrated the ability to capture
useful semantic meaning in these vector representations (Radford et al., 2019;
Touvron et al., 2023; Jiang et al., 2023). To ensure these models are capable
for retrieval, a secondary fine-tuning stage is used to encourage the model to
map similar texts to similar vectors (Ma et al., 2023; Jiang et al., 2023;
Weller et al., 2024). This is generally done via some form of supervised
contrastive training such as with InfoNCE (van den Oord et al., 2019). After
training, the models can be used to create a vector database of documents, and
rank them given a query representation by using a similarity function. Related
documents should have more similar representations to the query than those
unrelated. We experiment with two such models in this work: RepLLaMA (Ma et
al., 2023) and e5-Mistral (Wang et al., 2024).

#### RepLLaMA.

Ma et al. (2023) introduced RepLLaMA to demonstrate that state-of-the-art LLMs
could surpass the previous results achieved with smaller retrieval models,
especially when evaluated zero-shot on datasets not seen during training. They
construct RepLLaMA by fine-tuning LLaMA-2-7B (Touvron et al., 2023) on
approximately 500k labeled examples from the training split of the MS-MARCO
dataset (Bajaj et al., 2018).

#### e5-Mistral.

In a similar fashion, Wang et al. (2024) fine-tune the Mistral-7B LLM (Jiang
et al., 2023) using a combination of synthetic data, MS-MARCO, and multiple
other labeled datasets. The resulting e5-Mistral model achieves state-of-the-
art results on several text retrieval benchmarks.

In both cases, labeled data was needed to achieve the best results, and it is
unclear how to effectively incorporate the copious amount of unlabeled text
additionally available. For example, MS-MARCO contains almost 9 million
passages, but only a fraction of this data is used in training, due to the
limited number of associated queries available. In this work, we use RE-
AdaptIR to leverage this unlabeled data and improve the performance of both
RepLLaMA and e5-Mistral.

###  2.2 Reverse Engineered Adaptation

Fleshman and Van Durme (2024) introduced reverse engineered adaptation (RE-
Adapt) as a new method to efficiently update instruction-tuned models with
unlabeled data lacking the previously required instruction-tuning annotations
(Mishra et al., 2022; Wei et al., 2022; Ouyang et al., 2022). RE-Adapt works
by first isolating what has been learned from instruction-tuning by taking the
difference between the weights of the instruction-tuned and pretrained
versions of a model. This difference can be thought of as an adapter Houlsby
et al. (2019) or as a multi-task version of task-vectors (Ilharco et al.,
2023). Given this RE-Adapter $\Delta$, the pretrained weights $\Theta$ can be
fine-tuned with a new knowledge adapter $\Psi$ without impacting the previous
instruction-tuning. Finally, the model can be re-instantiated with weights
$\Theta+\alpha\Psi+\beta\Delta$ where $\alpha$ and $\beta$ are partial
adaptation scalars used to control the strength of fine-tuning (Fleshman and
Van Durme, 2024). The authors show that RE-Adapt improves the performance of
instruction-tuned models in the new domain while preserving or improving
performance out-of-domain.

##  3 Re-AdaptIR

In this work, we explore the effectiveness of RE-Adapt in the context of
information retrieval. While an instruction-tuned model still leverages the
pretraining capabilities of next-token prediction, most text retrieval models
do not. RepLLaMA and e5-Mistral both discard the next-token predictor from
their respective LLMs and fine-tune the model to produce a single vector
representation per document (Ma et al., 2023; Wang et al., 2024). It is
therefore unclear whether continued fine-tuning of the pretrained LLM with
next-token prediction will improve down-stream retrieval.

To answer this question, we first fine-tune the pretrained LLM on unlabeled
documents from a new domain. We then construct a RE-AdaptIR for the retrieval
model by discarding the pretrained next-token predictor weights as well as the
corresponding weights from the knowledge adapter. We can then follow the RE-
Adapt procedure using the remaining weights, and evaluate the readapted IR
model using queries from the new domain.

##  4 Experiments

We first replicate the shared text retrieval experiments conducted by Ma et
al. (2023) and Wang et al. (2024) and compare the base model performance
before and after applying RE-AdaptIR. We then conduct further analysis to
understand how different fine-tuning scenarios impact performance.

###  4.1 Datasets

Both RepLLaMA and e5-Mistral utilized MS-MARCO (Bajaj et al., 2018) as part of
their training data, and we include it in our evaluations to help measure any
benefits from using RE-AdaptIR in-domain. Specifically, RE-AdaptIR allows for
fine-tuning over the entire 8.84M passages, where only a subset of those
passages were used for retrieval training due to limited availability of
query-passage pairs (Ma et al., 2023; Wang et al., 2024). Additionally, we use
the same 13 public datasets from the BeIR IR benchmark (Thakur et al., 2021)
used by Ma et al. (2023) to assess RepLLaMA’s zero-shot performance across a
diverse set of IR tasks. Of these, we note that the training splits of FEVER
(Thorne et al., 2018), HotPotQA (Yang et al., 2018), NQ (Kwiatkowski et al.,
2019), and Quora (DataCanary et al., 2017) were also used by Wang et al.
(2024) in the training of e5-Mistral, providing more in-domain insight to our
experiments. These datasets are zero-shot for RepLLaMA, as is the remainder of
BeIR for both models. We use the same prompts used by Ma et al. (2023) and
Wang et al. (2024) for each dataset.

###  4.2 Adapters

We use parameter efficient fine-tuning (Mangrulkar et al., 2022) with DoRA
(Liu et al., 2024) to adapt the pretrained LLaMA-2-7B (Touvron et al., 2023)
and Mistral-7B (Jiang et al., 2023) LLMs. We train each adapter for a single
epoch on all passages from the dataset under evaluation. Specific training
details are included in Appendix A. As in Fleshman and Van Durme (2024), we
use a scalar of 0.5 with our knowledge adapters to minimize interference with
existing retrieval ability.

| RepLLaMA | e5-Mistral
---|---|---
Dataset | Base | RA | Base | RA
MS-MARCO | 46.5 | 46.1 | 36.5 | 40.1
FEVER | 84.0 | 83.8 | 85.1 | 87.7
HotPotQA | 67.2 | 67.6 | 72.5 | 73.4
NQ | 61.8 | 62.1 | 53.3 | 52.4
Quora | 80.1 | 82.8 | 85.6 | 88.0
Arguana | 52.3 | 52.6 | 52.0 | 59.0
Climate-FEVER | 30.8 | 30.4 | 24.9 | 31.4
DBPedia | 43.4 | 43.5 | 47.2 | 47.1
FiQA | 44.2 | 45.5 | 49.9 | 52.3
NFCorpus | 38.0 | 38.6 | 39.6 | 40.8
SCIDOCS | 17.7 | 18.3 | 18.6 | 18.7
SciFact | 74.5 | 76.3 | 71.4 | 73.3
TREC-COVID | 84.0 | 85.6 | 83.9 | 81.0
Touche-2020 | 27.5 | 27.0 | 29.0 | 30.1
Average | 53.7 | 54.3 | 53.5 | 55.4
Average Z-Shot | 54.3 | 54.9 | 46.3 | 48.2
Table 1: nDCG@10 across test splits for MS-MARCO and BeIR datasets. The
results highlighted in orange indicate the dataset’s train split was used for
training the corresponding model and are not zero-shot. Base is the unmodified
model and RA is the model RE-Adapted after fine-tuning on the domain.

###  4.3 Results

Our main results are compiled in Table 1. We see that RE-AdaptIR improves
results in the majority of cases, increasing the average zero-shot nDCG@10 by
0.6 and 1.9 points for RepLLaMA and e5-Mistral respectively. Importantly,
these performance gains required no additional labeled data and are achieved
by simply fine-tuning the pretrained model over the document database being
used for retrieval. The default partial adaptation scalar of 0.5 was used for
this experiment, but we note that optimizing this value per dataset does
improve results. While not applicable to our zero-shot analysis, in practice
this value can be set using withheld queries.

We notice the few cases where performance was reduced tend to occur with the
larger corpora. We plot this relationship in Figure 2 and do observe a
slightly negative correlation. This relationship is purely observational and
likely caused by latent topic or task diversity among the larger datasets used
in these experiments. For reference, we include the dataset sizes in Appendix
B.

$10^{4}$$10^{5}$$10^{6}$$10^{7}$$-2$$0$$2$$4$$6$Corpus Size$\Delta$nDCG@10RepLLaMAe5-Mistral Figure 2: The observed relationship between the corpus size and the change in performance when fine-tuning with RE-AdaptIR. RE-Adapted | Arguana | FiQA | NFCorpus | SciFact |
---|---|---|---|---|---
Model | w/o | w/ | both | w/o | w/ | both | w/o | w/ | both | w/o | w/ | both |
RepLLaMA | +0.5 | +0.2 | +0.3 | +1.0 | +1.2 | +1.3 | +0.3 | +0.5 | +0.6 | +0.1 | +0.5 | +1.8 |
e5-Mistral | +3.9 | +5.1 | +7.0 | +2.0 | +2.8 | +2.4 | +0.6 | +1.1 | +1.2 | +2.3 | +1.0 | +1.9 |
Table 2: Change in nDCG@10 over original retriever when pretrained model is
fine-tuned using only documents with (w/) or without (w/o) corresponding
queries in the test set, or with (both) subsets.

#### Are queried passages all that matter?

One reason the in-domain results could be better than baseline is because the
passages being queried for at test-time are included in the fine-tuning data,
although their corresponding queries are not. We test this hypothesis by
training two additional knowledge adapters, one which sees no test-time
passages, and one that sees only test-time passages. We compare the evaluation
results with the original adapter fine-tuned on all the passages (Table 2). We
find that neither of the subsets is always best, indicating that having in-
domain data is more important than specifically fine-tuning on the passages
being queried for at test-time.

#### Does any unlabeled data work?

Next, we explore the importance of using the domain specific data for fine-
tuning. We repeat our main experiment across the BeIR datasets, but using only
the knowledge adapter trained on MS-MARCO, our largest corpus. We compare the
resulting performance with the original RepLLaMA and e5-Mistral baselines as
well as the RE-Adapted models fine-tuned on the domain specific data (Table
3). We observe that additional fine-tuning with MS-MARCO improves RepLLaMA by
an average of 0.2 points but is still 0.5 points below the average performance
when using in-domain data. For e5-Mistral however, we see that the MS-MARCO
fine-tuning results in a significant increase of 2 points over baseline on
average, 0.3 points above what is achieved with in-domain data. The larger
gain with e5-Mistral is likely due to RepLLaMA’s use of MS-MARCO as the
majority of its training data, while e5-Mistral only used a subset to
supplement the otherwise synthetically generated data (Ma et al., 2023; Wang
et al., 2024). In both cases, the extra data improved the performance and
indicates that retrieval models can generally benefit from additional
unlabeled training using RE-AdaptIR.

| RepLLaMA | e5-Mistral
---|---|---
Dataset | Base | Domn | Base | Domn
FEVER | +0.1 | +0.3 | -0.1 | -2.7
HotPotQA | -0.4 | -0.8 | +0.5 | -0.4
NQ | -0.1 | -0.4 | +7.0 | +7.9
Quora | +2.0 | -0.7 | +2.4 | 0.0
Arguana | +0.8 | +0.5 | +10.1 | +3.1
Climate-FEVER | +1.0 | +1.4 | -1.1 | -7.6
DBPedia | +0.1 | 0.0 | +0.5 | +0.6
FiQA | +0.2 | -1.1 | +1.7 | -0.7
NFCorpus | 0.0 | -0.6 | +0.7 | -0.5
SCIDOCS | +0.1 | -0.5 | +0.6 | +0.5
SciFact | -0.9 | -2.7 | +2.1 | +0.2
TREC-COVID | -0.4 | -2.0 | +2.6 | +5.5
Touche-2020 | -0.1 | +0.4 | -0.5 | -1.6
Average | +0.2 | -0.5 | +2.0 | +0.3
Table 3: Change in nDCG@10 when RE-AdaptIR is applied with pretrained model
fine-tuned on MS-MARCO instead of the domain under evaluation. Base indicates
the change with respect to the original model, Domn the change with respect to
the model RE-Adapted on the evaluated domain.

##  5 Conclusion

In this work, we introduced RE-AdaptIR, an extension of RE-Adapt for using
unlabeled data to improve the zero-shot and in-domain performance of text
retrieval models. We demonstrated RE-AdaptIR improves two state-of-the-art
models: RepLLaMA and e5-Mistral. We find that fine-tuning on the documents
being queried for at test-time is not required, and still see increased
performance when they are excluded. RE-AdaptIR improved baseline performance
in two cases: one where in-domain data was used, and the other, using
additional unlabeled data related to the models’ original training corpus.
Combined, these results enforce the wide applicability of our approach, and
our findings ensure RE-AdaptIR is of immediate use to text-retrieval
practitioners.

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##  Appendix A Adapter Details

We trained DoRA adapters for all attention key, query, and value layers as
well as the up and down projection layers. All adapters used rank 32 with
alpha 64 and a LoRA dropout of 0.05.

We used a batch size of 4 using the AdamW optimizer with learning rate of
0.0002 with linear scheduling. Unless otherwise stated, all adapters were
trained over 1 epoch of the respective corpus with a max length of 1024 for
any example.

All training and evaluation was done using a single NVIDIA A100 GPU with 80GB
of memory.

##  Appendix B Corpus Sizes

Dataset | Corpus Size
---|---
MS-MARCO | 8.84M
Climate-FEVER | 5.42M
FEVER | 5.42M
HotPotQA | 5.23M
DBPedia | 4.63M
NQ | 2.68M
Quora | 523K
Touche-2020 | 382K
TREC-COVID | 171K
FiQA | 57K
SCIDOCS | 25K
ArguAna | 8.67K
SciFact | 5k
NFCorpus | 3.6K