# Suppressing classical noise in the accelerated geometric phase gate by
optimized dynamical decoupling

Da-tong Chen  School of Physics, Zhejiang University, Hangzhou 310027,
Zhejiang, China  Jun Jing<EMAIL_ADDRESS>School of Physics, Zhejiang
University, Hangzhou 310027, Zhejiang, China

###### Abstract

In the quantum-computation scenario, the geometric phase-gates are becoming
increasingly attractive for their intrinsic fault tolerance to disturbance.
With an adiabatic cyclic evolution, Berry phase appears to realize a geometric
transformation. Performing the quantum gates as many as possible within the
timescale of coherence, however, remains an inconvenient bottleneck due to the
systematic errors. Here we propose an accelerated adiabatic quantum gate based
on the Berry phase, the transitionless driving, and the dynamical decoupling.
It reconciles a high fidelity with a high speed in the presence of control
noise or imperfection. We optimize the dynamical-decoupling sequence in the
time domain under a popular Gaussian noise spectrum following the inversely
quadratic power-law.

##  I Introduction

By processing the data in a quantum-mechanical way, quantum computation out-
performs its classical counterpart in selected algorithms or tasks Shor
(1997); Grover (1997). The simulation of the coherent evolution of generic
quantum systems can be modelled by performing ordered sequences of high-
fidelity unitary operations Babbush  _et al._ (2014); Gerritsma  _et al._
(2010); Kivlichan  _et al._ (2018). The ubiquitous noises from uncontrollable
environment and imprecise control, however, are inevitable in any experimental
proposal, which pose challenges to the quality of quantum gate in terms of
fidelity and operation time Harper  _et al._ (2020); Song  _et al._ (2017).
The quantum geometric phase Berry (1984); Aharonov and Anandan (1987); Samuel
and Bhandari (1988); Wilczek and Zee (1984); Anandan (1988), which is
accumulated as the system is moving along the parametric path, has intrinsic
tolerant property against certain fluctuations during the trajectory. The
geometric quantum computation is therefore appealing and becoming feasible in
many experimental platforms, including the trapped ions Leibfried  _et al._
(2003); Cui  _et al._ (2019), the nuclear magnetic resonance Jones  _et al._
(2000); Feng  _et al._ (2013), and the superconducting circuits Abdumalikov Jr
_et al._ (2013); Xu  _et al._ (2018); Zhou  _et al._ (2021); Liang and Xue
(2022).

As a benchmark implementation of the Abelian geometric transformation, the
quantum gate based on the Berry phase requires that the quantum system
undergoes a desired adiabatic loop in the parametric space Berry (1984). The
process is intrinsically slow. Reducing the evolution time is in practice
demanded to avoid the accumulation of the decoherence influence. On the other
hand, however, speeding up always gives rise to the transition among the
instantaneous eigenstates of system, which is supposed to be “frozen” in the
adiabatic condition to maintain the gate fidelity. So that the competitive
coherence time and the high fidelity are seemingly contradictory for the
adiabatic geometric quantum gates. One of the compromise solutions to reduce
the evolution time is using the quantum gates based on the Aharonov-Anandan
phase Aharonov and Anandan (1987); Wang and Keiji (2001), which emerges in
arbitrary cyclic unitary evolutions. Alternatively, one can accelerate the
evolution process via the transitionless quantum driving (TQD) Berry (2009),
which is formulated by the ancillary Hamiltonian to avoid the level crossing
during the time evolution.

The TQD-accelerated quantum gate based on the Berry phase is still sensitive
to the systematic errors in the control Hamiltonian or driving parameters. A
significant feature of geometric quantum computation is its compatibility with
certain error-suppression techniques, e.g., the dynamical decoupling (DD)
Viola and Lloyd (1998); Witzel and Sarma (2007); Uhrig (2007); Pasini and
Uhrig (2010); Qin  _et al._ (2017), the unconventional geometric gates Zhu and
Wang (2003); Du  _et al._ (2006); Wang  _et al._ (2007), the decoherence-free
subspaces Feng  _et al._ (2009), the quantum error correction Oreshkov  _et
al._ (2009a, b), and the dynamical correction Liang and Xue (2022); Zhou  _et
al._ (2021), to maintain the fidelity of the adiabatic or nonadiabatic unitary
transformation. The dynamical decoupling was originated in the high-precision
magnetic resonance spectroscopy and has been applied to quantum control as a
long standing technique. A typical DD is to neutralize the effects from the
fluctuation noises by applying a sequence of inverse operations to a two-level
system Viola and Lloyd (1998). In this work, we aim to find an optimized
dynamical-decoupling sequence to protect the TQD-accelerated Berry-phase gate.

Particularly, we consider the effect from the Gaussian stochastic noises
Gardiner  _et al._ (2004) in the control parameters of our nonadiabatic
geometric quantum gates, which would deviate the output states from the noise-
free result. Under various resource of noises, we analyze the robustness of
the gate-fidelity for a general input state to estimate the deviation and then
apply the dynamical-decoupling sequences to improve the fidelity. In the time
domain, we analytically derive an optimized DD sequence under the Gaussian
color noise following the inversely quadratic power-law spectrum.

The rest of the work is arranged as follows. In Sec. II, we start from a semi-
classical Rabi model under parametric driving and obtain a general time-
dependent qubit Hamiltonian corrected by the counter-rotating interaction up
to the first order. This Hamiltonian is used to establish a set of more
accurate Berry phase gates accelerated by the transitionless quantum driving.
In Sec. III, we consider the stochastic fluctuations in various control
parameters and their damages to the fidelity of the preceding nonadiabatic
transformation in the free induction decay (FID). Then in Sec. IV, we apply
the DD technique to the Berry phase gate and discuss the results under the
inversely quadratic noise spectrum. The detailed derivation of the optimized
control sequence can be found in Appendix A. We discuss and summarize the
whole work in Sec. V.

##  II A more accurate effective Hamiltonian

We start with a Rabi model under parametric driving, in which a two-level
system (qubit) with energy-spacing $\omega_{a}$ is controlled by a driving
field with time-modulated frequency $\omega_{b}(t)$ and phase factor
$\varphi_{R}(t)$. The strength of the dipole-dipole interaction between the
qubit and the driving field (Rabi-frequency) is described by $\Omega_{R}(t)$.
The system Hamiltonian thus can be written as

$H(t)=\frac{\omega_{a}}{2}\sigma_{z}+\Omega_{R}(t)\cos\left[\Omega_{b}(t)+\varphi_{R}(t)\right]\sigma_{x},$
(1)

where $\Omega_{b}(t)\equiv\int_{0}^{t}ds\,\omega_{b}(s)$. It is not convenient
to construct a credible quantum gate directly from the original Hamiltonian in
Eq. (1) for the lack of a compact expression of the time evolution operator.
It is popular to see the application of the rotating-wave approximation (RWA)
in the previous treatments Gerry  _et al._ (2005); Leek  _et al._ (2007); Cui
_et al._ (2019). With respect to the unitary transformation
$R(t)=\exp[i\Omega_{b}(t)\sigma_{z}/2]$, one can find

$\displaystyle H^{\prime}(t)=R(t)H(t)R^{\dagger}(t)+i\dot{R}(t)R^{\dagger}(t)$
(2) $\displaystyle=\frac{1}{2}\big{\\{}[\omega_{a}-\omega_{b}(t)]\sigma_{z}$
$\displaystyle+\Omega_{R}(t)[\cos\varphi_{R}(t)+\cos(2\Omega_{b}(t)+\varphi_{R}(t))]\sigma_{x}$
$\displaystyle+\Omega_{R}(t)[\sin\varphi_{R}(t)-\sin(2\Omega_{b}(t)+\varphi_{R}(t))]\sigma_{y}\big{\\}}$
$\displaystyle\approx\frac{\omega_{a}-\omega_{b}(t)}{2}\sigma_{z}+\frac{\Omega_{R}(t)}{2}\left[e^{-i\varphi_{R}(t)}\sigma_{+}+h.c.\right],$

where the terms with the high-frequency $2\Omega_{b}(t)$ are omitted. The
error between the resultant Hamiltonian under RWA and the original one is thus
in the first order of $\mathcal{O}(\Omega_{R})$. $H^{\prime}(t)$ applies to
the dispersive regime of a sufficiently weak driving strength. The
contribution of the omitted counter-rotating terms becomes however significant
in the strong-coupling regime and demonstrates intriguing dynamical behaviors
Zheng  _et al._ (2008); Deng  _et al._ (2016); Lü and Zheng (2012); Jing  _et
al._ (2009).

To retain the first-order contribution from the counter-rotating terms, we
choose a different approach with a modified rotating-wave approximation Lü and
Zheng (2012). We come back to the Hamiltonian in Eq. (1) and apply a unitary
transformation with respect to

$S(t)=\exp\left\\{i\frac{\Omega_{R}(t)}{\omega_{a}+\omega_{b}(t)}\sin[\Omega_{b}(t)+\varphi_{R}(t)]\sigma_{x}\right\\}.$
(3)

The original Hamiltonian in the interaction picture is then rewritten as

$\displaystyle H_{0}(t)=S(t)H(t)S^{\dagger}(t)+i\dot{S}(t)S^{\dagger}(t)$ (4)
$\displaystyle=\frac{\omega_{a}}{2}\sigma_{z}+\Omega_{R}(t)\cos\left[\Omega_{b}(t)+\varphi_{R}(t)\right]\sigma_{x}$
$\displaystyle+\frac{\Omega_{R}(t)\omega_{a}}{\omega_{a}+\omega_{b}(t)}\sin\left[\Omega_{b}(t)+\varphi_{R}(t)\right]\sigma_{y}+\mathcal{O}(\Omega_{R}^{2})$
$\displaystyle-\frac{d}{dt}\left\\{\frac{\Omega_{R}(t)}{\omega_{a}+\omega_{b}(t)}\sin[\Omega_{b}(t)+\varphi_{R}(t)]\sigma_{x}\right\\}$
$\displaystyle\approx\frac{\omega_{a}}{2}\sigma_{z}+\frac{\Omega_{R}(t)\omega_{a}}{\omega_{a}+\omega_{b}(t)}\left[e^{-i\Omega_{b}(t)-i\varphi_{R}(t)}\sigma_{+}+h.c.\right],$

where we omitted the contribution up to the second-order of
$\mathcal{O}(\Omega_{R}^{2})$ and the first-order derivative of the driving
parameters with respect to time Leek  _et al._ (2007); Cui  _et al._ (2019)
under the assumptions that $|\dot{\omega}_{b}(t)|$, $|\dot{\varphi}_{R}(t)|$,
$|\dot{\Omega}_{R}(t)|\ll|\omega_{a}+\omega_{b}(t)|$.

Figure 1: The population dynamics of the qubit on $|+\rangle$ under various
Hamiltonians, in units of the Rabi frequency $\Omega_{R}$. The parameters are
set as $\omega_{a}=100\pi$ MHz, $\omega_{b}=60\pi$ MHz, and $\Omega_{R}=20\pi$
MHz.

Subsequently, in the rotating frame with respect to $R(t)$, one can find a
standard time-modulated Hamiltonian describing a qubit under an effective 3D
magnetic field, i.e.,

$\displaystyle H_{1}(t)=R(t)H_{0}(t)R^{\dagger}(t)+i\dot{R}(t)R^{\dagger}(t)$
(5)
$\displaystyle=\frac{\vec{B}(t)}{2}\cdot\vec{\sigma}=\frac{B_{0}(t)}{2}\vec{n}(t)\cdot\vec{\sigma},$

where
$\vec{n}(t)\equiv[\sin\theta(t)\cos\phi(t),\sin\theta(t)\sin\phi(t),\cos\theta(t)]$
parameterizes the direction of the magnetic field and $\vec{\sigma}$
represents the vector of Pauli operators. Using the driving parameters, we
have

$\left\\{\begin{aligned}
B_{0}(t)&=\sqrt{\left[\omega_{a}-\omega_{b}(t)\right]^{2}+\left[\frac{2\Omega_{R}(t)\omega_{a}}{\omega_{a}+\omega_{b}(t)}\right]^{2}},\\\
\phi(t)&=\varphi_{R}(t),\\\
\theta(t)&=\arctan\left[\frac{2\Omega_{R}(t)\omega_{a}}{\omega_{a}^{2}-\omega_{b}^{2}(t)}\right].\end{aligned}\right.$
(6)

The Hamiltonian in Eq. (5) adapts to a larger Rabi frequency $\Omega_{R}$ than
that in Eq. (2) by holding the first-order contribution from the counter-
rotating interaction. The distinction between these two approximated
Hamiltonians can be transparently illustrated by the Rabi oscillation of the
population on the state $|+\rangle\equiv(1,0)^{T}$ for a two-level system,
when the magnitudes of $\omega_{a}$, $\omega_{b}$, and $\Omega_{R}$ are chosen
in almost the same order. The blue solid line, the yellow dashed line, and the
green dot-dashed line in Fig. 1 represent the respective results under the
original Hamiltonian $H(t)$ in Eq. (1), the RWA Hamiltonian $H^{\prime}(t)$ in
Eq. (2), and the modified-RWA Hamiltonian $H_{1}(t)$. Much to our
anticipation, the rotating-wave interaction captures the sinusoid behavior in
quality, while losing nearly all the details of the dynamics. In contrast,
with the aid of the unitary transformation $S(t)$ in Eq. (3), the analytical
result by $H_{1}(t)$ is highly close to the numerical one by the original
Hamiltonian $H(t)$.

We then apply the effective Hamiltonian $H_{1}(t)$ to build a more accurate
quantum gate than $H^{\prime}(t)$. Universal single-qubit gates can be
implemented by virtue of the time-dependent $\theta(t)$ and $\phi(t)$ in Eq.
(6). $B_{0}$ is set as a constant number for simplicity. To avoid the
undesired transition among the instantaneous eigenstates of $H_{1}(t)$ with
accelerated time-dependence, one can add a counterdiabatic term following the
transitionless quantum driving Berry (2009) approach. The ancillary
Hamiltonian $H_{\rm CD}(t)$ can be expressed by

$\displaystyle H_{\rm
CD}(t)=\frac{1}{2}\left[\vec{n}(t)\times\dot{\vec{n}}(t)\right]\cdot\vec{\sigma}$
(7)
$\displaystyle=\frac{1}{2}\big{(}-\dot{\theta}\sin\phi-\dot{\phi}\sin\theta\cos\theta\cos\phi,$
$\displaystyle\dot{\theta}\cos\phi-\dot{\phi}\sin\theta\cos\theta\sin\phi,\dot{\phi}\sin^{2}\theta\big{)}\cdot\vec{\sigma},$

where $\vec{n}(t)$ is the unit vector of the magnetic field in Eq. (5). In Eq.
(7), the explicit $t$-dependence of quantities has been omitted for
simplicity, i.e., $\theta\equiv\theta(t)$ and $\phi\equiv\phi(t)$. The
corrected Hamiltonian with the counterdiabatic term is $H_{\rm
tot}(t)=H_{1}(t)+H_{\rm CD}(t)$, which could be diagonalized with the unitary
transformation $U_{0}(t)=\exp(i\theta\sigma_{y}/2)\exp(i\phi\sigma_{z}/2)$:

$\displaystyle H_{z}(t)=U_{0}(t)H_{\rm
tot}(t)U_{0}^{\dagger}(t)+i\dot{U}_{0}(t)U_{0}^{\dagger}(t)$ (8)
$\displaystyle=\frac{1}{2}\left(B_{0}-\dot{\phi}\cos\theta\right)\sigma_{z}.$

Rotating back to the concatenated rotating frame with respect to $S(t)$ and
$R(t)$, where $H_{1}(t)$ and $H_{\rm tot}(t)$ live, the time-evolution
operator reads,

$\displaystyle
U(t)=U_{0}^{\dagger}(t)U_{z}(t)=e^{-i\phi\sigma_{z}/2}e^{-i\theta\sigma_{y}/2}e^{-i\int_{0}^{t}dsH_{z}(s)}$
(9)
$\displaystyle=\begin{pmatrix}\cos\frac{\theta(t)}{2}e^{-\frac{i}{2}[\alpha(t)+\phi(t)]}&-\sin\frac{\theta(t)}{2}e^{\frac{i}{2}[\alpha(t)-\phi(t)]}\\\
\sin\frac{\theta(t)}{2}e^{-\frac{i}{2}[\alpha(t)-\phi(t)]}&\cos\frac{\theta(t)}{2}e^{\frac{i}{2}[\alpha(t)+\phi(t)]}\end{pmatrix}$

with $U_{z}(t)\equiv\exp[-i\int_{0}^{t}dsH_{z}(s)]$ and
$\alpha(t)=\int_{0}^{t}ds(B_{0}-\dot{\phi}\cos\theta)$. Under a proper
boundary condition, $U(t)$ can be used to realize any desired rotation or
qubit-gate. For example, under the setting that $\theta(T)=\pi$ and
$\alpha(T)-\phi(T)=\pi$, $U(T)$ realizes a Pauli-X gate up to a global phase.
In the following discussion, we are concerned with the phase shift in a cyclic
evolution. With $\theta(T)=2\pi$, the final time-evolution operator is found
to be in a diagonal form:

$U(T)=\begin{pmatrix}e^{-\frac{i}{2}[\alpha(T)+\phi(T)]}&0\\\
0&e^{\frac{i}{2}[\alpha(T)+\phi(T)]}\end{pmatrix},$ (10)

up to an unobservable global $\pi$-phase. In particular, when
$|\psi(0)\rangle=|\pm\rangle$ [$|-\rangle\equiv(0,1)^{T}$], one can derive a
cyclic process
$|\psi(T)\rangle=e^{i\pi\mp\frac{i}{2}[\alpha(T)+\phi(T)]}|\psi(0)\rangle=e^{i(\pi+\gamma_{\pm})}|\psi(0)\rangle$.
The total phase accumulated in this period is

$\gamma_{\pm}=\mp\frac{1}{2}\int_{0}^{T}dt\left[B_{0}-\dot{\phi}(\cos\theta-1)\right].$
(11)

The dynamical phase could be obtained by the corrected Hamiltonian $H_{\rm
tot}(t)$ and the eigenstates of the effective Hamiltonian $H_{1}(t)$. We have

$\displaystyle\gamma_{\pm}^{d}\equiv-\int_{0}^{T}dt\langle\psi(t)|H_{\rm
tot}(t)|\psi(t)\rangle$ (12)
$\displaystyle=-\int_{0}^{T}dt\langle\pm|U_{0}(t)H_{\rm
tot}(t)U_{0}^{\dagger}(t)|\pm\rangle$
$\displaystyle=-\int_{0}^{T}dt\langle\pm|\frac{1}{2}\left(-\dot{\phi}\sin\theta\sigma_{x}+\dot{\theta}\sigma_{y}+B_{0}\sigma_{z}\right)|\pm\rangle$
$\displaystyle=\mp\frac{1}{2}\int_{0}^{T}dtB_{0}=\mp\frac{1}{2}B_{0}T.$

The geometric phase is thus given by

$\gamma_{\pm}^{g}=\gamma_{\pm}-\gamma_{\pm}^{d}=\mp\frac{1}{2}\int_{0}^{T}dt\dot{\phi}(1-\cos\theta),$
(13)

which is exactly the solid angle in the Bloch sphere described by $\theta$ and
$\phi$ up to a scale of $-1/2$ Dong  _et al._ (2021). The evolution period $T$
could be shortened by TQD as long as the boundary conditions are satisfied.

Provided that the dynamical phase is completely cancelled by any control
technique, such as the dynamical decoupling Wang and Keiji (2001); Gregefalk
and Sjöqvist (2022) to be discussed later, the geometric phase in Eq. (13)
determines the final time evolution operator:

$U(T)\simeq\begin{pmatrix}e^{i\gamma_{+}^{g}}&0\\\
0&e^{i\gamma_{-}^{g}}\end{pmatrix}\simeq\begin{pmatrix}1&0\\\
0&e^{i\int_{0}^{T}dt\dot{\phi}(1-\cos\theta)}\end{pmatrix}.$ (14)

Then we can build a quantum geometric phase gate. Under the assumption
$\theta(t)=\omega t$, the period is $T=2\pi/\omega$ and $U(t)={\rm
diag}\\{1,e^{i\eta}\\}$, where $\eta=\gamma_{-}^{g}-\gamma_{+}^{g}$. Arbitrary
phase gate could be attained by adjusting $\phi$. For example, we can have a
Pauli-$Z$ gate when $\phi=\omega t/2$ and we have a $\pi/2$ gate when
$\phi=\omega t/4$. In general, an arbitrary input state
$|\psi(0)\rangle=\cos\alpha_{1}|+\rangle+\sin\alpha_{1}e^{i\alpha_{2}}|-\rangle$
would be transformed by the phase-gate to

$|\psi(T)\rangle=\cos\alpha_{1}|+\rangle+\sin\alpha_{1}e^{i\alpha_{2}}e^{i\eta}|-\rangle$
(15)

where $\alpha_{1}$ and $\alpha_{2}$ are real-number angles with
$\\{\alpha_{1},\alpha_{2}\\}\in[0,2\pi]$.

##  III The effects from classical noise

The geometric evolution in Eq. (15), which has been accelerated by the
counterdiabatic term, cannot be faithful in the presence of nonideal driving.
The existence of the classical noise gives rise to the deviation of the gate-
fidelity. Here we demonstrate how the noises in various control parameters
would be detrimental to the performance of quantum gates in the absence of
dynamical decoupling. Then in this section, we temporarily include the
dynamical phase in Eq. (12) that is exclusively determined by $B_{0}$. The
phase $\eta=\gamma_{-}^{g}-\gamma_{+}^{g}$ in Eq. (15) then becomes
$\eta=\gamma_{-}-\gamma_{+}$, i.e.,

$\eta=\int_{0}^{T}dt\left[B_{0}-\dot{\phi}(\cos\theta-1)\right].$ (16)

To measure the phase deviation induced by the classical noise on the quantum
gate, we use a gate-fidelity Cywiński  _et al._ (2008); Qin  _et al._ (2017);
Szańkowski  _et al._ (2017) defined as

$\mathcal{F}=\frac{1}{4\pi^{2}}\int_{0}^{2\pi}d\alpha_{1}\int_{0}^{2\pi}d\alpha_{2}\frac{M\left[\langle+|\rho(T)|-\rangle\right]}{\langle+|\psi(T)\rangle\langle\psi(T)|-\rangle},$
(17)

where $\rho(t)$ is the density matrix under the time evolution operator $U(t)$
with noisy parameters and the wavefunction $|\psi(T)\rangle$ describes the
ideal evolution in Eq. (15). $M[\cdot]$ means the ensemble average over the
random realizations of fluctuated control parameters, e.g., $B_{0}\rightarrow
B_{0}+\delta_{B}(t)$ and the arbitrary input states featured with $\alpha_{1}$
and $\alpha_{2}$.

In this work, the stationary Gaussian noise Jing  _et al._ (2017); Qin  _et
al._ (2017) $\delta_{\xi}(t)$ is assumed to follow the statistical properties

$\langle\delta_{\xi}(t)\rangle=0,\quad
C(t-s)=\langle\delta_{\xi}(t)\delta_{\xi}(s)\rangle=\frac{\Gamma\gamma}{2}e^{-\gamma|t-s|},$
(18)

where $\xi$ indicates the noise source, $\Gamma$ is the correlation intensity
of the noise, and $\gamma$ is the memory parameter. The Fourier transform of
the two-point correlation function $C(t-s)$ gives rise to an inverse-quadratic
spectral density,

$S(\omega)=\int_{-\infty}^{\infty}dte^{i\omega
t}C(t)=\frac{2\Gamma\gamma^{2}}{\omega^{2}+\gamma^{2}}.$ (19)

When $\gamma\rightarrow\infty$, $S(\omega)$ becomes structureless and
describes a Markovian or white noise; while when $\gamma\rightarrow 0$, it
describes a typical non-Markovian noise with a finite-memory capability.

In the rest part of this section, we calculate the gate-fidelity during the
free induction decay under the parametric fluctuations associated respectively
with the magnetic field intensity $B_{0}$ and the phase derivative
$\dot{\phi}$. Note these two parameters are separable in the expression of
quantum phases, allowing the individual addressing over different noise
resource. In comparison to both $B_{0}$ and $\dot{\phi}$, $\theta$ comes into
the phase in a cosine function, see e.g., Eq. (16), yielding a higher-order
contribution. Thus the effect from noisy $\theta$ could be omitted.

###  III.1 Fidelity under noisy magnetic field intensity

Figure 2: Landscape of the gate-fidelity $\mathcal{F}_{B}$ under the
stochastic $B_{0}$ in the parameter space of the strength-memory ratio
$\Gamma/\gamma$ and the running period $\gamma T$.

We first consider a fluctuated magnetic field $B_{0}$ in constructing the
phase gate, which is determined by the driving parameters in Eq. (5). With
respect to the unitary transformation $U_{0}(t)$, the nonideal counterdiabatic
corrected Hamiltonian in Eq. (8) becomes

$\tilde{H}_{z}(t)=\frac{1}{2}\left[B_{0}+\delta_{B}(t)-\dot{\phi}\cos\theta\right]\sigma_{z}.$
(20)

Consequently, the off-diagonal term of the final density matrix in Eq. (17)
can be obtained by

$\langle+|\rho(T)|-\rangle=\left\langle+\left|\tilde{U}^{\dagger}(T)\rho(0)\tilde{U}(T)\right|-\right\rangle$
(21)

where $\rho(0)\equiv|\psi(0)\rangle\langle\psi(0)|$ and $\tilde{U}(T)$ is

$\tilde{U}(T)=e^{-i\phi(T)\sigma_{z}/2}e^{-i\theta(T)\sigma_{y}/2}e^{-i\int_{0}^{T}ds\tilde{H}_{z}(s)}.$
(22)

Then we have

$\frac{\langle+|\rho(T)|-\rangle}{\langle+|\psi(T)\rangle\langle\psi(T)|-\rangle}=\exp\left[-i\int_{0}^{T}dt\delta_{B}(t)\right].$
(23)

Note this result is independent of $\alpha_{1}$ and $\alpha_{2}$, meaning the
noise effect on the gate-fidelity does not rely on the input states.
Substituting it into Eq. (17) and using the statistical properties in Eq.
(18), we have Gardiner  _et al._ (2004)

$\mathcal{F}_{B}=e^{-\frac{1}{2}\int_{0}^{T}dt\int_{0}^{T}dsC(t-s)}=e^{-\frac{\Gamma}{2\gamma}\left(\gamma
T+e^{-\gamma T}-1\right)}.$ (24)

The dependence of the gate-fidelity $\mathcal{F}_{B}$ on the dimensionless
cyclic period $\gamma T$ and the memory parameter $\Gamma/\gamma$ is plotted
in Fig. 2.

From both Eq. (24) and Fig. 2, a sufficiently large $\Gamma/\gamma$ or a
sufficiently small $\gamma$ gives rise to a nearly exponential decay, which is
consistent with a typical Markovian dynamics describing decoherence induced by
the white noise. While the non-exponential decay appears in the short-time
regime with $\gamma T\ll 1$, where up to the leading order the exponent of the
fidelity in Eq. (24) becomes quadratic to the running time:

$\mathcal{F}_{B}\approx\exp\left(-\frac{\Gamma\gamma}{4}T^{2}\right).$ (25)

Then the characteristic decoherence time $T_{2}$ can be obtained by the
definition $\ln\mathcal{F}_{B}(T_{2})=-1$ Cywiński  _et al._ (2008)

$T_{2}=\frac{2}{(\Gamma\gamma)^{1/2}}$ (26)

for the free induction decay. Clearly a high-level gate-fidelity can be
maintained in the presence of a weak and non-Markovian classical noise. It is
found when $\Gamma/\gamma\leq 1.0$, $\mathcal{F}_{B}$ is over $0.90$ at
$\gamma T=0.72$.

###  III.2 Fidelity under noisy control phase $\phi$

Figure 3: Landscape of the gate-fidelity $\mathcal{F}_{\phi}$ under the
stochastic $\dot{\phi}$ in the parameter space of the strength-memory ratio
$\Gamma/\gamma$ and the running period $\gamma T$. In (a) and (b), the noise-
free parameter $\theta$ is set as $\theta(t)=\omega t$ and $\theta(t)=\omega
t-\sin(\omega t)$, respectively.

Now we consider that the imperfect control over the phase $\phi$ in the
presence of random noise associated with its time-derivative $\dot{\phi}$,
which leads to $\phi(T)\rightarrow\phi(T)+\Delta_{\phi}(T)$ with
$\Delta_{\phi}(T)=\int_{0}^{T}dt\delta_{\phi}(t)$. Then in the rotating frame,
the nonideal Hamiltonian for the accelerated phase gate becomes,

$\tilde{H}_{z}(t)=\frac{1}{2}\left\\{B_{0}-\left[\dot{\phi}+\delta_{\phi}(t)\right]\cos\theta\right\\}\sigma_{z}.$
(27)

Using Eqs. (17), (21), and (27), it is straightforward to find for
$\theta=\omega t$ that

$\displaystyle\mathcal{F}_{\phi}=e^{-\frac{1}{2}\int_{0}^{T}dt\int_{0}^{T}ds\left[1-\cos\theta(t)\right]\left[1-\cos\theta(s)\right]C(t-s)}$
(28)
$\displaystyle=\exp\Big{\\{}-\frac{\Gamma}{4\gamma(4\pi^{2}+\gamma^{2}T^{2})^{2}}\big{[}32\pi^{4}e^{-\gamma
T}$ $\displaystyle+32\pi^{4}\left(-1+\gamma T\right)+20\pi^{2}\left(\gamma
T\right)^{3}+3\left(\gamma T\right)^{5}\big{]}\Big{\\}},$

Similar to $\mathcal{F}_{B}$ in Eq. (24), the fidelity $\mathcal{F}_{\phi}$ in
Eq. (28) is also a function of dimensionless parameters $\gamma T$ and
$\Gamma/\gamma$. An interesting observation is that under the short time
limit, i.e., $\gamma T\ll 1$, we have
$\mathcal{F}_{\phi}\approx\exp(-\Gamma\gamma T^{2}/4)$, the same as Eq. (25).
Then we obtain the same decoherence timescale $T_{2}$ as in Eq. (26). One can
hardly distinguish Fig. 2 and Fig. 3(a). It is found when $\Gamma/\gamma\leq
1.0$, $\mathcal{F}_{\phi}$ is over $0.90$ at $\gamma T=0.69$.

Equation (28) is however reminiscent of an improved scheme in constructing the
quantum phase-gate by virtue of the time-dependence of $\theta$. Rather than a
constant $\theta$ used in literature Gregefalk and Sjöqvist (2022); Wang and
Keiji (2001), one can reduce the magnitude of the integrand in Eq. (28) by
selecting a proper and experimentally accessible $\theta(t)$, provided that
the boundary condition is satisfied. For the exponent of the fidelity in Eq.
(28), the main contribution to the integral is around $t-s\approx 0$ according
to the exponential decay of the correlation function $C(t-s)$ [see e.g., Eq.
(18), and the monotonic or asymptotic decay behavior is popular for all the
stationary Gaussian noises]. Then we have
$\mathcal{F}_{\phi}\approx\exp\\{-1/2[\int_{0}^{T}dt(1-\cos\theta(t))]^{2}\\}$.
The fidelity can thus be enhanced to a certain extent by reducing the
magnitude of $\int_{0}^{T}dt[1-\cos\theta(t)]$. For example, one can replace
$\theta(t)=\omega t$ with $\theta(t)=\omega t-\sin(\omega t)$ that holds the
same boundary condition. The induced improvement in gate-fidelity can be
observed in Fig. 3(b). In contrast to Fig. 3(a), $\mathcal{F}_{\phi}$ is over
$0.90$ with $\Gamma/\gamma=3.1$ when $\gamma T=0.69$. And when
$\Gamma/\gamma=1.0$, the same high-level fidelity can be sustained until
$\gamma T=1.24$.

##  IV Suppress noise by dynamical decoupling

The noise analysis over both $\delta_{B}$ and $\delta_{\phi}$ renders the same
second-order behavior in the running time under the short-time limit, as
suggested by Eqs. (25) and (26). The gate-fidelity decay induced by the
classical noise on the control parameters can be suppressed by a sequence of
dynamical decoupling. As a developed technique, dynamical decoupling can
extend the coherence time in many experiments. It has been generalized into
various sequences of pulse to neutralize the influence of the environmental
noises Yang  _et al._ (2011). Spin echo (SE) Hahn (1950) presents the simplest
yet the original form in these pulse sequences, which consists of only one
$\pi$-pulse in the middle of the time evolution besides another one performed
in the end. Based on SE, the Carr-Purcell-Meiboom-Gill sequence (CPMG) Carr
and Purcell (1954); Meiboom and Gill (1958); Witzel and Sarma (2007) employs
two or more $\pi$-pulses. In general, the $n$-pulse version of CPMG Witzel and
Sarma (2007) can be described by a sequence of $t_{k}=(k-1/2)T/n$,
$k=1,2,\cdots,n$, that is obtained in the frequency domain. When the noise
spectrum has a hard cutoff Pasini and Uhrig (2010), i.e.,
$S(\omega)\sim\omega\Theta(\omega_{c}-\omega)$, where $\Theta(x)$ is the
Heaviside step function [$\Theta(x)=1$ when $x\geq 0$ and $\Theta(x)=0$ when
$x<0$] or has an exponential-decay cutoff, i.e.,
$S(\omega)\sim\exp(-\omega/\omega_{c})$, the Uhrig dynamical-decoupling (UDD)
sequence Uhrig (2007) is shown to be the most efficient scheme. It can reduce
the decoherence rate down to the $n$th order of the running time by using $n$
non-periodical pulses. In this section, we focus on the noise spectrum of the
magnetic field $\delta_{B}(t)$ following the inverse-quadratic power-law. It
is shown that CPMG is the optimized choice instead of UDD. The analysis can be
straightforwardly extended to the noisy $\dot{\phi}$.

###  IV.1 Spin echo on geometric phase

Figure 4: Landscape of the spin-echo gate-fidelity $\mathcal{F}_{\rm SE}$
under the stochastic $B_{0}$ in the parameter space of the strength-memory
ratio $\Gamma/\gamma$ and the running period $\gamma T$.

We choose SE as a warm-up example to illustrate the DD effect on the geometric
phase. The other pulse sequences can be analysed in a similar way. To focus on
the sequence itself, the pulses are considered to be ideal, i.e.,
instantaneous and with no error Uhrig (2007); Witzel and Sarma (2007). The SE
scheme is a concatenated process of two piecewise segments, i.e.,
$0\rightarrow t_{f}$ and $t_{f}\rightarrow T$, where $t_{f}=T/2$. A
$\pi$-pulse is inserted at the moment $t_{f}$ to switch the signs of the
eigenstates. Another $\pi$-pulse is imposed on the system in the end to
complete a cyclic process. Starting from the initial state
$|\psi(0)\rangle=|+\rangle$ and denoting the instantaneous eigenstates as
$|\pm(t)\rangle$, the whole process can be described by

$\displaystyle|\psi(0)\rangle=|+\rangle\rightarrow|\psi(t_{f}-0^{+})\rangle=e^{i\bar{\gamma}_{1}}\left|+(t_{f})\right\rangle\xrightarrow{\pi\,pulse}$
(29)
$\displaystyle|\psi(t_{f}+0^{+})\rangle=e^{i\bar{\gamma}_{1}}\left|-(t_{f})\right\rangle\rightarrow$
$\displaystyle\left|\psi(T-0^{+})\right\rangle=e^{i(\bar{\gamma}_{1}+\bar{\gamma}_{2})}\left|-(T)\right\rangle\xrightarrow{\pi\,pulse}$
$\displaystyle|\psi(T)\rangle=e^{i(\bar{\gamma}_{1}+\bar{\gamma}_{2})}\left|+(T)\right\rangle=e^{i(\pi+\bar{\gamma}_{1}+\bar{\gamma}_{2})}|+\rangle,$

where the boundary condition has been applied in the last equation.
$\bar{\gamma}_{1}$ and $\bar{\gamma}_{2}$ represent the quantum phases
generated in the first and the second segments, respectively. According to
Eqs. (10), (11), and (20), these two nonideal phases turn out to be

$\displaystyle\bar{\gamma}_{1}=-\frac{1}{2}\int_{0}^{T/2}dt\left[B_{0}+\delta_{B}(t)-\dot{\phi}^{\prime}(\cos\theta^{\prime}-1)\right],$
(30)
$\displaystyle\bar{\gamma}_{2}=\frac{1}{2}\int_{T/2}^{T}dt\left[B_{0}+\delta_{B}(t)-\dot{\phi}^{\prime}(\cos\theta^{\prime}-1)\right],$

where $\phi^{\prime}=\phi$ and $\theta^{\prime}=\theta$ during the first-half
period $t\in(0,t_{f})$ and $\phi^{\prime}=-\phi$ and $\theta^{\prime}=\theta$
during the second-half period $t\in(t_{f},T)$ due to the spin-echo effect. By
Eq. (30), the ideal dynamical phase proportional to $B_{0}$ vanishes in the
end of the running while the geometric phase is accumulated to realize a
desired transformation and it holds the same form as in Eq. (13). When the
initial state starts from $|\psi(0)\rangle=|-\rangle$, the signs of those
phases in Eq. (30) are reversed and the SE scheme still works. In any loop,
however, the dynamical phase associated with the accumulation of the
stochastic noise $\delta_{B}(t)$ during the two consecutive time-integrals has
not been completely eliminated.

Then we measure the gate-fidelity in the presence of the random magnetic field
under the spin echo scheme. The ratio of the off-diagonal elements of the
density matrix in Eq. (17) is found to be

$\frac{\langle+|\rho(T)|-\rangle}{\langle+|\psi(T)\rangle\langle\psi(T)|-\rangle}=e^{-i\left[\int_{0}^{T/2}dt\delta_{B}(t)-\int_{T/2}^{T}dt\delta_{B}(t)\right]}.$
(31)

On ensemble average, we have

$\displaystyle\mathcal{F}_{\rm
SE}=M\left[\exp\left(-i\int_{0}^{T/2}dt\delta_{B}(t)+i\int_{T/2}^{T}dt\delta_{B}(t)\right)\right]$
(32) $\displaystyle=\exp\left[-\frac{\Gamma}{2\gamma}\left(\gamma T-e^{-\gamma
T}-3+4e^{-\gamma T/2}\right)\right],$

as plotted in Fig. 4. In contrast to Fig. 2, one can find that under the SE
scheme, the high-level regime of the gate-fidelity is significantly enlarged.
When $\Gamma/\gamma=2.0$, $\mathcal{F}_{\rm SE}$ can be maintained over $0.98$
for $\gamma T=1.0$. In the short-time limit $\gamma T\ll 1$, we have

$\mathcal{F}_{\rm
SE}\approx\exp\left(-\frac{\Gamma\gamma^{2}}{24}T^{3}\right).$ (33)

In comparison to the quadratic dependence in Eq. (25), the exponent function
is reduced to the third-order of the running time of the geometric gate by the
spin echo effect. It is shown in Appendix A that for the inversely quadratic
power-law noise spectrum, one can further reduce the decay coefficient by more
DD operations, yet cannot reduce the exponent function to more higher orders
of the running interval $T$.

###  IV.2 CPMG on geometric phase

Generally, we apply $n$ $\pi$-pulses into the running period $(0,T)$ to
suppress the geometric-phase error. The whole process is divided into $n+1$
segments described by $0\rightarrow t_{1}$, $t_{1}\rightarrow t_{2}$,
$\cdots$, $t_{n}\rightarrow T$. We are desired to seek an optimized sequence
of the DD-operation moments $t_{k}$. Similar to Eq. (32), the gate-fidelity
can be represented by

$\mathcal{F}_{n}=M\left[e^{-i\int_{0}^{T}dt\delta_{B}(t)f(T;t)}\right],$ (34)

where $f(T;t)=\sum_{k=0}^{n}(-1)^{k}\Theta(t_{k+1}-t)\Theta(t-t_{k})$ with
$t_{0}=0$ and $t_{n+1}=T$, indicating the $\pi$-pulse-induced sign change of
the quantum phase. Moreover, to hold the same geometric phase as that in Eq.
(13) for the FID process, the phase parameters $\phi^{\prime}$,
$\theta^{\prime}$ in all the segments are set as $\phi^{\prime}=f(T;t)\phi$
and $\theta^{\prime}=\theta$.

The gate-fidelity in Eq. (34) can be represented by $\mathcal{F}_{n}\equiv
e^{-\chi(T)}$ Cywiński  _et al._ (2008); Szańkowski  _et al._ (2017), where
the decay function $\chi(T)$ is obtained by the noise spectrum $S(\omega)$ and
$\tilde{f}(\omega,T)=\int_{0}^{T}dte^{-i\omega t}f(T;t)$ [the Fourier
transform of $f(T;t)$],

$\displaystyle\chi(T)=\frac{1}{2}\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}|\tilde{f}(\omega,T)|^{2}S(\omega)$
(35)
$\displaystyle=\frac{1}{2}\int_{-\infty}^{\infty}\frac{d\omega}{\pi}\frac{F(\omega
T)}{\omega^{2}}S(\omega).$

Note in the second line of Eq. (35), a filter function $F(\omega
T)\equiv|\omega\tilde{f}(\omega,T)|^{2}/2$ appears to measure the effects of
the pulse sequence. Using the Taylor expansion around $\omega T=0$, we have

$\displaystyle\chi(T)$
$\displaystyle=\int_{-\infty}^{\infty}\frac{d\omega}{2\pi\omega^{2}}\Big{[}F(0)+F^{\prime}(0)\omega
T+F^{\prime\prime}(0)\frac{(\omega T)^{2}}{2!}$ (36)
$\displaystyle+F^{(3)}(0)\frac{(\omega T)^{3}}{3!}+\cdots+\Big{]}S(\omega).$

Under the general noise with a power-law spectrum $S(\omega)\sim
1/\omega^{m}$, $m\geq 2$, the high-order terms $F^{(n)}(0)$ with $n-m\geq 1$
would lead to the nonconvergent integration
$\int_{-\infty}^{\infty}d\omega\omega^{n-2-m}$. In contrast, if $S(\omega)$
has a hard cutoff at an upper-bound frequency $\omega_{c}$ or has an
exponential decay with frequency, such as
$S(\omega)\sim\exp(-\omega/\omega_{c})$, then the convergency could be hold at
any order. Without frequency cutoff, the UDD scheme Uhrig (2007) does not
necessarily supply an optimized sequence for polynomial spectrum.

Some existing works Szańkowski  _et al._ (2017); Cywiński  _et al._ (2008)
about DD have addressed the noise $\delta_{B}(t)$ with the statistical
properties in Eq. (18). Here we provide an alternative way in Appendix A to
optimize the pulse sequence in the time domain for the quadratic power-law
spectrum. We find the most efficient sequence can be analytically described by

$t_{k}=\frac{(k-1/2)}{n}T,\quad k=1,2,\cdots,n,$ (37)

which is exactly the CPMG$n$ sequence with $n\geq 2$. When $n=1$, it reduces
to the SE sequence. When $n=2$, it is also coincident to the UDD sequence.

Figure 5: The gate-fidelity under various DD sequences (SE, CPMG$n$, and
UDD$n$) in the presence of the Gaussian noise following the inverse-quadratic
power-law spectrum. (a) the gate dynamics versus $\gamma T$ when
$\Gamma/\gamma=4$; (b) the gate dependence on $\Gamma/\gamma$ when $\gamma
T=4$.

In Figs. 5(a) and (b), we show the gate-fidelities under various DD-sequence
in terms of the dynamics and their dependence on the memory parameter,
respectively. The solid line, circle-dotted line, plus-dotted line, star-
dotted line, cross-dotted line, right-triangle-dotted line, and left-triangle-
dotted line represent the fidelities under FID, SE, CPMG$2$, UDD$3$, CPMG$3$,
UDD$10$, and CPMG$10$, respectively, where $n$ denotes the number of
$\pi$-pulses. In both Figs. 5(a) and 5(b), the performance of the geometric
phase gate is steadily improved by inserting more periodical $\pi$-pulses into
the running period. We also demonstrate that CPMG$n$ outperforms UDD$n$ when
$n\geq 3$. It is found when $\Gamma/\gamma=4.0$ and $\gamma T=4.0$, the gate-
fidelity can be maintained over $0.90$ by CPMG$10$. By Eqs. (37) and (35), it
is found that up to the leading-order contribution the gate-fidelity becomes

$\mathcal{F}_{n}\approx\exp\left[-\frac{\Gamma}{24n^{2}\gamma}\left(\gamma
T\right)^{3}\right].$ (38)

As for the scaling of the decoherence time $T_{2}$ over the pulse number $n$,
we have

$T_{2}=\left(\frac{24}{\Gamma\gamma^{2}}\right)^{1/3}n^{2/3}.$ (39)

The power-law with $n^{2/3}$ implies that more DD pulses are required to
enhance the coherence time of our geometric phase gate.

##  V Discussion and conclusion

The analysis over the classical noise in this work is mainly based on the
Gaussian approximation. Then the two-point correlation function $C(t-s)$ is
sufficient to determine its impact on the decoherence behavior in the time
evolution. As pointed out in literature Szańkowski  _et al._ (2017); Cywiński
_et al._ (2008), random telegraph noise in the weak noise regime can be
described by Eq. (18) and the power-law noise can be described by a linear
combination of the similar correlation functions with various parameters.
Under the non-Gaussian assumption, the noise effect also involves with
multiple-point correlation function. It is thus not convenient to analytically
obtain $\mathcal{F}_{B}$ in Eq. (24), $\mathcal{F}_{\phi}$ in Eq. (28), and
$\mathcal{F}_{SE}$ in Eq. (32) to witness the effect of a non-Markovian noise
on the gate-fidelity. Although the non-Gaussian noise would reduce the
coherence time in FID, however, it has been shown to be more suppressed under
CPMG rather than UDD Szańkowski  _et al._ (2017). The non-Gaussian noise is
therefore negligible when applying our dynamical-decoupling scheme, that is
proved to be a CPMG, into the accelerated phase-gate construction.

In summary, we construct a quick-and-faith geometric phase gate by combining
the transitionless-quantum-driving approach and the dynamical decoupling
control into the Berry phase. The former is used to shorten the running time
as required by the adiabatic passage and the latter is used to neutralize the
classical noise on the control parameters. Our proposal is based on a semi-
classical Rabi model under a parametric driving. Using a modified
transformation to hold the first-order contribution from the counter-rotating
interaction, our effective Hamiltonian is superior to that under the
conventional rotating-wave approximation and adapts to a strong driving beyond
the dispersive regime. We analysis the gate-fidelity under the random
fluctuation or noise on the effective magnetic field and the control phase. In
the time domain, we find that the CPMG$n$ sequence is the most efficient DD
scheme against the inversely quadratic power-law noise to maintain a high-
level fidelity and we obtain the scaling behavior of the decoherence time
$T_{2}$ with respect to the DD operation number $n$. Our investigation
provides a systematic estimation over the errors in constructing the Berry-
phase gate caused by the classical noise. It is useful in optimizing the
performance of quantum gates under Gaussian noise following a power-law
spectrum.

In addition, combining the noninstantaneous DD pulses with the approximated
TQD approaches Hatomura and Takahashi (2021); Chandarana  _et al._ (2022)
would be a near-future target. The phase-gate fidelity is supposed to be more
robust when the counterdiabatic operation could be performed along the same
direction as the DD pulses.

## Acknowledgments

We acknowledge financial support from the National Science Foundation of China
(Grants No. 11974311 and No. U1801661).

##  Appendix A Optimized sequence for Gaussian noise of power-law spectrum

This appendix contributes to optimizing DD sequence to cancel the classical
Gaussian noise on the magnetic-field intensity. We provide a proof in the time
domain rather than that in the frequency domain Cywiński  _et al._ (2008). The
derivation starts from the gate-fidelity under $n$ pulses in Eq. (34). Using
the correlation function in Eq. (18), we have

$\displaystyle\mathcal{F}_{n}\left(\\{t_{k},k=1,2,\cdots,n\\}\right)=M\left[e^{-i\int_{0}^{T}dt\delta_{B}(t)f(T;t)}\right]$
(40)
$\displaystyle=\exp\left[-\frac{1}{2}\int_{0}^{T}dt\int_{0}^{T}dsC(t-s)f(T;t)f(T;s)\right].$

With the dimensionless parameters $x\equiv\gamma T$, $y\equiv\Gamma/\gamma$
and $\mu_{k}\equiv t_{k}/T$, the decay function
$\chi(x,y)\equiv-\ln\mathcal{F}_{n}$ can be expressed by

$\displaystyle\chi(x,y)=\frac{y}{2}\Big{[}x-1+(-1)^{n}e^{-x}$ (41)
$\displaystyle-2\sum_{k=0}^{n+1}\sum_{j=1}^{n}(-1)^{|k-j|}e^{-|\mu_{k}-\mu_{j}|x}\Big{]},$

where $\mu_{0}=0$ and $\mu_{n+1}=1$. Note expanding $\chi(x,y)$ about $x$ is
equivalent to expanding it about $T$. It is obtained that

$\chi(x,y)=\frac{y}{2}\left(C_{0}+C_{1}x+C_{2}x^{2}+C_{3}x^{3}+\cdots\right),$
(42)

where the coefficients of the first few orders are

$C_{0}=-1+(-1)^{n}-2\sum_{k=0}^{n+1}\sum_{j=1}^{n}(-1)^{|k-j|},$ (43a)
$C_{1}=1-(-1)^{n}+2\sum_{k=0}^{n+1}\sum_{j=1}^{n}(-1)^{|k-j|}|\mu_{k}-\mu_{j}|,$
(43b)
$C_{2}=\frac{1}{2}(-1)^{n}-\sum_{k=0}^{n+1}\sum_{j=1}^{n}(-1)^{|k-j|}|\mu_{k}-\mu_{j}|^{2},$
(43c)
$C_{3}=\frac{1}{6}(-1)^{n+1}+\frac{1}{3}\sum_{k=0}^{n+1}\sum_{j=1}^{n}(-1)^{|k-j|}|\mu_{k}-\mu_{j}|^{3}.$
(43d)

It is straightforward to verify that both the zero-order and the first-order
coefficients $C_{0}$ and $C_{1}$ are exactly $0$, irrespective to the sequence
arrangement $\\{\mu_{k}\\}$.

When $n$ is an even integer, the summation in the second-order coefficient
$C_{2}$ can be decomposed into three terms, being formally relevant to
$\mu_{k}^{2}$, $-2\mu_{k}\mu_{j}$, and $\mu_{j}^{2}$, respectively. By virtue
of $\mu_{0}=0$ and $\mu_{n+1}=1$, the $\mu_{k}^{2}$ or $\mu_{j}^{2}$ term is
proved to vanish by

$\sum_{k=0}^{n+1}\sum_{j=1}^{n}(-1)^{|k-j|}\mu_{k}^{2}=\sum_{k=0}^{n+1}\mu_{k}^{2}\sum_{j=1}^{n}(-1)^{|k-j|}=0.$
(44)

Equation (43c) can then be reduced to

$\displaystyle
C_{2}=\frac{1}{2}+2\sum_{k=0}^{n+1}\sum_{j=1}^{n}(-1)^{|k-j|}\mu_{k}\mu_{j}$
(45)
$\displaystyle=\frac{1}{2}+2\left[\sum_{k=1}^{n}(-1)^{k-1}\mu_{k}\right]^{2}+2\sum_{k=1}^{n}(-1)^{k-1}\mu_{k}$
$\displaystyle=\frac{1}{2}\left[1+\sum_{k=1}^{n}(-1)^{k-1}2\mu_{k}\right]^{2}.$

When $n$ is odd, the summations about $\mu_{k}^{2}$ and $\mu_{j}^{2}$ in
$C_{2}$ are simplified to

$\displaystyle\sum_{k=0}^{n+1}\mu_{k}^{2}\sum_{j=1}^{n}(-1)^{|k-j|}+\sum_{j=1}^{n}\mu_{j}^{2}\sum_{k=0}^{n+1}(-1)^{|k-j|}$
(46)
$\displaystyle=\sum_{k=0}^{n+1}\mu_{k}^{2}(-1)^{|k-1|}+\sum_{j=1}^{n}\mu_{j}^{2}(-1)^{j}$
$\displaystyle=\mu_{0}^{2}(-1)^{1}+\mu_{n+1}^{2}(-1)^{n}=-1.$

Then Eq. (43c) can be reduced to

$\displaystyle
C_{2}=-\frac{1}{2}-(-1)+2\sum_{k=0}^{n+1}\sum_{j=1}^{n}(-1)^{|k-j|}\mu_{k}\mu_{j}$
(47)
$\displaystyle=\frac{1}{2}+2\left[\sum_{k=1}^{n}(-1)^{k-1}\mu_{k}\right]^{2}-2\sum_{k=1}^{n}(-1)^{k-1}\mu_{k}$
$\displaystyle=\frac{1}{2}\left[-1+\sum_{k=1}^{n}(-1)^{k-1}2\mu_{k}\right]^{2}.$

Summarizing Eqs. (45) and (47), the condition of $C_{2}=0$ can be written as

$(-1)^{n}+2\sum_{k=1}^{n}(-1)^{k-1}\mu_{k}=0.$ (48)

It has a clear physical indication or consequence that after applying those
$\pi$-pulses at $\mu_{k}$, the dynamical phase in the absence of noise is thus
exactly eliminated to leave a geometric transformation.

To find the optimized sequence, we now study the extreme-value condition
$\partial_{\mu_{k}}C_{3}=0$, $k=1,2,\cdots,n$, under the constrains that
$0<\mu_{1}<\mu_{2}<\cdots<\mu_{n}<1$. By virtue of Eq. (48), we have

$\mu_{n}=\sum_{k=1}^{n-1}(-1)^{n-k-1}\mu_{k}+\frac{1}{2}.$ (49)

With Eq. (43d), we have for $k=n-1$ that

$\displaystyle\frac{\partial
C_{3}}{\partial\mu_{n-1}}=(\mu_{n-1}-\mu_{n})\Big{[}2-\mu_{n-1}-\mu_{n}$ (50)
$\displaystyle+(-1)^{n-1}(\mu_{n-1}+\mu_{n})$
$\displaystyle+2\sum_{k=1}^{n-2}(-1)^{n-k-1}(\mu_{n-1}+\mu_{n}-2\mu_{k})\Big{]}=0$

For $n$ is either even or odd, it gives rise to

$\mu_{n-1}=\frac{3}{2}\sum_{k=1}^{n-2}(-1)^{n-k-2}\mu_{k}+\frac{1}{4}.$ (51)

By a similar derivation with decreasing $k$, we can obtain a general formula

$\mu_{n-j}=\frac{2j+1}{j+1}\sum_{k=1}^{n-j-1}(-1)^{n-k-j-1}\mu_{k}+\frac{1}{2j+2}.$
(52)

In the end, it is found that $\mu_{1}=1/(2n)$. Then by iteratively using Eq.
(52), we can find the solution is exactly the CPMG$n$, i.e.,
$\mu_{k}=(k-1/2)/n$. And the minimum-value of the third-order coefficient
$C_{3}$ turns out to be $x^{3}/(12n^{2})$, which corresponds to the decay
behavior in Eq. (38) of the main text. The result also demonstrates that the
$C_{3}$ cannot be completely eliminated under arbitrary choice of $\mu_{k}$ or
$t_{k}$.

## References

  * Shor (1997) P. W. Shor,  _Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer,_SIAM J. Comput. 26, 1484 (1997).
  * Grover (1997) L. K. Grover,  _Quantum mechanics helps in searching for a needle in a haystack,_Phys. Rev. Lett. 79, 325 (1997).
  * Babbush  _et al._ (2014) R. Babbush, P. J. Love, and A. Aspuru-Guzik,  _Adiabatic quantum simulation of quantum chemistry,_Sci Rep 4, 1 (2014).
  * Gerritsma  _et al._ (2010) R. Gerritsma, G. Kirchmair, F. Zähringer, E. Solano, R. Blatt, and C. Roos,  _Quantum simulation of the dirac equation,_Nature 463, 68 (2010).
  * Kivlichan  _et al._ (2018) I. D. Kivlichan, J. McClean, N. Wiebe, C. Gidney, A. Aspuru-Guzik, G. K.-L. Chan, and R. Babbush,  _Quantum simulation of electronic structure with linear depth and connectivity,_Phys. Rev. Lett. 120, 110501 (2018).
  * Harper  _et al._ (2020) R. Harper, S. T. Flammia, and J. J. Wallman,  _Efficient learning of quantum noise,_Nat. Phys. 16, 1184 (2020).
  * Song  _et al._ (2017) C. Song, S.-B. Zheng, P. Zhang, _et al._ ,  _Continuous-variable geometric phase and its manipulation for quantum computation in a superconducting circuit,_Nat Commun 8, 1 (2017).
  * Berry (1984) M. V. Berry,  _Quantal phase factors accompanying adiabatic changes,_Proc. R. Soc. Lond. A 392, 45 (1984).
  * Aharonov and Anandan (1987) Y. Aharonov and J. Anandan,  _Phase change during a cyclic quantum evolution,_Phys. Rev. Lett. 58, 1593 (1987).
  * Samuel and Bhandari (1988) J. Samuel and R. Bhandari,  _General setting for berry’s phase,_Phys. Rev. Lett. 60, 2339 (1988).
  * Wilczek and Zee (1984) F. Wilczek and A. Zee,  _Appearance of gauge structure in simple dynamical systems,_Phys. Rev. Lett. 52, 2111 (1984).
  * Anandan (1988) J. Anandan,  _Non-adiabatic non-abelian geometric phase,_Phys. Lett. A 133, 171 (1988).
  * Leibfried  _et al._ (2003) D. Leibfried, B. DeMarco, V. Meyer, D. Lucas, M. Barrett, J. Britton, W. M. Itano, B. Jelenković, C. Langer, T. Rosenband, _et al._ ,  _Experimental demonstration of a robust, high-fidelity geometric two ion-qubit phase gate,_Nature 422, 412 (2003).
  * Cui  _et al._ (2019) J.-M. Cui, M.-Z. Ai, R. He, Z.-H. Qian, X.-K. Qin, Y.-F. Huang, Z.-W. Zhou, C.-F. Li, T. Tu, and G.-C. Guo,  _Experimental demonstration of suppressing residual geometric dephasing,_Sci. Bull. 64, 1757 (2019).
  * Jones  _et al._ (2000) J. A. Jones, V. Vedral, A. Ekert, and G. Castagnoli,  _Geometric quantum computation using nuclear magnetic resonance,_Nature 403, 869 (2000).
  * Feng  _et al._ (2013) G. Feng, G. Xu, and G. Long,  _Experimental realization of nonadiabatic holonomic quantum computation,_Phys. Rev. Lett. 110, 190501 (2013).
  * Abdumalikov Jr  _et al._ (2013) A. A. Abdumalikov Jr, J. M. Fink, K. Juliusson, M. Pechal, S. Berger, A. Wallraff, and S. Filipp,  _Experimental realization of non-abelian non-adiabatic geometric gates,_Nature 496, 482 (2013).
  * Xu  _et al._ (2018) Y. Xu, W. Cai, Y. Ma, X. Mu, L. Hu, T. Chen, H. Wang, Y. P. Song, Z.-Y. Xue, Z.-q. Yin, and L. Sun,  _Single-loop realization of arbitrary nonadiabatic holonomic single-qubit quantum gates in a superconducting circuit,_Phys. Rev. Lett. 121, 110501 (2018).
  * Zhou  _et al._ (2021) J. Zhou, S. Li, G.-Z. Pan, G. Zhang, T. Chen, and Z.-Y. Xue,  _Nonadiabatic geometric quantum gates that are insensitive to qubit-frequency drifts,_Phys. Rev. A 103, 032609 (2021).
  * Liang and Xue (2022) M.-J. Liang and Z.-Y. Xue,  _Robust nonadiabatic geometric quantum computation by dynamical correction,_Phys. Rev. A 106, 012603 (2022).
  * Wang and Keiji (2001) X. B. Wang and M. Keiji,  _Nonadiabatic conditional geometric phase shift with nmr,_Phys. Rev. Lett. 87, 097901 (2001).
  * Berry (2009) M. V. Berry,  _Transitionless quantum driving,_J. Phys. A: Math. Theor. 42, 365303 (2009).
  * Viola and Lloyd (1998) L. Viola and S. Lloyd,  _Dynamical suppression of decoherence in two-state quantum systems,_Phys. Rev. A 58, 2733 (1998).
  * Witzel and Sarma (2007) W. M. Witzel and S. D. Sarma,  _Multiple-pulse coherence enhancement of solid state spin qubits,_Phys. Rev. Lett. 98, 077601 (2007).
  * Uhrig (2007) G. S. Uhrig,  _Keeping a quantum bit alive by optimized $\pi$-pulse sequences, _Phys. Rev. Lett. 98, 100504 (2007).
  * Pasini and Uhrig (2010) S. Pasini and G. S. Uhrig,  _Optimized dynamical decoupling for power-law noise spectra,_Phys. Rev. A 81, 012309 (2010).
  * Qin  _et al._ (2017) X.-K. Qin, G.-C. Guo, and Z.-W. Zhou,  _Suppressing the geometric dephasing of berry phase by using modified dynamical decoupling sequences,_New J. Phys. 19, 013025 (2017).
  * Zhu and Wang (2003) S.-L. Zhu and Z. D. Wang,  _Unconventional geometric quantum computation,_Phys. Rev. Lett. 91, 187902 (2003).
  * Du  _et al._ (2006) J. Du, P. Zou, and Z. D. Wang,  _Experimental implementation of high-fidelity unconventional geometric quantum gates using an nmr interferometer,_Phys. Rev. A 74, 020302 (2006).
  * Wang  _et al._ (2007) Z. S. Wang, C. Wu, X.-L. Feng, L. C. Kwek, C. H. Lai, C. H. Oh, and V. Vedral,  _Nonadiabatic geometric quantum computation,_Phys. Rev. A 76, 044303 (2007).
  * Feng  _et al._ (2009) X.-L. Feng, C. Wu, H. Sun, and C. H. Oh,  _Geometric entangling gates in decoherence-free subspaces with minimal requirements,_Phys. Rev. Lett. 103, 200501 (2009).
  * Oreshkov  _et al._ (2009a) O. Oreshkov, T. A. Brun, and D. A. Lidar,  _Fault-tolerant holonomic quantum computation,_Phys. Rev. Lett. 102, 070502 (2009a).
  * Oreshkov  _et al._ (2009b) O. Oreshkov, T. A. Brun, and D. A. Lidar,  _Scheme for fault-tolerant holonomic computation on stabilizer codes,_Phys. Rev. A 80, 022325 (2009b).
  * Gardiner  _et al._ (2004) C. Gardiner, P. Zoller, and P. Zoller,  _Quantum noise: a handbook of Markovian and non-Markovian quantum stochastic methods with applications to quantum optics_ (Springer Science & Business Media, 2004).
  * Gerry  _et al._ (2005) C. Gerry, P. Knight, and P. L. Knight,  _Introductory quantum optics_ (Cambridge university press, 2005).
  * Leek  _et al._ (2007) P. J. Leek, J. M. Fink, A. Blais, R. Bianchetti, M. Göppl, J. M. Gambetta, D. I. Schuster, L. Frunzio, R. J. Schoelkopf, and A. Wallraff,  _Observation of berry’s phase in a solid-state qubit,_Science 318, 1889 (2007).
  * Zheng  _et al._ (2008) H. Zheng, S. Y. Zhu, and M. S. Zubairy,  _Quantum zeno and anti-zeno effects: Without the rotating-wave approximation,_Phys. Rev. Lett. 101, 200404 (2008).
  * Deng  _et al._ (2016) C. Deng, F. Shen, S. Ashhab, and A. Lupascu,  _Dynamics of a two-level system under strong driving: Quantum-gate optimization based on floquet theory,_Phys. Rev. A 94, 032323 (2016).
  * 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,_Phys. Rev. A 86, 023831 (2012).
  * Jing  _et al._ (2009) J. Jing, Z.-G. Lü, and Z. Ficek,  _Breakdown of the rotating-wave approximation in the description of entanglement of spin-anticorrelated states,_Phys. Rev. A 79, 044305 (2009).
  * Dong  _et al._ (2021) W. Dong, F. Zhuang, S. E. Economou, and E. Barnes,  _Doubly geometric quantum control,_PRX Quantum 2, 030333 (2021).
  * Gregefalk and Sjöqvist (2022) A. Gregefalk and E. Sjöqvist,  _Transitionless quantum driving in spin echo,_Phys. Rev. Applied 17, 024012 (2022).
  * Cywiński  _et al._ (2008) L. Cywiński, R. M. Lutchyn, C. P. Nave, and S. Das Sarma,  _How to enhance dephasing time in superconducting qubits,_Phys. Rev. B 77, 174509 (2008).
  * Szańkowski  _et al._ (2017) P. Szańkowski, G. Ramon, J. Krzywda, D. Kwiatkowski, and Ł. Cywiński,  _Environmental noise spectroscopy with qubits subjected to dynamical decoupling,_J. Phys.: Condens. Matter 29, 333001 (2017).
  * Jing  _et al._ (2017) J. Jing, C.-H. Lam, and L.-A. Wu,  _Non-abelian holonomic transformation in the presence of classical noise,_Phys. Rev. A 95, 012334 (2017).
  * Yang  _et al._ (2011) W. Yang, Z.-Y. Wang, and R.-B. Liu,  _Preserving qubit coherence by dynamical decoupling,_Front. Phys. China 6, 2 (2011).
  * Hahn (1950) E. L. Hahn,  _Spin echoes,_Phys. Rev. 80, 580 (1950).
  * Carr and Purcell (1954) H. Y. Carr and E. M. Purcell,  _Effects of diffusion on free precession in nuclear magnetic resonance experiments,_Phys. Rev. 94, 630 (1954).
  * Meiboom and Gill (1958) S. Meiboom and D. Gill,  _Modified spin‐echo method for measuring nuclear relaxation times,_Rev. Sci. Instrum. 29, 688 (1958).
  * Hatomura and Takahashi (2021) T. Hatomura and K. Takahashi,  _Controlling and exploring quantum systems by algebraic expression of adiabatic gauge potential,_Phys. Rev. A 103, 012220 (2021).
  * Chandarana  _et al._ (2022) P. Chandarana, N. N. Hegade, K. Paul, F. Albarrán-Arriagada, E. Solano, A. del Campo, and X. Chen,  _Digitized-counterdiabatic quantum approximate optimization algorithm,_Phys. Rev. Research 4, 013141 (2022).


$\displaystyle\leq{\mathbf{E}}\left[Y_{n}\right]-{\mathbf{E}}\left[\widetilde{M}_{n}\right]+{\mathbf{E}}\left[\widetilde{L}_{n}\right]$
$\displaystyle\leq(1+p-q){\mathbf{E}}\left[Z_{n-1}\right]-q{\mathbf{E}}\left[Z_{n-3}\right]+C_{7}q^{2}\max_{3\leq
k\leq 6}{\mathbf{E}}\left[Z_{n-k}\right]$
$\displaystyle\qquad-\frac{q}{2}{\mathbf{E}}\left[Z_{n-2}\right]+C_{5}q^{2}\max_{3\leq
k\leq
5}{\mathbf{E}}\left[Z_{n-k}\right]+4q^{2}(1+p)^{6}{\mathbf{E}}\left[Z_{n-2}\right]$
$\displaystyle\leq(1+p-q){\mathbf{E}}\left[Z_{n-1}\right]-\frac{q}{2}{\mathbf{E}}\left[Z_{n-2}\right]-q{\mathbf{E}}\left[Z_{n-3}\right]+C^{\prime}q^{2}\max_{2\leq
k\leq 6}{\mathbf{E}}\left[Z_{n-k}\right],$

where $C^{\prime}=C_{7}+C_{5}+2^{8}$. By (5) in Proposition 3.1, we have

${\mathbf{E}}\left[Z_{n-1}\right]\leq(1+p){\mathbf{E}}\left[Z_{n-2}\right]\quad\text{and}\quad{\mathbf{E}}\left[Z_{n-1}\right]\leq(1+p)^{2}{\mathbf{E}}\left[Z_{n-3}\right].$

Therefore

$\displaystyle{\mathbf{E}}\left[Z_{n}\right]$
$\displaystyle\leq(1+p-\tfrac{5}{2}q){\mathbf{E}}\left[Z_{n-1}\right]+\frac{q}{2}((1+p)-1){\mathbf{E}}\left[Z_{n-2}\right]$
$\displaystyle\qquad+q((1+p)^{2}-1){\mathbf{E}}\left[Z_{n-3}\right]+C^{\prime}q^{2}\max_{2\leq
k\leq 6}{\mathbf{E}}\left[Z_{n-k}\right],$

which completes the proof of (7), since we are assuming $q\leq p$.

Now, for $n\geq 4$, by (75), and then by (72) in Lemma 8.1 and (43) in Lemma
5.4,

$\displaystyle{\mathbf{E}}\left[Z_{n}\right]$
$\displaystyle\geq{\mathbf{E}}\left[Y_{n}\right]-{\mathbf{E}}\left[M_{n}\right]$
$\displaystyle\geq(1+p){\mathbf{E}}\left[Z_{n-1}\right]-q(1+p)^{4}\left({\mathbf{E}}\left[Z_{n-3}\right]+{\mathbf{E}}\left[Z_{n-4}\right]\right)-C_{3}(1+p)q^{2}{\mathbf{E}}\left[Z_{n-4}\right]$
$\displaystyle\qquad-\frac{q}{2}(1+p)^{4}{\mathbf{E}}\left[Z_{n-2}\right]$
$\displaystyle\geq(1+p){\mathbf{E}}\left[Z_{n-1}\right]-\tfrac{5}{2}q(1+p)^{4}\max_{2\leq
k\leq
4}{\mathbf{E}}\left[Z_{n-k}\right]-2C_{3}q^{2}{\mathbf{E}}\left[Z_{n-4}\right],$

which completes the proof of (8).

Finally, note that since for $n\geq 4$, the construction of $G^{\prime}_{n}$
from $G^{\prime}_{n-1}$ is the same as the construction of $G_{n}$ from
$G_{n-1}$, we have that when the graph process $\mathcal{G}(p,q)$ is replaced
with $\mathcal{G}^{\prime}(p,q)$, Lemma 5.1 holds for $n\geq 4$, (43) in Lemma
5.4 holds for $n\geq 5$, and (72) in Lemma 8.1 holds for $n\geq 7$ (because
(49) in Lemma 5.5 holds for $n\geq 6$, and (54) in Lemma 6.3 holds for $n\geq
6$). Therefore (8) holds for $m\geq 7$ when $Z_{l}$ is replaced with
$Z^{\prime}_{l}$ for each $l$, which completes the proof. ∎

It remains to prove Proposition 3.3. For $n\geq 1$, let $K_{n}$ denote the
number of deletions that occur when $\tilde{I}_{n}$ is constructed, i.e. let

$K_{n}:=|\\{ui:u\in I_{n-1},i\leq\xi_{u},i\notin\mathcal{C}_{u}\\}|=\sum_{u\in
I_{n-1}}(\xi_{u}-|\mathcal{C}_{u}|)=\sum_{u\in I_{n-1}}\xi_{u}-Y_{n}.$ (76)

Recall from (28) that $R_{n}:=|{\mathcal{R}}_{n}|$, where
${\mathcal{R}}_{n}:=\\{v\in I_{n}:|\\{w\in V_{n}:1\leq d_{G_{n}}(v,w)\leq
3\\}|=3\\}.$ We will use the following lower bound on $R_{n}$ to prove
Proposition 3.3.

######  Lemma 8.2.

For $n\geq 3$,

$\displaystyle R_{n}\geq T_{n}-\sum_{k=0}^{2}(K_{n-k}+2{\mathbf{1}}_{[n-k\geq
2]}M_{n-k}),$ (77)

where $T_{n}:=|\\{u\in I_{n-3}:\xi_{u}=\xi_{u1}=\xi_{u11}=1\\}|$.

###### Proof.

Suppose $u\in I_{n-3}$ with $\xi_{u}=\xi_{u1}=\xi_{u11}=1$. Suppose also that
$u1\in\mathcal{C}_{u}$, $u11\in\mathcal{C}_{u1}$, $u111\in\mathcal{C}_{u11}$,
and for each $v\in\\{u1,u11,u111\\}$ we have $|\sigma(v)|=1$, i.e.
$\\{v^{\prime}\in I_{|v|}:v^{\prime}\stackrel{{\scriptstyle
m}}{{\sim}}v\\}=\emptyset$. Then by our construction in Section 4.1, we have
$u111\in\mathcal{R}_{n}.$ Hence by the definitions of $M_{n}$ in (30) and
$K_{n}$ in (76),

$\displaystyle R_{n}$ $\displaystyle\geq|\\{u\in
I_{n-3}:\xi_{u}=\xi_{u1}=\xi_{u11}=1,\,u1\in\mathcal{C}_{u},\,u11\in\mathcal{C}_{u1},\,|\sigma(u1)|=1,\,|\sigma(u11)|=1\\}|$
$\displaystyle\qquad\qquad-2M_{n}-K_{n}$ $\displaystyle\geq|\\{u\in
I_{n-3}:\xi_{u}=\xi_{u1}=\xi_{u11}=1,\,u1\in\mathcal{C}_{u},|\sigma(u1)|=1\\}|$
$\displaystyle\qquad\qquad-2M_{n}-K_{n}-2M_{n-1}-K_{n-1}$
$\displaystyle\geq|\\{u\in
I_{n-3}:\xi_{u}=\xi_{u1}=\xi_{u11}=1\\}|-\sum_{k=0}^{2}(K_{n-k}+2{\mathbf{1}}_{[n-k\geq
2]}M_{n-k}).$

By the definition of $T_{n}$, this completes the proof. ∎

###### Proof of Proposition 3.3.

Take $n\geq 6$ and $0\leq q\leq p\leq 1$. Recall from the statement of Lemma
8.2 that we let $T_{n}:=|\\{u\in I_{n-3}:\xi_{u}=\xi_{u1}=\xi_{u11}=1\\}|$.
Since for $u\in I_{n-3}$, conditional on $\mathcal{F}_{n-3}$,
$\xi_{u},\xi_{u1},\xi_{u11}$ are i.i.d. with Poisson distribution with mean
$1+p$,

$\displaystyle{\mathbf{E}}\left[T_{n}\right]=((1+p)e^{-(1+p)})^{3}{\mathbf{E}}\left[Z_{n-3}\right]\geq
e^{-6}{\mathbf{E}}\left[Z_{n-3}\right],$ (78)

where the inequality follows since $0\leq p\leq 1$.

We will use Lemma 8.2 to obtain a lower bound on
${\mathbf{E}}\left[R_{n}\right]$; this will require an upper bound on
${\mathbf{E}}\left[K_{n-k}\right]+2{\mathbf{E}}\left[M_{n-k}\right]$ for
$0\leq k\leq 2$. Take $m\geq 4$; by (76) we have

$\displaystyle{\mathbf{E}}\left[K_{m}\right]$
$\displaystyle={\mathbf{E}}\left[\sum_{u\in
I_{m-1}}\xi_{u}\right]-{\mathbf{E}}\left[Y_{m}\right]$
$\displaystyle\leq(1+p){\mathbf{E}}\left[Z_{m-1}\right]$
$\displaystyle\qquad-((1+p){\mathbf{E}}\left[Z_{m-1}\right]-q(1+p)^{4}\left({\mathbf{E}}\left[Z_{m-3}\right]+{\mathbf{E}}\left[Z_{m-4}\right]\right)-C_{3}(1+p)q^{2}{\mathbf{E}}\left[Z_{m-4}\right])$

by (72) in Lemma 8.1 and since
${\mathbf{E}}\left[\xi_{u}|\mathcal{F}_{m-1}\right]=1+p$ for $u\in I_{m-1}$.
By (43) in Lemma 5.4, it follows that

$\displaystyle{\mathbf{E}}\left[K_{m}\right]+2{\mathbf{E}}\left[M_{m}\right]$
$\displaystyle\leq
q(1+p)^{4}({\mathbf{E}}\left[Z_{m-3}\right]+{\mathbf{E}}\left[Z_{m-4}\right]+qC_{3}{\mathbf{E}}\left[Z_{m-4}\right]+{\mathbf{E}}\left[Z_{m-2}\right])$
$\displaystyle\leq 16(3+C_{3})p\max_{2\leq k\leq
4}{\mathbf{E}}\left[Z_{m-k}\right]$

since $q\leq p\leq 1$. Therefore, since
${\mathbf{E}}\left[Z_{n-2}\right]\leq(1+p){\mathbf{E}}\left[Z_{n-3}\right]\leq
2{\mathbf{E}}\left[Z_{n-3}\right]$ by (5) in Proposition 3.1, we have

$\displaystyle\sum_{k=0}^{2}({\mathbf{E}}\left[K_{n-k}\right]+2{\mathbf{E}}\left[M_{n-k}\right])$
$\displaystyle\leq 48(3+C_{3})p\max_{2\leq k\leq
6}{\mathbf{E}}\left[Z_{n-k}\right]\leq 96(3+C_{3})p\max_{3\leq k\leq
6}{\mathbf{E}}\left[Z_{n-k}\right].$ (79)

By (77) in Lemma 8.2 together with (78) and (79), it follows that

${\mathbf{E}}\left[R_{n}\right]\geq
e^{-6}{\mathbf{E}}\left[Z_{n-3}\right]-96(3+C_{3})p\max_{3\leq k\leq
6}{\mathbf{E}}\left[Z_{n-k}\right],$

and by setting $2\eta=e^{-6}$ and $p_{2}=(192e^{6}(3+C_{3}))^{-1}$, this
establishes (9).

Finally, since for $n\geq 4$, the construction of $G^{\prime}_{n}$ from
$G^{\prime}_{n-1}$ is the same as the construction of $G_{n}$ from $G_{n-1}$,
we have that when the graph process $\mathcal{G}(p,q)$ is replaced with
$\mathcal{G}^{\prime}(p,q)$, Lemma 8.2 holds for $n\geq 6$, (72) in Lemma 8.1
holds for $n\geq 7$, (43) in Lemma 5.4 holds for $n\geq 5$, and (5) in
Proposition 3.1 holds. It follows that for $n\geq 9$, when $R_{n}$, $Z_{n-3}$
and $Z_{n-k}$ are replaced with $R^{\prime}_{n}$, $Z^{\prime}_{n-3}$ and
$Z^{\prime}_{n-k}$, (9) holds, which completes the proof. ∎

## Acknowledgements

We thank Joel Spencer for suggesting cousin merging in relation to a branching
process to recover the coefficients of the critical probability for hypercube
percolation. Part of this research was undertaken at the 2018 and 2019
Bellairs Workshops on Probability, held at the Bellairs Research Institute of
McGill University. LE was partially supported by PAPIIT TA100820. FS was
partially supported by the project AI4Research at Uppsala University and by
the Wallenberg AI, Autonomous Systems and Software Program (WASP) funded by
the Knut and Alice Wallenberg Foundation.

## References

  * [1] Ajtai, M., J. Komlós and E. Szemerédi. _Largest random component of a $k$-cube_. Combinatorica 2(1) (1982), 1–7.
  * [2] Athreya, K.B. and P.E. Ney. “Branching Processes”, Math. Surveys 7, Springer-Verlag, Berlin, 1972.
  * [3] Bollobás, B. _The evolution of the cube_. North-Holland Mathematics Series, (75) (1983), 91–97.
  * [4] Bollobás, B. _The evolution of sparse graphs_. In “Graph Theory and Combinatorics: Proc. Cambridge Combinatorial Conf. in honour of Paul Erdős”, pp. 35–57, Academic Press, New York, 1984.
  * [5] Bollobás, B. and Y. Kohayakawa. _Percolation in high dimensions._ European Journal of Combinatorics 15(2) (1994), 113-125.
  * [6] Bollobás B., Y. Kohayakawa and T. $\L$uczak. _The evolution of random subgraphs of the cube_. Random Structures $\&$ Algorithms 3(1) (1992), 55–90.
  * [7] Bollobás, B. and O. Riordan. “Percolation”, Cambridge University Press, Cambridge, 2006.
  * [8] Borgs, C., J.T. Chayes, R. van der Hofstad, G. Slade and J. Spencer. _Random subgraphs of finite graphs: I. The scaling window under the triangle condition_. Random Structures $\&$ Algorithms 27(2) (2005), 137–184.
  * [9] Borgs, C., J.T. Chayes, R. van der Hofstad, G. Slade and J. Spencer. _Random subgraphs of finite graphs: II. The lace expansion and the triangle condition_. The Annals of Probability 33(5) (2005), 1886–1944.
  * [10] Borgs, C., J.T. Chayes, R. van der Hofstad, G. Slade and J. Spencer. _Random subgraphs of finite graphs: III. The phase transition for the $n$-cube_. Combinatorica 26(4) (2006), 395–410.
  * [11] Erdős, P. and J. Spencer. _Evolution of the $n$-cube._ Computers and Mathematics with Applications 5(1) (1979), 33–39.
  * [12] Federico, L., R. van der Hofstad, F. den Hollander and T. Hulshof. _Expansion of percolation critical points for Hamming graphs._ Combinatorics, Probability and Computing 29(1) (2020), 68–100.
  * [13] Hara, T. and G. Slade. _The self-avoiding-walk and percolation critical points in high dimensions_. Combinatorics, Probability and Computing 4(3) (1995), 197–215.
  * [14] Heydenreich, M. and R. van der Hofstad. “Progress in high-dimensional percolation and random graphs” (CRM Short Courses). Springer, 2017.
  * [15] Heydenreich, M. and K. Matzke. _Expansion for the critical point of site percolation: the first three terms._ arXiv:1912.04584 (2019).
  * [16] Heydenreich, M. and K. Matzke. _Critical site percolation in high dimension._ Journal of Statistical Physics (181)3 (2020), 816–853.
  * [17] van der Hofstad, R. and A. Nachmias. _Unlacing hypercube percolation: a survey_. Metrika 77(1) (2014), 23–50.
  * [18] van der Hofstad, R. and A. Nachmias. _Hypercube percolation_. J. Eur. Math. Soc. 19(3) (2017), 725–814.
  * [19] van der Hofstad, R. and G. Slade. _Asymptotic expansions in $n^{-1}$ for percolation critical values on the $n$-cube and $\mathbb{Z}^{n}$_. Random Structures $\&$ Algorithms 27(3) (2005), 331–357.
  * [20] van der Hofstad, R. and G. Slade. _Expansion in $n^{-1}$ for percolation critical values on the n-cube and $\mathbb{Z}^{n}$: the first three terms_. Combinatorics, Probability and Computing 15(5) (2006), 695–713.
  * [21] Janson, S. and L. Warnke. _On the critical probability in percolation_. Electronic Journal of Probability 23 (2018).
  * [22] Nachmias, A. and Y. Peres. _Critical random graphs: diameter and mixing time_. Annals of Probability 36(4) (2008), 1267–1286.
  * [23] Pittel, B., J. Spencer and N. Wormald. _Sudden emergence of a giant $k$-core in a random graph_. Journal of Combinatorial Theory, Series B 67(1) (1996), 111–151.
  * [24] Riordan, O. _The k-core and branching processes_. Combinatorics, Probability and Computing 17(1) (2008), 111-136.

##  Appendix A Appendix

######  Lemma A.1.

Let $H=(V,E)$ be a simple, non-empty graph with $t$ vertices and $m$ edges.
Let $\ell$ be the number of directed paths of length two in $H$. Then
$m-\ell<t$.

###### Proof.

If $m=0$ then we have $m-\ell\leq 0<t$. Now assume that $m\geq 1$. For every
vertex $v\in V$, denote the degree of $v$ by $d_{v}$. It is straightforward to
see that $2m=\sum_{v\in V}d_{v}$ and $\ell=\sum_{v\in V}d_{v}(d_{v}-1)$.
Recall that $|V|=t$ and $m\geq 1$; the result follows since

$\displaystyle t+\ell=\sum_{v\in V}(1+d_{v}(d_{v}-1))\geq\sum_{v\in
V}d_{v}=2m>m.$

∎

###  A.1 Natural couplings

$\emptyset$$1$$2$$11$$12$$21$$22$$23$ (a) Ulam-Harris tree. Each vertex $u$ in
generations 0 and 1 has $\xi^{\prime}_{u}$ offspring in the next generation
linked with black edges, and $\xi_{u}-\xi^{\prime}_{u}$ offspring linked with
red edges. Dotted lines between pairs of vertices $u$ and $v$ in generation 2
indicate that $\mu_{\\{u,v\\}}=1$.

$\emptyset$$1$$2$$11$$12$$21$$22$ (b) Realisation of $G_{2}(p^{\prime},q)$

$\emptyset$$1$$2$$11$$21$$22$ (c) Realisation of $G_{2}(p,q)$

Figure A.1: The number of individuals in the second generation is smaller in
the graph process $\mathcal{G}(p,q)$ than in the graph process
$\mathcal{G}(p^{\prime},q)$, while $p>p^{\prime}>-1$.

As mentioned in the introduction, we now briefly discuss natural couplings of
the graph processes with different values of $p$ and $q$, using our Ulam-
Harris construction in Section 4. Take $p>p^{\prime}>-1$ and $q^{\prime}>q>0$.
Let $(\xi_{u})_{u\in\mathcal{U}}$ and $(\xi^{\prime}_{u})_{u\in\mathcal{U}}$
be families of i.i.d. random variables where $\xi_{u}$ has Poisson
distribution with mean $1+p$ for each $u$, and $\xi^{\prime}_{u}$ has Poisson
distribution with mean $1+p^{\prime}$ for each $u$, coupled in such a way that
$\xi_{u}\geq\xi^{\prime}_{u}$ $\forall u\in\mathcal{U}$. Let
$(\delta_{u,v})_{u,v\in\mathcal{U}}$, $(\mu_{\\{u,v\\}})_{u\neq
v\in\mathcal{U}}$ be independent families of i.i.d. random variables with
Bernoulli distribution with mean $q$, and let
$(\delta^{\prime}_{u,v})_{u,v\in\mathcal{U}}$,
$(\mu^{\prime}_{\\{u,v\\}})_{u\neq v\in\mathcal{U}}$ be independent families
of i.i.d. random variables with Bernoulli distribution with mean $q^{\prime}$,
coupled in such a way that $\delta_{u,v}\leq\delta^{\prime}_{u,v}$ and
$\mu_{\\{u,v\\}}\leq\mu^{\prime}_{\\{u,v\\}}$ $\forall u,v\in\mathcal{U}$.

Construct the graph process $\mathcal{G}(p,q)$ according to the Ulam-Harris
construction in Section 4, using the random variables
$(\xi_{u})_{u\in\mathcal{U}}$, $(\delta_{u,v})_{u,v\in\mathcal{U}}$ and
$(\mu_{\\{u,v\\}})_{u\neq v\in\mathcal{U}}$. Similarly, construct the graph
process $\mathcal{G}(p^{\prime},q)$ using the random variables
$(\xi^{\prime}_{u})_{u\in\mathcal{U}}$, $(\delta_{u,v})_{u,v\in\mathcal{U}}$
and $(\mu_{\\{u,v\\}})_{u\neq v\in\mathcal{U}}$, and construct the graph
process $\mathcal{G}(p,q^{\prime})$ using the random variables
$(\xi_{u})_{u\in\mathcal{U}}$, $(\delta^{\prime}_{u,v})_{u,v\in\mathcal{U}}$
and $(\mu^{\prime}_{\\{u,v\\}})_{u\neq v\in\mathcal{U}}$. We illustrate
possible realisations of these graph processes in Figures A.1 and A.2, showing
that these couplings do not trivially imply monotonicity of the survival
probability in $p$ or $q$.

$\emptyset$$1$$2$$11$$21$$22$$211$$221$$222$$223$ (a) Ulam-Harris tree. Each
vertex $u$ in generations 0, 1 and 2 has $\xi_{u}$ offspring in the next
generation. Black dotted lines between pairs of vertices $u$ and $v$ indicate
that $\mu_{\\{u,v\\}}=1$; red dotted lines indicate that
$\mu^{\prime}_{\\{u,v\\}}=1$ and $\mu_{\\{u,v\\}}=0$.

$\emptyset$$1$$2$$11$$22$$221$$222$$223$ (b) Realisation of
$G_{3}(p,q^{\prime})$

$\emptyset$$1$$2$$11$$21$$22$$211$$223$ (c) Realisation of $G_{3}(p,q)$

Figure A.2: The number of individuals in the third generation is smaller in
the graph process $\mathcal{G}(p,q)$ than in the graph process
$\mathcal{G}(p,q^{\prime})$, while $q^{\prime}>q>0$.