# Protecting quantum gates from arbitrary single- and two-qubit errors

Chunfeng Wu<EMAIL_ADDRESS>Science, Mathematics and Technology,
Singapore University of Technology and Design, 8 Somapah Road, Singapore
487372, Singapore  Gangcheng Wang  School of Physics and Center for Quantum
Sciences, Northeast Normal University, Changchun 130024, China  Xun-Li Feng
Science, Mathematics and Technology, Singapore University of Technology and
Design, 8 Somapah Road, Singapore 487372, Singapore

###### Abstract

We explore the protection of quantum gates from arbitrary single- and two-
qubit noises with properly designed dynamical decoupling pulses. The proposed
dynamical decoupling method is a concatenation of a sequence of pulses formed
by $\sigma_{x}$, $\sigma_{x}\sigma_{x}$ with another sequence constructed by
$\sigma_{z}$, $\sigma_{z}\sigma_{z}$. The concatenation of the two sequences
results in desired pulses to fight agianst any single- and two-qubit errors.
The success of our method relies on the ability to adjust system parameters or
interaction terms, which can be achieved in different physical systems,
including trapped ions and superconducting qubits. We finally explore the
performance of our method numerically with the above-mentioned errors that are
changing at any moment and show the preferred protection offered by the
method. Therefore, our method is a timely step forward in preserving quantum
gates at the level of physical qubits.

Fault-tolerant quantum computation requires to scale up to enough qubits and
implement quantum gates with sufficiently small errors in physical systems [1,
2, 3, 4, 5, 6]. These significant challenges are expected to be overcome step
by step with growing noisy intermediate small-scale quantum (NISQ) devices [7,
8]. To realize scalable quantum computation in NISQ devices, the system
interactions should be adjustable in order to execute quantum gates on target
qubits. The desired controllability in system parameters may induce additional
difficulties besides decoherence in manipulating the system evolution for
achieving specific quantum gates, because of unavoidable fluctuations in the
systerm parameters [9, 10].

Dynamical decoupling (DD) plays an effective role in mitigating errors or
decoherence during system evolution [11, 12, 13, 14, 15]. The technique aims
to approximately kick out errors or decoherence via suitable pulses applied in
sequence under certain conditions. Compared with quantum error-correction
codes, DD is also an active method to battle with errors or decoherence but
with a moderate resouce of DD pulses. This merit makes DD practically useful
in NISQ devices. In commonly explored physical systems, most of the errors
during system evolution are due to single- and two-qubit noises when qubit-
qubit couplings are demanded. In the literature, fighting against aribitrary
single-qubit noises by DD approach has been extensively investigated both
theoretically and experimentally [11, 12, 13, 14, 15]. Meanwhile, it has been
directed towards mitigating arbitrary two-qubit noises with DD technique in
physical systems as well [16, 17, 9, 10, 18]. In Refs. [16, 17], it was shown
that system operators can generally be protected by complicated levels of
nested DD pulses, but without discussing the details of eliminating arbitrary
two-qubit errors or decoherence in physical systems. In 2016, the Viola group
explored the supression of multi-qubit dephasing by comparatively short DD
sequences [9]. Generally, the DD techniques cannot be readily implemented in
actual experiments since the DD pulses may remove desired system interactions
when fighting against errors or decoherence. It is thus of practical
importance to make the DD techniques workable in actual experments to battle
with both single- and two-qubit errors. Recently, special DD pulses invovling
two-qubit interactions were designed to minimize the negative effect of
specific two-qubit errors by adjusting system parameters in a system of
superconducting transmon qubits in Ref. [10]. One year later, the Lidar group
investigated the mitigation of ZZ coupling in superconducting qubits [18, 19]
and the Kim group studied the supression of typical crosstalk in implementing
the Mølmer-Sørensen (MS) gate with trapped ions [20].

In this work, we explore the supression of arbitrary single- and two-qubit
errors or decoherence through DD approach with experimental achievable
Hamiltonians. Generally, arbitrary single- and two-qubit errors are described
by the error operators in the set of
$\mathcal{E}=\\{\sigma_{x,y,z}^{i,j},\sigma_{x}^{i}\sigma_{x}^{j},\sigma_{x}^{i}\sigma_{y}^{j},\sigma_{y}^{i}\sigma_{x}^{j},\sigma_{x}^{i}\sigma_{z}^{j},\sigma_{z}^{i}\sigma_{x}^{j},\sigma_{y}^{i}\sigma_{y}^{j},\sigma_{y}^{i}\sigma_{z}^{j},\sigma_{z}^{i}\sigma_{y}^{j},\sigma_{z}^{i}\sigma_{z}^{j}\\}$.
We first study one DD sequence constructed by $\sigma_{x}^{i,j}$ and
illustrate that the errors in the subset
$\mathcal{E}_{1}=\\{\sigma_{y,z}^{i}$, $\sigma_{y,z}^{j}$,
$\sigma_{x}^{i}\sigma_{y}^{j}$, $\sigma_{y}^{i}\sigma_{x}^{j}$,
$\sigma_{x}^{i}\sigma_{z}^{j}$, $\sigma_{z}^{i}\sigma_{x}^{j}$,
$\sigma_{y}^{i}\sigma_{y}^{j}$, $\sigma_{y}^{i}\sigma_{z}^{j}$,
$\sigma_{z}^{i}\sigma_{y}^{j}$, $\sigma_{z}^{i}\sigma_{z}^{j}\\}$ can be
eliminated approximately. Next we investigate another DD sequence built by
$\sigma_{z}^{i,j}$ to supress the errors in the subset of
$\mathcal{E}_{2}=\\{\sigma_{x}^{i,j},\sigma_{x}^{i}\sigma_{x}^{j}\\}$. The
concatenation of the two DD sequences can thus fight against any errors in the
full set as $\mathcal{E}=\mathcal{E}_{1}+\mathcal{E}_{2}$, and this result is
consistent with the nested Uhrig dynamical decoupling (NUDD) explored in Ref.
[16, 17]. During the process of applying DD pulses, system parameters should
be controllable such that required Hamiltonian terms will not be affected by
the DD pulses. The requirement is experimentally achievable in commonly
studied physical systems and as a result, a universal set of protected quantum
gates are realizable with current experimental techniques based on our method.
We next explore the performance of our method by modelling all the above-
mentioned errors to be time-dependent in numerical calculations. The
perfomance is dependent on the number of DD pulses and the more the number of
DD pulses, the better the performance is. But in practical experiments, the
number of DD pusles is finite. We then show with a moderate number of
repeating the concatenated DD sequences, the explored quantum gates can be
largely saved from the errors. It is reasonable to expect that the performance
can be further improved given more DD pusles with the development of
experimental techniques.

We study the first DD sequence constructed by $\sigma_{x}^{i,j}$ to fight
against the errors included in the subset $\mathcal{E}_{1}$. The desired
sequence is of the form,
$\big{(}\sigma_{x}^{j}[\cdot]\sigma_{x}^{j}\big{)}\big{(}\sigma_{x}^{i}[\cdot]\sigma_{x}^{i}\big{)}\big{(}\sigma_{x}^{i}\sigma_{x}^{j}[\cdot]\sigma_{x}^{i}\sigma_{x}^{j}\big{)}[\cdot]$,
where $[\cdot]$ represents evolution operator over a period of $\tau$. We
analyze the system evolution with DD pusles step by step in the following.

  * •

The evolution $[\cdot]$ carries the errors as $\\{\sigma_{y,z}^{i}$,
$\sigma_{y,z}^{j}$, $\sigma_{x}^{i}\sigma_{y}^{j}$,
$\sigma_{y}^{i}\sigma_{x}^{j}$, $\sigma_{x}^{i}\sigma_{z}^{j}$,
$\sigma_{z}^{i}\sigma_{x}^{j}$, $\sigma_{y}^{i}\sigma_{y}^{j}$,
$\sigma_{y}^{i}\sigma_{z}^{j}$, $\sigma_{z}^{i}\sigma_{y}^{j}$,
$\sigma_{z}^{i}\sigma_{z}^{j}\\}$.

  * •

$\big{(}\sigma_{x}^{i}\sigma_{x}^{j}[\cdot]\sigma_{x}^{i}\sigma_{x}^{j}\big{)}$
converts the errors to $\\{-\sigma_{y,z}^{i}$, $-\sigma_{y,z}^{j}$,
$-\sigma_{x}^{i}\sigma_{y}^{j}$, $-\sigma_{y}^{i}\sigma_{x}^{j}$,
$-\sigma_{x}^{i}\sigma_{z}^{j}$, $-\sigma_{z}^{i}\sigma_{x}^{j}$,
$\sigma_{y}^{i}\sigma_{y}^{j}$, $\sigma_{y}^{i}\sigma_{z}^{j}$,
$\sigma_{z}^{i}\sigma_{y}^{j}$, $\sigma_{z}^{i}\sigma_{z}^{j}\\}$.

  * •

$\big{(}\sigma_{x}^{i}[\cdot]\sigma_{x}^{i}\big{)}$ converts the errors to
$\\{-\sigma_{y,z}^{i}$, $\sigma_{y,z}^{j}$, $\sigma_{x}^{i}\sigma_{y}^{j}$,
$-\sigma_{y}^{i}\sigma_{x}^{j}$, $\sigma_{x}^{i}\sigma_{z}^{j}$,
$-\sigma_{z}^{i}\sigma_{x}^{j}$, $-\sigma_{y}^{i}\sigma_{y}^{j}$,
$-\sigma_{y}^{i}\sigma_{z}^{j}$, $-\sigma_{z}^{i}\sigma_{y}^{j}$,
$-\sigma_{z}^{i}\sigma_{z}^{j}\\}$.

  * •

$\big{(}\sigma_{x}^{j}[\cdot]\sigma_{x}^{j}\big{)}$ converts the errors to
$\\{\sigma_{y,z}^{i}$, $-\sigma_{y,z}^{j}$, $-\sigma_{x}^{i}\sigma_{y}^{j}$,
$\sigma_{y}^{i}\sigma_{x}^{j}$, $-\sigma_{x}^{i}\sigma_{z}^{j}$,
$\sigma_{z}^{i}\sigma_{x}^{j}$, $-\sigma_{y}^{i}\sigma_{y}^{j}$,
$-\sigma_{y}^{i}\sigma_{z}^{j}$, $-\sigma_{z}^{i}\sigma_{y}^{j}$,
$-\sigma_{z}^{i}\sigma_{z}^{j}\\}$.

As a whole,
$\big{(}\sigma_{x}^{j}[\cdot]\sigma_{x}^{j}\big{)}\big{(}\sigma_{x}^{i}[\cdot]\sigma_{x}^{i}\big{)}\big{(}\sigma_{x}^{i}\sigma_{x}^{j}[\cdot]\sigma_{x}^{i}\sigma_{x}^{j}\big{)}[\cdot]$
removes the errors in the subset $\mathcal{E}_{1}$ approximately. To eliminate
the errors in the subset $\mathcal{E}_{2}$, we utilize the second DD sequence
in the form of
$\big{(}\sigma_{z}^{j}[\cdot]\sigma_{z}^{j}\big{)}\big{(}\sigma_{z}^{i}[\cdot]\sigma_{z}^{i}\big{)}\big{(}\sigma_{z}^{i}\sigma_{z}^{j}[\cdot]\sigma_{z}^{i}\sigma_{z}^{j}\big{)}[\cdot]$.
Only with the errors in the subset $\mathcal{E}_{2}$, we find

  * •

The evolution $[\cdot]$ carries the errors as $\\{\sigma_{x}^{i,j}$,
$\sigma_{x}^{i}\sigma_{x}^{j}\\}$.

  * •

$\big{(}\sigma_{z}^{i}\sigma_{z}^{j}[\cdot]\sigma_{z}^{i}\sigma_{z}^{j}\big{)}$
converts the errors to $\\{-\sigma_{x}^{i,j}$,
$\sigma_{x}^{i}\sigma_{x}^{j}\\}$.

  * •

$\big{(}\sigma_{z}^{i}[\cdot]\sigma_{z}^{i}\big{)}$ converts the errors to
$\\{-\sigma_{x}^{i}$, $\sigma_{x}^{j}$, $-\sigma_{x}^{i}\sigma_{x}^{j}\\}$.

  * •

$\big{(}\sigma_{z}^{j}[\cdot]\sigma_{z}^{j}\big{)}$ converts the errors to
$\\{\sigma_{x}^{i}$, $-\sigma_{x}^{j}$, $-\sigma_{x}^{i}\sigma_{x}^{j}\\}$.

Therefore, the errors in the subset $\mathcal{E}_{2}$ can be effectively
mitigated by the second DD sequence roughly. Finally, the concatenation of the
two DD sequences gives us
$\big{(}\sigma_{z}^{j}[\cdot\cdot]\sigma_{z}^{j}\big{)}\big{(}\sigma_{z}^{i}[\cdot\cdot]\sigma_{z}^{i}\big{)}\big{(}\sigma_{z}^{i}\sigma_{z}^{j}[\cdot\cdot]\sigma_{z}^{i}\sigma_{z}^{j}\big{)}[\cdot\cdot]$,
where
$[\cdot\cdot]=\big{(}\sigma_{x}^{j}[\cdot]\sigma_{x}^{j}\big{)}\big{(}\sigma_{x}^{i}[\cdot]\sigma_{x}^{i}\big{)}\big{(}\sigma_{x}^{i}\sigma_{x}^{j}[\cdot]\sigma_{x}^{i}\sigma_{x}^{j}\big{)}[\cdot]$.
The nested DD sequence can approximately get rid of all the errors in the set
$\mathcal{E}=\mathcal{E}_{1}+\mathcal{E}_{2}$. Moreover, we can reduce the
number of DD pulses by simplifying the nested DD sequence as
$\sigma_{z}^{j}[\cdot\cdot]\sigma_{z}^{j}\sigma_{z}^{i}[\cdot\cdot]\sigma_{z}^{j}[\cdot\cdot]\sigma_{z}^{i}\sigma_{z}^{j}[\cdot\cdot]$,
where
$[\cdot\cdot]=\sigma_{x}^{j}[\cdot]\sigma_{x}^{j}\sigma_{x}^{i}[\cdot]\sigma_{x}^{j}[\cdot]\sigma_{x}^{i}\sigma_{x}^{j}[\cdot]$.
It is clearly shown that to complete one full DD cycle, there are 16 steps of
system evolution and so, the total evolution time is $T=16\tau$.

Our scheme is possibly implementable in physical systems, with tunable system
parameters such that desired gate Hamiltonians will not be canceled with DD
pulses applied. One possible gate Hamiltonian is of the below form,

$\displaystyle
H_{1}=\delta\sigma_{z}^{j}+\Omega(e^{-i\phi}\sigma_{+}^{j}+e^{i\phi}\sigma_{-}^{j})+J(\sigma_{+}^{j}\sigma_{-}^{k}+\sigma_{-}^{j}\sigma_{+}^{k}),$
(1)

where $\delta$ is the detuning of the qubit frequency from the frequency of
external driving, $\Omega$ is the Rabi frequency of external driving, $\phi$
is the phase of external driving,
$\sigma_{\pm}^{j,k}=\frac{1}{2}(\sigma_{x}^{j,k}\pm i\sigma_{y}^{j,k})$, and
$J$ is the qubit-qubit coupling strength. The desired interaction can be
implemented in various physical systems with rotating-wave approximation
applied, including cold atoms [21], trapped ions [22, 23] and superconducting
transmon qubits [24, 25, 26], etc. The parameters $\delta$, $\Omega$, $\phi$
and $J$ are adjustable experimentally.

With gate Hamiltonian (1), single-qubit rotations about $x$ and $y$ axes can
be achieved by choosing $\delta=0$, $J=0$ and tuning $\phi=0$ or $\pi$ for $x$
axis and $\phi=\frac{\pi}{2}$ or $\frac{3\pi}{2}$ for $y$ axis. While single-
qubit rotations about $z$ axis is realizable by making $\Omega=0$ and $J=0$.
For two-qubit entangling gates, one can select $\delta=0$ and $\Omega=0$ to
keep the coupling term only.

In what follows, we consider a physical system of transmon qubits to
demonstrate the implementation of quantum gates protected by our scheme
against arbitrary single- and two-qubit errors. We study one unmodulated qubit
(qubit 1) coupled to another frequency-modulated qubit (qubit 2) with external
drives. The Hamiltonian in the interaction picture can be written as [24, 25,
26],

$\displaystyle H_{\rm trans}$ $\displaystyle=$
$\displaystyle\delta\sigma_{z}^{1,2}+\Omega(e^{-i\phi}\sigma_{+}^{1,2}+e^{i\phi}\sigma_{-}^{1,2})$
(2) $\displaystyle+$ $\displaystyle
gJ_{1}(\beta)(\sigma_{+}^{1}\sigma_{-}^{2}e^{-i\varphi}+\sigma_{-}^{1}\sigma_{+}^{2}e^{i\varphi}),$

where $g$ is the coupling strength, $J_{1}(\beta)$ is the Bessel function of
the first kind when $n=1$, $\beta$ is the ratio of amplitude and frequency of
the periodic frequency modulation, and $\varphi$ is the phase of the periodic
frequency modulation. The $H_{\rm trans}$ is of the same form as $H_{1}$, in
which $J=gJ_{1}(\beta)$.

In the system, the performance of the quantum gates proteced by our DD scheme
can be demonstrated by the following numerical results. In the presence of
errors or decoherence, we assume the total Hamiltonian is written as
$H_{T}=H_{S}+H_{e}$, where $H_{S}$ and $H_{e}$ are system Hamiltonian and the
stochastic error term. In our calculations, $H_{S}$ is $H_{\rm trans}$ and
$H_{e}$ is given by

$\displaystyle H_{e}$ $\displaystyle=$
$\displaystyle\delta_{1,2}^{x}\sigma_{x}^{1,2}+\delta_{1,2}^{y}\sigma_{y}^{1,2}+\delta_{1,2}^{z}\sigma_{z}^{1,2}+\delta_{12}^{xx}\sigma_{x}^{1}\sigma_{x}^{2}+\delta_{12}^{yy}\sigma_{y}^{1}\sigma_{y}^{2}$
(3) $\displaystyle+$
$\displaystyle\delta_{12}^{zz}\sigma_{z}^{1}\sigma_{z}^{2}+\delta_{12}^{xy}\sigma_{x}^{1}\sigma_{y}^{2}+\delta_{12}^{yx}\sigma_{y}^{1}\sigma_{x}^{2}+\delta_{12}^{xz}\sigma_{x}^{1}\sigma_{z}^{2}+\delta_{12}^{zx}\sigma_{z}^{1}\sigma_{x}^{2}$
$\displaystyle+$
$\displaystyle\delta_{12}^{yz}\sigma_{y}^{1}\sigma_{z}^{2}+\delta_{12}^{zy}\sigma_{z}^{1}\sigma_{y}^{2},$

where $\delta_{1,2}^{x,y,z}$ and $\delta_{12}^{uv}(u,v=x,y,z)$ describe the
time-dependent strength of stochastic errors in different directions and
couplings. To model the time-dependance, we create random numbers with respect
to time (800 sets of values over a period of $T$) from a uniform distribution
$[2\pi\times 1,2\pi\times 10]$ MHz when solving the differential equation with
ode45 solver.

For single-qubit gates, it is easy to apply the protection by DD approach as
we only need to manipulate one qubit. According to the literature [18, 19],
XY4 or Periodic DD (PDD) sequence is able to remove all errors described by
$\sigma_{x,y,z}$ on the qubit with DD pusles applied. Therefore, it is not
necessary to use the DD scheme proposed here since it is more complicated. We
focus on fighting against errors or decoherence when exploring the
implementation of two-qubit gates. To realize an entangling gate
$U_{3}=e^{-i\gamma(\sigma_{+}^{1}\sigma_{-}^{2}+\sigma_{-}^{1}\sigma_{+}^{2})}$
(where $\gamma=JT=\pi/4$), we select system parameter as $J=2\pi\times 10$
MHz. For a two-qubit gate, XY4 or PDD is not able to remove all errors any
more and our DD scheme takes turn. Some of the DD pulses change desired
interaction from $\sigma_{+}^{1}\sigma_{-}^{2}+\sigma_{-}^{1}\sigma_{+}^{2}$
to $\sigma_{+}^{1}\sigma_{+}^{2}+\sigma_{-}^{1}\sigma_{-}^{2}$ and some of the
DD pusles change $J$ to $-J$. The changes certainly ruin the implementation of
the entangling gates. Therefore, we need to tune the coupling term
correspondingly to avoid the unwanted changes. In the former case, we control
the frequency modulation to get
$\sigma_{+}^{1}\sigma_{+}^{2}+\sigma_{-}^{1}\sigma_{-}^{2}$, which will be
changed to desired interaction with certain DD pusles applied. In the latter
case, we adjust $\varphi$ to obtain a negative value of $J$, which will then
be converted back to $J$ by some DD pulses. We list out required coupling in
Table 1 (a) for all the 16 steps of system evolution. In Table 1 (a), we use
the symbol “*” to indicate at that step, the coupling is set to be
$\sigma_{+}^{1}\sigma_{+}^{2}+\sigma_{-}^{1}\sigma_{-}^{2}$ and without the
symbol, the coupling is of the form
$\sigma_{+}^{1}\sigma_{-}^{2}+\sigma_{-}^{1}\sigma_{+}^{2}$. With the
specified couplings, our DD scheme is workable in the physical system as
demonstrated by the following numerical calculations. We randomly choose 50
initial states and calculate average fidelity of the gate when $\gamma=\pi/4$
in the absence or presence of errors (without DD and with DD), respectively.
Specially, we simluate the DD pulses by unitary evolutions over a short time
period and take errors occurred to DD pulses into account in our calculations
with $\sigma_{x}\propto e^{-i(\frac{\pi}{2}+\zeta)\sigma_{x}}$ and
$\sigma_{z}\propto e^{-i(\frac{\pi}{2}+\zeta)\sigma_{z}}$, where $\zeta$ is
selected from Gaussian distribution (1) with a mean of $\frac{\pi}{500}$ and a
standard deviation of $\frac{\pi}{500}$ and from Gaussian distribution (2)
with a mean of $\frac{\pi}{200}$ and a standard deviation of $\frac{\pi}{200}$
to describe different stochastic errors happened to the DD pulses. The
numerical results are summarized in Table 2. Here, we consider the the case of
getting $\sigma_{+}^{1}\sigma_{-}^{2}+\sigma_{-}^{1}\sigma_{+}^{2}$ and
$\sigma_{+}^{1}\sigma_{+}^{2}+\sigma_{-}^{1}\sigma_{-}^{2}$ ideally by
rotating-wave approximation. If include the approximation errors, the
fideilities will be a bit lower. It is clearly demonstrated in Table 2, the
two-qubit gate is ruined by the errors without DD pulses applied, and it can
be largely saved according to our DD scheme. Moreover, the success of the DD
scheme requires very small errors happened in DD pulses, see numerical results
in the case with superscript (1). Given the DD pulses with increased value of
$\zeta$ as shown in the case with superscript (2), the DD scheme still can
offer protection against the errors, though it does not perform as well as it
does in the case with superscript (1). Here, we only complete one full DD
cycle with 16 steps. Further improvement of the average fidelity can be
achieved by increasing the number of repeating the DD cycle.

Steps | $U_{3}$
---|---
1st $[\cdot]$ | $J$
2nd $[\cdot]$ | $J$
3rd $[\cdot]$ | $J*$
4th $[\cdot]$ | $J*$
5th $[\cdot]$ | $J$
6th $[\cdot]$ | $J$
7th $[\cdot]$ | $J*$
8th $[\cdot]$ | $J*$
9th $[\cdot]$ | $-J$
10th $[\cdot]$ | $-J$
11th $[\cdot]$ | $-J*$
12th $[\cdot]$ | $-J*$
13th $[\cdot]$ | $-J$
14th $[\cdot]$ | $-J$
15th $[\cdot]$ | $-J*$
16th $[\cdot]$ | $-J*$
(a)

Steps | $U_{e}^{1}$
---|---
1st $[\cdot]$ | $J^{\prime}$
2nd $[\cdot]$ | $J^{\prime}$
3rd $[\cdot]$ | $-J^{\prime}$
4th $[\cdot]$ | $-J^{\prime}$
5th $[\cdot]$ | $J^{\prime}$
6th $[\cdot]$ | $J^{\prime}$
7th $[\cdot]$ | $-J^{\prime}$
8th $[\cdot]$ | $-J^{\prime}$
9th $[\cdot]$ | $J^{\prime}$
10th $[\cdot]$ | $J^{\prime}$
11th $[\cdot]$ | $-J^{\prime}$
12th $[\cdot]$ | $-J^{\prime}$
13th $[\cdot]$ | $J^{\prime}$
14th $[\cdot]$ | $J^{\prime}$
15th $[\cdot]$ | $-J^{\prime}$
16th $[\cdot]$ | $-J^{\prime}$
(b)

Steps | $U_{e}^{2}$
---|---
1st $[\cdot]$ | $J^{\prime}$
2nd $[\cdot]$ | $J^{\prime}$
3rd $[\cdot]$ | $J^{\prime}$
4th $[\cdot]$ | $J^{\prime}$
5th $[\cdot]$ | $J^{\prime}$
6th $[\cdot]$ | $J^{\prime}$
7th $[\cdot]$ | $J^{\prime}$
8th $[\cdot]$ | $J^{\prime}$
9th $[\cdot]$ | $-J^{\prime}$
10th $[\cdot]$ | $-J^{\prime}$
11th $[\cdot]$ | $-J^{\prime}$
12th $[\cdot]$ | $-J^{\prime}$
13th $[\cdot]$ | $-J^{\prime}$
14th $[\cdot]$ | $-J^{\prime}$
15th $[\cdot]$ | $-J^{\prime}$
16th $[\cdot]$ | $-J^{\prime}$
(c)

Steps | $U_{e}^{3}$
---|---
1st $[\cdot]$ | $J^{\prime}$
2nd $[\cdot]$ | $-J^{\prime}$
3rd $[\cdot]$ | $-J^{\prime}$
4th $[\cdot]$ | $J^{\prime}$
5th $[\cdot]$ | $-J^{\prime}$
6th $[\cdot]$ | $J^{\prime}$
7th $[\cdot]$ | $J^{\prime}$
8th $[\cdot]$ | $-J^{\prime}$
9th $[\cdot]$ | $J^{\prime}$
10th $[\cdot]$ | $-J^{\prime}$
11th $[\cdot]$ | $-J^{\prime}$
12th $[\cdot]$ | $J^{\prime}$
13th $[\cdot]$ | $-J^{\prime}$
14th $[\cdot]$ | $J^{\prime}$
15th $[\cdot]$ | $J^{\prime}$
16th $[\cdot]$ | $-J^{\prime}$
(d)

Table 1: Required system parameters for different quantum gates at every step of system evolution to make the DD scheme compatible, where (a) is for gate Hamiltonian (1) for two-qubit gates with our DD, (b) is for gate Hamiltonian (4) with ZZ coupling, (c) is for the gate Hamiltonian (4) with XX coupling, and (d) is for gate Hamiltonian (4) with ZX coupling Gate | Average Fidelity
---|---
w/o DD (w/ iDD) | w/o DD (w/ DD${}^{(1)})$ | w/o DD (w/ DD${}^{(2)})$
$U_{3}|_{\gamma=\pi/4}$ | $0.2496\;(0.9967)$ | $0.2290\;(0.9949)$ | $0.2282\;(0.9807)$
$U_{e}^{1}|_{\gamma=\pi/4}$ | $0.1808\;(0.9923)$ | $0.1602\;(0.9886)$ | $0.1519\;(0.9776)$
Table 2: The fidelities of two-qubit quantum gates in the absence (presence)
of errors and DD pulses with Hamiltonian $H_{\rm trans}$, where iDD is short
for ideal DD, and superscript (1) or (2) indicates the case with $\zeta$
selected from Gaussian distribution (1) or (2), correspondingly.

Other possible gate Hamiltonians with qubit-qubit coupling only are given as

$\displaystyle
H_{2}=J^{\prime}\sigma_{z}^{j}\sigma_{z}^{k}\;\text{or}\;J^{\prime}\sigma_{x}^{j}\sigma_{x}^{k}\;\text{or}\;J^{\prime}\sigma_{z}^{j}\sigma_{x}^{k},$
(4)

where $J^{\prime}$ is the coupling strength. Here, we neglect the terms for
realizing single-qubit gates as they can be preserved by XY4 or PDD. The type
of interaction can be found in superconducting charge qubits [27, 28],
transmons [29, 30] and fluxoniums [31, 32], etc. From gate Hamiltonian (4),
two-qubit entangling gates $U_{e}^{i}$ ($i=1,2,3$ and $e$ stands for
entangling) are achievable dependent on the interaction
$J^{\prime}\sigma_{z}^{j}\sigma_{z}^{k}$ or
$J^{\prime}\sigma_{x}^{j}\sigma_{x}^{k}$ or
$J^{\prime}\sigma_{z}^{j}\sigma_{x}^{k}$. Based on the results in the
literature, the system parameters may be fully or partially controllable in
different systems. To make our DD scheme compatible in experiments, we require
the tunablity of the system parameters from positive to negative. Similarly,
we list out desired system parameters to keep gate Hamiltonian (4) for two-
qubit gates with DD pulses applied in Table 1 (b), (c) and (d), where (b) is
for interaction $J^{\prime}\sigma_{z}^{j}\sigma_{z}^{k}$, (c) is for
interaction $J^{\prime}\sigma_{x}^{j}\sigma_{x}^{k}$ and (d) is for
interaction $J^{\prime}\sigma_{z}^{j}\sigma_{x}^{k}$. In the table,
$U_{e}^{1}=e^{-i\xi\sigma_{z}^{1}\sigma_{z}^{2}}$,
$U_{e}^{2}=e^{-i\xi\sigma_{x}^{1}\sigma_{x}^{2}}$ and
$U_{e}^{3}=e^{-i\xi\sigma_{z}^{1}\sigma_{x}^{2}}$, where $\xi$ is dependent of
the value of $J^{\prime}$ and evolution time. The coupling term may take one
of the forms or include two of them in some physical systems, because of
crosstalk. As our DD scheme can diligently eliminate aribrary two-qubit
errors, unwanted crosstalk term can also be kicked out by DD pulses without
changing the sign of the coupling strength of the crosstalk term. It may
happen that in some physical system, two-qubit gates cannot be executable in a
protective way according to our scheme with current experimental techniques
when there is a lack of desired tunability in the coupling strength. We then
aniticipate further development from the perspective of experiments.

We next consider ZZ coupling as an example to explore the protection offered
by our DD scheme through numerical calculations. The action of DD pulses
sometimes changes the sign of the ZZ coupling strength. To remain the
interaction for realizing a two-qubit gate, we need to adjust the sign as
demonstrated in Table 1 (b). In the example, we investigate the average
fidelity of gate $U_{e}^{1}$ with $\xi=J^{\prime}T=\pi/4$ and
$J^{\prime}=2\pi\times 10$ MHz by randomly choosing 50 initial states, without
errors and with errors (without DD and with DD). The numerical results are
obtained by completing one whole DD cycle of 16 steps, and are outlined in
Table 2 too. We find the two-qubit gate is in ruins with the errors in the
absence of DD pulses and it can be preserved when DD pusles are present.
Similarly, the performance of the DD scheme relies on desirably small errors
happened to the DD pulses, described by the value of $\zeta$ and the
preservation offered is expected to be enhanced if we repeat the whole DD
cycle multiple times.

To summarize, we present one DD scheme to battle with arbitrary single- and
two-qubit errors. The DD scheme is based on the concatenation of the two DD
sequences. To make the DD scheme compatible with realizing quantum gates in
physical systems, we require tunable system parameters such that desired gate
Hamiltonians will not be eliminated by the DD pulses. We investigate the
implementation of the DD scheme based on gate Hamiltonian (1), which can be
achieved experimentally in cold atoms, trapped ions and superconducting qubits
with controllable system parameters. Specifically, we demonstrate that two-
qubit quantum gates are excellently preserved against arbitrary single- and
two-qubit errors by considering a physical system of transmon qubits.
Moreover, we also show that the DD scheme is workable with gate Hamiltonian
(4), that is experimentally attainable in superconducting charge qubits, flux
qubits and fluxoniums. With the current experimental techniques, however, it
may not be possible to obtain excellently protected quantum gates since some
of the parameters in (4) may not be adjustable as desired by the DD scheme. We
expect to see further development in actual experiments to realize
controllable parameters in near future. Our DD scheme provides an efficient
and practical way to mitigate arbitrary single- and two-qubit errors that are
the main source of noises in executing quantum gates, and so it is a timely
contribution to improving NISQ devices.

This research is supported by the National Research Foundation, Singapore and
A*STAR under its Quantum Engineering Programme (NRF2021-QEP2-02-P03). G.W. is
supported by the Fundamental Research Funds for the Central Universities
(Grant No. 2412020FZ026).

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