††thanks: These two authors contributed equally to this work.††thanks: These
two authors contributed equally to this work.

# Qudit Dynamical Decoupling on a Superconducting Quantum Processor

Vinay Tripathi  Department of Physics & Astronomy, University of Southern
California, Los Angeles, CA 90089, USA  Center for Quantum Information Science
& Technology, University of Southern California, Los Angeles, CA 90089, USA
Noah Goss  Quantum Nanoelectronics Laboratory, Department of Physics,
University of California at Berkeley, Berkeley, CA 94720, USA  Applied Math
and Computational Research Division, Lawrence Berkeley National Lab, Berkeley,
CA 94720, USA  Arian Vezvaee  Center for Quantum Information Science &
Technology, University of Southern California, Los Angeles, CA 90089, USA
Department of Electrical & Computer Engineering, University of Southern
California, Los Angeles, California 90089, USA
Long B. Nguyen  Quantum Nanoelectronics Laboratory, Department of Physics,
University of California at Berkeley, Berkeley, CA 94720, USA  Applied Math
and Computational Research Division, Lawrence Berkeley National Lab, Berkeley,
CA 94720, USA  Irfan Siddiqi  Quantum Nanoelectronics Laboratory, Department
of Physics, University of California at Berkeley, Berkeley, CA 94720, USA
Applied Math and Computational Research Division, Lawrence Berkeley National
Lab, Berkeley, CA 94720, USA  Daniel A. Lidar  Department of Physics &
Astronomy, University of Southern California, Los Angeles, CA 90089, USA
Center for Quantum Information Science & Technology, University of Southern
California, Los Angeles, CA 90089, USA  Department of Electrical & Computer
Engineering, University of Southern California, Los Angeles, California 90089,
USA  Department of Chemistry, University of Southern California, Los Angeles,
California 90089, USA

###### Abstract

Multi-level qudit systems are increasingly being explored as alternatives to
traditional qubit systems due to their denser information storage and
processing potential. However, qudits are more susceptible to decoherence than
qubits due to increased loss channels, noise sensitivity, and crosstalk. To
address these challenges, we develop protocols for dynamical decoupling (DD)
of qudit systems based on the Heisenberg-Weyl group. We implement and
experimentally verify these DD protocols on a superconducting transmon
processor that supports qudit operation based on qutrits $(d=3)$ and ququarts
$(d=4)$. Specifically, we demonstrate single-qudit DD sequences to decouple
qutrits and ququarts from system-bath-induced decoherence. We also introduce
two-qudit DD sequences designed to suppress the detrimental cross-Kerr
couplings between coupled qudits. This allows us to demonstrate a significant
improvement in the fidelity of time-evolved qutrit Bell states. Our results
highlight the utility of leveraging DD to enable scalable qudit-based quantum
computing.

Multilevel quantum systems, also known as qudits [1], offer potentially
superior computational capabilities and denser information encoding relative
to traditional qubit-based schemes [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13,
14, 15, 16, 17, 18, 19, 20, 21]. In addition, qudits enable resource-efficient
fault-tolerant quantum computation [22, 23, 24] and the exploration of complex
novel quantum applications [25, 26, 27, 28, 29] with reduced resource
requirements. However, in superconducting devices, qudits are more susceptible
to low-frequency noise and correlated errors, which pose significant
challenges [30]. Addressing these requires the development of scalable
strategies for the mitigation and suppression of decoherence [31]. Dynamical
decoupling (DD) [32, 33, 34, 35, 36, 37] is a powerful technique designed to
enhance the fidelity of quantum states by employing carefully timed control
pulses. It has been used to effectively decouple superconducting qubits from
environmental noise [38, 39, 40] and unwanted crosstalk [41, 42, 43, 44]. DD
has been studied across a broad spectrum of qubit-based systems, but its
experimental application to qudits has been limited primarily to trapped ions
and nitrogen-vacancy ensembles [45, 46, 47, 48], and very recently to enhance
the fidelity of a qutrit-assisted three-qubit Toffoli gate on an IBM transmon
device [49].

Figure 1: Schematic illustration of two transmon qudits with quantized energy
levels affected by relaxation and dephasing errors, along with the qudit-qudit
cross-Kerr couplings $\alpha_{ij}$.

Figure 2: Experimental results showing the fidelity of the qudit uniform
superposition state $|+\rangle_{d}$ as a function of total time under free
evolution (No DD) and $dX_{D}$, for (a) qutrits and (b) ququarts. The minimum
pulse interval is $\tau_{\min}=120$ ns. For qutrits, we implement $1$, $2$,
and $3$ repetitions of the $3X_{3}$ sequence, with corresponding pulse
intervals $3\tau$, $3\tau/2$, and $\tau$, respectively. The total evolution
time is always $9\tau$. Universal DD is a sequence of $9$ pulses formed by
cycling over the entire HWG, applied once with a pulse interval of $\tau$ and
$\tau_{\min}=180$ ns. For ququarts, we similarly implement $1$, $2$, $3$, and
$4$ repetitions of the $4X_{4}$ sequence, with corresponding pulse intervals
of $4\tau$, $2\tau$, $\frac{4}{3}\tau$, and $\tau$ respectively, where the
total time is always $16\tau$.

In this work, we present a general DD framework tailored for qudits and
experimentally demonstrate its effectiveness using coupled superconducting
transmon circuits [50] operated as qutrits $(d=3)$ and ququarts $(d=4)$. Our
framework employs the Heisenberg-Weyl (HW) group, which has found many uses in
the study of $d$-dimensional quantum systems [51, 52]. We present DD sequences
for universal noise suppression and also introduce a single-axis DD sequence
designed to suppress the prevalent $1/f$ dephasing noise that plagues
superconducting qudits. We then introduce a multi-qudit DD sequence designed
to suppress unwanted cross-Kerr interactions between coupled qudits (see Fig.
1), which stand in the way of scaling superconducting qudit systems [10, 30].
Using our DD sequences, we additionally report a significant enhancement in
preserving the fidelity of a qutrit Bell state over time. This work serves as
a proof-of-concept demonstration of the efficacy and scalability of active
refocusing techniques in qudit systems and provides a stepping stone toward
operating large-scale high-dimensional architectures.

Qudit dynamical decoupling theory.— Building on the general symmetrization
ideas of Refs. [33, 34], the theory of qudit DD was developed in Refs. [53,
54, 55, 56]. We briefly introduce essential terminology and present a detailed
review in the Supplementary Materials (SM), where we also generalize the
theory.

The decoupling group $\mathcal{G}_{d}$ is a set of unitary transformations
(pulses) $g_{j}$ acting purely on the system:
$\mathcal{G}_{d}=\left\\{g_{0},\cdots,g_{K}\right\\}$, where $g_{0}$ is the
$d$-dimensional identity operator $I$. Under the instantaneous and ideal pulse
assumptions, cycling over all elements of the group yields the following DD
pulse sequence [33, 34]:

$U(T)=\prod_{j=0}^{K}g_{j}^{\dagger}f_{\tau}g_{j}.$ (1)

Here, $\tau$ is pulse interval, $T=|\mathcal{G}_{d}|\tau=(K+1)\tau$ is the
total time taken by the sequence, and $f_{\tau}=e^{-i\tau H}$ is the free-
evolution unitary, where $H$ is the total Hamiltonian of the system and the
bath. A universal DD sequence for a qubit ($d=2$) is obtained by choosing the
decoupling group as the Pauli group $\mathcal{G}_{2}=\\{I,X,Y,Z\\}$, for which
$U(T)$ simplifies into the well-known XY4 sequence
$U(4\tau)=Yf_{\tau}Xf_{\tau}Yf_{\tau}Xf_{\tau}$ [57].

For $d>2$, we instead use the decoupling group to be the Heisenberg-Weyl group
(HWG) of order $d^{2}$, which generalizes the Pauli group. The HWG is
generated by the following shift and phase operators:

$X_{d}\equiv\sum_{k=0}^{d-1}|{(k+1)\bmod d}\rangle\\!\langle k|,\,\quad
Z_{d}\equiv\sum_{k=0}^{d-1}\gamma_{d}^{k}|{k}\rangle\\!\langle k|,$ (2)

where $\gamma_{d}=e^{2\pi i/d}$ is the $d$th root of unity. The remaining HWG
elements are given by
$\Lambda_{\alpha\beta}=(-\sqrt{\gamma_{d}})^{\alpha\beta}X_{d}^{\alpha}Z_{d}^{\beta}$
where $\alpha,\beta\in\mathbb{Z}_{d}=\\{0,1,2,...,d-1\\}$.

The dominant decoherence mechanism in transmon qutrits and ququarts is
dephasing due to $1/f$ noise [58], which has been connected to charge
fluctuations and higher level charge sensitivity [50, 59]. Thus, for single
qudits, we focus on single-axis DD sequences consisting only of the shift
operator and its powers, i.e., the decoupling group formed by the HW subgroup
$\\{X_{d}^{k}\\}_{k=0}^{d-1}$. Note that $(X_{d}^{k})^{\dagger}=X_{d}^{d-k}$.
Thus, cycling over these operators, we obtain
$U(T)=(X_{d}^{1}f_{\tau}X_{d}^{d-1})...(X_{d}^{d-2}f_{\tau}X_{d}^{2})(X_{d}^{d-1}f_{\tau}X_{d})(If_{\tau}I)$.
Simplifying, this becomes the sequence $dX_{d}\equiv
X_{d}f_{\tau}X_{d}f_{\tau}X_{d}...X_{d}f_{\tau}$.

Single qudit $dX_{d}$ experiment.— We conduct all our experiments on a
superconducting transmon qudit processor with $d=3$ and $4$; other parameters
are detailed in the SM. Since DD sequences are particularly effective against
low-frequency noise [60], and superconducting circuits are especially
susceptible to such noise when higher excited states are targeted [10], we
focus primarily on the $dX_{d}$ sequence family. The underlying cycle operator
$X_{d}$ is compiled using $2(d-1)$ native $\sqrt{\sigma^{x}_{s}}$ subspace
rotations where $s\in\\{(0,1),(1,2),..,(d-1,d)\\}$, and
$\sigma^{x}_{(i,j)}=|{i}\rangle\\!\langle j|+|{j}\rangle\\!\langle i|$ is the
Pauli-$x$ operator between levels $i$ and $j$. Fig. 2 presents our
experimental single-qudit $dX_{d}$ results. Free evolution (no DD) corresponds
to the preparation of a uniform qudit superposition state
$|+\rangle_{d}\equiv(|0\rangle+\dots+|d-1\rangle)/\sqrt{d}$, waiting for a
specified delay time, unpreparing the state, and finally measuring the qudit.
Assuming ideal preparation, unpreparation, and measurement, the fidelity of
the superposition state $|+\rangle_{d}$ is the probability of finding the
qudit back in the $|0\rangle$ state. We then repeat the experiment with the
$dX_{d}$ sequence applied during the delay time and study its impact on the
state fidelity.

Fig. 2(a) presents the results for the qutrit experiments. Crucially, all the
DD curves exhibit an improvement over the free evolution (No DD) experiment,
confirming the effectiveness of our DD sequences in suppressing decoherence.
In more detail, for each total time $T=9\tau$ we conducted a free evolution
experiment and four DD experiments: $1$, $2$, or $3$ repetitions of $3X_{3}$,
and universal qutrit DD (the full order-$9$ HWG), with respective pulse
intervals of $\tau_{1}=3\tau$, $\tau_{2}=\frac{3}{2}\tau$, $\tau_{3}=\tau$,
and $\tau_{\text{univ.}}=\tau$. DD theory predicts that for instantaneous,
ideal pulses, state preservation fidelity increases monotonically as the pulse
interval decreases for a fixed total evolution time [61, 62]. Moreover,
universal DD is expected to outperform single-axis DD. Our results exhibit the
opposite of both expectations: the single repetition $1\times 3X_{3}$
experiment, with the longest pulse interval $\tau_{1}=3\tau$, yields the
highest fidelity, while the universal sequence, with the shortest interval
$\tau$ (as for $3\times 3X_{3}$) yields the lowest DD fidelity. The reason for
these results is likely to be the presence of coherent pulse errors, which
accumulate more detrimentally the longer the pulse sequence, and whose effect
overwhelms the benefit of shorter pulse intervals [63, 64]. While $X_{3}$
gates can be decomposed in terms of four native $\sqrt{\sigma^{x}_{s}}$ gates,
the remaining HW pulses require six native $\sqrt{\sigma^{x}_{s}}$ gates, so
that $\tau_{\min}=180$ ns for universal DD compared to $\tau_{\min}=120$ ns
for the $3X_{3}$ sequences. This additional opportunity for the accumulation
of coherent errors explains why the universal sequence underperforms the
$3\times 3X_{3}$ sequence. The superior performance of the $3X_{3}$ sequences
also confirms that the dominant source of noise is dephasing.

Similar improvements are observed with DD for ququarts, as shown in Fig. 2(b).
Since ququarts are more susceptible to charge noise due to the involvement of
the third excited state [50], the free evolution fidelity is significantly
lower than in the qutrit case, and the improvement with DD is even more
pronounced. Note that the difference between the DD sequences is much smaller
than in the qutrit case, except for the $4\times 4X_{4}$ case at times $<25\
\mu$s. This could be attributed to the stronger $1/f$ dephasing suppression
effect of the $4X_{4}$ sequences relative to the $3X_{3}$ sequences, which
outweighs the accumulation of pulse errors. These ququart results highlight
the effectiveness of DD in higher dimensions and its critical role in
suppressing noise in more complex systems.

Figure 3: Experimental results showing suppression of cross-Kerr interactions
using the CKDD sequence. (a) Fidelity of the qutrit superposition state
$|+\rangle_{3}$, with CKDD (solid) and without (dashed). CKDD is a
$9\tau$-long sequence, with $\tau_{\min}=120$ ns. (b) Fidelity of the ququart
superposition state $|+\rangle_{4}$, with CKDD (solid) and without (dashed).
CKDD is a $16\tau$-long sequence, with $\tau_{\min}=180$ ns. As in Fig. 2,
longer evolution times correspond to a single repetition of CKDD with
increased $\tau$. CKDD removes the cross-Kerr oscillations and improves the
fidelity in both cases, converging to the fully mixed state fidelity baseline
(dashed horizontal line) more slowly than the free evolution (No DD) curves.
See text for further details.

Cross-Kerr suppressing DD (CKDD).— Having shown significant improvements with
single-qudit DD against decoherence, we now deploy DD to mitigate qudit
crosstalk. Recent work has demonstrated the efficacy of DD in suppressing
coherent crosstalk errors in qubit systems [41, 42, 65]. In our fixed linear
coupling qudit processor, single-qudit operations suffer from always-on
crosstalk, which is a generalization of the $ZZ$ interaction between transmon
qubits [66]. This type of crosstalk is commonly referred to as cross-Kerr
interactions, which describe the spectator-state-dependent shifts of the
relevant qudit transition frequencies. For two coupled qubits, in the lab
frame where $H_{\rm 2q}=\sum_{i,j=0}^{1}E_{ij}|{ij}\rangle\\!\langle ij|$ (in
the eigenbasis), $\mathrm{Tr}(ZZH_{\rm 2q})=E_{11}+E_{00}-E_{10}-E_{01}$. To
model cross-Kerr interactions, consider the rotating frame that nullifies all
bare transmon energy terms, leaving only the diagonal interaction terms. Then
the Hamiltonian for the two-coupled transmon qudits simplifies to [10, 12]:

$\displaystyle H_{\rm CK}$
$\displaystyle=\sum_{i,j=1}^{d-1}\alpha_{ij}|{ij}\rangle\\!\langle ij|.$ (3)

Here, $\alpha_{ij}=\omega_{ij}+\omega_{00}-\omega_{i0}-\omega_{0j}$ (taking
$\hbar=1$), where $i,j\in\mathbb{Z}_{d}$, are the qudit frequency shifts (see
Fig. 1).

Although these cross-Kerr interactions, along with off-resonantly applied
microwave drives, have been shown to facilitate entangling operations [12,
10], even minor cross-Kerr interactions during idle periods can introduce
significant coherent errors. Building on the qudit DD formalism developed
above, we now propose a DD sequence for coupled qudits. This DD sequence
effectively suppresses all the cross-Kerr interactions, thereby enhancing
system stability and operational fidelity.

The evolution operator due to $dX_{d}$ applied only to the first qudit is
given by $U^{(1)}_{d\tau}\equiv dX_{d}\otimes I$. By concatenating this
sequence with the same $dX_{d}$ sequence applied to the second qudit, we
obtain the total evolution $U_{d^{2}\tau}\equiv U^{(2)}_{d\tau}\circ
U^{(1)}_{d\tau}$, i.e.,

$\displaystyle U_{d^{2}\tau}=(I\otimes X_{d})U^{(1)}_{d\tau}(I\otimes
X_{d})\cdots U^{(1)}_{d\tau}(I\otimes X_{d})U^{(1)}_{d\tau}.$ (4)

Concatenation of DD sequences was originally introduced in order to obtain
high-order suppression [67]; here, it serves the purpose of staggering the
sequences on the two qudits, thus generalizing the idea of robust qubit-
crosstalk suppression via staggering [42]. We show in the SM that
$U_{d^{2}\tau}=e^{i\theta}I\otimes I+O(T^{2})$, where $\theta$ is a global
phase, and $T=d^{2}\tau$. Thus, we expect that, to first order in the pulse
interval, the $d^{2}\tau$-long _cross-Kerr DD_ (CKDD) sequence in Eq. 4
suppresses all crosstalk between two coupled qudits.

Figure 4: Top: Fidelity of the qutrit Bell state over time with and without
CKDD. See text for details. Bottom: real (upper) and imaginary (lower)
components of quantum state tomography results for the final time point in the
top plot (red circles). Ideally,
$\Re(\rho_{ik,jl})=\frac{1}{3}\delta_{ik}\delta_{jl}$ and
$\Im(\rho_{ik,jl})=0$ for $i,j,k,l\in\mathbb{Z}_{3}$, where $\rho$ is the
density matrix. Left: free evolution. The nine red-colored bars at the ideal
positions are of varying magnitude and some contain large imaginary
components, indicating deviations from the ideal qutrit Bell state. Right:
with CKDD. In contrast, the nine red-colored bars are nearly uniform in height
are have negligible imaginary components, indicating proximity to the ideal
qutrit Bell state. Note that here CKDD is a $9\tau$-long sequence with
$\tau_{\min}=180~{}$ns, the $X_{3}$ gate duration; this differs from the
previous figures where the $X_{3}$ gate duration is $120~{}$ns due to a
different calibration.

Experimental validation of CKDD.— To validate the CKDD sequence, we conduct
experiments on coupled transmon qutrits and ququarts. Fig. 3(a) shows the
results for a linear chain of three coupled qutrits, where we prepare the set
of nine initial states $\ket{i}\otimes\ket{+}_{3}\otimes\ket{j}$,
$i,j\in\mathbb{Z}_{3}$. We trace out the states of the left and right
“spectator” qutrits, and display the fidelity of the middle (main) qutrit’s
state with respect to $|+\rangle_{3}$ as a function of delay time. As depicted
by the dashed light-color curves, different initial spectator states exhibit
distinct curves that oscillate at different frequencies. The high frequency
oscillations and the differences between the curves are attributable to the
cross-Kerr interactions between the main qutrit and the two spectators. Next,
we apply the CKDD sequence as $U^{(2)}_{d\tau}\circ(U^{(1)}_{d\tau}\otimes
U^{(3)}_{d\tau})$, that is, an inner sequence where $dX_{d}$ is applied
simultaneously to the spectators, and an outer sequence where $dX_{d}$ is
applied to the main qutrit. The spectators’ pulses are synchronized, so the
sequence still takes a total time of $d^{2}\tau$. The resulting solid, bold-
color curves exhibit a higher fidelity and none of variation of the free
evolution curves, highlighting the efficacy of the CKDD sequence in
suppressing cross-Kerr interactions and stabilizing the system dynamics for
different initial spectator states.

In Fig. 3(b), we present the fidelity results for two coupled ququarts, for
the set of four initial states $\ket{i}\otimes\ket{+}_{4}$,
$i\in\mathbb{Z}_{4}$ for the spectator. We again trace out the spectator
state. Similarly to the qutrit case, under free evolution we observe different
curves corresponding to different states of the spectator, but when CKDD is
applied, all four curves exhibit a similar exponential decay, indicating
suppression of the cross-Kerr interactions. The CKDD sequence consists of the
ququart shift operators $X_{4}$ comprising three $\sigma^{x}_{i,i+1}$ gates in
the two-level subspaces spanned by $\\{\ket{i},\ket{i+1}\\}$,
$i\in\mathbb{Z}_{4}$. In the presence of large cross-Kerr interactions,
driving the two-level subspaces is prone to large detuning errors. This
results in somewhat lower CKDD fidelities for the ququart experiments compared
to qutrits. Despite this, the suppression of the cross-Kerr interactions is
clearly evident from the results, thus providing a proof-of-principle
demonstration of the scalability of the CKDD protocol to higher dimensions.

Qutrit entanglement preservation via CKDD.— To demonstrate the effectiveness
of CKDD beyond the preservation of product states, we prepare the qutrit Bell
state $(\ket{00}+\ket{11}+\ket{22}/\sqrt{3}$ and measure its fidelity over
time with and without CKDD. The preparation of this state involves a qutrit
controlled-phase (CZ) gate [12]. We employ quantum state tomography to compute
the fidelity of the Bell state. As depicted in Fig. 4, the experimental
results for free evolution (No DD) and with CKDD applied to both qutrits
contrast sharply. In the absence of DD, the state fidelity suffers
significantly due to strong coherent errors arising from large cross-Kerr
interactions. The fidelity drops to near zero in $\sim 1\ \mu$s, meaning that
the state evolves to an orthogonal qutrit Bell state, then oscillates around
the fully mixed state fidelity baseline of $1/9$ (dashed horizontal line),
highlighting the importance of suppressing cross-Kerr interactions. In
contrast, with CKDD we observe a marked improvement; the oscillations are
nearly eliminated, and the fidelity remains $>50\%$ even after $10~{}\mu$s.
The state tomography histograms in Fig. 4 further highlight CKDD’s ability to
maintain the integrity of qutrit Bell states.

Conclusions and outlook.— Building on the theory of qudit DD [53, 54, 55, 56],
we have demonstrated the suppression of decoherence in transmon-based qutrits
and ququarts. Our experimental results exhibit a substantial improvement in
the preservation of the fidelity of superposition states of such qudit
systems. Beyond decoherence, a significant challenge in scaling
superconducting qudit systems arises from the persistent cross-Kerr
interactions between coupled qudits. To address this, we introduced cross-Kerr
DD as a protocol aimed at suppressing these spurious interactions. Our
experimental results demonstrate that the CKDD sequence successfully
suppresses cross-Kerr interactions in both qutrits and ququarts, which
suggests that CKDD can be employed in higher-dimensional systems as well.
Furthermore, we have shown that CKDD significantly improves the fidelity of
maximally entangled qutrit states.

Our findings broaden the scope of dynamical decoupling used in the service of
the suppression of decoherence and crosstalk beyond the traditional setting of
qubits to qudit systems. While our focus here was on transmons, our findings
can be directly applied to other quantum computing platforms with access to
qudits, such as fluxonium systems operated at half flux [68, 69]. This
addition to the quantum noise suppression toolkit will hopefully benefit the
development of scalable qudit-based quantum processors.

Acknowledgement— V.T., A.V. and D.A.L. were supported by the ARO MURI grant
W911NF-22-S-0007 and by the National Science Foundation Quantum Leap Big Idea
under Grant No. OMA-1936388. N.G. was supported by funding from the National
Science Foundation under Grant No. 2210391. L.B.N. acknowledges funding from
the Office of Advanced Scientific Computing Research, Testbeds for Science
program, Accelerated Research in Quantum Computing Program, Office of Science
of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231.

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Supplemental Material: Qudit Dynamical Decoupling on a Superconducting Quantum
Processor

##  Appendix A Experimental Device Characterization

The superconducting qudit device employed in this work consists of 8 fixed-
frequency transmon qudits coupled together by coplanar waveguide resonators in
a ring topology. For all the experiments in this work, we employ a susbset of
the device consisting of a $3$-qudit line. We report the basic single qudit
parameters of the subset of the device used in this work in Table. 1. For
further, more extensive characterization of the device, including gate
fidelities and readout fidelities, see Refs. [19, 30, 12].

Parameters | Q1 | Q2 | Q3
---|---|---|---
$\omega_{01}/2\pi$ (GHz) | 5.333 | 5.396 | 5.572
$\omega_{12}/2\pi$ (GHz) | 5.061 | 5.124 | 5.303
$\omega_{23}/2\pi$ (GHz) | 4.757 | 4.821 | 5.005
Avg. $T_{1}^{01}$ ($\mu$s) | 50(4) | 49(4) | 60(5)
Avg. $T_{1}^{12}$ ($\mu$s) | 35(2) | 35(4) | 31(8)
Avg. $T_{1}^{23}$ ($\mu$s) | 24(4) | 26(3) | 23(4)
Avg. $T_{2e}^{01}$ ($\mu$s) | 78(5) | 85(9) | 90(6)
Avg. $T_{2e}^{12}$ ($\mu$s) | 57(4) | 57(4) | 56(9)
Avg. $T_{2e}^{23}$ ($\mu$s) | 26(2) | 27(2) | 24(3)
Table 1: Transition frequencies $\omega_{ij}=(E_{j}-E_{i})/\hbar$ up to $d=4$
of the qudits employed in our DD experiments. Additionally, we provide the
two-level subspace mean $T_{1}$ and $T_{2}$ echo times of the device
calculated from 100 repetitions of each coherence experiment.

$Q_{2}$ was used for both single-qudit DD experiments reported in Fig. 2 of
the main text.

##  A.0.1 Qudit Cross-Kerr Coupling Rates

As discussed in the main text, the fixed-linear coupling between
superconducting qudits generates a longitudinal hybridization mediated largely
by the higher levels present in each transmon well. These longitudinal
interactions impart entangling phases to all qudit states $\ket{i,j}$
($i,j\in\mathbb{Z}_{d}$) and lead to significant coherent errors. In the
doubly rotating frame of the two-qudit system, the effective cross-Kerr
Hamiltonian defines the accumulation of all non-local entangling phases and is
given by $H_{\rm CK}=\sum_{i,j=1}^{d-1}\alpha_{ij}|{ij}\rangle\\!\langle ij|$
[10, 12] [Eq. 3 in the main text]. In the cross-Kerr DD experiments reported
in Fig. 3 of the main text, we demonstrate that local pulses are sufficient to
provide effective suppression of the $d=3$ and $d=4$ cross-Kerr interaction as
well as of the system-bath coupling.

##  A.0.2 $d=3$ cross-Kerr DD Experiment

For our $d=3$ cross-Kerr DD experiment [see Fig. 3(a) of the main text], we
prepared the $\ket{+}_{3}$ on $Q_{2}$ using a qutrit Hadamard gate and
simultaneously prepared the two spectator qutrits $Q_{1}$ and $Q_{3}$ in
states $\ket{i,j}$ ($i,j\in\mathbb{Z}_{3}$). We then allowed the system to
time-evolve with and without our cross-Kerr DD sequence, and assessed the
time-evolved state fidelity by remapping $\ket{+}_{3}$ to $\ket{0}$ via a
final qutrit Hadamard gate. The relevant four qutrit cross-Kerr rates for this
experiment measured via conditional Ramsey experiments are presented in Table
2.

Parameters$/2\pi$ |  $Q_{1}$-$Q_{2}$ |  $Q_{2}$-$Q_{3}$
---|---|---
$\alpha_{11}$ (MHz) | 0.112 | 0.212
$\alpha_{12}$ (MHz) | 0.623 | 0.465
$\alpha_{21}$ (MHz) | -0.515 | -0.162
$\alpha_{22}$ (MHz) | 0.341 | 0.615
Table 2: The qutrit cross-Kerr crosstalk rates present in the spectator DD
experiment in Fig. 3(a) in the main text.

##  A.0.3 $d=4$ cross-Kerr DD Experiment

For our $d=4$ cross-Kerr DD experiment [see Fig. 3(b) of the main text], we
prepared the $\ket{+}_{4}$ state on $Q_{2}$ using a ququart Hadamard gate and
simultaneously prepared the spectator qudit $Q_{1}$ in $\ket{i}$
($i\in\mathbb{Z}_{4}$). We then allowed the system to time-evolve with and
without our cross-Kerr DD sequence, and assessed the time-evolved state
fidelity by remapping $\ket{+}_{4}$ to $\ket{0}$ via a final ququart Hadamard
gate. The relevant nine ququart cross-Kerr rates for this experiment measured
via conditional Ramsey experiments are presented in Table 3.

Parameters$/2\pi$ |  $Q_{1}$-$Q_{2}$
---|---
$\alpha_{11}$ (MHz) | 0.112
$\alpha_{12}$ (MHz) | 0.623
$\alpha_{13}$ (MHz) | 0.021
$\alpha_{21}$ (MHz) | -0.515
$\alpha_{22}$ (MHz) | 0.341
$\alpha_{23}$ (MHz) | 0.730
$\alpha_{31}$ (MHz) | 0.226
$\alpha_{32}$ (MHz) | -0.442
$\alpha_{33}$ (MHz) | 0.345
Table 3: The ququart cross-Kerr crosstalk rates present in the spectator DD
experiment in Fig. 3(b) in the main text.

##  Appendix B General theory of qudit dynamical decoupling

Consider the total time-independent Hamiltonian of a system coupled to a bath
$H=H_{S}\otimes I_{B}+I_{S}\otimes H_{B}+H_{SB}$ where $H_{S}$, $H_{B}$, and
$H_{SB}$ are the Hamiltonian terms associated with the system, the bath, and
the system-bath interaction, respectively. Here, $H_{S}$ represents undesired
system terms such as crosstalk and stray local fields. Pulses are applied to
the system via an additional, time-dependent control Hamiltonian.
Correspondingly, the _decoupling group_ $\mathcal{G}_{d}$ is defined by a set
of unitary transformations $g_{j}$ acting purely on the system:
$\mathcal{G}_{d}=\left\\{g_{0},\cdots,g_{K}\right\\}$, where $g_{0}$ is the
$d$-dimensional identity operator $I$. Under the instantaneous (zero width)
and ideal (error-free) pulse assumptions, cycling over all elements of the
group yields the following DD pulse sequence [33, 34]:

$U(T)=\prod_{j=0}^{|\mathcal{G}_{d}|-1}g_{j}^{\dagger}f_{\tau}g_{j}=e^{-iTH^{\prime}}+\mathcal{O}(T^{2}).$
(5)

Here, $\tau$ is the pulse interval (the time between consecutive pulses),
$T=|\mathcal{G}_{d}|\tau$ is the total time taken by the sequence, and
$f_{\tau}=e^{-i\tau H}$ is the free-evolution unitary. The effective
Hamiltonian at the end of the sequence is

$H^{\prime}=H^{\prime}_{SB}+H^{\prime}_{S}+H_{B},$ (6)

where

$H^{\prime}_{SB}=\mathcal{P}_{\mathcal{G}_{d}}(H_{SB})\ ,\quad
H^{\prime}_{S}=\mathcal{P}_{\mathcal{G}_{d}}(H_{S}).$ (7)

Here

$\mathcal{P}_{\mathcal{G}_{d}}(\Omega)=\frac{1}{|\mathcal{G}_{d}|}\sum_{j=0}^{|\mathcal{G}_{d}|-1}g_{j}^{\dagger}\Omega
g_{j}$ (8)

is the projection of the operator $\Omega$ into the commutant of
$\mathcal{G}_{d}$, i.e., the set of operators that commute with every element
of $\mathcal{G}_{d}$. Crucially, this projection can be made proportional to
$I$ or even vanish via a proper choice of $\mathcal{G}_{d}$. When
$H^{\prime}_{SB}=H^{\prime}_{S}=0$, we call $\mathcal{G}_{d}$ and the
corresponding DD sequence universal. For example, a universal DD sequence for
a qubit ($d=2$) is obtained by choosing the decoupling group as the Pauli
group $\mathcal{G}_{2}=\mathcal{P}$, which leads to the well-known XY4
sequence $U=Yf_{\tau}Xf_{\tau}Yf_{\tau}Xf_{\tau}$ [57].

For $d\geq 2$, we instead use the corresponding Heisenberg-Weyl group (HWG) of
order $d^{2}$, which generalizes the Pauli group. We define shift and phase
operators as in Eq. 2 of the main text, repeated here for convenience:

$X_{d}\equiv\sum_{k=0}^{d-1}|{(k+1)\bmod d}\rangle\\!\langle k|,\,\quad
Z_{d}\equiv\sum_{k=0}^{d-1}\gamma_{d}^{k}|{k}\rangle\\!\langle k|,$ (9)

where $\gamma_{d}=e^{2\pi i/d}$ is the $d$’th root of unity.

Note that $X_{d}^{d}=Z_{d}^{d}=I$, and that $X_{d}$ and $Z_{d}$ are generally
non-Hermitian but are both unitary for all $d$ and hence satisfy

$(X_{d}^{\dagger})^{\alpha}X_{d}^{\beta}=X_{d}^{\beta-\alpha},\quad(Z_{d}^{\dagger})^{\alpha}Z_{d}^{\beta}=Z_{d}^{\beta-\alpha}.$
(10)

In particular,

$X_{d}^{\dagger}=X_{d}^{-1}=X_{d}^{d-1}\ ,\quad
Z_{d}^{\dagger}=Z_{d}^{-1}=Z_{d}^{d-1}.$ (11)

$X_{d}$ and $Z_{d}$ are the generators of the HWG, whose elements are

$\Lambda_{\alpha\beta}=(-\sqrt{\gamma_{d}})^{\alpha\beta}X_{d}^{\alpha}Z_{d}^{\beta},$
(12)

where $\alpha,\beta\in\mathbb{Z}_{d}$. For $d=2$, the HWG trivially reduces to
the Pauli group.

For the two generators $X_{d}$ and $Z_{d}$, we have

$\displaystyle X_{d}Z_{d}$ $\displaystyle=\sum_{k=0}^{d-1}e^{2\pi
ik/d}|{k+1\bmod d}\rangle\\!\langle k|$ (13a) $\displaystyle Z_{d}X_{d}$
$\displaystyle=\sum_{k=0}^{d-1}e^{2\pi i(k+1)/d}|{k+1\bmod d}\rangle\\!\langle
k|,$ (13b)

i.e.,

$Z_{d}X_{d}=\gamma_{d}X_{d}Z_{d}.$ (14)

Similarly, we find:

$Z_{d}^{\dagger}X_{d}=\gamma_{d}^{-1}X_{d}Z_{d}^{\dagger}.$ (15)

Using Eqs. 14 and 15, we can show that

$\displaystyle Z_{d}^{\beta}X_{d}^{\alpha}$
$\displaystyle=\gamma_{d}^{\alpha\beta}X_{d}^{\alpha}Z_{d}^{\beta}$ (16a)
$\displaystyle(Z_{d}^{\dagger})^{\beta}X_{d}^{\alpha}$
$\displaystyle=\gamma_{d}^{-\alpha\beta}X_{d}^{\alpha}(Z_{d}^{\dagger})^{\beta}.$
(16b)

###### Proof.

For $\alpha=\beta=1$, Eq. 16 reduces to Eq. 14. Consider $\beta\geq 2$:

$Z_{d}^{\beta}X_{d}=\gamma_{d}Z_{d}^{\beta-1}X_{d}Z_{d}=\dots=\gamma_{d}^{\beta}X_{d}Z_{d}^{\beta}.$
(17)

When $\alpha\geq 2$:

$\displaystyle Z_{d}^{\beta}X_{d}^{\alpha}$
$\displaystyle=(Z_{d}^{\beta}X_{d})X_{d}^{\alpha-1}=\gamma_{d}^{\beta}X_{d}(Z_{d}^{\beta}X_{d})X_{d}^{\alpha-2}$
(18a)
$\displaystyle=\gamma_{d}^{2\beta}X_{d}^{2}Z_{d}X_{d}^{\alpha-2}=\dots=\gamma_{d}^{\alpha\beta}X_{d}^{\alpha}Z_{d}^{\beta}.$
(18b)

Eq. 16b follows analogously. ∎

This means that the commutation relations for two arbitrary HW operators are

$\displaystyle\Lambda_{\alpha\beta}\Lambda_{\mu\nu}$
$\displaystyle=\gamma_{d}^{\beta\mu-\alpha\nu}\Lambda_{\mu\nu}\Lambda_{\alpha\beta}$
(19a) $\displaystyle\Lambda_{\alpha\beta}^{\dagger}\Lambda_{\mu\nu}$
$\displaystyle=\gamma_{d}^{\alpha\nu-\beta\mu}\Lambda_{\mu\nu}\Lambda_{\alpha\beta}^{\dagger}.$
(19b)

Unless $\alpha=\beta=0$, the HW operators are non-Hermitian for $d\geq 3$

$\displaystyle\Lambda_{\alpha\beta}^{{\dagger}}$
$\displaystyle=e^{-i\pi\alpha\beta\frac{d+1}{d}}(X_{d}^{\alpha}Z_{d}^{\beta})^{\dagger}$
(20a)
$\displaystyle=(-\sqrt{\gamma_{d}})^{-\alpha\beta}(Z_{d}^{\dagger})^{\beta}(X_{d}^{\dagger})^{\alpha}\neq\Lambda_{\alpha\beta},$
(20b)

but unitary for all $d$:

$\Lambda_{\alpha\beta}^{{\dagger}}\Lambda_{\alpha\beta}=(Z_{d}^{\dagger})^{\beta}(X_{d}^{\dagger})^{\alpha}X_{d}^{\alpha}Z_{d}^{\beta}=I,$
(21)

where we used the unitarity of $X_{d}$ and $Z_{d}$. Combining unitarity with
Eq. 19a, we obtain the identity

$\Lambda_{\alpha\beta}^{{\dagger}}\Lambda_{\mu\nu}\Lambda_{\alpha\beta}=\gamma_{d}^{\alpha\nu-\beta\mu}\Lambda_{\mu\nu},$
(22)

which will prove to be crucial below for demonstrating that the HWG is a
universal DD group.

The operators $\\{\Lambda_{\alpha\beta}\\}_{\alpha,\beta\in\mathbb{Z}_{d}}$
form an irreducible, unitary, and projective representation of the HWG over
the $d$-dimensional system Hilbert space when $d$ is a prime power ($d=p^{k}$
for prime number $p$ and positive integer $k$). This implies, by Schur’s
Lemma, that both $H^{\prime}_{SB}$ and $H^{\prime}_{S}$ are proportional to
$I$ or vanish. Therefore, the unitary $U(T)$ defined in Eq. 5 reduces (up to a
global phase) to the identity operation on the system, i.e., the condition for
first-order decoupling is satisfied. The latter (or an equivalent one using
group character tables) was the argument used in Refs. [53, 54, 55, 56]; going
beyond the case of prime powers, we now show that, in fact, first-order
decoupling holds for arbitrary $d$.

The HW operators also form an operator basis for the $d$-dimensional system
Hilbert space. Thus, we can expand $H=H_{S}\otimes I_{B}+H_{SB}+I_{S}\otimes
H_{B}$ as

$H=\sum_{\mu,\nu=0}^{d^{2}-1}\Lambda_{\mu\nu}\otimes B_{\mu\nu},$ (23)

where $B_{\mu\nu}$ is either zero, proportional to $I_{B}$ (to account for
$H_{S}\otimes I_{B}$), or is a non-identity bath operator. The term with
$\mu=\nu=0$ corresponds to the pure-bath term $I_{S}\otimes H_{B}$.

Now recall that cycling over the decoupling group $\mathcal{G}_{d}$ yields Eq.
5. Choosing the decoupling group as the HWG $\\{\Lambda_{\alpha\beta}\\}$
means that the effective Hamiltonian becomes

$\displaystyle H^{\prime}$ $\displaystyle=\mathcal{P}_{\mathcal{G}_{d}}(H)$
(24a)
$\displaystyle=\frac{1}{d^{2}}\sum_{\alpha,\beta=0}^{d^{2}-1}\Lambda_{\alpha\beta}^{{\dagger}}\sum_{\mu,\nu=0}^{d^{2}-1}\Lambda_{\mu\nu}\Lambda_{\alpha\beta}\otimes
B_{\mu\nu}$ (24b)
$\displaystyle=\frac{1}{d^{2}}\sum_{\mu,\nu=0}^{d^{2}-1}f_{\mu\nu}\Lambda_{\mu\nu}\otimes
B_{\mu\nu},$ (24c)

where, using Eq. 22,

$f_{\mu\nu}=\sum_{\alpha,\beta=0}^{d^{2}-1}\gamma_{d}^{\alpha\nu-\beta\mu}.$
(25)

Let us now show that

$f_{\mu\nu}=d^{4}\delta_{\mu 0}\delta_{\nu 0}.$ (26)

Intuitively, this follows from the zero-sum property of the roots of unity:
$\sum_{k=0}^{d-1}\gamma_{d}^{k}=0$.

###### Proof.

Note that $f_{\mu\nu}=h^{*}_{\mu}h_{\nu}$, and

$\displaystyle h_{\nu}$
$\displaystyle=\sum_{\alpha=0}^{d^{2}-1}\gamma_{d}^{\nu\alpha}=\sum_{k=0}^{d-1}\sum_{j=0}^{d-1}e^{2\pi
i(kd+j)\nu/d}$ (27a) $\displaystyle=\sum_{k=0}^{d-1}\sum_{j=0}^{d-1}(e^{2\pi
i\nu/d})^{j}=dS,$ (27b)

where $S=\sum_{j=0}^{d-1}\omega^{j}$ and $\omega=e^{2\pi i\nu/d}$. If $\nu\neq
0$ then $\omega\neq 1$ is a $d$’th root of unity (since $\omega^{d}=1$).
Multiplying both sides by $\omega-1$ yields
$(\omega-1)S=\sum_{j=0}^{d-1}\omega^{j+1}-\sum_{j=0}^{d-1}\omega^{j}$. The
terms $\omega,\omega^{2},\ldots,\omega^{d-1}$ cancel out, leaving
$(\omega-1)S=\omega^{d}-1=0$. Since $\omega\neq 1$ we can divide both sides by
$\omega-1$, giving $S=0$. If $\nu=0$ then $S=d$. ∎

Combining Eqs. 24 and 26, we finally have

$H^{\prime}=I_{S}\otimes H_{B},$ (28)

i.e., $H^{\prime}_{SB}=H^{\prime}_{S}=0$, leaving only the pure-bath term.
This proves that the HWG is a universal decoupling group.

We numerically confirm this universality in Fig. 5. For various dimensions
$2\leq d\leq 10$, we consider a system-bath Hamiltonian containing all the HW
operators with randomized coefficients (i.e., a classical bath). We then apply
the corresponding universal sequence and compare the fidelity of the resulting
unitary to the identity operator $I$ in each case. Given that the errors in
the unitary evolution under DD are suppressed to the first order [Eq. 5],
i.e., leaving the leading order term $\mathcal{O}(T^{2})$ where
$T\propto\tau$, we expect the fidelity to scale as $\mathcal{O}(\tau^{4})$.
This is confirmed in Fig. 5.

Figure 5: Numerical simulation of the the universality of the DD sequence
generated by cycling over the HWG for $2\leq d\leq 10$. Here we plot the
infidelity of the resulting unitary evolution as a function of the pulse
interval time $\tau$. Since the sequence is expected to cancel the errors to
the first order $\mathcal{O}(\tau^{2})$, the infidelity should scale as
$\mathcal{O}(\tau^{4})$, as confirmed by our simulations.

##  Appendix C Application to single-axis, pure dephasing noise

We utilize the formalism developed above to analyze the single-axis noise
problem (i.e., pure dephasing) as a special case. This is the basis for the
transmon-based qutrit and ququart experiments we present in the main text,
where dephasing is the dominant source of decoherence. In this single-axis
scenario, the system-bath interaction component of Eq. 23 reduces to

$H_{SB}^{Z}=\sum_{\nu=1}^{d-1}\Lambda_{0\nu}\otimes B_{\nu},$ (29)

where $\Lambda_{0\nu}=Z_{d}^{\nu}$ [Eq. 12].

We could choose the full HWG as a decoupling group, but since the $Z_{d}$-type
HW operators commute with $H_{SB}^{Z}$, we need only consider the pure
$X_{d}$-type decoupling operators $\Lambda_{\alpha 0}=X_{d}^{\alpha}$, which
satisfy the commutation relations Eq. 19 non-trivially. I.e., for pure qudit
dephasing the relevant decoupling group is the order-$d$ HWG subgroup
$\mathcal{G}_{d}^{X}=\\{\Lambda_{\alpha
0}\\}_{\alpha=0}^{d-1}=\\{I,X_{d},X_{d}^{2},\cdots,X_{s}^{d-1}\\}$.

To show that $\mathcal{G}_{d}^{X}$ decouples $H_{SB}^{Z}$, we observe, using
Eq. 22, that the effective Hamiltonian is

$\displaystyle(H_{SB}^{Z})^{\prime}$
$\displaystyle=\mathcal{P}_{\mathcal{G}^{X}_{d}}(H_{SB}^{Z})$ (30a)
$\displaystyle=\frac{1}{d}\sum_{\alpha=0}^{d-1}\Lambda_{\alpha
0}^{{\dagger}}\sum_{\nu=1}^{d-1}\Lambda_{0\nu}\Lambda_{\alpha 0}\otimes
B_{\nu}$ (30b)
$\displaystyle=\frac{1}{d}\sum_{\nu=1}^{d-1}f_{\nu}\Lambda_{0\nu}\otimes
B_{\nu},$ (30c)

where

$f_{\nu}=\sum_{\alpha=0}^{d-1}\gamma_{d}^{\nu\alpha}=d\delta_{\nu 0},$ (31)

and the last equality is again due to zero-sum property of the $d$’th root of
unity. It follows that $(H_{SB}^{Z})^{\prime}=0$, i.e., $\mathcal{G}_{d}^{X}$
is a decoupling group for qudit dephasing.

##  Appendix D Qutrit ($d=3$) dynamical decoupling and $3X_{3}$

For concreteness, we now illustrate the results above by giving the explicit
form of the HW subgroup $\mathcal{G}_{d}^{X}$ for the case of qutrit
dephasing. This is the simplest non-trivial example going beyond qubits.

The generators of the qutrit HWG are $X_{3}$ and $Z_{3}$, which can be seen as
generalizations of the Pauli matrices $\sigma_{x}$ and $\sigma_{z}$,
respectively. Setting $\omega\equiv\gamma_{3}=e^{2\pi i/3}$ (the cube root of
unity), the shift and phase operators are

$\displaystyle\Lambda_{10}=X_{3}=\begin{pmatrix}0&0&1\\\ 1&0&0\\\
0&1&0\end{pmatrix}\ ,\quad\Lambda_{01}=Z_{3}=\begin{pmatrix}1&0&0\\\
0&\omega&0\\\ 0&0&\omega^{2}\end{pmatrix}$ (32a)
$\displaystyle\Lambda_{20}=X_{3}^{2}=\begin{pmatrix}0&1&0\\\ 0&0&1\\\ 1&0&0\\\
\end{pmatrix}\ ,\quad\Lambda_{02}=Z_{3}^{2}=\begin{pmatrix}1&0&0\\\
0&\omega^{2}&0\\\ 0&0&\omega\end{pmatrix}.$ (32b)

Their action on the qutrit computational basis states $|m\rangle$ ($m=0,1,2$)
is $X_{3}|m\rangle=|(m+1)\bmod 3\rangle$ and
$Z_{3}|m\rangle=\omega^{m}|m\rangle$. Note that $O^{-1}=O^{\dagger}=O^{2}$ and
$O^{3}=I$ for $O=X_{3},Z_{3},X_{3}^{2}$, and $Z_{3}^{2}$.

Their commutation properties follow from Eq. 16:

$\displaystyle X_{3}Z_{3}=\omega^{2}Z_{3}X_{3},$ $\displaystyle\quad
X_{3}^{2}Z_{3}=\omega Z_{3}X_{3}^{2},$ (33a) $\displaystyle
X_{3}Z_{3}^{2}=\omega Z_{3}^{2}X_{3},$ $\displaystyle\quad
X_{3}^{2}Z_{3}^{2}=\omega^{2}Z_{3}^{2}X_{3}^{2}.$ (33b)

Since $X_{3}^{\dagger}=X_{3}^{2}$, we can write

$\displaystyle(X_{3}^{2})^{\dagger}Z_{3}X_{3}^{2}=\omega^{2}Z_{3},$
$\displaystyle\quad X_{3}^{\dagger}Z_{3}X_{3}=\omega Z_{3},$ (34a)
$\displaystyle(X_{3}^{2})^{\dagger}Z_{3}^{2}X_{3}^{2}=\omega Z_{3}^{2},$
$\displaystyle\quad X_{3}^{\dagger}Z_{3}^{2}X_{3}=\omega^{2}Z_{3}^{2},$ (34b)

which is a special case of Eq. 22. The decoupling group is
$\mathcal{G}_{3}^{X_{3}}=\\{I,X_{3},X_{3}^{2}\\}$, which suppresses dephasing
due to the system-bath interaction $H_{SB}^{Z}$ with $d=3$ [Eq. 29]. For
example, consider free evolution subject to a term of the form $Z_{3}\otimes
B$; writing out the DD sequence Eq. 5 explicitly, we have, for $T=3\tau$:

$\displaystyle U(T)$
$\displaystyle=\big{(}If_{\tau}I\big{)}\big{(}(X_{3}^{2})^{\dagger}f_{\tau}X_{3}^{2}\big{)}\big{(}X_{3}^{\dagger}f_{\tau}X_{3}\big{)}$
(35a) $\displaystyle=e^{-i\tau Z_{3}\otimes
B}e^{-i\tau(X_{3}^{2})^{\dagger}Z_{3}X_{3}^{2}\otimes B}e^{-i\tau
X_{3}^{\dagger}Z_{3}X_{3}\otimes B}$ (35b)
$\displaystyle=e^{-i\tau(Z_{3}\otimes B)}e^{-i\tau(\omega^{2}Z_{3}\otimes
B)}e^{-i\tau(\omega Z_{3}\otimes B)}$ (35c)
$\displaystyle=e^{-i\tau(1+\omega^{2}+\omega)Z_{3}\otimes
B}+\mathcal{O}(T^{2})$ (35d) $\displaystyle=I+\mathcal{O}(T^{2}),$ (35e)

i.e., suppression to second order of dephasing due to $Z_{3}\otimes B$.
Replacing $Z_{3}\otimes B$ with $Z_{3}^{2}\otimes B^{\prime}$ simply
rearranges the order of the roots of unity, yielding $1+\omega+\omega^{2}=0$
instead of $1+\omega^{2}+\omega=0$ in Eq. 35d, with the same outcome.

Note that since $(X_{3}^{2})^{\dagger}=X_{3}=X_{3}^{2}X_{3}^{\dagger}$, this
DD sequence in fact reduces to three equidistant pulses of type $X_{3}$:

$U(T)=X_{3}f_{\tau}X_{3}f_{\tau}X_{3}f_{\tau}$ (36)

which is the reason we called it $3X_{3}$ in the main text. We could have
equivalently used the sequence consisting of three $X_{3}^{2}$ pulses.

##  Appendix E Proof of first order suppression of cross-Kerr coupling by the
CKDD sequence

The goal of the CKDD sequence is to suppress the cross-Kerr coupling
Hamiltonian Eq. 3:

$H_{\rm CK}=\sum_{i,j=1}^{d-1}\alpha_{ij}|{i}\rangle\\!\langle
i|\otimes|{j}\rangle\\!\langle j|.$ (37)

Since the HWG is an operator basis, and in particular its phase elements
$\\{Z_{d}^{\alpha}\\}_{\alpha=0}^{d-1}$ are a basis for diagonal operators, we
can expand each diagonal term in $H_{\rm CK}$ as

$|{i}\rangle\\!\langle i|=\sum_{k=0}^{d-1}c_{ik}Z_{d}^{k},$ (38)

where

$c_{ik}=\frac{1}{d}\mathrm{Tr}[(Z_{d}^{\dagger})^{k}|{i}\rangle\\!\langle
i|]=\frac{1}{d}\gamma_{d}^{-ik}.$ (39)

This is a consequence of Eq. 10 and the zero-sum property:
$\mathrm{Tr}[(Z_{d}^{\dagger})^{k}Z_{d}^{l}]=\mathrm{Tr}(Z_{d}^{l-k})=\sum_{m=0}^{d-1}\gamma_{d}^{m(l-k)}=d\delta_{kl}$.
Thus, we can rewrite the cross-Kerr Hamiltonian as

$H_{\rm CK}=\sum_{k,l=0}^{d-1}\zeta_{kl}Z_{d}^{k}\otimes Z_{d}^{l},$ (40)

where

$\zeta_{kl}=\sum_{i,j=1}^{d-1}c_{ik}c_{jl}\alpha_{ij}=\frac{1}{d^{2}}\sum_{i,j=1}^{d-1}\gamma_{d}^{-(ik+jl)}\alpha_{ij}.$
(41)

The CKDD sequence $U_{d^{2}\tau}$ [Eq. 4] consists of an inner $X_{d}$-type
sequence applied to the first qudit and an outer $X_{d}$-type sequence applied
to the second qudit. Namely, $U_{d^{2}\tau}\equiv U^{(2)}_{d\tau}\circ
U^{(1)}_{d\tau}=(I_{d}\otimes X_{d})U^{(1)}_{d\tau}(I_{d}\otimes X_{d})\cdots
U^{(1)}_{d\tau}(I_{d}\otimes X_{d})U^{(1)}_{d\tau}$, where
$U^{(1)}_{d\tau}\equiv dX_{d}\otimes I_{d}$ and $U^{(2)}_{d\tau}\equiv
I_{d}\otimes dX_{d}$.

Generalizing from the single qudit dephasing case, the effect of the inner
$U^{(1)}_{d\tau}$ sequence is to project $H_{\rm CK}$ into the commutant of
$\mathcal{G}^{X}_{d}\otimes I_{d}$, i.e.,

$\displaystyle H^{\prime}_{\rm CK}$
$\displaystyle=\mathcal{P}_{\mathcal{G}^{X}_{d}\otimes I_{d}}(H_{\rm CK})$
(42a)
$\displaystyle=\frac{1}{d}\sum_{\alpha=0}^{d-1}X_{d}^{\alpha{\dagger}}\otimes
I_{d}\sum_{k,l=0}^{d-1}\zeta_{kl}Z_{d}^{k}X_{d}^{\alpha}\otimes Z_{d}^{l}$
(42b)
$\displaystyle=\frac{1}{d}\sum_{k,l=0}^{d-1}f_{k}\zeta_{kl}Z_{d}^{k}\otimes
Z_{d}^{l},$ (42c)

where $f_{k}=\sum_{\alpha=0}^{d-1}\gamma_{d}^{k\alpha}=d\delta_{k0}$, just as
in Eq. 31. Thus, the effect of the inner sequence is to leave just the
identity term on the first qudit:

$H^{\prime}_{\rm CK}=\sum_{l=0}^{d-1}\zeta_{0l}I_{d}\otimes Z_{d}^{l}.$ (43)

The outer sequence then removes the remaining dephasing terms:

$\displaystyle H^{\prime\prime}_{\rm CK}$
$\displaystyle=\mathcal{P}_{I_{d}\otimes\mathcal{G}^{X}_{d}}(H^{\prime}_{\rm
CK})$ (44a) $\displaystyle=\frac{1}{d}\sum_{\alpha=0}^{d-1}I_{d}\otimes
X_{d}^{\alpha{\dagger}}\sum_{l=0}^{d-1}\zeta_{0l}I_{d}\otimes
Z_{d}^{l}X_{d}^{\alpha}$ (44b)
$\displaystyle=\frac{1}{d}\sum_{l=0}^{d-1}f_{l}\zeta_{0l}I_{d}\otimes
Z_{d}^{l}=\zeta_{00}I_{d^{2}}.$ (44c)

Consequently, it follows from Eq. 5 that
$U_{d^{2}\tau}=e^{-i\zeta_{00}T}I_{d}\otimes I_{d}+\mathcal{O}(T^{2})$, where
$T=d^{2}\tau$ and $\zeta_{00}=\frac{1}{d^{2}}\sum_{i,j=1}^{d-1}\alpha_{ij}$.
This proves that the CKDD sequence achieves first-order decoupling of the
cross-Kerr interaction.

Note that applying simultaneous $X_{d}$-type sequences to both qudits does not
work. I.e., using $\mathcal{G}^{X}_{d}\otimes\mathcal{G}^{X}_{d}$ as the
decoupling group results instead of Eq. 42 in the projection

$\displaystyle H^{\prime}_{\rm CK}$
$\displaystyle=\mathcal{P}_{\mathcal{G}^{X}_{d}\otimes\mathcal{G}^{X}_{d}}(H_{\rm
CK})$ (45a)
$\displaystyle=\frac{1}{d^{2}}\sum_{\alpha=0}^{d-1}\sum_{k,l=0}^{d-1}\zeta_{kl}X_{d}^{\alpha{\dagger}}Z_{d}^{k}X_{d}^{\alpha}\otimes
X_{d}^{\alpha{\dagger}}Z_{d}^{l}X_{d}^{\alpha{\dagger}}$ (45b)
$\displaystyle=\frac{1}{d^{2}}\sum_{k,l=0}^{d-1}\zeta_{kl}Z_{d}^{k}\otimes
Z_{d}^{l}\sum_{\alpha=0}^{d-1}\gamma_{d}^{(k+l)\alpha},$ (45c)

and $\sum_{\alpha=0}^{d-1}\gamma_{d}^{(k+l)\alpha}=d\delta_{k+l,d}$, i.e.,
terms of the form $Z_{d}^{k}\otimes Z_{d}^{d-k}$ are not suppressed. This is a
generalization of the qubit case, where it is well known that simultaneous
$X$-type sequences do not cancel crosstalk [41].