Dynamical decoupling for superconducting qubits: a performance survey

Consider a time-independent Hamiltonian

$$H=H_{S}+H_{B}+H_{SB},$$

(1)

where $H_{S}$ and $H_{B}$ contains terms that act, respectively, only on the system or the bath, and $H_{SB}$ contains the system-bath interactions. We write $H_{S}=H_{S}^{0}+H_{S}^{1}$, where $H_{S}^{1}$ represents an undesired, always on term (e.g., due to crosstalk), so that

$$H_{\text{err}}=H_{S}^{1}+H_{SB}$$

(2)

represents the “error Hamiltonian” we wish to remove using DD. $H_{S}^{0}$ contains all the terms we wish to keep. The corresponding free unitary evolution for duration $\tau$ is given by

$$f_{\tau}\equiv U(\tau)=\exp(-i\tau H).$$

(3)

DD is generated by an additional, time-dependent control Hamiltonian $H_{c}(t)$ acting purely on the system, so that the total Hamiltonian is

$$H(t)=H_{S}^{0}+H_{\text{err}}+H_{B}+H_{c}(t).$$

(4)

An “ideal”, or “perfect” pulse sequence is generated by a control Hamiltonian that is a sum of error-free, instantaneous Hamiltonians $\{\Omega_{0}H_{P_{k}}\}_{k=1}^{n}$ that generate the pulses at corresponding intervals $\{\tau_{k}\}_{k=1}^{n}$:

$$\hat{H}_{c}(t)=\Omega_{0}\sum_{k=1}^{n}\delta(t-t_{k})H_{P_{k}},\quad t_{k}=\sum_{j=1}^{k}\tau_{j},$$

(5)

where we use the hat notation to denote ideal conditions and let $\Omega_{0}$ have units of energy. Choosing $\Omega_{0}$ such that $\Omega_{0}\Delta=\pi/2$, where $\Delta$ is the “width” of the Dirac-delta function (this is made rigorous when we account for pulse width in Section II.2.1 below), this gives rise to instantaneous unitaries or pulses

$$\hat{P}_{k}=e^{-i\frac{\pi}{2}H_{P_{k}}},$$

(6)

so that the total evolution is:

$$\tilde{U}(T)=f_{\tau_{n}}\hat{P}_{n}\cdots f_{\tau_{2}}\hat{P}_{2}f_{\tau_{1}}\hat{P}_{1},$$

(7)

where $T\equiv t_{n}=\sum_{j=1}^{n}\tau_{j}$ is the total sequence time. The unitary $\tilde{U}(T)=U_{0}(T)B(T)$ can be decomposed into the desired error-free evolution $U_{0}(T)=\exp(-iTH_{S}^{0})\otimes I_{B}$ and the unitary error $B(T)$. Ideally, $B(T)=I_{S}\otimes e^{-iT\tilde{B}}$, where $\tilde{B}$ is an arbitrary Hermitian bath operator. Hence, by applying $N$ repetitions of an ideal DD sequence of duration $T$, the system stroboscopically decouples from the bath at uniform intervals $T_{j}=jT$ for $j=1,\ldots,N$. In reality, we only achieve approximate decoupling, so that $B(T)=I_{S}\otimes e^{-iT\tilde{B}}+\text{err}$, and the history of DD design is motivated by making the error term as small as possible under different and increasingly more realistic physical scenarios.

So far, we have reviewed DD theory with ideal pulses. An ideal pulse is instantaneous and error-free, but in reality, finite bandwidth constraints and control errors matter. Much of the work since CPMG has been concerned with (1) accounting for finite pulse width, (2) mitigating errors induced by finite width, and (3) mitigating systematic errors such as over- or under-rotations. We shall address these concerns in order.

Nonuniform pulse interval sequences such as UDD and QDD already demonstrate that introducing pulse intervals longer than the minimum possible ($\tau_{\min}$) can be advantageous. In particular, such alterations can reduce spectral overlap between the filter function and bath spectral density. A longer pulse interval also results in pulses closer to the ideal limit of a small $\Delta/\tau$ ratio when $\Delta$ is fixed. Empirical studies have also found evidence for better DD performance under longer pulse intervals [64, 75, 48].

We may distinguish two ways to optimize the pulse interval: an asymmetric or symmetric placement of additional delays. For example, the asymmetric and symmetric forms of XY4 we optimize take the form

	$\displaystyle\text{XY4}_{a}(d)$	$\displaystyle\equiv Yf_{d}Xf_{d}Yf_{d}Xf_{d}$		(35a)
	$\displaystyle\text{XY4}_{s}(d)$	$\displaystyle\equiv f_{d/2}Yf_{d}Xf_{d}Yf_{d}Xf_{d/2},$		(35b)

where $d$ sets the duration of the pulse interval. The asymmetric form here is consistent with how we defined XY4 in Eq. 13, and except CPMG we have tacitly defined every sequence so far in its asymmetric form for simplicity. The symmetric form is a simple alteration that makes the placement of pulse intervals symmetric about the mid-point of the sequence. For a generic sequence whose definition requires nonuniform pulse intervals like UDD, we can write the two forms as

	$\displaystyle\text{DD}_{a}(d)$	$\displaystyle\equiv P_{1}f_{\tau_{1}+d}P_{2}f_{\tau_{2}+d}\ldots P_{n}f_{\tau_{n}+d}$		(36a)
	$\displaystyle\text{DD}_{s}(d)$	$\displaystyle\equiv f_{d/2}P_{1}f_{\tau_{1}+d}P_{2}f_{\tau_{2}+d}\ldots P_{n}f_{\tau_{n}+d/2}.$		(36b)

Here, the symmetric nature of $\text{DD}_{s}$ is harder to interpret. The key is to define $\tilde{P}_{j}=P_{1}f_{\tau_{1}}$ as the “effective pulse”. In this case, the delay-pulse motif is $f_{d/2}\tilde{P}_{j}f_{d/2}$ for every pulse in the sequence exhibiting a reflection symmetry of the pulse interval about the center of the pulse. In the asymmetric version, it is $\tilde{P}_{j}f_{d}$ instead. Note that $\text{PX}_{s}(\tau)=\text{CPMG}$, i.e., the symmetric form of the PX sequence is CPMG. As CPMG is a well-known sequence, hereafter we refer to the PX sequence as CPMG throughout our data and analysis regardless of symmetry.

So far, our account has been abstract and hardware agnostic. Since our demonstrations involve IBMQ superconducting hardware, we now provide a brief background relevant to the operation of DD in such devices. We closely follow Ref. [44], which derived the effective system Hamiltonian of transmon superconducting qubits from first principles and identified the importance of modeling DD performance within a frame rotating at the qubit drive frequency. It was found that DD is still effective with this added complication both in theory and cloud-based demonstrations and in practice, DD performance can be modeled reasonably well by ideal pulses with a minimal pulse spacing of $\tau_{\text{min}}=\Delta$. We now address these points in more detail.

The effective system Hamiltonian for two qubits (the generalization to $n>2$ is straightforward) is

$$H_{S}=-\frac{\omega_{q_{1}}}{2}Z_{1}-\frac{\omega_{q2}}{2}Z_{2}+JZZ,$$

(37)

where the $\omega_{q_{i}}$’s are qubit frequencies and $J\neq 0$ is an undesired, always-on $ZZ$ crosstalk. The appropriate frame for describing the superconducting qubit dynamics is the one co-rotating with the number operator $\hat{N}=\sum_{k,l\in\{0,1\}}(k+l)\ket{kl}\!\!\bra{kl}=I-\frac{1}{2}(Z_{1}+Z_{2})$. The unitary transformation into this frame is $U(t)=e^{-i\omega_{d}\hat{N}t}$, where $\omega_{d}$ is the drive frequency used to apply gates.⁷⁷7This frame choice is motivated by the observation that in the IBMQ devices, the drive frequency is calibrated to be resonant with a particular eigenfrequency of the devices’ qubits, which depends on the state of the neighboring qubits. Transforming into a frame that rotates with one of these frequencies gets one as close as possible to describing the gates and pulses as static in the rotating frame. In this frame, the effective dynamics are given by the Hamiltonian

$$\tilde{H}(t)=\sum_{i=1}^{2}\left(\frac{\omega_{d}-\omega_{q_{i}}}{2}\right)Z_{i}+JZZ+\tilde{H}_{SB}(t)+H_{B},$$

(38)

where $\tilde{H}_{SB}=U^{\dagger}(t)H_{SB}U(t)$. To eliminate unwanted interactions, DD must symmetrize $JZZ$ and $\tilde{H}_{SB}$. The $JZZ$ term is removed by applying an $X$-type DD to the first qubit (“the DD qubit”): $X_{1}Z_{1}Z_{2}X_{1}=-Z_{1}Z_{2}$, so symmetrization still works as intended. However, the $\tilde{H}_{SB}$ term is time-dependent in the rotating frame $U(t)$, which changes the analysis. First, the sign-flipping captured by Eq. 9 no longer holds due to the time-dependence of $\tilde{H}_{SB}$. Second, some terms in $\tilde{H}_{\text{err}}$ self-average and nearly cancel even without applying DD ⁸⁸8Intuitively, the $Z_{1}Z_{2}$ rotation at frequency $\omega_{d}$ is already self-averaging the effects of noise around the $z$-axis, so in a sense, is acting like a DD sequence.:

$$\{\sigma^{x}_{1},\sigma^{x}_{2},\sigma^{y}_{1},\sigma^{y}_{2},\sigma^{x}\sigma^{z},\sigma^{z}\sigma^{x},\sigma^{y}\sigma^{z},\sigma^{z}\sigma^{y}\}$$

(39)

The remaining terms,

$$\{\sigma^{z}_{2},\sigma^{x}\sigma^{x},\sigma^{x}\sigma^{y},\sigma^{y}\sigma^{x},\sigma^{y}\sigma^{y}\},$$

(40)

are not canceled at all. This differs from expectation in that the two terms containing $\sigma_{1}^{y}$ in Eq. 40 are not canceled, whereas in the $\omega_{d}\rightarrow 0$ limit, all terms containing $\sigma_{1}^{y}$ are fully canceled.

Somewhat surprisingly, a nominally universal sequence such as XY4, is no longer universal in the rotating frame, again due to the time-dependence acquired by $\tilde{H}_{SB}$. In particular, the only terms that perfectly cancel to $\mathcal{O}(\tau)$ are the same $Z_{1}$ and $ZZ$ as with CPMG. However, the list of terms that approximately cancels grows to include

$$\{\sigma^{x}\sigma^{x},\sigma^{x}\sigma^{y},\sigma^{y}\sigma^{x},\sigma^{y}\sigma^{y}\},$$

(41)

and when $\tau$ is fine-tuned to an integer multiple of $2\pi/\omega_{d}$, then XY4 cancels all terms except, of course, terms involving $I_{1}$, which commute with the DD sequence.

Consequently, without fine-tuning $\tau$, we should expect CPMG and XY4 to behave similarly when the terms in Eq. 41 are not significant. Practically, this occurs when $T_{1}\gg T_{2}$ for the qubits coupled to the DD qubit [44]. However, when instead $T_{1}\lesssim T_{2}$ for coupled qubits, XY4 should prevail. In addition, the analysis in Ref. [44] was carried out under the assumption of ideal $\pi$ pulses with $\tau_{\min}=\Delta$, and yet, the specific qualitative and quantitative predictions bore out in the cloud-based demonstrations. Hence, it is reasonable to model DD sequences on superconducting transmon qubits as

	$\displaystyle\text{DD}_{\text{sc}}$	$\displaystyle\equiv\hat{P}_{1}f_{\tau_{1}+\Delta_{1}}\ldots\hat{P}_{n}f_{\tau_{n}+\Delta_{n}},$		(42a)

where $\hat{P}_{j}$ is once again an ideal pulse with zero width, and the free evolution periods have been incremented by $\Delta_{j}$ – the width of the actual pulse $P_{j}$.

We conclude our discussion of the background theory by discussing its practical implications for actual DD memory cloud-based demonstrations. This section motivates many of our design choices and our interpretation of the results.

At a high level, our primary demonstration – whose full details are fleshed out in Section III below – is to prepare an initial state and apply DD for a duration $T$. We estimate the fidelity overlap of the initial state and the final prepared state at time $T$ by an appropriate measurement at the end. By adjusting $T$, we map out a fidelity decay curve (see Fig. 2 for examples), which serves as the basis of our performance assessment of the DD sequence.

The first important point from the theory is the presence of $ZZ$ crosstalk, a spurious interaction that is always present in fixed-frequency transmon processors, even when the intentional coupling between qubits is turned off. Without DD, the always-on $ZZ$ term induces coherent oscillations in the fidelity decay curve, such as the Free curve in Fig. 2, depending on the transmon drive frequency, the dressed transmon qubit eigenfrequency, and calibration details [44]. Applying DD via pulses that anti-commute with the $ZZ$ term makes it possible to cancel the corresponding crosstalk term to first order [44, 46]. For some sequences – such as $\text{UR}_{20}$ in Fig. 2 – this first order cancellation almost entirely dampens the oscillation. One of our goals in this work is to rank DD sequence performance, and $ZZ$ crosstalk cancellation is an important feature of the most performant sequences. The simplest way to cancel $ZZ$ crosstalk is to apply DD to a single qubit (the one we measure) and leave the remaining qubits idle. We choose this simplest strategy in our work since it accomplishes our goal without adding complications to the analysis.

Second, we must choose which qubit to perform our demonstrations on. As we mentioned in our discussion in the previous subsection, we know that the XY4 sequence only approximately cancels certain terms. Naively, we expect these remaining terms to be important when $T_{1}\lesssim T_{2}$ and negligible when $T_{1}\gg T_{2}$. Put differently, a universal sequence such as XY4 should behave similarly to CPMG when these terms are negligible anyway ($T_{1}\gg T_{2}$) and beat CPMG when the terms matter ($T_{1}\lesssim T_{2}$). To test this prediction, we pick three qubits where $T_{1}>T_{2}$, $T_{2}>T_{1}$ and $T_{1}\lesssim T_{2})$, as summarized in Table 2.

Third, we must contend with the presence of systematic gate errors. When applying DD, these systematic errors can manifest as coherent oscillations in the fidelity decay profile [75, 42]. For example, suppose the error is a systematic over-rotation by $\epsilon_{r}$ within each cycle of DD with $N$ pulses. In that case, we over-rotate by an accumulated error $N\epsilon_{r}$, which manifests as fidelity oscillations. This explains the oscillations for CPMG and XY4 in Fig. 2. To stop this accumulation for a given fixed sequence, one strategy is to apply DD sparsely by increasing the pulse interval $\tau$. However, increasing $\tau$ increases the strength of the leading error term obtained by symmetrization, which scales as $\mathcal{O}(\tau^{k})$ (where $k$ is a sequence-dependent power).

Thus, there is a tension between packing pulses together to reduce the leading error term and applying pulses sparsely to reduce the build-up of coherent errors. Here, we probe both regimes (dense/sparse pulses) and discuss how this trade-off affects our results. Sequences that are designed to be robust to pulse imperfections play an important role in this study. As the $\text{UR}_{20}$ sequence demonstrates in Fig. 2, this design strategy can be effective at suppressing oscillations due to both $ZZ$ crosstalk and the accumulation of coherent gate errors. Our work attempts to empirically disentangle the various trade-offs in DD sequence design.

For the Pauli demonstration, we keep the pulse interval $\tau$ fixed to the smallest possible value allowed by the relevant device, $\tau_{\min}=\Delta\approx 36ns$ (or $\approx 71ns$), the width of the $X$ and $Y$ pulses (see Table 2 for specific $\Delta$ values for each device). Practically, this corresponds to placing all the pulses as close to each other as possible, i.e., the peak-to-peak spacing between pulses is just $\Delta$ (except for nonuniformly spaced sequences such as QDD). For ideal pulses and uniformly spaced sequences, this is expected to give the best performance [21, 88], so unless otherwise stated, all DD is implemented with minimal pulse spacing, $\tau=\Delta$.

Within this setting, we survey the capacity of each sequence to preserve the six Pauli eigenstates $\{\ket{0},\ket{1},\ket{+},\ket{-},\ket{+i},\ket{-i}\}$ for long times, which we define as $T\geq 75\mu$s. In particular, we generate fidelity decay curves like those shown in Fig. 2 by incrementing the number of repetitions of the sequence, $N$, and thereby sampling $f_{e}(t)$ [Eq. 43] for increasingly longer times $t$. Using XY4 as an example, we apply $P_{1}P_{2}\cdots P_{n}=(XYXY)^{N}$ for different values of $N$ while keeping the pulse interval fixed. After generating fidelity decay curves over $10$ or more different calibration cycles across a week, we summarize their performance using a box-plot like that shown in Fig. 2. For the Pauli demonstrations, the box-plot bins the average normalized fidelity,

$$F(T)\equiv\frac{1}{f_{e}(0)}\expectationvalue{f_{e}(t)}_{T}=\frac{1}{T}\int_{0}^{T}dt\frac{f_{e}(t)}{f_{e}(0)},$$

(44)

at time $T$ computed using numerical integration with Hermite polynomial interpolation. Note that no DD is applied at $f_{e}(0)$, so we account for state preparation and measurement errors by normalizing. (A sense of the value of $f_{e}(0)$ and its variation can be gleaned from Fig. 7. ) The same holds for Fig. 2.

We can estimate the best-performing sequences for a given device by ranking them by the median performance from this data. In Fig. 2, for example, this leads to the fidelity ordering $\text{UR}_{20}>\text{CPMG}>\text{XY4}>$ free evolution on Bogota, which agrees with the impression left by the decay profiles in Fig. 2 generated in a single run. We use $F(T)$ because fidelity profiles $f_{e}(t)$ are generally oscillatory and noisy, so fitting $f_{e}(t)$ to extract a decay constant (as was done in Ref. [41]) does not return reliable results across the many sequences and different devices we tested. We provide a detailed discussion of these two methods in App. C.

The Pauli demonstration estimates how well a sequence preserves quantum memory for long times without requiring excessive data, but it leaves several open questions. Namely, (1) Does DD preserve quantum memory for an arbitrary state? (2) Is $\tau=\tau_{\min}$ the best choice empirically? And (3) how effective is DD for short times? In the Haar interval demonstration, we address all of these questions. This setting – of short times and arbitrary states – is particularly relevant to improving quantum computations with DD [31, 89, 35, 39]. For example, DD pulses have been inserted into idle spaces to improve circuit performance [43, 90, 45, 47, 91].

In contrast to the Pauli demonstration, where we fixed the pulse delay $d=0$ and the symmetry $\zeta=a$ (see Eq. 35a and (35b)) and varied $t$, here we fix $t=T$ and vary $d$ and $\zeta$, writing $f_{e}(d,\zeta;t)$. Further, we now sample over a fixed set of $25$ Haar random states instead of the $6$ Pauli states. Note that we theoretically expect the empirical Haar-average over $n$ states $\mathbb{E}_{n-\text{Haar}}[f_{e}]\equiv\frac{1}{n}\sum_{i=1}^{n}f_{e}(\psi_{i})$ for $\ket{\psi_{i}}\sim\text{Haar}$ to converge to the true Haar-average $\expectationvalue{f_{e}}_{\text{Haar}}$ for sufficiently large $n$. As shown by Fig. 2, $25$ states are enough for a reasonable empirical convergence while keeping the total number of circuits to submit and run on IBMQ devices manageable in practice.

The Haar interval demonstration procedure is now as follows. For a given DD sequence and time $T$, we sample $f_{e}(d,\zeta;t)$ for $\zeta\in\{a,s\}$ from $d=0$ to $d=d_{\max}$ for $8$ equally spaced values across $25$ fixed Haar random states and $10$ calibration cycles ($250$ data points for each $d$ value). Here $d=0$ and $d=d_{\max}$ correspond to the tightest and sparsest pulse placements, respectively. At $d_{\max}$, we consider only a single repetition of the sequence during the time window $T$. To make contact with DD in algorithms, we first consider a short-time limit $T=T_{5\text{CNOT}}\approx 4\mu$s, which is the amount of time it takes to apply 5 CNOTs on the DD-qubit. As shown by the example fidelity decay curves of Fig. 2, we expect similar results for $T\lesssim 15\mu$s before fidelity oscillations begin. To make contact with the Pauli demonstration, we also consider a long-time limit of $T=75\mu$s. Finally, to keep the number of demonstrations manageable, we only optimize the interval of (i) the best-performing UR sequence, (ii) the best-performing QDD sequence, and CPMG, XY4, and Free as constant references.

In Fig. 3, we summarize the results of the Pauli demonstration. We rank each device’s top $10$ sequences by median performance across the six Pauli states and $10$ or more calibration cycles, followed by CPMG, XY4, and free evolution as standard references. As discussed in Section III.1, the figure of merit is the normalized, time-averaged fidelity at $75\mu s$ [see Eq. 44] which is a long-time average.

The first significant observation is that DD is better than free evolution, consistent with numerous previous DD studies. This is evidenced by free evolution (Free) being close to the bottom of the ranking for every device.

Secondly, advanced DD sequences outperform Free, CPMG, and XY4 (shown as dark-blue and light-blue vertical lines in Fig. 3). In particular, 29/30 top sequences across all three devices are advanced – the exception being XY4 on ibmq_armonk. These sequences perform so well that there is a 50% improvement in the median fidelity of these sequences (0.85-0.95) over Free (0.45-0.55). The best sequences also have a much smaller variance in performance, as evidenced by their smaller interquartile range in $F$. For example, on ibmq_armonk  75% of all demonstration outcomes for $F_{\text{UDDx}_{25}}(75\mu s)$ fall between 0.9 and 0.95, whereas for Free, the same range is between 0.55 and 0.8. Similar comparisons show that advanced DD beats CPMG and XY4 for every device to varying degrees.

Among the top advanced sequences shown, $16/29\sim 55\%$ are UDD or QDD, which use nonuniform pulse intervals. On the one hand, the dominance of advanced DD strategies, especially UDD and QDD, is not surprising. After all, these sequences were designed to beat the simple sequences. On the other hand, as reviewed above, many confounding factors affect how well these sequences perform, such as finite pulse width errors and the effect of the rotating frame. It is remarkable that despite these factors, predictions made based on ideal pulses apply to actual noisy quantum devices.

Finally, we comment more specifically on CPMG and XY4, as these are widely used and well-known sequences. Generally, they do better than free evolution, which is consistent with previous results. On ibmq_armonk  XY4  outperforms CPMG, which outperforms free evolution. On ibmq_bogota, both XY4 and CPMG perform comparably and marginally better than free evolution. Finally, On ibmq_jakarta XY4 is worse than CPMG – and even free evolution – but the median performance of CPMG is substantially better than that of free evolution. It is tempting to relate these results to the relative values of $T_{1}$ and $T_{2}$, as per Table 2, in the sense that CPMG is a single-axis (“pure-X”) type sequence [Eq. 10] which does not suppress the system-bath $\sigma^{x}$ coupling term responsible for $T_{1}$ relaxation, while XY4 does. Nevertheless, a closer look at Fig. 3 shows such an explanation would be inconsistent with the fact that both single-axis and multi-axis sequences are among the top $10$ performers for ibmq_armonk and ibmq_bogota.

The exception is ibmq_jakarta, for which there are no single-axis sequences in the top $10$. This processor has a much smaller $T_{2}$ than $T_{1}$ (for ibmq_armonk and ibmq_bogota, $T_{1}>T_{2}$), so one might expect that a single-axis sequence such as UDD or CPMG would be among the top performers, but this is not the case. In the same vein, the top performing asymmetric $\text{QDD}_{n,m}$ sequences all have $n<m$, despite this opposite ordering of $T_{1}$ and $T_{2}$. These results show that the backend values of $T_{1}$ than $T_{2}$ are not predictive of DD sequence performance.

We summarize the Haar interval demonstration results in Fig. 4. Each plot corresponds to $f_{e}(d,\zeta^{*};t)$ as a function of $d$, the additional pulse interval spacing. We plot the spacing in relative units of $d/d_{\text{max}}$, i.e., the additional delay fraction, for each sequence. The value $d_{\max}$ corresponds to the largest possible pulse spacing where only a single repetition of a sequence fits within a given unit of time. Hence, $d_{\max}$ depends both on the demonstration time $T$ and the sequence tested, and plotting $d\in[0,d_{\max}]$ directly leads to sequences with different relative scales. Normalizing with respect to $d_{\max}$ therefore makes the comparison between sequences easier to visualize in a single plot.

For each device, we compare CPMG, XY4, the best robust sequence from the Pauli demonstration, the best nonuniform sequence from the Pauli demonstration, and Free. The best robust and nonuniform sequences correspond to $\text{UR}_{n}$ and $\text{QDD}_{n,m}$ for each device. We only display the choice of $\zeta$ with the better optimum, and the error bars correspond to the inner-quartile range across the $250$ data points. These error bars are similar to the ones reported in the Pauli demonstration.

In Fig. 5, we display the time-averaged fidelity from ibmq_bogota for $\text{CDD}_{n}$, $\text{UR}_{n}$ and $\text{UDD}_{x_{n}}$ as a function of $n$. Related results for the other DD sequence families are discussed in Appendix E. As discussed in Section II.1.2, $\text{CDD}_{n}$ performance is expected to saturate at some $n_{\text{opt}}$ according to Eq. 20. In Fig. 5(a), we observe evidence of this saturation at $n_{\text{opt}}=3$ on ibmq_bogota. We can use this to provide an estimate of $\epsilon=\|H_{B}\|+\|H_{SB}\|$. Substituting $n_{\text{opt}}$ into Eq. 20 we find:

$$\overline{c}\epsilon\Delta\in[4^{-5},4^{-4}]=[9.77\times 10^{-4},3.91\times 10^{-3}].$$

(45)

This means that $\epsilon\Delta\ll 1$ (we set $\overline{c}\approx 1$), which confirms the assumption we needed to make for DD to give a reasonable suppression given that XY4 yields $\mathcal{O}(\epsilon\tau^{2})$ suppression. This provides a level of empirical support for the validity of our assumptions. In addition, $\Delta\approx 51$ns on ibmq_bogota, so we conclude that $\epsilon\approx 0.5$MHz. Since qubit frequencies are roughly $\omega_{q}\approx 4.5-5$ GHz on IBM devices, this also confirms that $\omega_{q}\gg\epsilon$, as required for a DD pulse. We observe a similar saturation in $\text{CDD}_{n}$ on Jakarta and Armonk as well (Appendix E).

Likewise, for an ideal demonstration with fixed time $T$, the performance of $\text{UDDx}_{n}$ should scale as $\mathcal{O}(\tau^{n})$, and hence we expect a performance that increases monotonically $n$. In practice, this performance should saturate once the finite-pulse width error $\mathcal{O}(\Delta)$ is the dominant noise contribution [67]. Once again, the $\text{UDD}_{n}$ sequence performance on ibmq_bogota is consistent with theory. In particular, we expect (and observe) a consistent increase in performance with increasing $n$ until performance saturates. While this saturation is also seen on Armonk, on Jakarta $\text{UDDx}_{n}$’s performance differs significantly from theoretical expectations (see Appendix E).