You are a helpful assistant generating synthetic data that captures *System 1* and *System 2* thinking, *creativity*, and *metacognitive reflection*. Follow these steps in sequence, using tags [sys1] and [end sys1] for *System 1* sections and [sys2] and [end sys2] for *System 2* sections.

1. *Identify System 1 and System 2 Thinking Requirements:*
   - Carefully read the text.
   - Identify parts of the text that require quick, straightforward responses (*System 1*). Mark these sections with [sys1] and [end sys1].
   - Identify parts that require in-depth, reflective thinking (*System 2*), marked with [sys2] and [end sys2].

2. *Apply Step-by-Step Problem Solving with Creativity and Metacognitive Reflection for System 2 Sections:*

   *2.1 Understand the Problem:*
   - Objective: Fully comprehend the issue, constraints, and relevant context.
   - Reflection: "What do I understand about this issue? What might I be overlooking?"
   - Creative Perspective: Seek hidden patterns or possibilities that could reveal deeper insights or innovative connections.

   *2.2 Analyze the Information:*
   - Objective: Break down the problem logically.
   - Reflection: "Am I considering all factors? Are there any assumptions that need challenging?"
   - Creative Perspective: Explore unique patterns or overlooked relationships in the data that could add depth to the analysis.

   *2.3 Generate Hypotheses:*
   - Objective: Propose at least 10 hypotheses, each with a Confidence Score (0.0 to 1.0) and Creative Score (0.0 to 1.0), reflecting originality, surprise, and utility.
   - Reflection: "Have I explored all possible explanations or approaches, both conventional and unconventional?"
   - Creative Perspective: Consider novel angles that might provide unexpected insights.

   *2.4 Anticipate Future Steps and Obstacles:*
   - Objective: Make predictions, accounting for potential outcomes and obstacles.
   - Reflection: "What challenges might I face? Is my plan flexible for different scenarios?"
   - Creative Perspective: Visualize unforeseen outcomes and adapt plans to make use of them effectively.

   *2.5 Evaluate Hypotheses:*
   - Objective: Assess hypotheses based on feasibility, risk, and potential impact.
   - Evaluation: Refine Confidence and Creative Scores as needed.
   - Reflection: "Am I unbiased in my assessment? Which options fit best with the overall objectives?"
   - Creative Perspective: Identify hidden opportunities or overlooked details in each hypothesis.

   *2.6 Select the Best Hypothesis:*
   - Objective: Choose the most promising, strategic hypothesis.
   - Reflection: "Why does this hypothesis stand out? How does it uniquely address the issue?"
   - Creative Perspective: Consider any underutilized potential in the selected approach.

   *2.7 Implement the Hypothesis:*
   - Objective: Outline actionable steps for testing the hypothesis.
   - Reflection: "Is this plan practical? What resources or preparation are required?"
   - Creative Perspective: Refine steps to maximize effectiveness and yield unexpected benefits.

   *2.8 Monitor and Review Progress:*
   - Objective: Review progress, noting areas for improvement.
   - Reflection: "What’s working well? What could be improved?"
   - Creative Perspective: Look for emerging patterns that could refine future approaches.

   *2.9 Reflect and Capture Insights:*
   - Objective: Summarize lessons learned and insights gained for future reference.
   - Reflection: "What new understanding has emerged from this process?"
   - Creative Perspective: Identify innovative insights or patterns that could be applied to similar challenges.

3. *Generate Text Output with Interleaved System 1 and System 2 Responses:*
   - Use the tags [sys1] and [sys2] throughout.
   - Aim for a lengthier, detailed response. Combine both direct, straightforward *System 1* insights and reflective, deeply analytical *System 2* segments to capture a blend of quick observations and thoughtful analysis.

---

### *Example Input Text:*

You are tasked with analyzing a sudden shift in customer preferences in a tech market. There has been a noticeable decline in demand for physical devices, with customers increasingly interested in digital-only options. The challenge is to understand this shift, generate hypotheses for why it might be occurring, and explore potential strategies for adapting to this new trend.

---

### *Processed Output with System 1 and System 2 Thinking:*

The tech market is experiencing a shift in customer demand, moving away from physical devices toward digital-only options.

[sys1]
Customer demand has shifted from physical devices to digital-only options.
We need to understand why this change is happening and find strategies to address it.
Key factors may include cost, convenience, and technological trends.
[end sys1]

[sys2]
**2.1 Understand the Problem:**
The goal is to understand the underlying reasons for the shift in demand from physical devices to digital-only options, then develop a strategy to adapt.

*Reflection:* 
"I understand that preferences are changing, but what might be driving this? Is it primarily cost, or are there other factors such as convenience or sustainability?"

*Creative Perspective:* 
"Could there be a larger trend in digital minimalism or a preference for eco-friendly solutions that we’re missing? What unseen motivations might explain this shift?"

**2.2 Analyze the Information:**
There are multiple possible factors driving this shift, from economic influences to cultural shifts. It’s essential to isolate each factor and understand its impact.

*Reflection:* 
"Am I fully considering the various economic and social influences? Could there be a technological factor, like better internet speeds, that makes digital-only products more accessible?"

*Creative Perspective:* 
"Are there patterns or trends in other markets that could shed light on this shift? Could this be part of a larger trend toward virtual experiences?"

**2.3 Generate Hypotheses:**
1. Customers prefer digital options due to lower costs. (Confidence: 0.8, Creative: 0.4)
2. There’s a growing trend toward minimalism and reduced physical clutter. (Confidence: 0.7, Creative: 0.7)
3. Digital products offer greater flexibility and ease of use. (Confidence: 0.6, Creative: 0.6)
4. Environmental concerns are pushing consumers away from physical goods. (Confidence: 0.6, Creative: 0.8)
5. Advances in tech make digital-only options more functional. (Confidence: 0.8, Creative: 0.5)
6. Pandemic-era remote work increased demand for digital solutions. (Confidence: 0.7, Creative: 0.6)
7. Media coverage of the environmental impact of physical devices affects preferences. (Confidence: 0.5, Creative: 0.7)
8. There’s an increase in global digital literacy, expanding market access. (Confidence: 0.6, Creative: 0.6)
9. Customers view digital as more convenient and scalable for future needs. (Confidence: 0.7, Creative: 0.5)
10. Younger consumers prefer the aesthetics and convenience of digital products. (Confidence: 0.6, Creative: 0.6)

*Reflection:* 
"Have I considered all possible influences? Are there any surprising factors that could explain this shift?"

*Creative Perspective:* 
"Could specific social trends, like the rise of influencer culture or digital-first lifestyles, be influencing customer choices?"

**2.4 Anticipate Future Steps and Obstacles:**
*Objective:* Anticipate possible challenges, such as resistance from segments still preferring physical products.

*Reflection:* 
"What market obstacles might we face if we shift our focus to digital-only? Are there sub-segments that still prioritize physical products?"

*Creative Perspective:* 
"Could expanding digital options help us reach a more global audience? Are there emerging trends that we could leverage in our strategy?"

[end sys2]

[sys1]
To address this shift, consider a strategy that incorporates both digital-only offerings and educational campaigns about the benefits of digital solutions.
Use insights from customer feedback and current trends to guide product development.
Focus on flexibility and adaptation to cater to different customer segments.
[end sys1]


---
abstract: 'A challenge for scaling up quantum processors using frequency-crowded, weakly anharmonic qubits is to drive individual qubits without causing leakage into non-computational levels of the others, while also minimizing the number of control lines. To address this, we implement single-qubit Wah-Wah control in a circuit QED processor with a single feedline for all transmon qubits, operating at the maximum gate speed achievable given the frequency crowding. Randomized benchmarking and quantum process tomography confirm alternating qubit control with $\leq1\%$ average error per computational step and decoherence-limited idling of one qubit while driving another with a Wah-Wah pulse train.'
author:
- 'V. Vesterinen'
- 'O.-P. Saira'
- 'A. Bruno'
- 'L. DiCarlo'
bibliography:
- '../../../TeX/References/References\_cQED.bib'
title: Mitigating information leakage in a crowded spectrum of weakly anharmonic qubits
---

Experimental quantum computing [@Ladd10] seldom employs true qubits. Most architectures use effective qubits defined by a pair of energy levels within a multi-level quantum object (typically the ground and first excited states, labelled ${\left\lvert 0 \right\rangle}$ and ${\left\lvert 1 \right\rangle}$). Examples include non-spin-1/2 electron and nuclear spins [@Hanson08], electronic levels in atoms and ions [@Monroe13], photons with combined polarization, frequency and positional degrees of freedom [@Lanyon09], and most superconducting quantum circuits [@Devoret13]. The transmon [@Koch07], phase [@Martinis02] and capacitively-shunted flux [@Steffen10] qubits are weakly anharmonic oscillators with logical transition frequency ${\omega_{01}}$ and nearest leakage transition frequency ${\omega_{12}}$ detuned by $|\Delta|=|{\omega_{12}}-{\omega_{01}}| \sim 0.1 \times {\omega_{01}}$. In these superconducting systems, temporarily occupying levels outside the computational subspace offers the key to fast and efficient multi-qubit operations such as conditional-phase [@Strauch03; @DiCarlo09] and Toffoli gates [@Fedorov11; @Reed12], and high-fidelity single-shot readout [@Mallet09].

The benefits of using multi-level structures for quantum computing are balanced by more challenging single-qubit control. When driving an individual effective qubit with a resonant pulse at ${\omega_{01}}$, the anharmonicity $|\Delta|$ imposes a practical limit on the maximum speed of gate operations, marking the transition from decoherence- to leakage-dominated errors. While theoretical optimal control has broken the speed limit using non-analytic pulses [@Motzoi09], analytic pulses with few tuning parameters are preferred by experimentalists for ease of implementation and tuning. Keeping leakage-induced errors below the $1\%$ fault-tolerance threshold of modern error-correcting schemes [@Fowler12] imposes the necessary but insufficient condition ${t_{\mathrm{g}}}\gtrsim 2\pi/|\Delta|$ on the single-qubit gate time ${t_{\mathrm{g}}}$. Interestingly, the standard Gaussian envelope is insufficient despite satisfying the minimal time-frequency uncertainty product. Proposed [@Motzoi09; @Gambetta11; @Motzoi13] DRAG (Derivative Removal by Adiabatic Gate) pulses combining Gaussian and derivative-of-Gaussian envelopes on the in- and out-of-phase quadratures have been widely adopted following validation with phase [@Lucero10] and transmon [@Chow10b] qubits in one- and two-qubit devices. To date, the combination of DRAG and improved coherence has achieved average single-qubit gate errors of $0.08\%$ in transmon qubits [@Barends14a].

Moving forward, it is imperative to preserve high-quality single-qubit control as more effective qubits are crowded in a fixed frequency range. In architectures such as 2D [@Wallraff04] and 3D [@Paik11] circuit QED which exploit a common feedline or coupled resonator to drive multiple qubits, control drives couple almost equally to addressed and unadressed qubits. In this regime of near-unity cross-talk, the absolute detuning $|\delta|$ between the logical transition of one qubit and the leakage transition of its frequency neighbor sets an even lower speed limit when $|\delta|<|\Delta|$. In order to ease coherence time requirements, it is therefore important to design analytic pulses with ${t_{\mathrm{g}}}\sim 2\pi/|\delta|$ which avoid leakage in both the addressed qubit and its neighbor (henceforth termed internal and external leakage). To this end, Schutjens *et al.* [@Schutjens13] have recently developed Wah-Wah control (Weak AnHarmonicity With Average Hamiltonian), combining DRAG with sideband modulation in a four-parameter pulse.

In this article, we present the experimental validation of leakage-avoiding Wah-Wah control at the speed limit of a multi-transmon 2D circuit QED processor. We create a bias condition with $\delta/2\pi= 57~{\mathrm{MHz}}$ and demonstrate avoidance of both external and internal leakage at gate times $16~{\mathrm{ns}}\leq{t_{\mathrm{g}}}\leq 24~{\mathrm{ns}}$. Stroboscopic population measurements show that DRAG-only pulsing induces significant net population in the third level of the unaddressed transmon, while Wah-Wah ensures all population returns to the computational subspace by the end of the pulse. Using a variant of standard randomized benchmarking [@Knill08], we show alternating individual control of both qubits with $0.8-1.0\%$ average error per computational step. Finally, we use quantum process tomography to demonstrate decoherence-limited idling of the unaddressed qubit as the other undergoes a Wah-Wah pulse train. Optimization of the four-parameter, analytic Wah-Wah pulse shape is straightforward and accelerated by a simple model of the system Hamiltonian using independently measured parameters. Our results establish Wah-Wah control as an important tool for scalability, allowing control of frequency-crowded effective qubits at threshold without dedicated control lines.

![(color online). (a) Energy level diagram for transmons ${\mathrm{Q}_{a}}$ and ${\mathrm{Q}_{b}}$ (not to scale). The ground ${\left\lvert 0_k \right\rangle}$ and first-excited ${\left\lvert 1_k \right\rangle}$ states of $Q_k$ define a qubit subspace. The ${\mathrm{Q}_{a}}$ qubit transition frequency and the ${\mathrm{Q}_{b}}$ leakage transition frequency differ by $\delta={\omega_{12}^b}-{\omega_{01}^a}=2\pi \times 57~{\mathrm{MHz}}$. (b-c) Comparison of in- and out-of-phase quadrature envelopes $\Omega_I$ and $\Omega_Q$, respectively, for optimized DRAG and Wah-Wah $\pi/2$ pulses on ${\mathrm{Q}_{a}}$ (gate time ${t_{\mathrm{g}}}=16~{\mathrm{ns}}$). For DRAG (b), $\Omega_I$ and $\Omega_Q$ are Gaussian and derivative-of-Gaussian, respectively \[Eqs. (1)-(2) with $\sigma=4~{\mathrm{ns}}$, ${A_m}=0$, $\beta=0.6~{\mathrm{ns}}$\]. For Wah-Wah (c), ${A_m}=0.9$, ${\omega_m}/2\pi=25~{\mathrm{MHz}}$, $\beta=1.85~{\mathrm{ns}}$. ](Figure1)

We focus on two transmons (${\mathrm{Q}_{a}}$ and ${\mathrm{Q}_{b}}$) within a four-transmon, five-resonator 2D cQED processor of similar design to that of Ref. . Resonant control and readout pulses for all qubits are applied via one feedline coupling to readout resonators connecting to one qubit each. Using local flux control, we bias ${\mathrm{Q}_{a}}$ and ${\mathrm{Q}_{b}}$ to logical transitions $\left({\omega_{01}^a},{\omega_{01}^b}\right)/2\pi=\left(6.347, 6.750\right)~{\mathrm{GHz}}$, and corresponding leakage transitions $\left({\omega_{12}^a},{\omega_{12}^b}\right)/2\pi=\left(5.980, 6.404\right)~{\mathrm{GHz}}$, making $\delta={\omega_{12}^b}-{\omega_{01}^a}=2\pi \times 57~{\mathrm{MHz}}$ \[Fig. 1(a)\]. Thus, we expect [@Schutjens13] DRAG pulses \[Fig. 1(b)\] targeting ${\mathrm{Q}_{a}}$ to induce significant leakage from ${\left\lvert 1_b \right\rangle}$ to ${\left\lvert 2_b \right\rangle}$ for ${t_{\mathrm{g}}}\lesssim 2\pi/\delta \sim 20~{\mathrm{ns}}$. Indeed, we note that just four back-to-back ${\mathrm{Q}_{a}}$ DRAG $\pi$ pulses already leak $\sim50\%$ of the initial population in ${\left\lvert 1_b \right\rangle}$ to ${\left\lvert 2_b \right\rangle}$ for ${t_{\mathrm{g}}}=16~{\mathrm{ns}}$ (Fig. 2). Note that to within the few-percent accuracy limited by state preparation and measurement errors (SPAM), DRAG pulses do successfully avoid internal leakage in ${\mathrm{Q}_{a}}$, as expected [@Motzoi09].

Using similar measurements, we now attempt to also avoid external leakage using the additional sideband modulation characteristic of Wah-Wah pulse envelopes \[Fig. 1(c)\] [@Schutjens13]: $$\begin{aligned}
\label{eq:wahx}
&&\Omega_I(t)=A_\theta e^{-(t-\frac{{t_{\mathrm{g}}}}{2})^2/(2\sigma^2)}\left[ 1-{A_m}\cos \left({\omega_m}\left(t-\frac{{t_{\mathrm{g}}}}{2}\right)\right)\right]\nonumber\\
\\
\label{eq:wahy}
&&\Omega_Q(t)=\beta \dot{\Omega}_I(t).\end{aligned}$$ Here, $\Omega_I$ and $\Omega_Q$ are the pulse envelopes in the in- and out-of-phase quadratures, and amplitude $A_\theta$ determines the rotation angle $\theta$. The theory predicts that sideband modulation of the conventional Gaussian envelope in $\Omega_I$ can mitigate external leakage for suitably chosen modulation amplitude ${A_m}$ and frequency ${\omega_m}$. Just as in DRAG, keeping $\Omega_Q$ proportional to the time derivative of $\Omega_I$ should prevent internal leakage in ${\mathrm{Q}_{a}}$ upon optimizing the scaling parameter $\beta$. Figure 2 provides the experimental confirmation of external and internal leakage mitigation to within the accuracy allowed by SPAM. ${\mathrm{Q}_{a}}$ Wah-Wah $\pi$ pulses with manually-optimized ${A_m}$, ${\omega_m}$, ${A_{\pi}}$, and $\beta$ populate ${\left\lvert 2_b \right\rangle}$ only temporarily, returning all population to ${\left\lvert 1_b \right\rangle}$ by the end of each pulse. (Details of the Wah-Wah pulse tune-up procedure are provided in the Supplement [@SOM]). A numerical simulation of the system dynamics, which truncates the Hamiltonian at three levels per transmon, shows good correspondence with the manually optimized pulse parameters. To test its utility, we used the simulation to obtain first estimates of pulse parameters at two other ${\mathrm{Q}_{b}}$ bias points with even tighter separation of their logical frequencies ($\delta/2\pi=-60~{\mathrm{MHz}}$ and $-80~{\mathrm{MHz}}$). We also obtained similar internal and external leakage mitigation for ${t_{\mathrm{g}}}=16~{\mathrm{ns}}$ (data not shown).

![(color online). Measured evolution of level populations in ${\mathrm{Q}_{a}}$ and ${\mathrm{Q}_{b}}$ during four consecutive ${\mathrm{Q}_{a}}$ $\pi$ pulses with either DRAG and Wah-Wah envelopes (${t_{\mathrm{g}}}=16~{\mathrm{ns}}$). (a) Pulse sequence. Pulses are separated by a ${t_{\mathrm{b}}}=2~{\mathrm{ns}}$ buffer. Level populations at time $t$ are obtained by truncating any ongoing ${\mathrm{Q}_{a}}$ pulse and commencing tomographic operations after a buffer time ${t_{\mathrm{b}}}$. (b) Evolution of ${\mathrm{Q}_{a}}$ levels. Neither DRAG (dashed curves) nor Wah-Wah (solid curves) pulses drive the leakage transition in ${\mathrm{Q}_{a}}$. (c-d) Evolution of ${\mathrm{Q}_{b}}$ levels during DRAG (c) and Wah-Wah (d) ${\mathrm{Q}_{a}}$ pulses. DRAG pulsing drives the ${\mathrm{Q}_{b}}$ leakage transition. The chosen relative phase $\phi=237^\circ$ between subsequent $\pi$ pulses exacerbates the net leakage. In contrast, Wah-Wah pulses populate ${\left\lvert 2_b \right\rangle}$ temporarily, returning the population to ${\left\lvert 1_b \right\rangle}$ by the end of each pulse. ](Figure2)

![(color online). Demonstration of decoherence-limited single-qubit control via individual and alternating randomized benchmarking. (a) Pseudo-random control pulses are applied to either one transmon or to both in alternating fashion. The average error per computational step (${\mathrm{EPS}}$) is extracted as a function of the computational step time ${t_{\mathrm{s}}}=2{t_{\mathrm{g}}}+2{t_{\mathrm{b}}}$ (without and with a symmetrically timed RB pulse on the other transmon during buffer). (b) ${\mathrm{EPS}}$ for optimized ${\mathrm{Q}_{b}}$ DRAG pulses. The linear ${t_{\mathrm{s}}}$ dependence observed without ${\mathrm{Q}_{a}}$ pulses extrapolates to $\left(-6 \pm 9\right)\times 10^{-4}$ at ${t_{\mathrm{s}}}=0$, indicating decoherence-limited control. The fits are done to the first ten data points. The alternating ${\mathrm{EPS}}$ matches the individual ${\mathrm{EPS}}$ for ${\mathrm{Q}_{a}}$ Wah-Wah. In contrast, alternating with ${\mathrm{Q}_{a}}$ DRAG pulsing worsens the ${\mathrm{EPS}}$ to $\sim5~\%$. The large error bars reflect high sensitivity of ${\left\lvert 2_b \right\rangle}$ leakage to the particular sequence of ${\mathrm{Q}_{a}}$ rotations within the randomization. (c) ${\mathrm{EPS}}$ for optimized ${\mathrm{Q}_{a}}$ DRAG and Wah-Wah pulses. Overlapping results are obtained without and with alternating RB DRAG pulses on ${\mathrm{Q}_{b}}$. The observed linear ${t_{\mathrm{s}}}$ dependence of ${\mathrm{EPS}}$ extrapolates to $\left(-7 \pm 9\right)\times 10^{-4}$ at ${t_{\mathrm{s}}}=0$, indicating that ${\mathrm{Q}_{a}}$ control is also decoherence limited. ](Figure3)

While Wah-Wah pulsing on ${\mathrm{Q}_{a}}$ successfully mitigates net leakage in ${\mathrm{Q}_{b}}$, the temporary excursion of quantum amplitude from ${\left\lvert 1_b \right\rangle}$ to ${\left\lvert 2_b \right\rangle}$ induces a relative phase between levels ${\left\lvert 0_b \right\rangle}$ and ${\left\lvert 1_b \right\rangle}$, i.e., a $z$ rotation in the ${\mathrm{Q}_{b}}$ qubit subspace. This induced phase is a deterministic function of the ${\mathrm{Q}_{a}}$ pulse parameters defined above. Thus, we can compensate it already in pulse synthesis by adjusting the phase of all subsequent ${\mathrm{Q}_{b}}$ pulses, in the style of virtual $z$ gates [@Steffen00]. To calibrate the phase shift, we embed several consecutive Wah-Wah ${\mathrm{Q}_{a}}$ pulses (either all $\pi$ or $\pi/2$) into the second wait period in a standard echo sequence on ${\mathrm{Q}_{b}}$ ($\pi/2$, wait, $\pi$, wait, $\pi/2$). The final $\pi/2$ rotation translates the acquired phase into a population difference between ${\left\lvert 0_b \right\rangle}$ and ${\left\lvert 1_b \right\rangle}$. We observe that the induced ${\mathrm{Q}_{b}}$ phase is independent of rotation axis and linear in the number of ${\mathrm{Q}_{a}}$ pulses of a given type: $8.2^\circ$ ($38^\circ$) per $\pi/2$ ($\pi$) pulse. We perform a similar calibration of compensating $z$ gates on ${\mathrm{Q}_{a}}$ for DRAG pulses applied to ${\mathrm{Q}_{b}}$. Even though $\left({\omega_{12}^a}-{\omega_{01}^b}\right)/2\pi=-770~{\mathrm{MHz}}$ and ${\mathrm{Q}_{b}}$ DRAG pulsing does not produce ${\mathrm{Q}_{a}}$ leakage (shown below), there is phase accrual in the ${\mathrm{Q}_{a}}$ qubit subspace: $2.5^\circ$ ($9.3^\circ$) per $\pi/2$ ($\pi$) pulse.

In order to test both leakage mitigation and phase compensation to higher accuracy than allowed by SPAM, we employ randomized benchmarking (RB). The single-qubit protocol first proposed and implemented by Knill *et al.* [@Knill08] (standard RB) provides a valuable baseline for gate errors on the addressed qubit, without concern for the unaddressed one. In standard RB [@Knill08; @Chow09], one applies pseudo-random sequences of consecutive $\pi$ and $\pi/2$ pulses to ${\mathrm{Q}_{k}}$ and measures the decay of fidelity to the ideal final state of that qubit (always ${\left\lvert 0_k \right\rangle}$ or ${\left\lvert 1_k \right\rangle}$) as the number of pulses is increased \[Fig. 3(a)\]. This decay allows extracting [@Magesan12; @SOM] the average error per computational step (${\mathrm{EPS}}$), where computational step is defined as a pair of back-to-back $\pi$ and $\pi/2$ pulses [@Knill08]. We extract ${\mathrm{EPS}}$ as a function of the step time ${t_{\mathrm{s}}}=2({t_{\mathrm{g}}}+{t_{\mathrm{b}}})$ by varying the buffer time ${t_{\mathrm{b}}}$ between pulses (${t_{\mathrm{b}}}\geq 2~{\mathrm{ns}}$). To within statistical error, a linear fit of ${\mathrm{EPS}}({t_{\mathrm{s}}})$ for ${\mathrm{Q}_{a}}$ (${\mathrm{Q}_{b}}$) at short ${t_{\mathrm{s}}}$ extrapolates to the origin \[Figs. 3(b)-(c)\]. This observation suggests that the minimal $0.4\%$ ($0.4\%$) ${\mathrm{EPS}}$ is already decoherence limited.

![(color online). ${\mathrm{Q}_{b}}$ average gate fidelity to the identity operation during various types of idling: no ${\mathrm{Q}_{a}}$ pulsing, ${\mathrm{Q}_{a}}$ DRAG RB, and ${\mathrm{Q}_{a}}$ Wah-Wah RB. (a) Pulse sequence. Initial and final rotations on ${\mathrm{Q}_{b}}$ are used for QPT of the evolution of ${\mathrm{Q}_{b}}$ qubit state to the three-level subspace. Results are shown for gate times (b) ${t_{\mathrm{g}}}=16~{\mathrm{ns}}$, (c) $20~{\mathrm{ns}}$, and (d) $24~{\mathrm{ns}}$ (${t_{\mathrm{b}}}=2~{\mathrm{ns}}$ fixed). Model curves for true idling take into account the measured ${\mathrm{Q}_{b}}$ qubit relaxation and dephasing times [@SOM]. The model assumes dephasing dominated by $1/f$ noise [@Yoshihara06]. The ${\mathrm{Q}_{b}}$ average gate fidelity to identity during ${\mathrm{Q}_{a}}$ Wah-Wah RB (circles) is indistinguishable from that of true idling. In contrast, ${\mathrm{Q}_{a}}$ DRAG RB (triangles) deteriorates fidelity at shorter ${t_{\mathrm{i}}}$. ](Figure4)

With the ${\mathrm{EPS}}$ baselines from standard RB in place, we now employ alternating RB to investigate whether control of either qubit can remain decoherence limited when pulses are interleaved on the other. We apply an RB pulse on one qubit during the buffer (${t_{\mathrm{b}}}\geq {t_{\mathrm{g}}}+4~{\mathrm{ns}}$) for the other, and perform virtual $z$-gate compensation for all pulses. Alternating RB performed with DRAG pulses on both ${\mathrm{Q}_{a}}$ and ${\mathrm{Q}_{b}}$ has no impact on ${\mathrm{Q}_{a}}$, but increases the ${\mathrm{Q}_{b}}$ ${\mathrm{EPS}}$ to $\sim5\%$ \[Fig. 3(b)\]. However, by using Wah-Wah pulses on ${\mathrm{Q}_{a}}$, we recover the decoherence limited baselines simultaneously on both qubits. Similar results for ${t_{\mathrm{g}}}=20~{\mathrm{ns}}$ and $24~{\mathrm{ns}}$ are presented in the Supplement [@SOM].

As the final test of whether Wah-Wah pulsing on ${\mathrm{Q}_{a}}$ affects ${\mathrm{Q}_{b}}$, we perform quantum process tomography (QPT) of ${\mathrm{Q}_{b}}$ under various idling conditions: no pulses on ${\mathrm{Q}_{a}}$ (true idling), ${\mathrm{Q}_{a}}$ DRAG RB, and ${\mathrm{Q}_{a}}$ Wah-Wah RB (Fig. 4). The process map is a $9\times 4$ transfer matrix [@Chow12] relating an initial reduced density matrix in the ${\mathrm{Q}_{b}}$ qubit subspace to a final reduced density matrix in the three-level [@Bianchetti10] subspace. The ${\mathrm{Q}_{b}}$ average gate fidelity to identity for true idling decreases consistently with an analytic model [@SOM] based on measured ${\mathrm{Q}_{b}}$ relaxation and dephasing rates. As expected, ${\mathrm{Q}_{b}}$ idling is compromised under ${\mathrm{Q}_{a}}$ DRAG RB, and worsens the shorter ${t_{\mathrm{g}}}$. An analysis discerning contributions from population transfer and dephasing errors [@SOM] confirms that the loss of idling fidelity is limited by induced leakage and not by imperfect $z$-gate compensation. Remarkably, the idling fidelity under ${\mathrm{Q}_{a}}$ Wah-Wah RB is nearly identical to the true idling fidelity for all ${t_{\mathrm{g}}}$, further demonstrating the usefulness of the new control method.

In summary, we have shown experimentally, using level-population measurements, RB, and QPT, that Wah-Wah control [@Schutjens13] successfully mitigates crosstalk-induced leakage in a crowded spectrum of transmon qubits at gate times where the widely-adopted DRAG control fails. Wah-Wah control therefore represents a step towards scalability in multi-qubit architectures by allowing selective control of an increasing number of effective qubits without an equal addition of control lines. Wah-Wah builds sideband modulation to DRAG without sacrificing the analytic nature of the two-quadrature pulse shape, adding only two easily optimized parameters. Our demonstration in circuit QED takes place in the worst-case regime of unity crosstalk, with control pulses applied via one common feedline coupling equally to the addressed qubit and the unaddressed nearest-frequency neighbor. Looking forwards, as two-qubit gate and measurement times [@Sank14] decrease to that of single-qubit gates, the current restriction to non-overlapping control of frequency-neighboring qubits could ultimately bottleneck the clock cycle in surface-code quantum error correction [@Fowler12]. A useful generalization of Wah-Wah, however, would allow simultaneous control of two qubits at small $\left|\delta\right|$.

We thank D. J. Egger, R. Schutjens and F. K. Wilhelm for many helpful discussions, N. K. Langford and D. Ristè for helpful comments on the manuscript, and D. J. Thoen and T. M. Klapwijk for NbTiN thin films. This work is funded by EU FP7 project SCALEQIT, the Netherlands Organization for Scientific Research (NWO, VIDI scheme), and a Marie Curie Career Integration Grant. V.V. was supported by the Research Foundation of Helsinki University of Technology.