The research culture at Berkeley shaped our understanding that the purpose of a small academic quantum laboratory is to delve into the fundamental elements of relevant technologies, laying the groundwork for future developments. In the realm of solid-state systems, this can be broken down into two general directions: (i) the development of innovative hardware inherently shielded from unavoidable environmental noise, and (ii) the creation of techniques to facilitate interactions among these high-coherence systems. Observations in our field have shown that manipulating interqubit interactions through flux modulation inevitably makes the qubits vulnerable to flux noise during operations. On the other hand, fixed-frequency systems have been consistently reported to have record coherence times. We thereby pinpointed the investigation of extensible interactions between fixed-frequency qubits using microwave drives as a key objective. At the time, all-microwave gates were used in the field to implement cross-resonance CX and differential ac Stark shift CZ gates [1,2]. Both at Berkeley and in my prior group at Maryland, we observed that increasing the drive amplitudes beyond a certain limit caused the qubits to act unpredictably, a phenomenon we struggled to comprehend at first.

I had the good fortune of beginning my postdoctoral appointment at the same time as Yosep Kim, with whom I quickly established a strong and lasting friendship. Yosep, having completed his Ph.D. in quantum optics, brought a deep understanding of quantum operations and characterization. Shortly after his arrival, he developed an innovative method to implement the iToffoli gate, involving the simultaneous application of non-commuting operations on qubits [3]. This approach was successfully put into practice with the assistance of Alexis Morvan, another exceptional researcher who later joined Google Quantum AI as a research scientist. Alexis, known for his academic acumen, also played a unique role in our group. He was fondly regarded as the keeper of the 'brother code,' often reminding us to adhere to a communal schedule: lunch at noon, tea at 4 p.m., and occasionally, social drinks after work. Given our Berkeley location, these tea breaks naturally evolved into boba breaks. This led to the creation of a 'boba economy' within our group, where assistance on projects was often reciprocated with boba treats. Consequently, many afternoons were filled with lively discussions about our research progress and ideas.

On a sunny and breezy afternoon in April 2021, following an extended measurement session in the lab, Yosep and I were walking home when he proposed that I join his project. The idea was to spend more time together, engaging in enjoyable work — a proposal too tempting to pass up. In response, I shared my aspirations and thoughts about novel superconducting architectures, beginning with ideas about a fluxonium quantum processor. While fine-tuning the iToffoli gate, Yosep identified a geometric phase error stemming from a noticeable ZZ interaction. This error surfaced when we increased the amplitude of the cross-resonance drive. Following our initial implementation of the gate, we contemplated an alternative approach: substituting the ZX interaction with ZZ. This adjustment aimed to circumvent the geometric phase error, a move that could potentially evolve into a distinct project on its own.

In the course of implementing non-commuting operations alongside dynamical ZZ, we encountered an intriguing phenomenon: the drives induced significant energy shifts, resulting in Z phases accumulating on the qubits. Concurrently, I was engaged in a theoretical exploration of strategies for operating a fluxonium quantum processor that would be both robust and scalable [4]. This led me to develop a range of simulation routines. My interest was particularly piqued by a theoretical concept proposed by my friend and colleague, Ziwen Huang. Ziwen's proposal highlighted that both XY and ZZ interactions could be independently adjusted in flux-modulated Floquet fluxonium qubits [5].

This theoretical insight, combined with the dynamics we had previously observed in transmon devices, spurred me to investigate the impact of a similar set of drive parameters on the qubits. Specifically, I observed that by carefully selecting appropriate drive amplitudes and frequencies, we could adjust the qubits to demonstrate exchange interaction. This finding was later independently validated by Yosep for general coupled two-level systems through the use of quantum process tomography. The presence of XY coupling in a fixed-frequency platform with constant coupling was particularly stimulating, leading us to shift our focus and begin investigating Floquet-engineered interactions more intensively.

Building on our earlier measurement outcomes, Yosep adeptly adjusted the parameters to facilitate the exchange interaction, which manifested as coherent oscillations between the single-excitation states of the two qubits. However, we encountered our first major challenge during the characterization of this operation through quantum process tomography. We found that the extracted Pauli Transfer Matrix (PTM) significantly deviated from the ideal model.

It was then that we identified the issue of differential phase evolution. Due to the distinct frequencies of the qubits, the energy-exchange interaction introduced a time-dependent phase to each qubit. This aspect was initially overlooked during the preparation of the initial states. To address this, we modified the sequence to introduce the two-qubit unitary consistently at the same time in each run. This simple yet effective adjustment allowed us to compensate for the phase differences subsequently, enabling us to measure PTMs with much higher fidelities. Armed with our refined approach and high-fidelity readout capabilities, tomography then became a powerful and reliable tool for characterizing both the XY and ZZ interactions in our system. This advancement marked a significant step forward in our understanding and control of these quantum interactions.

Randomized benchmarking has emerged as a key technique for providing SPAM (State Preparation and Measurement) error-free assessments of gate fidelities. Despite our encouraging PTM results, we recognized the necessity of accurately evaluating the robustness of our interactions using this established benchmarking tool. This required meticulous tracking of all local phases introduced during the process and correctly accounting for them at the end of any arbitrary gate sequence. This task presented a significant challenge, especially as Yosep departed from the group shortly after the completion of the iToffoli project, leaving me less acquainted with the intricacies of the measurement framework.

Thankfully, Akel came to our aid, offering invaluable guidance through the layers of the sequence compiler and transpiler. Together with Yosep's participation over Zoom, we were able to confidently adapt these tools, leading to the development of an effective phase tracking routine within the cycle benchmarking framework. Crucially, Akel also introduced us to the covariance matrix adaptation (CMA) technique, which we subsequently employed to optimize our gate fidelities. This approach significantly bolstered our ability to fine-tune and assess our quantum gates, marking a pivotal advancement in our research. One night, we set the optimization and characterization processes to unfold silently in the lab's dim light. The next morning, I awoke to a sight that made my heart skip a beat: the iSWAP gate fidelity had surpassed the 99% threshold. For a moment, I held my breath, wondering if it was just a dream. A year had passed since Yosep extended his hand in partnership, and now, finally, we had achieved results worthy of publication.

While exploring the extensibility of the protocol, we made an intriguing discovery: it was also applicable for implementing interactions between qutrits (three-level quantum systems). Fortunately, our group had already established a robust framework for measuring and characterizing qutrits. With Noah's assistance, we were able to characterize and refine the interaction between qutrit states effectively. Leveraging this capability, we successfully implemented the three-qubit controlled-controlled-Z (CCZ) gate.

To further corroborate the high fidelity of our CCZ gate, Brian introduced us to cross-entropy benchmarking. This advanced method provided additional validation for the performance of our gate, solidifying our confidence in the results we had achieved. This collaborative effort exemplified the dynamic and adaptive nature of our research, demonstrating how collective expertise and quick problem-solving can lead to significant advancements in our field. 

As Yosep was finalizing the Floquet theoretical framework with assistance from our collaborators at Chapman University, I realized that the gate we had implemented was experiencing finite leakage. Responding promptly, I connected to the measurement setup remotely while attending the Applied Superconductivity Conference in Hawaii, then corrected the sequence and fine-tuned the gate, achieving the level of fidelity that we later reported in our paper. Our puzzle was finally completed, and I spent the following week polishing the manuscript during the rainy days in Hilo. One fine morning, as the faint light bathed the distant Mauna Kea in a golden glow, I gave the manuscript one last proofread over a cup of coffee. In awe of the surrounding landscape and reflective of our collaborative journey, I look at each paragraph as an echo of the past few years of working on the project. With a deep breath and a mix of emotions, I sent it off to the other authors for their approval, marking both an end and a new beginning.

With the quasi-energy treatment, Floquet theory is particularly suited to describe the quantum electrodynamics of strongly driven systems, and we found our theoretical solutions to match those in Ziwen's paper. Besides the discovery of the extensible gates, our work emphasizes the importance of adiabaticity, which is so far not well understood. During the gate tune-up, we observed clear nonadiabatic effects corresponding to strong drive amplitudes, which agree well with the unpredictable behaviors mentioned above. These can be later explained using the prior theory framework by our colleague Chunqing Deng [6], which elucidates the outstanding mystery. Given the universality of these concepts across various platforms, we are optimistic that our experimental contributions will pave the way for further investigations into strongly driven solid-state systems. Currently, our group is actively working on multi-chromatic driving protocols and diabatic gates, which we hope to further transform the research landscape of superconducting qubits.

Growing up, I was captivated by the world of remote-controlled toys and the thrill of communication through walkie-talkies. This early fascination laid the foundation for my interest in microwave technologies and their role in shaping a wireless future. My curiosity about the seamless convenience enabled by these technologies, coupled with a growing interest in integrated circuits, naturally steered me towards research on superconducting qubits and circuit quantum electrodynamics. To this day, I remain profoundly intrigued by our ability to manipulate these macroscopic devices into exhibiting quantum mechanical behaviors using microwaves. I am optimistic that as our colleagues have been enhancing their proficiency in using lasers to precisely manipulate neutral atoms, we can similarly make strides in controlling artificial atoms by refining our techniques with microwaves.

References
[1] Xiong, H. et al. Arbitrary controlled-phase gate on fluxonium qubits using differential ac Stark shifts. Phys. Rev. Research 4, 023040 (2022)
[2] Mitchell, B. K. et al. Hardware-efficient microwave-activated tunable coupling between superconducting qubits. Phys. Rev. Lett. 127, 200502 (2021).
[3] Kim, Y. et al. High-fidelity three-qubit iToffoli gate for fixed-frequency superconducting qubits. Nat. Phys. 18, 783 (2022).
[4] Nguyen, L. B. et al. Blueprint for a high-performance fluxonium quantum processor. PRX Quantum 3, 037001 (2022).
[5] Huang, Z. et al. Engineering dynamical sweet spots to protect qubits from 1/f noise. Phys. Rev. Appl. 15, 034065 (2021).
[6] Deng, C. et al. Dynamics of a two-level system under strong driving: quantum-gate optimization based on Floquet theory. Phys. Rev. A 94, 032323 (2016).