Optimizing multi-tone microwave pulses via phase selection for quantum computing applications

Multi-tone microwave signals find utility in advanced quantum computing operations, such as frequency-multiplexed qubit manipulation^1–12 and simultaneous execution of two-qubit gates.^13–19 For instance, in the context of frequency-multiplexed qubit control, as shown in Fig. 1(a), a multi-tone microwave signal is employed where each microwave signal component is resonant with its targeted qubit. Employing frequency-multiplexed qubit control is expected to reduce the number of microwave cables required to address individual qubits. This reduction can provide a solution to associated challenges such as the cooling capacity and the limited space of dilution refrigerators.²⁰ 

Successful implementation of frequency-multiplexed qubit control demands careful allocation of qubit frequencies to avoid unwanted excitations and AC-Stark shifts.² Utilizing a qubit control device with a bandwidth of $B$ to manage $N$ qubits, the strategic setting of the frequency difference $Δω$ between qubits to $B/N$ maximizes the multiplicity within the available bandwidth and effectively mitigates the aforementioned issues.² 

However, a challenge arises from the periodic nature of the microwave frequencies used to drive the qubits. This periodicity leads to interference among the microwave signals, resulting in the amplification of voltage peaks, as illustrated in Fig. 1(b). Typically, the generation of such multi-tone microwave signals utilizes a qubit controller based on an arbitrary waveform generator (AWG). The AWG commonly features waveform memory and a digital-to-analog converter (DAC). However, interference among signal components can cause signal amplitudes to exceed the AWG’s operational range, consequently causing the qubit controller to fail to generate the desired microwave signal and potentially compromising the realization of the desired quantum operation.

For instance, according to a previous work, achieving a Rabi frequency of 10 MHz in spin qubits is expected to require a microwave output of about $−$16 dBm.⁴ However, multiplexing $N$ qubits leads to an amplification of the multi-tone microwave signal’s amplitude by approximately a factor of $N$ due to interference among each microwave component. Given that the output capacity of a typical AWG used for qubit control is limited to a few dBm or less, this constrains the maximum number of qubits that can feasibly be multiplexed to around ten.

In this work, we introduce a method specifically designed to mitigate interference issues encountered in multi-tone microwave signal generation. Our approach utilizes phase selection to suppress voltage peaks resulting from signal interference. Specifically, we propose employing a crest factor reduction algorithm,²¹ which assigns specific phases that minimize the peaks caused by interference among microwave signals at certain frequency intervals.

We calculate the multi-tone signal using specific parameters.²² To simplify, we set the amplitude $a i$ to 1 for all components of the multi-tone signal; however, the method should fundamentally be applicable to other amplitude values as well. Additionally, we assume that the amplitudes are tuned such that each qubit has the same Rabi frequency. It is also important to note that our scheme is not limited to simultaneous excitation; we can selectively excite specific qubits by setting the amplitude $a i$ to 0 for non-targeted components. Assuming a base frequency of $ω 0/2π=5$ GHz and a frequency increment of $Δω/2π=0.1$ GHz, we consider all phases $ϕ i$ to be zero. With $σ=15$ and $N=30$⁠, the resulting signal is depicted in Fig. 2 (blue curve). As can be seen in Fig. 2, peaks in amplitude are observed, which result from the interference among the microwave signals. Such waveforms are actually used in quantum control experiments.^23,24

We then explore the use of a crest factor reduction algorithm. In this study, we specifically consider applying the Schroeder algorithm²⁵ to suppress peaks caused by interference. Note that, to date, numerous crest factor reduction algorithms have been explored, and other algorithms could also be applicable to our proposal.

The inset of Fig. 2 illustrates the effect of the Schroeder algorithm on voltage suppression compared to the constant phase setting across different numbers of frequency components. It is evident that as the number of frequency components increases, peak voltage suppression is enhanced, demonstrating the effectiveness of the Schroeder algorithm in reducing signal interference.

In this paper, we propose the application of the crest factor reduction algorithm to quantum gate operations. While we focus on this specific application, it is also conceivable to apply similar existing techniques. A comparison with these techniques will be addressed in future research.

Here, we explore the two potential applications where our proposal can be beneficial. It should be noted that a detailed discussion of the quantum dynamics is beyond the scope of this article. Our focus is on demonstrating how our proposed method can be applied to practical applications.

The first application is frequency-multiplexed qubit control. In this context, we consider realizing a single-qubit gate operation in a frequency-multiplexed manner. Note that two-qubit gate operations are considered to be implemented without the use of microwaves. Superconducting qubits²⁶ and spin qubits²⁷ are the prime candidates for the physical implementation of this application.

The second application is the simultaneous execution of cross-resonance (CR) gates. The CR gate is one of the commonly employed two-qubit gates in superconducting qubits.^30–32 The CR gate is realized by irradiating the control qubit with a microwave that resonates with the target qubit. Furthermore, by utilizing microwaves that carry multiple frequencies, it is possible to implement CR gates simultaneously between different qubits. Previous implementations have achieved multi-qubit gates by applying drive signals to control qubits that share a common target qubit.^17,19 In this paper, as illustrated in Fig. 3(a), we explore the simultaneous implementation of CR gates, where a single control qubit is shared by multiple neighboring target qubits.

As illustrated in Fig. 3(a), by applying a multi-tone microwave signal composed of frequency components resonant with neighboring target qubits to a control qubit, it is possible to execute CR gates concurrently. The CR gate, on the other hand, requires a significantly larger microwave amplitude compared to single-qubit gates. This requirement stems from the strength of CR gate interaction, represented by $Ω sg/Δ$⁠. Here, $Ω s$ is the amplitude of the external drive, $g$ represents the coupling strength between the qubits, and $Δ$ denotes the detuning between them. Therefore, a microwave with a magnitude proportionally larger, as dictated by qubit coupling strength and its detuning, is necessary. This requirement becomes particularly critical in scenarios where there is substantial detuning between the qubits (i.e., in a far-detuned regime), necessitating much larger microwave amplitudes. Nonetheless, the interference from such intense microwave signals can quickly surpass the AWG’s operational limits. As a result, our proposed method provides significant advantages in scenarios with considerable qubit detuning, mitigating the challenges associated with using large signal amplitudes.

In conclusion, we have developed a method to mitigate interference among the components of multi-tone microwave signals. By selecting the phases of these signals using a crest-factor reduction algorithm, we can significantly suppress peak amplitudes.

Our calculations reveal that using the crest factor reduction algorithm significantly reduces the peak voltage amplitude of the multi-tone signal. Specifically, in the case studied here, the amplitude of a 30-frequency-multiplexed signal is suppressed to less than 10 times that of a single-tone microwave signal. Without crest factor reduction, the amplitude would be amplified by approximately 30 times.

Note that, in this work, we have considered a scenario where qubit frequency allocation is equally distributed. However, our method can be adapted with some modifications to accommodate scenarios where the frequency of each qubit is not equally distributed. By considering Eq. (1) as the objective function of $θ i$ and by finding the set of phases that minimize the amplitude as given by Eq. (1), it is possible, in principle, to implement our method.

Our proposed method is especially useful in applications that employ multi-tone microwave signals. It enables frequency-multiplexed qubit control, a key element for the development of large-scale quantum computers. Moreover, our approach allows for the simultaneous execution of the CR gate, thereby enabling the efficient execution of quantum algorithms and logical qubit encoding. These capabilities highlight the significant role our method plays in advancing quantum computing by enabling complex operations and improving scalability. Finally, our method is potentially applicable to various types of quantum systems beyond the spin and superconducting qubits discussed.

This research was supported by JST COINEXT (Grant No. JPMJPF2014) and JST Moonshot R $&$D (Grant Nos. JPMJMS2067 and JPMJMS226A). K.O. acknowledges support from JST, PRESTO (Grant No. JPMJPR23F2). M.N. acknowledges support from MEXT Q-LEAP (Grant No. JP-MXS0118068682).

The authors have no conflicts to disclose.

R. Ohira: Conceptualization (equal); Data curation (lead); Formal analysis (equal); Investigation (lead); Methodology (lead); Supervision (equal); Visualization (lead); Writing – original draft (lead); Writing – review & editing (equal). R. Matsuda: Conceptualization (equal); Formal analysis (equal). H. Shiomi: Conceptualization (equal); Formal analysis (equal). K. Ogawa: Conceptualization (equal); Writing – review & editing (equal). M. Negoro: Conceptualization (equal); Funding acquisition (lead); Supervision (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.