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Abstract
With the rapid development of superconducting quantum computing and the implementation of surface code, large-scale quantum computing is emerging as an urgent demand. In a superconducting computing system, the qubit is maintained in a cryogenic environment to avoid thermal excitation. Thus, the transmission of control signals, which are generated at room temperature, is needed. Typically, the transmission of these signals to the qubit relies on a coaxial cable wiring approach. However, in a large-scale computing system with hundreds or even thousands of qubits, the coaxial cables will pose great space and heat load to the dilution refrigerator. Here, to tackle this problem, we propose and demonstrate a direct-modulation-based optical transmission line. In our experiment, the average single-qubit XEB error and control error are measured as 0.139% and 0.014% separately, demonstrating the feasibility of the optical wiring approach and paving the way for large-scale superconducting quantum computing.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
In recent years, significant progress has been made in the development of quantum computing systems [1–10]. In particular, superconducting quantum computing systems have emerged as a highly promising approach due to their relatively long coherence times, low noise levels, and fast circuit design capabilities [11]. A major breakthrough in this area has been the recent demonstration of quantum advantage through systems [12] such as Sycamore with 53 and 70 superconducting qubits [4,13], and Zuchongzhi with 66 superconducting qubits [14]. These developments mark a new era in the simultaneous manipulation of large numbers of qubits with high precision, which is critical for the advancement of noisy intermediate-scale quantum (NISQ) technology and the realization of error-corrected logic qubits through the use of surface code [14]. Already, a wide range of potential near-term applications, including quantum chemistry [15–17], many-body physics [18,19], and quantum machine learning [20,21], are being explored. The realization of surface codes has demonstrated the feasibility of error correction [22,23], making it imperative to scale up superconducting systems further.
In a superconducting quantum computing system, the operating temperature of the quantum processor must be maintained below tens of millikelvin (mK) in order to initialize the qubit to the ground state and minimize the risk of thermal excitations that can affect the accuracy of the computation [24]. Typically, a dilution refrigerator (DR) [24] is employed to produce such a cryogenic environment. In the most prevalent application scenarios, the microwave signals utilized for initializing, controlling, and extracting information from the qubits are generated outside the DR [25,26]. These signals are then transmitted to the qubits via coaxial cables. However, as the quantity of qubits increases, the number of control lines grows accordingly, making it more and more difficult to accommodate all the cables within the limited space and heat budget of the DR.
One promising solution is to generate the microwave signal inside the DR and directly apply it to the qubit. A typical implementation is the single flux quantum (SFQ) digital logic [27], which only requires a few transmission lines from room temperature to cryostat below 4 K. In an SFQ digital logic, a single flux is generated and triggered by the clock, so as to control the qubit state precisely. However, due to issues such as quasi-particle poisoning, the fidelity of a single-qubit gate for an SFQ-controlled superconducting system typically can only achieve 95% [28–31]. Such fidelity is much lower than that controlled by conventional room-temperature electronics systems, which is typically over 99.9% [32,33].
Another promising solution is to use optical transmission as a microwave carrier to replace the coaxial transmission [34–36]. Compared with conventional coaxial transmission, optical transmission is more advantageous in large-scale superconducting quantum computing. Firstly, thermal conductivity at 50 K of fiber, which is about 2 $W/m \cdot K$ [37], is typically three orders of magnitude lower than that of coaxial cable, which is approximately 1000 $W/m \cdot K$ [24]. Besides, benefit from advances of near-infrared photons being used as a carrier for generating and manipulating microwave signals [38,39], the bandwidth of optical transmission can be much broader. Moreover, the diameter of a single-mode optical fiber is typically $250~\mu m$, which is one order of magnitude lower than that of a coaxial cable. Furthermore, the loss of photonic transmission at the C band is typically as low as 0.2 $dB$/$km$. Thus, the optical transmission attenuation will be much lower than that of coaxial cable. In recent years, significant progress has been made in the implementation of optical transmission [34], in which external modulation was adopted to generate the optical signal.
Here, we design a direct-modulation-based optical transmission method. Our method could be easily implemented at a low cost as fewer optical components are needed. Besides, the microwave signals could be applied directly on the laser diode chip, making it a better choice for large-scale applications. Furthermore, direct modulation of optical signals is widely applied in the field of optical communication [40,41]. On the other hand, in a superconducting system, one promising all-microwave nonadiabatic Controlled-Z gate can achieve high fidelity up to 99.54% [42,43], as the typically used pulse signals are all replaced by microwave signals. Thus, all-microwave optical transmission is applied here. The feasibility of controlling a superconducting qubit precisely with the optical transmission method is discussed in detail, including the attenuation and phase noises. The overall attenuation of the transmission line and phase noise of the reconverted control signals are 25 $dB$ and about −140 $dBc/Hz$ at an offset frequency of 10 $MHz$, respectively. The average single-qubit XEB error and control error are measured as 0.139% and 0.014% separately, demonstrating the feasibility of the optical wiring approach.
2. Experimental setup
The diagram of the conventional coaxial cable wiring approach could be illustrated in Fig. 1(a). The microwave signals used to initialize, control, and read out the state of the qubit are generated in room-temperature electronics and then transmitted into the DR through coaxial cables. Attenuators are installed on different plates inside the DR to adjust the power of the signals and reduce their noise. Meanwhile, as the resonant frequency of the qubit is approximately 4 – 8 $GHz$, in the mixing chamber, 8 $GHz$ low-pass filters are installed to filter out high-frequency noise.
Fig. 1. Schematics of the wiring approach. (a) Scheme of conventional coaxial cable wiring. Microwave signals are directly transmitted to the qubit by coaxial cables. (b) scheme of optical transmission wiring. Microwave signals are firstly modulated as optical signals, then transmitted into DR, and lastly reconverted into microwave signals within DR. LD, laser diode; DR, dilution refrigerator; PD, photodiode.
In Fig. 1(b), we depict the workflow of our optical transmission line. Firstly, microwave signals are generated by room-temperature electronics and then fed into the laser diode (LD) to generate a photonic signal in an amplitude-modulation way. A direct current (DC) drive is applied to the LD to ensure that the LD is working at the linear region. Next, the photons carrying the microwave signal are attenuated and sent into the DR, where the photons are measured by an InGaAs photodiode (PD) to be reconverted to the microwave signals again, which will then be transmitted to the qubit by coaxial cables.
In the photon detection process, the photon signal is converted into a photocurrent as a result of the photoelectric effect. Thus, lower optical frequency with lower photon energy is beneficial for reducing the heat load of the PD. Besides, considering that the loss of an optical fiber reaches a minimum at the C band, a working wavelength of the optical transmission of 1490 nm is adopted, at which most of the optical elements designed for the C band will work well. Moreover, the working frequency of the microwave signal is also set at a frequency range of 4 $\sim$ 8 $GHz$. To modulate such a high-frequency signal, an LD with a dynamic frequency range of 10 $GHz$, in which the central frequency of the photon could be applied on demand, is adapted to apply the modulation.
The performance of the optical transmission is affected by several aspects, among which the transmission loss and phase noise induced in the transmission plays the most important role. To control the qubit, a microwave signal with too low power could not excite a state transition, causing more errors and lowering the fidelity. Thus, the transmission loss needs to be carefully controlled. Besides, if the induced phase noise during optical transmission is too high, causing floor leverage that is too high, the fidelity of controlling the qubit will be significantly impacted. Considering these aspects, in the design of the optical transmission system, to evaluate the performance and feasibility of the transmission system, we mainly focus on the overall optical transmission attenuation and phase noise induced in the transmission.
3. Results and benchmarking
3.1 Overall optical transmission attenuation
In our scheme, the coaxial cables from the room temperature to the cryostat are replaced by optical transmission lines. Thus, it is important to evaluate the attenuation of the optical transmission line. The attenuation from the room-temperature generated microwave signals to the reconverted microwave signals can be evaluated as parameter $S_{21}$. We denote the slope efficiency of the laser diode and the responsivity of the photodiode as $k_1$ and $k_2$, and the parameter $S_{21}$ can be put as:
The value of $S_{21}$ is measured by a vector network analyzer (VNA). Firstly, the VNA generates and feeds a microwave signal with a fixed power into the optical link. Next, the output of the optical link is collected and sent to the VNA to measure its power. At last, $S_{21}$ is calculated as the power difference between the generated and collected microwave signals. In our experimental demonstration, the measured $S_{21}$ is about −24 to −26 $dB$, as shown in Fig. 2, fairly matches our estimation. In experiments, the qubit operates at a specific and fixed frequency. Therefore, the variations of $S_{21}$ within the frequency range of 4 to 8 $GHz$ will not impact the control of the qubit.
Fig. 2. The measured attenuation ($S_{21}$) of the optical transmission link under different microwave input powers with a typical LD driving current of 40 $mA$. Within the working frequency of 4 – 8 $GHz$, the optical transition attention is about −25 $dB$, which fairly matches the estimation.
3.2 Phase noise
To evaluate the impact of the induced phase noise during the optical transmission on the fidelity of the qubit, a simulation could be made by measuring the phase noise of the reconverted microwave signal. Physically, the phase noise is a rapid, random fluctuation of the phase of a signal. Typically, a random phase fluctuation of the signal will cause a random fluctuation in both the frequency and time domain of the signal. Thus, the phase noise can be measured in terms of both the frequency domain and the time domain. Denote the phase noise as $\mathcal {L}$, the fidelity of the single-qubit gate as $\mathcal {F}$, in the frequency domain, the phase noise and fidelity could be put as [44]:
where $P_{noise}$ and $P_{signal}$ denote the power of the noise and the signal, respectively, $\Delta f$ denotes the frequency offset. $\chi (\tau )$ denotes the error integral, expressed as a product of the noise power spectral density and filter transfer functions, and can be expressed as:At room temperature, after reconverting the microwave from the PD, the phase noise is measured with a fixed typical LD driving current of 40 $mA$, while varying the frequency and power of the input microwave signals using a signal source analyzer. In the experimental setup, the phase noise is assessed at microwave signal frequencies of 4 and 5 $GHz$, with power levels of 5 and 10 $dBm$, respectively, as depicted in Fig. 3(a). It is clear that at an offset of 10 $MHz$, the phase noise is about −135 to −145 $dBc$/$Hz$. The simulated fidelity could be calculated using Eq. (4) as illustrated in Fig. 3(b). As the evolution time increases, the qubit error accumulates. At a typical time period of about 10 $ms$, the infidelity of a single-qubit gate induced from the phase noise is lower than 0.0001%. Typically, the decoherence time ($T_1$) of superconducting qubits, which represents their natural evolution time, is in the range of a few hundred nanoseconds. This indicates that the phase noise resulting from the reconverted microwave signal has a limited impact on the fidelity of single-qubit gates.
Fig. 3. The measured phase noise and simulated XEB error. (a) The measured phase noise curve of the reconverted microwave signal with a fixed typical LD driving current of 40 $mA$, while varying the frequency and power of the input microwave signals. (b) The simulated infidelity of a single-qubit gate induced by the phase noise data is calculated from Eq. (4). It is clear that the induced infidelity of the phase noise is less than 0.0001% for a typical evolution time of 10 $ms$.
3.3 Benchmarking single-qubit gate performance
For convenience and simplicity, the evaluation of the optical transmission overall attenuation and the phase noise of the reconverted signal are both performed under room-temperature conditions. The experimental results of these two items match our theoretical analysis fairly well. We then apply the optical transmission to our superconducting computing system called hangzhou to benchmark the single-qubit gate performance. The laser diode is driven with a current of 45 $mA$, generating the optical signal, which is then transmitted into the DR. Next, the optical signal is detected at the 1 $K$ plate to generate the photocurrent, which is lastly applied to control the qubit.
Here, we use cross entropy benchmarking (XEB) [45] to benchmark the single-qubit gate performance of a qubit. An average single-qubit gate Pauli XEB error and control error of 0.139% and 0.014% are measured 20 times of repeated experiment, as shown in Fig. 4. The average single-qubit gate Pauli error using our optical link is about the same as that using conventional coaxial cable wiring, which is typically about 0.14% [14], demonstrating the feasibility of the optical transmission approach.
Fig. 4. Single-qubit gate performance benchmark. The Pauli XEB error (blue line) is a summation of the control error (green dot-dashed line) and the speckle purity benchmarking (SPB) error (orange dashed line), which is a measure of decoherence of the qubit. The average single-qubit gate Pauli XEB error and control error of about 0.139% and 0.014% are measured, respectively, demonstrating the feasibility of the optical transmission approach.
4. Conclusions
In this experiment, an optical transmission line is proposed and demonstrated. The measured transmission attenuation, the phase noise, and the single-qubit XEB error show that the optical transmission works as well as the conventional coaxial-cable wiring approach, demonstrating the feasibility of the optical transmission scheme. The next step will be focused on the integration of optical transmission lines with tens to thousands of qubits into the superconducting computing system.
Funding
Youth Innovation Promotion Association of the Chinese Academy of Sciences (2022460, 2023574); Chinese Academy of Sciences; Open Research Fund from State Key Laboratory of High Performance Computing of China; China Postdoctoral Science Foundation (2021M700315); Anhui Initiative in Quantum Information Technologies; Shanghai Sailing Program (21YF1452500); Shanghai Rising-Star Program (21QA1409600, 23QA1410000); National Natural Science Foundation of China (12174374, 12274464); Shanghai Municipal Science and Technology Major Project (2019SHZDZX01); The Innovation Program for Quantum Science and Technology (2021ZD0300200).
Acknowledgments
This work was supported by The Innovation Program for Quantum Science and Technology, Shanghai Municipal Science and Technology Major Project, the National Natural Science Foundation of China, Shanghai Rising-Star Program, the Shanghai Sailing Program, the Chinese Academy of Sciences, Anhui Initiative in Quantum Information Technologies, China Postdoctoral Science Foundation, the Open Research Fund from State Key Laboratory of High Performance Computing of China, and the CAS Project for Young Scientists in Basic Research. Y.-H. Li and M. Gong were supported by the Youth Innovation Promotion Association of CAS.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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