1. Introduction

Trapped ions are highly promising candidates for realizing quantum computing processors since they can offer longer coherence time compared to other platforms and their reproducibility and scalability [1–3]. Fast and highly accurate state measurement is one of the pivotal steps in quantum information technology [4–6]. The state measurement of ion-based qubits is carried out by applying a laser beam whose frequency is resonant with a certain energy level of the ions and collecting state-dependent fluorescence. By observing the emitted photons, one can discriminate between the $|{1}\rangle$ state that scatters photons from the cycling transition and the $|{0}\rangle$ state that does not emit any photons. Developing a large-scale quantum information processor with multiple ions requires spatially resolved photon detectors such as multi-channel photomultiplier tubes (PMTs) [7–10] or an electron-multiplying charge-coupled device (EMCCD) camera [11–14] to determine the individual quantum states of the multiple qubits.

A multi-channel PMT is a good tool for fast measurement of the quantum states of multiple qubits and it is commonly used. Its ultra-fast response enables straightforward analysis of the signals since a single photon almost clearly corresponds to a single measured signal. However, its extreme sensitivity can result in electrically-induced inter-channel crosstalk and spurious detection from unpredictable cosmic rays from the background [8,12]. These spurious signals cannot be distinguished from the actual signals, which are one of the main factors in fidelity degradation. Furthermore, since the detection area of each channel of the multi-channel PMT is fixed, each channel of the detector needs to be optically mapped to each qubit. This results in increased difficulty during the initial installation phase and hinders the flexibility of the spacing of trapped ions.

An EMCCD camera can be a suitable solution to address the problems described above. An EMCCD is a charge-coupled-device (CCD) camera integrated with additional electron-multiplying (EM) gain registers [15]. These gain registers transfer induced photoelectrons and multiply them by impact ionization on each clock, which enables one to detect a very small signal at the single photon level. Since the EMCCD provides two-dimensional information, it contains more spatial information than PMT-based readout. This spatially-resolved information enables the identification of the source ions of each emitted photon. Similar to PMTs, the high sensitivity and gain of EMCCD give rise to clock-induced-charge (CIC) noise that cannot be discriminated from the actual signals [16]. However, the EMCCD is more robust against these false signals because they are spatially distributed [12]. On the other hand, due to the random occurrence of impact ionization in the electron-multiplying (EM) gain registers, the final count of electrons, when it is read out by an analog-digital converter, entails some level of uncertainty. Combined with readout noise from complex circuits (modeled as white noise), this uncertainty broadens the histogram of the EMCCD signals. Consequently, the broadened signals hinder the proper interpretation of them and obscure the threshold value of each photon [17].

When performing state measurement of multiple qubits, crosstalk from adjacent qubits should be taken into account since it causes measurement errors [8,12]. This crosstalk can occur due to the scattered photons from neighboring qubits [18,19] or electrically-induced false detection [8]. To address these crosstalk problems, applying different Zeeman shifts to each ion with a strong field gradient [14,20] and detecting photons with high quantum efficiency [18] have been studied.

Machine learning techniques have become a rapidly developing field with a wide range of applications. Recently, many approaches applying machine learning techniques to the quantum system such as trapped ions [8,21], superconducting qubits [22–24], and quantum dots [25–27] have been reported to improve measurement fidelity. Both studies with trapped ions applied machine-learning techniques to PMT-based state measurements for ¹⁷¹Yb⁺ ions, utilizing time-bin analysis of photon arrival time. One study integrated deep neural networks achieved real-time state measurement with 99.5% fidelity, but this study is limited to a single ¹⁷¹Yb⁺ ion [21]. In another study, convolutional neural networks (CNNs) were employed, leading to improved measurement fidelity and robustness to crosstalk, achieving 90.87${\pm} $0.05% mean measurement fidelity for five ¹⁷¹Yb⁺ ions. [8].

CNN is a widely used deep learning model that is well-known for its outstanding ability to extract meaningful features, especially from image data [28]. They typically consist of convolutional layers, pooling layers, and fully connected (FC) layers. The convolutional layers generate feature maps that contain object information, the pooling layers sample features from the feature maps and gradually diminish the size of the inputs, and the FC layers connect the final feature maps to the outputs. CNNs have been applied in a plethora of areas such as image and video processing, object detection, natural language processing, speech recognition, and many other fields [29].

In this paper, a ResNet-based CNN model [30] is applied to the single-shot EMCCD images. Unlike PMT-based studies [8,21], this work solely utilizes the spatial information of the EMCCD images without time information. The filtering and feature extraction capability of CNNs is expected to exploit not only the values of the pixels but also the shape of the images of the ions when determining the quantum state. Additionally, CNNs automatically learn the spatial correlations of local features in data, which can help overcome various noises [31,32] such as CIC noise. In addition, the ResNet-based model preserves input data information through shortcut connections.

A practical method is used for acquiring multi-qubit data without individual control of multiple ions. The method is implemented by replacing ¹⁷¹Yb⁺ in $|{0}\rangle$ with ¹⁷⁰Yb⁺, an isotope of the qubit ion ¹⁷¹Yb ⁺. The qubit ions that are prepared in $|{1}\rangle$ by a global microwave scatter lots of photons during the state detection. On the other hand, the isotope ions rarely interact with the detection beam and remain dark. Using this method, EMCCD images of all 16 possible multi-qubit states of the 4 trapped ions are obtained with low state preparation error.

The results of the application of the ResNet-based CNN model are compared with those of the conventional two methods: a threshold method and a maximum likelihood estimation (MLE) method [12]. In addition, to test the robustness of the proposed method against long-term ion image drift, the ion images are slightly shifted laterally and similar tests are performed with ions imaged onto the different areas of the EMCCD. The results show that applying CNN architecture to the EMCCD data not only increases the state detection fidelity for multiple qubits but is also robust against the long-term drift of the ion image. Additionally, simultaneous Rabi oscillations of four ¹⁷¹Yb⁺ ions are measured to confirm that the model correctly determines the quantum state of each ion by observing the coherence of the oscillations.

2. Background

When performing a state measurement with trapped ions, state change induced by off-resonant transitions during the measurement is one of the main causes of measurement errors. This state change can happen in two cases: one is from $|{1}\rangle$ to $|{0}\rangle$, and the other is from $|{0}\rangle$ to $|{1}\rangle$. If an off-resonant transition from $|{1}\rangle$ occurs and the excited electron decays to $|{0}\rangle$ before any photons are detected by the detector, it is impossible to determine the quantum state correctly. Similarly, if an off-resonant transition from $|{0}\rangle$ occurs and the spin flips to $|{1}\rangle$, then the ion emits lots of photons by the cycling transition, making accurate quantum state identification impossible.

Figure 1 illustrates simplified ¹⁷¹Yb⁺ energy levels. The two hyperfine levels ${^2}{S_{1/2}}\,|F = 0,\; {m_F} = 0\rangle$ and ${^2}{S_{1/2}}\,|F = 1,\; {m_F} = 0\rangle$ are encoded as $|{0}\rangle$ and $|{1}\rangle$, respectively. The detection beam drives a cycling transition between ${^2}{S_{1/2}}$ $|F = 1\rangle$ and ${^2}{P_{1/2}}$ $|F = 0\rangle$. As shown in Fig. 1, since the dipole transition between ${^2}{S_{1/2}}$ $|F = 0\rangle$ and ${^2}{P_{1/2}}$ $|F = 0\rangle$ is forbidden by the selection rule, the off-resonant transition from $|{1}\rangle$ requires a 2.1 GHz detuning, whereas that from $|{0}\rangle$ requires a 14.7 GHz detuning, which combines the two hyperfine splittings of the ${^2}{S_{1/2}}$ and ${^2}{P_{1/2}}$ levels. Potential decays to the ²D_3/2 manifold (0.05%) are repumped to the ²S_1/2 manifold using a 935-nm laser.

 

Fig. 1. Simplified energy levels of ¹⁷¹Yb ⁺. Two hyperfine levels of ${^2}{S_{1/2}}$ are encoded as $|1 \rangle =|F = 1,\,{m_F} = 0\rangle$ and $|0 \rangle =|F = 0,\,{m_F} = 0\rangle$. The separation of the two levels is 12.6 GHz with forbidden electrical dipole transition. Due to the selection rule, the off-resonant transition from $|{1}\rangle$ requires a 2.1 GHz detuned transition, while a 14.7 GHz detuned transition is necessary for the off-resonant transition from $|{0}\rangle$. Possible decays to the ²D_3/2 manifolds during the measurement are repumped by the 935-nm laser as illustrated.

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The relatively small value of the hyperfine splitting of ${^2}{P_{1/2}}$ levels at 2.1 GHz imposes a fundamental limit on the state measurements for ¹⁷¹Yb⁺ ions. This limits the measurement fidelity to less than 99.9% for a typical optical setup where photon collection efficiency ranges from 0.01 to 0.02 for UV light [33]. Moreover, due to the state change to $|{0}\rangle $, the measurement fidelity for $|1\rangle $ tends to saturate beyond a certain detection time, depending on the experimental setup [34]. On the contrary, the large detuning from |0〉 for off-resonant transition makes background noise or dark counts of the detector the dominant error sources, which increases the error for $|0 \rangle$ as the detection time increases [34].

Due to the challenge of interpreting signals based on the number of photons using the EMCCD, we employed a PMT to set the detection time. The measurement fidelity of a single ion was measured using a PMT and a threshold of 0.5 photons was applied to determine the $|1\rangle $ state. The measurement fidelity of the $|1\rangle $ state reached saturation at 250 µs, with a mean of 21.23${\pm} $6.82 measured photons. We presumed that the amount of information to determine the $|1\rangle $ state also saturates at that time. Further increasing the detection time was not preferred since it only increases the error from the $|0\rangle $ state. As a result, we set the detection time of the EMCCD to 250 µs accordingly.

To calculate the detection error probability, dark state pumping rate, and bright state pumping rate were measured as described in Ref. [34]. According to Ref. [35], the photon scattering rate of the $|{1}\rangle$ state can be maximized by adjusting the polarization of the detection beam and Zeeman splitting of the hyperfine levels. When the polarization angle is set to $\arccos \left( {1/\sqrt 3 } \right)\; $ with respect to the quantization axis and the Zeeman splitting is set to one-half of the Rabi frequency of the Rabi oscillation between ${^2}{S_{1/2}}$ $|F = 1\rangle$ and ${^2}{P_{1/2}}$ $|F = 0\rangle$, the rate that the initial state is pumped to a different state can be approximated as [34]:

Another significant source of measurement errors is the noise from the EMCCD camera. Since the amplification of the photoelectrons in the EM gain register stochastically occurs [12], the number of electrons at the end of the gain registers has some ambiguity, resulting in broadened histograms of the EMCCD signals. This resulting broadening makes it challenging to determine threshold values based on the number of photons. Moreover, the photoelectron signals are spatially distributed within the region of interest (ROI) of the ions, requiring spatial analysis when the state measurement is performed. However, since the measurement is carried out within the shot-noise limit, the spatial distribution of signals can be arbitrary in each measurement, making it difficult to set the optimal ROI for ions. Therefore, reducing measurement errors caused by EMCCD noise requires improvements in the analysis algorithm.

Two conventional methods used for state measurements with EMCCD are the threshold method and the MLE method [12]. These methods rely on histograms of signals, which can be obtained through simulations based on pixel characterizations or from extensive measurement data. Consequently, the performance of these methods depends on pixel characteristics. The threshold method is a simple approach that compares the sum of the values within an ROI to a threshold value, which is determined by minimizing overlaps between signal histograms of the $|{0}\rangle$ and the $|{1}\rangle$ state [17]. If the sum falls below the threshold, then the state is determined to be $|{0}\rangle$; otherwise, it is $|{1}\rangle$. On the other hand, the MLE method utilizes probability distributions of the signals from the data histograms. If the probability of an image being in $|{1}\rangle $ is higher than that of being in $|{0}\rangle$ the state is determined to be $|{1}\rangle$; otherwise, it is $|{0}\rangle$.

The performance of these conventional methods can be adversely affected by any misalignment or mechanical drift in the optical system. Their effectiveness relies on specific pixel characteristics within the ROI for determining the state of the ion, and any changes in the pixel positions can negatively impact their performance. The long-term drift of ion images can be caused by various factors, such as potential changes by electrical drift [36,37], drift in the optical system caused by temperature or humidity change, mechanical stress on the optical mounts, or aging of the optical components [38,39].

To overcome this limitation, measurement methods can be developed that do not depend on fixed ROIs and consider both the pixel values and the shape of the ion images, similar to the CNN approach. Section 5 compares the vulnerability of the conventional methods with that of CNN for ion state measurement by testing the performance of the data that is obtained from different pixels on the EMCCD with intentional shifts in the imaging system.

3. Setup

3.1 Experimental setup

The experimental setup for the state detection of trapped ions is shown in Fig. 2, which employs a surface-electrode ion trap as described in Ref. [40]. The ion trap is situated inside an ultra-high vacuum (UHV) chamber. The setup involves the use of lasers with three different wavelengths. A 399-nm laser is used for the isotope-selective ionization of neutral ytterbium atoms from an oven. A 369-nm laser is employed for ion cooling and quantum state measurement. Additionally, a 935-nm laser is utilized to repump the electrons of ions that occasionally decay from the cycling transition to the ${^2}{D_{3/2}}$ manifold.

 

Fig. 2. The simplified diagram of the experimental setup. A microwave horn antenna installed near UHV produces a microwave to control the quantum states of the trapped ions globally. The emitted photons from trapped ions are collected by an imaging lens (NA = 0.6) and detected by an EMCCD which is placed at the focus of the imaging lens.

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In addition, to simultaneously trap two different isotopes of ytterbium ions, two independent 369-nm lasers are combined using a polarized-beam splitter (PBS) before entering the vacuum chamber. Each laser is equipped with an independent acousto-optic modulator (AOM), and the two lasers were activated in turn to identify the locations of isotopes along the ion chain.

To apply microwave radiation to the trapped ions, an antenna horn is installed near a viewport of the UHV chamber, which is carefully aligned to deliver maximum power to the ions. A magnetic field perpendicular to the trap surface is introduced to establish the quantization axis for the trapped ions and destabilize coherent dark states of hyperfine levels to maximize the photon scattering for state measurement [35].

A custom-designed diffraction-limited imaging lens (Photon Gear 15470-S) with a numerical aperture (NA) of 0.6 collects photons emitted by the trapped ions. The emitted photons collected by the imaging lens are focused onto the sensor of the EMCCD camera (Andor DU-897). The imaging system magnifies the object by a factor of 8.7.

To mitigate measurement noise, the temperature of the EMCCD is maintained at -90°C to suppress the CIC noise and thermally induced charge noise. The EMCCD offers two capturing modes: external triggering mode and internal triggering mode. In the external triggering mode, an external controller determines the capture timing, while in the internal triggering mode, the registers of the EMCCD determine when to capture. Due to the lack of time for the cleaning process, the external triggering mode tends to have more noise than the internal triggering mode [12]. Hence, the internal clock of the EMCCD is synchronized with the FPGA controller within 10 ns and the experiments are conducted using the internal triggering mode [17].

To evaluate the background noise, the EMCCD was enclosed in a black box, and the dark count was measured. The exposure time was fixed at 250 µs, and 200,000 data points were collected. Following the model described in Ref. [17], the false signal generation rate was calculated to be an average of 0.020 ± 0.004 per pixel for a single shot. Using this value as a baseline, the influence of the background was measured without the black box. The measured rate was 0.028 ± 0.005 per pixel, which indicates that the background noise from ambient light contributes only 0.008 false signals per pixel.

3.2 CNN architecture

Among the various CNN models, ResNet [30,41] is chosen as the base structure for our model. ResNet is known for its shortcut connection, which bypasses convolutional operations and adds the original information $\textrm{x}$ to the output of the convolutional operations $F(x )$ as shown in Fig. 3. This design aims to preserve the original information of the input and facilitate efficient gradient flow during backpropagation, speeding up the training process. The shortcut connection is expected to help overcome crosstalk by retaining the state information of the neighboring ions. When adjacent ions are in the $|{1}\rangle$ state, the emitted photons from those ions can affect the value of the pixels that contain the current ion, raising the total value. By transferring the state information of the neighboring ions through the shortcut connection, the influence of adjacent ions can be considered when determining the state of the target ion.

 

Fig. 3. The base unit structure of the ResNet. The overall CNN architecture consists of cascaded base structures. The input x corresponds to the $w \times h$ data, where w and h represent the width and height of the EMCCD data or a feature map within the model. The skip connection (denoted by a dashed line) bypasses the convolutional operations $F(x )$ for the input x and adds it to the output $F(x )$, which preserves the information of the input data. For the activation functions after convolutional operations, rectified linear unit (ReLU) is selected for a faster training process and fewer computational resources [42].

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Figure 4 shows the simplified representation of the designed model. Based on the ResNet structure, several factors are taken into account during the design of the model. Firstly, average pooling layers are chosen instead of max pooling layers since max pooling layers tend to preserve the CIC noise that has high signal intensities, resulting in lower measurement fidelity when the prepared state is $|{0}\rangle$. On the other hand, average pooling layers average out the CIC noise, particularly when the background values are low. Our experiments have shown that models with average pooling layers outperform those with max-pooling layers (Supplement 1). To minimize the number of parameters in the model while maximizing performance, the total number of layers is determined at the point where the performance saturates, resulting in a total of 51 layers. Secondly, setting the number of outputs to the number of possible states would exponentially increase the required number of outputs as ${2^n}$, where n is the number of ions. To ensure scalability and reduce the number of parameters in the FC layers, the number of outputs was set to be equal to the number of qubits. Lastly, since the qubit states are quantized, the model should output only 0 or 1. To achieve this, a custom sigmoid layer was added to each output of the FC layers, which gradually increased the slope of the sigmoid during training epochs. The custom sigmoid function is as follows:

 

Fig. 4. Simplified ResNet-based CNN structure. The raw $36 \times 36$ EMCCD images are provided to the model as input. The feature extraction stage contains convolutional layers and activation functions and average pooling layers. The convolutional layers filter the input images, the pooling layers reduce the spatial size of the inputs. The shortcut connections in the convolutional layers preserve the input information. The number of outputs is equal to the number of ions. Otherwise, lots of parameters and outputs are necessary as the number of ions increases. The activation function at the last layer is a customized sigmoid function, which ideally outputs quantized values only.

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When training the model with the custom sigmoid function, the hyperparameter a should be assigned with caution. If the slope of the function increases too rapidly due to a large value of a, the model tends to converge to a local minimum and predicts every state as 0.5 regardless of the input data. Conversely, if a is exceedingly small, the learning process becomes inefficient due to the low value of the loss function. This inefficient learning can lead the model to get stuck at one of the local minima, thus high performance cannot be achieved. The model used in this paper sets $a = 0.3$ and $b ={-} 0.4$.

During the acquisition of EMCCD data, only a small region of the EMCCD sensor is read to quickly transfer data using cropped sensor mode. We used a 36 × 36 pixel area out of the entire 512 × 512 pixel region, which includes the ion chain. As a result, the input data provided to the CNN model is an image with a size of $36 \times 36$.

The first convolutional layers consist of 64 filters and 128 filters corresponding to the batch normalization and the convolution layers shown in Fig. 4, which expands the $36 \times 36$ image to $36 \times 36 \times 128$ feature maps. Subsequently, the image size is halved after each pooling layer, while the depth is doubled. At the end of the feature extraction stage illustrated in Fig. 4, the model produces a $9 \times 9 \times 512$ array. This array is then utilized in the classification stage to map the quantum state of each ion.

For training the model, the model receives EMCCD data as input and the quantum state as the ground truth output. It is important to note that no ROI information is provided to the model. The model automatically learns the pixels that contain information about the ions from multiple training data. Therefore, if a well-labeled dataset is available, the CNN method does not require additional pixel-dependent analysis steps, unlike other conventional methods.

To prove that the skip connections help to improve the measurement fidelity, it is tested the performance of the model with and without the skip connections. The result showed that the model shows better performance with the skip connections (Supplement 1).

4. Experiment

The individual multi-qubit control is usually performed using a multi-channel AOM which can individually address the control beam to each ion [10] or by applying a strong magnetic field gradient to vary the energy levels of each ion [14]. However, our current experimental setup cannot control the individual quantum states of multiple ions with high fidelity. To circumvent this limitation, we have developed a practical method that utilizes the isotope of the qubit ion to generate images of multiple qubits using a global microwave.

In this method, the desired multi-qubit states are prepared by combining the qubit ions and the isotope ions. The qubit ions are prepared in the $|{0}\rangle$ states using optical pumping, and their states are then spin-flipped to the $|{1}\rangle$ states with high fidelity through a global microwave pulse. Then the qubit ions scatter lots of photons when exposed to the detection beam during the state detection. Conversely, the isotope ions barely interact with the detection beam remaining dark and representing the $|{0}\rangle$ states of the qubit ions.

Among the isotopes of ytterbium, ¹⁷⁰Yb⁺ is an ideal candidate for mimicking the $|{0}\rangle$ state of ¹⁷¹Yb ⁺. Table 1 illustrates the resonant frequencies between ${^2}{S_{1/2}}$ and ${^2}{P_{1/2}}$ of different isotopes, showing that ¹⁷⁰Yb⁺ has the farthest frequency from the detection beam. Moreover, the mass of ¹⁷⁰Yb⁺ differs from ¹⁷¹Yb⁺ by only one neutron, which is 0.58% of the mass of the qubit ion. The maximum position difference caused by replacing the ¹⁷¹Yb⁺ ions with ¹⁷⁰Yb⁺ is calculated to be 0.006 pixels for the four-ions case following the equations in Ref. [43].

Table 1. Isotopes of ytterbium ions and their transition frequency

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To ensure that the isotope can effectively imitate the $|{0}\rangle$ state of the qubit ion, we calculate the error induced by off-resonant transitions. Although the frequency difference (5.5 GHz) between the resonant frequency of ¹⁷⁰Yb⁺ and the detection beam is smaller than the frequency difference (14.7 GHz) for the case of ¹⁷¹Yb⁺ in the $|{0}\rangle$ state, the subsequent calculations confirm the probability of the off-resonant transition remains negligible.

The rate of the off-resonant transition for ¹⁷⁰Yb⁺ induced by the detection beam can be calculated using the transition equation, which is applicable when the detuning is large [34]:

The estimated probability of detecting more than zero photons from the off-resonant transitions of ¹⁷⁰Yb⁺ is calculated to be 0.06%. However, this calculated value does not directly correspond to the actual measurement error since the detection of a single photon does not lead to a guaranteed determination of the state as $|{1}\rangle$. On the other hand, even in the case where no ions are trapped, the error of 0.08 ± 0.02% per ion was measured using the trained CNN model. Therefore, we considered the error arising from the background to be more significant than the error caused by the off-resonant transitions of ¹⁷⁰Yb ⁺. Consequently, this validates the simulation of the $|{0}\rangle$ states using ¹⁷⁰Yb⁺ ions.

Before acquiring data for multi-qubit states, several factors associated with a microwave-based control have been checked. The resonant frequency of the clock state of the ¹⁷¹Yb⁺ ion, which corresponds to the magnetic-dipole-induced transition between ${^2}{\textrm{S}_{1/2}}\,|0,\; 0\rangle$ and ${^2}{\textrm{S}_{1/2}}\,|1,\; 0\rangle$, is then measured as shown in Fig. 5. The Rabi frequency at resonance is also measured to enable the preparation of an arbitrary state of the ion with high fidelity. The microwave has a wavelength of 2.3 cm, while the length of the ion chain is less than 20 µm. Thus, any strength or phase mismatch of the microwave among the ions is considered negligible when the microwave is radiated to multiple ions.

 

Fig. 5. Simplified energy level diagram of ¹⁷⁰Yb⁺ and ¹⁷¹Yb ⁺. The bold solid line represents the frequency of the detection beam and the thin solid lines indicate hyperfine splittings of ¹⁷¹Yb⁺, the bold dashed line represents the energy between ²S_1/2 and ²P_1/2 of ¹⁷⁰Yb⁺, and the dotted lines denote the detuning frequency for off-resonant transitions.

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The data acquisition sequence for multi-qubit states proceeds as follows: four ytterbium ions are trapped with a combination of ¹⁷⁰Yb⁺ and ¹⁷¹Yb⁺ to prepare the desired multi-qubit state. After the qubit ions are initialized by optical pumping, a microwave is applied through the antenna horn for half a period of Rabi oscillation to excite the qubit ions (${F_{\textrm{spin flip}}} = $99.98 ± 0.03%). The qubit ions are then prepared in the $|{1}\rangle$ state, while the isotope ions remain unaffected. Finally, the detection beam, with a resonant frequency between ${^2}{\textrm{S}_{1/2}}\,|1,\; 0\rangle$ and ${^2}{\textrm{P}_{1/2}}\,|0,\; 0\rangle$ of ¹⁷¹Yb⁺, is radiated. The ¹⁷¹Yb⁺ qubit ions emit photons, while the ¹⁷⁰Yb⁺ isotope ions remain dark. The procedure allows for the acquisition of labeled multi-qubit state measurement data, which is used for training and validating the CNN model. The experimental sequence is repeated over 65,000 times for each of the 16 possible states. During the data acquisition process, we periodically examined the arrangement of the isotopes to monitor any accidental swapping.

To evaluate the performance of the trained model and its robustness against long-term ion image drift, we adjusted the imaging lens intentionally to capture ion images using different areas of the sensor. Note that the pixels in the new region typically had slightly different characteristics compared to the pixels in the previously used crop area and there were minor variations in spacing, which is believed to originate from the small change of aberration.

When we cropped a new region of $36 \times 36$ pixels outside the original active area, we made sure that the ion chain will be imaged onto the center of the region so that the obtained images remain similar to those from previous measurements. The misalignments of the ion images compared to the previous images in the cropped region were quantitatively determined by fitting their images using a 2D Gaussian function, revealing an average shift of 0.21 pixels in the $x$-direction and 0.37 pixels in the $y$-direction. With the adjusted imaging system, the same acquisition procedure was repeated to capture similar data with different pixel areas.

Finally, to verify that the ResNet-based CNN model accurately determines the state of each ion, simultaneous Rabi oscillations were measured by trapping four ¹⁷¹Yb⁺ ions and applying a microwave. Since the wavelength of the microwave is much larger than the length of the ion chain, the Rabi oscillations of all four ions should be in phase. The observed coherence of the oscillation confirmed that the numbers of $|{1}\rangle$ states were measured following the expected statistics. Therefore, it can be concluded that the model accurately determines the individual quantum states of the ions.

5. Results

Figure 6 presents the measured multi-qubit EMCCD images. Figure 6(a) shows the averaged entire dataset which is utilized to determine the optimal ROI for each ion necessary for both the threshold method and the MLE method. Figure 6(b) shows exemplary data from all 16 possible states.

 

Fig. 6. Measured EMCCD data. The images are zoomed in to 16 × 16 pixels, which contain the ion images from the original 36 × 36 images for better visualization. (a) averaged total acquisition data. The red outline represents the optimal ROI for the threshold method and the white outline indicates the optimal ROI for the MLE method; The distance between the three marked pixels represents the distance in the object plane. (b) representative single-shot images of all possible 16 states of four ¹⁷¹Yb⁺ ions. The acquired data shows that the images of the ions have unexpected patterns since the acquisition is carried out in the shot noise limit. These arbitrary patterns can be a hint of discriminating crosstalk and false signals from the real photoelectron signals, which are not considered when threshold and MLE methods are applied.

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This section compares the results of the two conventional EMCCD methods with those of the machine learning-based CNN method. In addition, the robustness of these detection methods to the long-term drift in the optical image is assessed by applying them to newly obtained data without calibration for drift compensation. Finally, the simultaneous Rabi oscillations of four trapped ¹⁷¹Yb⁺ ions are measured to demonstrate that the CNN model accurately determines the state of each ion by showing the coherence of the oscillation.

In this work, two different types of fidelity definitions are used for a fair comparison: The mean measurement fidelity (MMF) is employed, as defined in Ref. [8], which is defined as:

5.1 Threshold method

The basic threshold method determines the quantum state of an ion by comparing the sum of measured pixel values within the ROI to a threshold value. If the sum is lower than the threshold, the state is determined as $|{0}\rangle$; otherwise, it is determined as $|{1}\rangle$.

To evaluate the fidelity of the threshold method, we applied the procedure described in Ref. [17] to our EMCCD data as follows: Using the average of all the acquisition data as shown in Fig. 6(a), the pixels within the image of each ion are sorted in decreasing order of signal intensity, which can ensure that the most informative pixels are given priority during the analysis. As an initial ROI for the first ion, we start with only a single pixel having the highest signal value, and the optimal threshold value is found for the initial ROI. Then, the procedure continues by incrementally adding pixels to the ROI and finding the threshold value until the highest fidelity for the first ion is achieved. The same procedure is repeated for the rest of the ions to obtain an optimal ROI for each ion, and they can be found in Fig. 6(a). The measurement fidelity of each ion depending on the number of ROI pixels, can be found in Supplement 1.

This simple threshold method can achieve a reasonable value of fidelity (MMF of 89.90 ± 0.22%), but we found that the fidelity can be improved by an iterative method introduced in Ref. [17]. The iterative method considers the crosstalk effect when determining the multi-qubit states, by using different optimal sets of ROI and threshold value depending on the states of the neighboring ions.

The iterative method starts by assuming the initial state as $|{0000}\rangle $ and performs the state measurement based on this initial state. If the measurement result does not agree with the initial state, the output state is taken as the new input trial state, and the state measurement is repeated with the parameters for the new trial state. This process is iterated until the measurement result agrees with the input trial state. Consequently, this method employs different parameters for the state measurement depending on the crosstalk, leading to improved measurement fidelity.

Figure 7 shows the measurement results for all the possible multi-qubit states. The MMF is calculated as 91.38 ± 0.34%, while the MIMF is 97.54 ± 0.54%. The uncertainties of the fidelities are calculated statistically.

 

Fig. 7. The MMF of each labeled state of 4 qubit ions; the MMFs (MIMFs) of each method are calculated as 91.38 ± 0.34%(97.51 ± 0.54%), 96.86 ± 0.09%(99.13 ± 0.08%), and 98.32 ± 0.10%(99.53 ± 0.14%) for the threshold method, MLE method, and CNN, respectively. The errors of the fidelities are statistically calculated.

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5.2 Maximum likelihood estimation method

The MLE method determines the quantum state of the ion based on the likelihood of the signals within the ion image. If the ion image of the obtained data has a higher likelihood of being generated by an ion in $|{1}\rangle$ than in $|{0}\rangle$, then the state is determined as $|{1}\rangle$, and vice versa. The probability distribution of each pixel value corresponding to $|{0}\rangle$ or $|{1}\rangle$ can be calculated from the histograms of pixel values. The histograms of the pixel values can be obtained from the extensive experimental data or the simulation as well [17]. In this work, the histograms were mainly obtained from the experimental data. However, occasionally when the number of data is not sufficient, the histograms were interpolated based on the theoretical model described in Ref. [17]. The method is explained in Supplement 1.

The likelihood from each quantum state can be calculated as follows:

The process of determining the optimal ROI for each ion is performed similarly to the threshold method. Starting from the pixel with the highest signal intensity, pixels are progressively added to the ROI to identify the configuration that maximizes the measurement fidelity. The optimal ROI for MLE can be also found in Fig. 6(a). The measurement fidelity of each ion, with respect to the number of pixels in the ROI, can be found in Supplement 1.

Similar to the threshold method, the iterative method described in Ref. [17] was applied to address crosstalk issues. The MMF of the MLE method is calculated as 96.86 ± 0.09% and the MIMF is calculated as 99.13 ± 0.08%. The uncertainties of the fidelities are calculated statistically.

5.3 ResNet-based CNN model

Although the experiment is usually performed with a fixed imaging system, small drifts of the ion image might occur over time. These small drifts in the ion image can lead to a decrease in measurement fidelity. The conventional methods of state measurement using EMCCD are vulnerable to this kind of drift since these methods rely on pixel-specific characterization methods [17]. However, neural networks like CNNs exhibit generalization, enabling them to exhibit good performance not only on the training data but also on unseen data that is similar to the training dataset [45]. Therefore, the CNN model used in this work is expected to maintain its performance even though some drift on the ion image occurs.

In a typical computer vision area with machine learning techniques, it is common to augment the training dataset by including artificially generated translated data during the training process of the model to mitigate the adverse effects of slight translations on performance [46]. Therefore, when the model was trained, adding simulated random translation data to the training dataset can be helpful. The translation is simulated within the ROIs where the potential crosstalk from neighboring ion ROIs was minimized.

However, when generating simulation data for data augmentation, a simple translation, where all ions are moved equally in the same direction, was ineffective for the dataset measured in the different region. We assumed that the simple translation represents the same characteristics as the experimental data because it was generated based on the experimental data. Therefore, to ensure that the simulation data has different characteristics from the experimental data, we randomly translated the center of each ion up to 0.3 pixels independently for every image generation.

The total number of experimental data was 1,078,000, with half of the data used for training the model and 20% used as validation data to prevent overfitting. The remaining data were used to calculate the measurement fidelity. Additionally, the number of generated simulation data for random translation is 20,000 for each state and these simulation data were added to the training dataset.

The MMF of the CNN model is calculated as 98.32 ± 0.10%, while the MIMF is calculated as 99.53 ± 0.14%. The errors are statistically calculated. We believe that the reason for the better performance of the CNN model compared to other methods is attributed to its ability to consider both the amount of the signal intensity and the shape of ion images simultaneously.

Based on the results of the CNN model, we calculated crosstalk. The crosstalk is defined in this work as the increase in the measurement error for the $|{0}\rangle$ state caused by the adjacent $|{1}\rangle$ state. To calculate crosstalk, using the $|{0000}\rangle $ state as a baseline, we evaluated the increase in measurement error for the target ion when neighboring ions were in the |1〉 state. For instance, to compute the crosstalk of the first ion (left-most ion), we compared the error of the first ion in the states $|{0100}\rangle $, $|{0101}\rangle $, $|{0110}\rangle $, and $|{0111}\rangle $ to the $|{0000}\rangle $ state. Then we averaged the increased errors. The calculated error increased by crosstalk for each ion is 0.08 ± 0.05%, 0.11 ± 0.04%, 0.10 ± 0.05%, and 0.07 ± 0.05%, respectively. The uncertainty of each crosstalk is calculated statistically.

The CNN model can achieve an MIMF of 99.19 ± 0.28% when trained with the 1% of the training dataset. It is interesting to note that it still shows reasonably good performance compared with the conventional methods. In the case of dealing with a large number of ions ($\textrm{N}$), it becomes time-consuming to acquire a training dataset as the possible quantum states increase exponentially with ${2^\textrm{N}}$. In such cases, reducing the size of the training dataset can be an option to address this issue. Although there may be a trade-off in performance, the machine learning-assisted method can still achieve high enough performance.

5.4 Simultaneous Rabi oscillation of 4 qubits

To verify whether the state detection for each ion is accurately made, simultaneous Rabi oscillations of four ¹⁷¹Yb⁺ ions by global microwave were measured using EMCCD. The oscillation results for each ion, shown in Fig. 8, clearly show that the 4 ions oscillate in phase. The measured Rabi oscillations are fitted to the following sine squared function:

 

Fig. 8. Simultaneous Rabi oscillations of four trapped ¹⁷¹Yb⁺ ions are measured to verify whether the CNN model determines the individual state more accurately compared to other methods. The result clearly shows that the qubit state of ions oscillates by the duration of the microwave. The mean Rabi oscillation result of 4 ions (a) shows that they oscillate in phase. Individual Rabi oscillation result is plotted in (b) and the plots are zoomed in to show the difference in amplitudes of each method. The errors in the fitted amplitudes of Rabi oscillations are obtained from the covariance matrix of the fitted parameters.

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The offset of the fitted sine squared function can be considered as an equivalent value of the detection error of the $|{0}\rangle$ state, which is on the order of 10⁻⁴. Thus, the amplitude of the oscillations is nearly equivalent to the measurement fidelity of the $|{1}\rangle$ state. The amplitudes of measurement results are consistent with the MIMF values, indicating that the CNN model accurately measures the quantum state of each ion.

5.5 Application to data collected with a shifted imaging system

Table 2 shows the MIMF values for the three different methods when applied to the data acquired with a shifted imaging system to test robustness against ion image drift. It should be noted that when the threshold method and MLE method were applied to the new data, the previous ROIs optimized for the original datasets cannot be reused due to the different ion spacing. Therefore, to make a fair comparison with the CNN result, we adjusted the positions of the ROIs by aligning their centers with the corresponding centers in the new data.

Table 2. Comparison of MIMF when applied to the data with a shifted imaging system. The errors of fidelities are statistically calculated.

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In contrast, since the CNN model does not require ROI information in the input data, no adjustment was made when new data was provided to the CNN model. Despite a slight decrease in performance, the fidelity remained reasonably high, indicating the robustness of the CNN model to different ion spacing of ions and pixel characteristics. This robustness is an advantage of the CNN method over the conventional methods in terms of the small drift of the ion images, ensuring reliable measurements over time.

6. Discussion and conclusion

The multi-qubit quantum states of four trapped ions were prepared with high fidelity using the bright state of the qubit ion ¹⁷¹Yb⁺ in $|{1}\rangle$ and the dark state represented by the isotope ¹⁷⁰Yb⁺ ion. These multi-qubit states were then measured using EMCCD and analyzed using a CNN model which is one of the popular machine-learning techniques. The CNN model is built based on ResNet architecture since the shortcut connection appropriately preserves the original information including the presence of adjacent ions. The measurement results obtained based on conventional methods are compared with those obtained by employing this machine-learning-assisted method. The machine-learning-assisted method reduced errors of MIMF by an average of 46% compared to the MLE method.

In a typical ion trap setup with multiple ¹⁷¹Yb⁺ ions for quantum computing, the ion spacing of an ion chain is generally determined by the secular frequencies from the requirement of two-qubit entangling gates [7,9], and therefore the ion spacing is generally limited to a few micrometers. The limitation of the short ion spacing leads to overlaps of ion images due to the discretized pixels of EMCCD, and a traditional approach to this problem is to use higher magnification to improve the separation of ion images, but this change inevitably leads to a lower signal-to-noise ratio (SNR) per pixel. If the exposure time is increased to compensate for the low SNR, the probability of bit flip error also increases in proportion to the exposure time for the case of ¹⁷¹Yb⁺ ions due to relatively large off-resonant transition probability.

To address these challenges, another option is to utilize moderate or low image magnification like this work. By opting for a lower magnification, a better SNR per pixel can be achieved within the limited detection time. To address the overlaps of the ion images, the shape information of the detected ion image can be utilized. Therefore, machine learning was employed to conduct state measurements, taking advantage of these techniques.

The experiments with the shifted imaging system demonstrated the generalized performance of the CNN model and its robustness to long-term drift. The result shows that the CNN model is robust to the long-term drift of the optical image compared with other methods, achieving an MIMF of 99.15 ± 0.08% without any pre-processing.

Furthermore, simultaneous Rabi oscillations of the four ¹⁷¹Yb⁺ ions were measured to validate the accuracy of the CNN model in determining the quantum state of each ion. The results were consistent with the individual measurement fidelity of each ion.