Learning Informative Latent Representation for Quantum State Tomography

Thank you for your constructive and insightful comments for helping us improve our paper. Please see below our response to your specific questions.

Q1: The idea of using transformer self-attention layers for QST is not strongly motivated, and hence not theoretically sound to me.

A1: Thank you very much for your comments. Compared with other networks (such as fully connected networks and convolutional neural networks), the transformer enables to characterize the correlation between different measurements. Like sentences in natural language, entangled quantum states feature long-range correlations among their constituent qubits. Results in revised Figure 4 could verify the superiority of the transformer structure than FCN and CNN. To make it more readable, we made the following modifications:

(page 3, revised version)

Inspired by the potential of attention layers to capture long-range correlation among their constituent qubits (Cha et. al., 2021), we design a transformer-based autoencoder with a pre-training technique to reconstruct quantum states using the intermediate latent extracted from imperfect data.

Q2: The model scales poorly with the number of qubits due to the exponential number of operators in a complete set of QST measurements, so the contribution is limited.

A2: Our method aims to deal with the imperfect measurement data usually encountered in practical applications. We admit that the proposed method does not aim to solve the exponential growth in the number of measurement operators. However, our method has demonstrated its advantage over other methods under limited measurement copies or incomplete measurements.

To demonstrate the effectiveness of our proposed method, we have implemented experiments on the IBM quantum devices \{ibmq\_{manila}} (Qiskit-IBM-Q only supports up to 5 qubits), with results summarized in Table 2 (see Page 8, Section 4.2 in the main text). Although our ILR model is trained using simulated data that do not consider the noise model in real quantum devices, the results show excellent robustness of our ILR model in practical applications compared with LRE and the IBM-built method.

In addition, we have presented the performance of predicting properties for higher qubits using two-qubit Pauli measurements on nearest-neighbor qubits, with $(n-1)*36$ measurement types, which does not introduce exponential scaling. The results in Table 1 and Table 2 (in the main text) demonstrate the scalable ability of the highly informative latent representation in predicting properties under resource-constrained conditions.

Q3: The latent representation contains a mixture of encoded features and raw input features.

A3: Figure 3 provides a diagrammatic sketch that may lead to confusion. In reality, the symbol $\mathbf{O}$ undergoes initial embedding through a linear layer before being added to the encoder feature. This process is elaborated on page 5, where it states, "In the decoding phase, we linearly embed the missed operators $(\mathbf{O} \setminus \mathbf{\widetilde{O}}) \in \mathbb{R}^{m\times 2d^2}$, which are omitted in the encoder input, as remedy tokens added to $\hat{\mathbf{E}}$." Therefore, it's important to note that $\mathbf{O}$ does not represent raw input features; instead, it is embedded, aligning with the standard approach in transformer-based models.

I would like to thank the authors for their efforts on clarifying the questions and improving the paper. Most of my concerns have been addressed. However, I am still not convinced of the motivation and soundness of the proposed approach, so I will raise the score to 5.

Some minor points:

• Q3: Do I understand it correctly that the missed operators are each embedded to dimension $2d^2$ and linearly projected to L-dim vector? If so, it means the original operators and missed operators are not rendered equally, and this may still lead to minor theoretical issues.

-Q5: I think the same model structure can still be used to estimate the density matrix without having the missing operators in the decoder (correct me if I am wrong). I ask the question expecting to see how the model performs in comparison to pre-training (naively predicting p from f for existing operators) and prediction without involving any missing operators, because that indicates the benefit of introducing missing operators in decoder. Similarly, the pre-trained task is different from the training task, so I think it may be helpful to compare it against pre-training with the target of estimating $\rho$ or $\mu$ (or even both) to see why it makes sense.

Thank you for your constructive and insightful comments for helping us improve our paper. Please see below our response to your specific questions.

Q1: The scalability of the proposed model for large-scale quantum systems.

A1: Our method aims to deal with the imperfect measurement data usually encountered in practical applications. We admit that the proposed method does not aim to solve the exponential growth in the number of measurement operators. However, our method has demonstrated its advantage over other methods under limited measurement copies or incomplete measurements. In addition, we have presented the performance of predicting properties for higher qubits using two-qubit Pauli measurements on nearest-neighbor qubits, with $(n-1)*36$ measurement types, which does not introduce exponential scaling. The results in Table 1 and Table 2 (in the main text) demonstrate the scalable ability of the highly informative latent representation in predicting properties under resource-constrained conditions.

Q2: The experiments about QST in this paper are limited to 2-qubit and 4-qubit quantum states. Even for 4-qubit pure states, and no operators are masked, the reconstruction fidelity is approximately $1-e^{-2} \sim 0.89$, which is not high.

A2: Thank you very much for your comments. There seems to have confusion about the definition of log of infidelity. In this work, all of the values regarding log of infidelity are actually obtained using $\log_{10}(\cdot)$ rather than $\log_e(\cdot)$. In that case, the current value for -2, actually corresponds to the fidelity value of $1-10^{-2} \sim 0.99$. By comparsion, the other methods achieve values about $1-10^{-1} \sim 0.9$. In addition, when masking 40% of the operators, our method achieves the result better than other methods under no masking operators. Those results in Fig. 4 (b) demonstrate the superiority of our method in dealing with few measurement copies.

To make it more readable, we have made the following modifications:

(page 4, revised version)

The log of infidelity is measured as $\bar{F}=\log_{10}(1-F(\rho,\hat{\rho}))\in \mathbb{R}$,

Q3: The proposed model lacks novelty in some sense. While it incorporates a transformer architecture, the fundamental encoder-decoder framework closely resembles those found in existing references [1] and [2] for quantum state tomography and quantum state learning.

A3: Thank you very much for your comments. There is a significant difference between our method and the other two methods.

Compared with the method in [1], our method aims to learn a latent representation from imperfect measured data, which can be subsequently applied to different tasks, while their method is specifically designed for reconstructing density matrices. In addition, their method introduces a discriminator to distinguish between fake data and real data, while we introduce a pre-training strategy together with a shared transformer autoencoder and decoders. Compared with GAN which is difficult to train, our approach adopts a pre-training strategy to achieve a good performance in the fine-tuning stage.

Compared with the method in [2], our method is a general approach that can be applied to different tasks. Their method aims to predict measured statistics of unseen measurements, which still require further steps to obtain the density matrix or properties of quantum states. Owing to the different goals, our method and the method in [2] exhibit distinctions regarding the latent representation, the architecture, and the training procedure.

(1) Latent representations: Their method aims to learn a lower-dimensional representation of quantum states to predict measured statistics of new measurements. Our method aims to take advantage of the expressive ability of highly informative representation in the latent space to characterize quantum states under resource-constrained conditions.

(2) Network architectures: Their method utilizes an FCN-based representation network and an LSTM-based generation network to learn the measured statistics of new measurements, while our method utilizes a transformer-encoder with global reception fields to extract intrinsic features from imperfect data and a transformer-decoder to characterize quantum states in different levels.

(3) Training procedures: Their method trains the combination of the representation network and the generation network as a whole, while our method trains the model within two stages, firstly pre-training and then fine-tuning, with the shared parts trained twice. This is a common practice in the ML community to make the best use of data to bring in improved generalization for different tasks.

Thank you for your constructive and insightful comments for helping us improve our paper. Please see below our response to your specific questions.

Q1: The statement about informative measurement is not entirely precise. This structural aspect should be included in the introduction with two references.

A1: Thank you very much for your constructive comments. There exist some effective methods that are tailored for special quantum states. Our method is designed for QST with limited measurements that may happen in practical QST scenarios and can be applied to general quantum states without any restrictions. Following your valuable comments, we have made the following modifications:

(page 1, revised version)

Generally, to uniquely identify a quantum state, the measurements must be informatively complete to provide all the information about $\rho$~(Jeˇzek et al., 2003), except for the special case when density matrices of quantum states are low-rank (Haah et al., 2017) or take the form of matrix product operators (Qin et al., 2023).

Q2: it is advisable to use the notation $2^n$ instead of just $d$. It would be beneficial to introduce the concepts of Hermitian, positive semidefinite (PSD) structure, and unit trace in the density matrix earlier in the section for improved clarity.

A2: Thank you very much for your constructive comments. We have made the following modifications:

(page 3, revised version)

QST is a process of extracting useful information about a quantum state based on a set of measurements. For a $n$-qubit quantum system with dimension $d=2^n$, the full mathematical representation of a quantum state can be described as a density matrix $\rho\in \mathbb{C}^{d\times d}$.

(page 1, revised version)

In the physical world, the full description of a quantum state can be represented as a density matrix $\rho$, i.e., a positive-definite Hermitian matrix with unit trace $\rho=\rho^{\dagger}$, $\textup{Tr}(\rho)=1$.

Q3: In Figure 2, due to the missing definition of the number of qubits, for readers without any quantum background, it is hard to compute the total number of density matrices.

A3: Thank you very much for your constructive comments. We have included the definition of the number of qubits in Section 3.2, with the following modifications:

(page 3, revised version)

QST is a process of extracting useful information about a quantum state based on a set of measurements. For a $n$-qubit quantum system with dimension $d=2^n$ , the full mathematical representation of a quantum state can be described as a density matrix $\rho\in \mathbb{C}^{d\times d}$.

In Figure 2, we perform QST on 2-qubit systems based on cube measurements with a total of 36 measurements. To make it more readable, we have made the following modifications:

(page 4, revised version)

Specifically, we observe a sharp decrease in infidelity as the number of copies increases, while masking 8 operators among the original complete cube measurements (with a total of 36 measurements) results in a large increase in infidelity. The underlying reasons might be that the noise addition and information loss in the imperfect measurement data result in a diminished representation of quantum states.

Q4: A systematic examination involving a wider spectrum of advanced deep learning methods is imperative. Especially GAN-based method [2]

A4: Thank you very much for your constructive comments. To demonstrate the effectiveness of our method, we have implemented two additional deep learning methods, the convolutional neural networks (CNN)-based QST [1] and GAN-based method [2] (PRL), with results provided in Figure 4 (page 8, revised version). Moreover, we include ILR without employing the pretraining strategy to ensure a fair comparison with other methods that do not utilize this specific strategy. In these ill-posed scenarios, the transformer-based autoencoder architecture outperforms FCN and CNN architectures. Moreover, the incorporation of a pre-training strategy significantly enhances the performance of the autoencoder architecture. Following your comments, we have made the following modifications:

(page 7, revised version)

Here, we implement the FCN (Ma et al., 2021), LRE (Qi et al., 2013), and MLE (Jeˇzek et al., 2003), CNN-based method (Lohani et al., 2020) and GAN-based QST method (Ahmed et al., 2021 for comparison.

Q5: Compared with Zhu et al. in 2022, about GHZ and W states with 8/12 qubits.

A5: We would like to compare these two methods within the same scenario. However, despite both methods considering GHZ and W states, their applications and settings differ, making it challenging to establish uniformity. Due to the challenges associated with implementing their method in our specific scenario, accomplishing a fair reconstruction of their methods will demand additional time and effort on our part. We are still working to address this aspect.

Before posting the comparison result, we currently analyze the differences between the two methods. Both our work and [Zhu et al. 2022]'s method explore the idea of extracting hidden features from imperfect measurement data to learn quantum states. However, their method aims to learn a lower-dimensional representation of quantum states from measured statistics and operators, and our method aims to learn an informative latent representation from noisy and incomplete measurements that helps solve practical problems.

Specifically, the two methods exhibit the following distinctions in their implementation:

1. Latent representations: Their method aims to learn a lower-dimensional representation of quantum states to predict measured statistics of new measurements. Our method aims to take advantage of the expressive ability of highly informative representation in the latent space to characterize quantum states from imperfect measurement data.

2. Network architectures: Their method utilizes an FCN-based representation network and an LSTM-based generation network to learn the measured statistics of new measurements, while our method utilizes a transformer-encoder with global reception fields to extract intrinsic features from imperfect data and a transformer-decoder to characterize quantum states in different levels.

3. Training procedures: Their method trains the combination of the representation network and the generation network as a whole, while our method trains the model within two stages, firstly pre-training and then fine-tuning, with the shared parts trained twice. This is a common practice in the ML community to make the best use of data to bring in improved generalization for different tasks.

[1] Sanjaya Lohani et al. Machine learning assisted quantum state estimation. Machine Learning: Science and Technology, 1(3): 035007, 2020.

[2] Ahmed Shahnawaz et al. Quantum state tomography with conditional generative adversarial networks. Physical Review Letters 127.14 (2021): 140502

[3] Yan Zhu et al. Flexible learning of quantum states with generative query neural networks. Nature Communications, 13 (1):1–10, 2022.

We kindly request that you inform us if there are any other concerns that might have been overlooked, or if you have new ones.

Thank you for your constructive and insightful comments for helping us improve our paper. Please see below our response to your specific questions.

Q1: The motivation behind designing the auto-decoder structure is not entirely clear.

A1: Thank you very much for your comments. We drew inspiration from the implementation of mask autoencoder, but focusing on different fields and having different start points. The motivation behind this is that in practical applications, QST may be challenged by incomplete measurement settings or limited measurement copies to obtain inaccurate statistics. For example, in solid-state systems, the process associated with measuring one copy of a quantum state can be time-consuming, and implementing a sufficient number of measurement operators requires complex and costly experimental setups. Given a coherence time (beyond which quantum states may change), the total copies of identical states for measurements may be constrained. In this scenario, the measurement data may be a complete but inaccurate frequency vector (a few copies assigned to each measurement operator ) or an accurate but incomplete frequency vector (sufficient copies assigned to partial measurement operators that are experimentally easy to generate). The two factors are collectively referred to as imperfect scenarios (i.e., ill-posed problems) in this paper, which hinder the precise reconstruction of a quantum state (see Fig. 2, page 4 in the main text). Recognizing the representation reduction posed by the ill-posed problem, we turn to a powerful neural architecture to extract a latent representation that may compensate for imperfect data. Specifically, we propose a transformer-based autoencoder to address the challenge of imperfect measurement data in practical QST applications.

Q2: The use of a state decoder to predict state properties appears to introduce confusion.

A2: In this work, we aim to solve two problems using a unified framework, with the pre-training process implemented (see Fig. 3., page 5 in the main text). The pre-training process can retrieve high-quality frequencies from imperfect data to enhance the expressiveness of the latent representation. Then, based on the informative latent representation, we introduce a state decoder to reconstruct density matrices and a property decoder to predict properties, respectively. For each task, its decoder is fine-tuned separately, with the pre-trained part i.e., the encoder part shared, which can reduce the model size. We also implement experiments in Appendix E to verify that directly using a property predictor can obtain better performance than calculating the properties from the reconstructed density matrices.

Additionally, we consider that in practical applications, implementing a sufficient number of measurement operators requires complex and costly experimental setups, and measuring large copies of a quantum state can be time-consuming. The masked operation serves the purpose of diminishing the total sets of measurement operators, while the pre-training strategy aims to minimize the number of measurement shots required for each state. In fact, reducing the number of measurement copies may demonstrate the effective performance of our method.

Q3: It remains uncertain whether the purported contributions and advantages can be effectively realized in practical applications.?

A3: Our method is designed for practical applications that suffer from limited measurement resources, i.e., limited copies or incomplete measurements that are usually encountered in practical applications. To demonstrate the effectiveness of our proposed method, we have implemented experiments on the IBM quantum devices $ibmq\_{manila}$, with results summarized in Table 2 (see Page 8, Section 4.2 in the main text). Although our ILR model is trained using simulated data that do not consider the noise model in real quantum devices, the results show excellent robustness of our ILR model in practical applications compared with LRE and the IBM-built method.

Thanks for the reply. While measured statistics enable both full and partial estimation, theoretical investigations reveal an exponential gap in the required number of measurements for achieving full versus partial estimation within a tolerable error. It is implausible for deep learning models to bridge this gap unless: 1) the authors claim that deep learning can efficiently and accurately simulate the dynamics of quantum computers; or 2) the explored states are highly restrictive, resulting in a simple low-dimensional representation (as seen in the numerical simulation). This is why I suggested the authors adopt the setting in [Zhu et al. 2022] to assess the proposed method's performance in learning a class of ground states around 50 qubits. I am willing to reconsider my evaluation if the authors provide compelling evidence demonstrating that deep learning can achieve a polynomial number of training examples and measurements per training example with respect to the qubit number in that scale. Btw, I never discount the use of pretraining models. My primary concern centers on the specific focus on two downstream tasks.

Dear Reviewer aFck,

I hope this message finds you well.

Thanks for your dedicated time to review our work. We have taken your comments seriously and have made the necessary explanations based on your suggestions. With the imminent closure of the discussion period, time is of the essence.

Given that our work received your’s less favorable scores in certain aspects, we worked diligently for a continuous span of seven days during the rebuttal period, demonstrating our unwavering commitment to addressing every facet of your feedback.

We kindly request your esteemed attention to our rebuttal response. Your responsible and positive endorsement holds paramount significance during this critical phase.

Wishing you all the best and awaiting your valued evaluation.

Best regards,

Authors

Thank you for your response. While I recognize your attempt to employ a single model for various quantum tasks, I remain unconvinced about employing one model for two markedly distinct types of tasks. Specifically, in quantum state tomography, the learning model needs to capture comprehensive information about the quantum state. In contrast, predicting the properties of quantum states may only require capturing partial information. Due to this concern, I maintain my score at 3.