Comment

We appreciate the detailed feedback provided by the reviewers. We acknowledge the concerns and have addressed them comprehensively in our revised manuscript. Below, we respond specifically to the main points raised.

Discussion on error parameters

We have conducted comprehensive numerical experiments in Section 4.3, fully considering the influence of $\delta$, $\epsilon$, and the tomography strategy. These experiments were repeated five times to examine numerical stability and ensure the reliability of the results. We have expanded the parameter range and corrected a programming error identified during the execution of $l_{\infty}$ tomography. The revised results now include plots explicitly showing the relationship between model performance and tomography and computing errors. These plots demonstrate that the "multiple reuse" strategy maintains its advantage up to the point where the model fails.

Numerical Results with Transformer Depths

We acknowledge that the numerical experiments primarily focus on the impact of tomography errors rather than varying the transformer depths. the theoretical basis for our claim lies in the ability of the "multiple reuse" strategy to mitigate information loss, which is critical for constructing deeper networks, as discussed in Section 2. We will consider including numerical experiments varying transformer depths in future work to further substantiate this claim.

The Use of a Single Random Seed

In the revised manuscript, we have repeated all experiments five times and reported the mean and standard deviation to ensure robustness and account for randomness. This change addresses the concern about numerical stability and provides a clearer picture of the results' reliability.

Scope of the "Multiple Reuse" Strategy

We appreciate the suggestion to demonstrate the "multiple reuse" strategy on a lightweight architecture and simpler tasks. While this would provide a broader validation of its general applicability, we chose to focus on transformers due to their widespread use and have addressed some unique challenges with the nonlinear computations in quantum implementations. Nonetheless, we think that the "multiple reuse" strategy can be generalized to other architectures, and we plan to explore this direction in future work.

Long-Term Applicability and Practical Implementation Challenge

We acknowledge that several quantum technologies required by our approach—such as QRAM, amplitude encoding, and $l_\infty$ tomography—are not yet practically accessible. We have highlighted these limitations in the revised manuscript's Limitations section. While it is uncertain when fault-tolerant quantum computers will become available, we believe that the techniques we propose for linear computation and intermediate measurement optimization are valuable and could be applied to other frameworks beyond QViTs.

Comment

We appreciate the detailed feedback provided by the reviewers. We acknowledge the concerns and have addressed them comprehensively in our revised manuscript. Below, we respond specifically to the main points raised.

Riview of current methods

We have provided a review of current methods in the third paragraph of the Introduction, covering approaches based on variational networks and quantum linear algebra. These references include an overview of related works in the context of quantum transformers and residual connections in quantum neural networks.

Difference between the 'Multiple reuse' strategy with the available quantum residual connection.

While residual connections have been implemented in quantum neural networks, our proposed "Multiple Reuse" strategy is distinct. It leverages quantum arithmetic operations and a ladder quantum circuit design to enhance model performance. Unlike conventional quantum residual connections, which intuitively involve adding up the amplitudes of quantum states (e.g., arXiv:2401.15871), our approach integrates additional computational structures to ensure better propagation of information and improved training efficiency.

Comment

We appreciate the detailed feedback provided by the reviewers. We acknowledge the concerns and have addressed them comprehensively in our revised manuscript. Below, we respond specifically to the main points raised.

Novelty of the Work

First, we propose a Measurement Reuse strategy that effectively mitigates the issue of information loss caused by intermediate measurements. Second, the execution flow of our proposed quantum transformer goes beyond quantum linear algebra. We alternate between digital encoding and amplitude encoding during computations: the digital encoding calculations are performed using quantum arithmetic operations, while the amplitude encoding calculations leverage quantum linear algebra. This alternation optimizes the success rate of non-unitary operations executed via quantum linear algebra, which typically decreases exponentially with the number of layers. These innovations highlight the unique contributions of our work.

Addressing Residual Connections and Computational Challenges

The core idea of our Measurement Reuse strategy ensures an uninterrupted information pathway throughout the model's execution, despite the presence of intermediate measurements. Residual connections, in our context, can be viewed as two pathways: intermediate measurements are applied to only one pathway, enabling multiple reuses of the measured information. Our approach is distinct from the method discussed in arxiv:2401.15871, which remains within the framework of variational quantum circuits and cannot be directly applied to our quantum transformer model. This distinction is now clarified in the revised manuscript.

Theoretical Explanation and Justification

As outlined above, our Measurement Reuse strategy ensures that there is always an uninterrupted information pathway during the execution of the quantum transformer model. All intermediate measurement information can propagate through this pathway and be reused effectively. Figure 1 in the manuscript provides a visual illustration of this concept. By addressing the information loss issue caused by intermediate measurements as the model depth increases, our approach enables the training of deeper models. This directly supports the feasibility and advantages of our proposed quantum transformer in achieving deeper quantum model training.

Comment

We appreciate the detailed feedback provided by the reviewers. We acknowledge the concerns and have addressed them comprehensively in our revised manuscript. Below, we respond specifically to the main points raised.

Clarification of Quantum Transformer Implementation

Our quantum transformer alternates between digital encoding and amplitude encoding during computations. Digital encoding computations are executed using quantum arithmetic operations, while amplitude encoding computations utilize quantum linear algebra. The definitions of digital encoding and amplitude encoding are provided in the main text. The execution of the quantum transformer includes quantum-classical data conversion following the quantum attention layer. Specifically, the amplitude-encoded state output from the attention layer is sampled and stored in QRAM. The QRAM is then queried to prepare operations in digital encoding form, and computations proceed in digital encoding until the next attention layer. This process is elaborated in Appendix D of the revised manuscript.

Performance Impact of Quantum-Classical Data Conversion

We acknowledge that quantum-classical data conversion affects the performance of our proposed quantum transformer. To address this, we optimized the complexity of intermediate measurements in two ways:
First, we employed the $l_{\infty}$ sampling algorithm, whose complexity scales logarithmically with the dimension $n$ of the quantum state.
Second, we introduced the Measurement Reuse strategy, which mitigates information loss caused by intermediate measurements. This enables the model to remain trainable even when sampling errors are relatively high. These optimizations reduce the overall impact of quantum-classical data conversion on complexity.

Implementation on a Real Quantum Computer:

Our algorithm is designed for fault-tolerant quantum computers, which are not yet available for practical use. Therefore, it cannot currently be demonstrated on NISQ devices. We acknowledge this limitation and have added a discussion in the revised manuscript's final section. Additionally, details about our simulation implementation have been included in the updated text.

Quantum Noise Considerations

Since our algorithm is intended for execution on fault-tolerant quantum computers, quantum error correction techniques in such systems theoretically suppress errors caused by quantum noise. Thus, we did not explicitly address quantum noise in this work. However, we did consider statistical errors introduced by intermediate measurements and computational precision errors in quantum arithmetic operations. The impact of these errors on the performance of the quantum transformer has been validated through numerical experiments.

Pre-trained Model Dataset

The pre-trained QViT model was trained on the ImageNet-21k dataset. This information has been clarified in the revised manuscript.

Application Scope Beyond Image Classification

Thank you for this valuable suggestion. In this work, we primarily focused on image classification. Expanding the application of our quantum transformer to natural language processing is a promising direction, and we plan to explore this in future work.