Paper: Node2ket: Efficient High-Dimensional Network Embedding in Quantum Hilbert Space (ICLR-2024)

Reviewer comment (Reproducibility):
There are gaps for actually making this algorithm "quantum-friendly," as it is generally not easy to load classical information into a quantum device.

Author response:
A1: Thanks for your feedback. We would like to emphasize that our approach is so far a classical one, aligning with the principles of quantum mechanics and potentials to be run on a quantum machine. We will provide a more detailed explanation below.

We agree with the reviewer's observation that amplitude encoding theoretically allows the use of $n$ quantum bits to encode $q=p^n$ dimensional inputs, but often requires deep quantum circuits, where $p$ denotes the dimension of a single quantum bit. This implies an exponential increase in circuit depth with the input dimension $q$, demanding all quantum bits to be entangled and be well-connected physically [1].

In contrast, in our paper, all quantum bits in the quantum system are organized into $C$ columns each containing $k$ qubits. **It is only required that each subsystem consisting of $k$ quantum bits has the capability to generate entanglement, while the $C$ subsystems are independent of each other.** Additionally, we appropriately configure parameters to ensure $q=(p^k)^C$, meaning we in fact use a subspace of the entire Hilbert space.

As a result, the depth of the circuit for data loading is **independent of $q$ and only depends on $C$**, which is adjustable. It is possible to leverage circuit depth affordable by NISQ by restricting $C$ as a small value (e.g. 8 in our paper). We hold the view that this demonstrates the quantum-friendly in terms of data loading in our approach.

[1] Plesch M, Brukner Č. Quantum-state preparation with universal gate decompositions[J]. Physical Review A, 2011, 83(3): 032302.

Reviewer comment (Novelty):
What is the significance of the embeddings being designed for "quantum Hilbert space"? To me, the relationship to quantum computation is not clear and seems more like an afterthought. Quantum computers are known to be good at problems with certain structure, and it's not clear to me what structure is being leveraged here (and what benefits are obtained as a result).

Author response:
A1: Thanks for your feedback. We would like to emphasize that our approach is so far a classical one, aligning with the principles of quantum mechanics and potentials to be run on a quantum machine. We will provide a more detailed explanation below.

We agree with the reviewer's observation that amplitude encoding theoretically allows the use of $n$ quantum bits to encode $q=p^n$ dimensional inputs, but often requires deep quantum circuits, where $p$ denotes the dimension of a single quantum bit. This implies an exponential increase in circuit depth with the input dimension $q$, demanding all quantum bits to be entangled and be well-connected physically [1].

In contrast, in our paper, all quantum bits in the quantum system are organized into $C$ columns each containing $k$ qubits. **It is only required that each subsystem consisting of $k$ quantum bits has the capability to generate entanglement, while the $C$ subsystems are independent of each other.** Additionally, we appropriately configure parameters to ensure $q=(p^k)^C$, meaning we in fact use a subspace of the entire Hilbert space.

As a result, the depth of the circuit for data loading is **independent of $q$ and only depends on $C$**, which is adjustable. It is possible to leverage circuit depth affordable by NISQ by restricting $C$ as a small value (e.g. 8 in our paper). We hold the view that this demonstrates the quantum-friendly in terms of data loading in our approach.

[1] Plesch M, Brukner Č. Quantum-state preparation with universal gate decompositions[J]. Physical Review A, 2011, 83(3): 032302.

Reviewer comment (Theory):
The fact that pure quantum states are normalized leads to the constraint $\|\mathbf{x}_i\|=1$. Is there any consideration for eliminating a global phase, which I assume would affect the embedding? Also, I wonder what embedding might be developed for more general quantum states, such as mixed states.

Author response:
A2: Though intuitively constraining $\|\mathbf{x}_i\|=1$ would erase part of information, **the erased information could be toxic for embedding learning**:

As proved by [3], with the skip-gram embedding loss, as the training goes on, the embedding model will converge at:

$$\lim\_{t\rightarrow \infty} \mathbf{x}\_i^{(t)\top} \mathbf{x}\_j^{(t)} = +\infty, \\\\ \lim\_{t\rightarrow \infty} \|\mathbf{x}\_i^{(t)} - \mathbf{x}\_j^{(t)}\| = 0.$$

where the nodes i and j are a positive node pair, and t denotes the training iteration. The results lead to:

$$\lim\_{t\rightarrow \infty} \|\mathbf{x}\_i^{(t)}\|^2 + \|\mathbf{x}\_i^{(t)}\|^2 = \lim_{t\rightarrow \infty} 2\mathbf{x}\_i^{(t)\top} \mathbf{x}\_j^{(t)} +\|\mathbf{x}\_i^{(t)} - \mathbf{x}\_j^{(t)}\|^2 = +\infty,$$

which indicates that the modulus of some embeddings would go infinity. By constraining $\|\mathbf{x}\_i\|=1$, the above problem would fade automatically. Also, some works [2] show that the constraint empirically works.

**Moreover, the most importantly, to make the node2ket program runnable, the constraint $\|\mathbf{x}\_i\|=1$ is empirially a must** as shown in the Table 11 (the row ''Remove constraint i)''), after removing which the program would crash.

[2] Meng, Yu, et al. "Spherical text embedding." Advances in neural information processing systems 32 (2019).

[3] H. Xiong, J. Yan and Z. Huang, "Learning Regularized Noise Contrastive Estimation for Robust Network Embedding," in IEEE Transactions on Knowledge and Data Engineering, vol. 35, no. 5, pp. 5017-5034, 1 May 2023, doi: 10.1109/TKDE.2022.3148284.

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A3: As the quantum states go general, the difficulty of training them on classic machines will greatly increase. So far the scheme of representing embeddings by product state is the one that simultaneously can be run by classical machines with super efficiency and also can be well interpreted with quantum mechanics. A ''slightly'' entangled state might be more classical-friendly than a mixed state. 

Thanks for your suggestions and we would consider trying that on a real quantum machine.

Reviewer comment (Writing):
Forgive my ignorance, but what is the definition of positive node pairs and negative nodes? Does it just mean the inner product is positive or negative?

Author response:
A4: These are concepts from contrastive learning, which, given a batch of samples, aims to embed positive pairs of samples close but negative pairs of samples distant in the embedding space. In the embedding learning, such positive sample pairs are called ''positive node pairs'', and for a node in the node pair, we would sample some nodes (e.g. by random sampling from the whole networks) to form negative node pairs, and we call these nodes ''negative nodes''. 

Thanks for your suggestion and we will make it more clear in the updated version.

Reviewer comment (Presentation):
Compared with Fig. 1, can the authors provide a more succinct plot to convey the main idea of the paper? Fig. 1 is still a bit busy which is useful yet a more direct illustration in the beginning of the paper is welcomed. Something like Fig. 2 is better.

Author response:
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### **Q3: Fig. 1 is still a bit busy.**

A3: Thanks for the suggestion, we will modify the paper accordingly.  

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Reviewer comment (Experiments):
Can the approach be useful for solving combinatorial problems especially for large-scale ones? As there is little attributes need to be considered in these cases thus it seems suited to the proposed methods?

Author response:
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### **Q2: Potentials in solving CO problems.** 

A2: Indeed solving CO problems with the network embedding technique is what I am trying to do right now. For example, in solving the very large traveling salesman problem, especially in a non-Euclidean space (which means the distances are not derived from Euclidean coordinates), even the strongest heuristic LKH will face the challenge of a prohibitively large searching space. I believe that NE which has a long history in dimension reduction [3] will play a strong role in shrinking the search space. 

[3] Yan S, Xu D, Zhang B, et al. Graph embedding and extensions: A general framework for dimensionality reduction[J]. IEEE transactions on pattern analysis and machine intelligence, 2006, 29(1): 40-51.