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On the quantum simulation of complex networks
by Duarte Magano, João Moutinho, Bruno Coutinho
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Submission summary
| Authors (as registered SciPost users): | Duarte Magano |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2212.06126v2 (pdf) |
| Date submitted: | 2023-02-10 11:29 |
| Submitted by: | Magano, Duarte |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
Quantum walks provide a natural framework to approach graph problems with quantum computers, exhibiting speedups over their classical counterparts for tasks such as the search for marked nodes or the prediction of missing links. Continuous-time quantum walk algorithms assume that we can simulate the dynamics of quantum systems where the Hamiltonian is given by the adjacency matrix of the graph. It is known that such can be simulated efficiently if the underlying graph is row-sparse and efficiently row-computable. While this is sufficient for many applications, it limits the applicability for this class of algorithms to study real world complex networks, which, among other properties, are characterized by the existence of a few densely connected nodes, called hubs. In other words, complex networks are typically not row-sparse, even though the average connectivity over all nodes can be very small. In this work, we extend the state-of-the-art results on quantum simulation to graphs that contain a small number of hubs, but that are otherwise sparse. Hopefully, our results may lead to new applications of quantum computing to network science.
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Reports on this Submission
Report
Overall, I think this is a well-written paper that tackles a fundamental problem and makes modest progress toward it by cleverly finding a solvable subproblem and solving it with a careful combination of known techniques. I recommend acceptance.
The detailed report is attached.
Anonymous Report 1 on 2023-6-6 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2212.06126v2, delivered 2023-06-06, doi: 10.21468/SciPost.Report.7311
Strengths
Nice extension of already known results.
Weaknesses
Too technical.
Report
This manuscript focuses on improved quantum algorithms to deal with
sparse complex network which may have few nodes densely connected
with the rest of the network.
On a technical level, the manuscript considers a very specific case and extend some results,
valid for sparse network, such that, also for slightly more complex network,
one can reduce the complexity of the computation to a poly-logarithmic scaling with the number of nodes.
As far as I understood, the idea is that the adjacency matrix A (Hamiltonian) can be split in such a way that
A = G + (-Am + Ah + Ar), with G easy to exponentiate. Actually G, if collecting all the hubs in the first row(columns), has a very simple structure. Basically spectral decomposition of matrix G is trivial (Appendix A is overcomplicated in my personal opinion).
Actually, the real novelty of the entire proposed approach is based on a trivial decomposition
together with a trivial understanding that exp(-i G t) can be easily prepared
via a series of local quantum operations which properly scale with the number of qubits.
All that summarised in Sec. 5. Then, finally, as already honestly mentioned by the authors,
the problem of solving the time dependent Schroedinger equation in the interaction picture has been
already addressed by Low and Wiebe in [25]. In practice, the only thing that they have still to ensure is an efficient block encoding of the interaction representation of Am(t), Ar(t), and Ah(t).
I would say that this manuscript provides a good description of the
underlying problem, with a reasonable selection of references.
However, from my personal point of view, it is hard to follow for a non specialist.
Although the findings are new and I have generally no complaints concerning the derivations,
I believe that they should be of interest to the experts in the field.
In addition, I have some concerns regarding the assumptions related to the amplitude amplification:
as a generalisation of Grover’s search, I was wandering whether it is not efficient because encoding the classical information to quantum data is slow and makes the overall process inefficient.
Moreover, a still open problem in this approach remain the usual assumption of having access to a series of
“Oracles”; in the specific case OA, OL, OH. In this sense, this could be the mayor limitation of the entire algorithm.
To conclude, I am sorry to say that I cannot argue that these findings, although interesting,
are of a sufficient magnitude, or are likely to be of sufficiently broad interest,
to warrant publication in SciPost Physics.
I would rather suggest that these results would fit better in SciPost Physics Core.
Finally some minor remarks:
1) Between eq (6) and eq(7), there is probably an unwonted repetition “.in the i-th row of A.”
2) In Sec. 6 : “Schödinger” —> “Schrödinger”