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Implementing discrete-time quantum walks on multi-dimensional arbitrary graphs in circuit quantum electrodynamics
by Qi-Ping Su , Chen-Hui Peng, Li Yu, Wei Feng, Guo-Qiang Zhang and Chui-Ping Yang
Submission summary
| Authors (as registered SciPost users): | QiPing Su |
| Submission information | |
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| Preprint Link: | scipost_202211_00003v1 (pdf) |
| Date submitted: | 2022-11-01 14:50 |
| Submitted by: | Su, QiPing |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
One of the final goals in quantum information science is to achieve large-scale quantum computing. Circuit QED (quantum electrodynamics) provides one of the best platforms for quantum computing. However, previous research focuses on implementing discrete-time quantum walks (DTQWs) on one-dimensional simple graphs based on circuit QED, which can only be used to realize quantum computing with a small size. To implement large-scale quantum computing, it becomes necessary and urgent to realize DTQWs on multi-dimensional graphs with arbitrary structures. We here propose a general protocol for realizing DTQWs on multi-dimensional arbitrary graphs based on circuit QED, where each graph node can have a different number of connected neighbor nodes. As an application, we numerically simulate a Grover walk search algorithm in a cubic graph. With decoherence considered, our simulation results fit well with the theoretical results. The protocol is universal and can be extended to accomplish the same task in a wide range of physical systems, which consist of natural or artificial atoms and optical or microwave cavities. This work paves an avenue to realize DTQWs on multi-dimensional arbitrary graphs, which could have broad applications in large-scale quantum computing and quantum simulation.
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Report
The Authors described a protocol for implementing discrete-time quantum walks (DTQWs) on multi-dimensional (including 1D) graphs. The implementation is described for circuit-QED systems with superconducting qudits and cavities, such that the qudits correspond to the nodes of a given graph, and the cavities serve as the edges connecting these nodes. To show the usefulness of the protocol for implementing quantum algorithms, the Authors conducted simulations of a Grover search algorithm on a cubic graph with eight elements. Their numerical results based on 500 random simulations suggest that the target element can be found with a probability approaching 1 even by assuming experimentally feasible values of relevant parameters. These include: the coupling strength (of $g/2\pi=100$MHz) between the neighboring qudits, the Rabi frequency (of $\Omega/2\pi=100$MHz), detuning (of $\Delta/2\pi=100$GHz), relaxation time $T_1=5\mu s$, and dephasing time $T_2=T_1/2$, which are feasible using the current circuit QED technologies (but seemingly using also other quantum technologies).
(1) I think the theory of quantum walks itself is applied properly. However, possible problems may be in the implementation of QED itself on many qudits. This is crucial, because the manuscript reports (as its main result) a proposal of a circuit-QED implementation, rather than a new theoretical fundamental result.
The analysis of the applied gate operations is quite general, as given in the subsections on Process I (bottom of page 4) and Process II (bottom of page 5). Thus, it seems that the operations are not limited to circuit-QED implementations. In my opinion, similar values of the above-mentioned parameters are also experimentally feasible for trapped ions or other systems allowing to experimentally reach the ratio of $g/\Omega=1$, as assumed in Fig. 5. Anyway, I would suggest to clarify the issue whether it is possible or not (see also the comments below) to use other platforms by applying the transformations described in the above-mentioned subsections.
(2) The assumed qudits must have many levels (more than 3) and the operations require many sequences and a very high precision. I do not know to what extent this is currently achievable, but the Authors should clearly address this issue in the manuscript.
(3) The Authors mentioned that their method allows for walks on arbitrary graphs, but they only discussed the construction of a single element from which the whole graph can be assembled. It is like discussing only two qubits together with 1- and 2-qubit gates, and then saying that a quantum computer can be assembled from those. Although this is mathematically correct, but extremely challenging concerning any physical implementations. Considering the complexity of single operations (see the previous point), the whole system will be even more prone to imperfections.
(4) Although the Authors consider a simulation for Grover's search walk and consider simple noise models, this analysis is done too superficially in my opinion.
(5) I feel that the paper in its present form does not discuss a realistic circuit-QED implementation. The main result is effectively a decomposition of abstract unitarity operations, which are needed to realize a quantum walk, into sequences that can be theoretically realized in various platforms.
The discussion of realistic constraints and noise effects is very limited. Indeed, its considered via a general Lindblad master equation (in Appendix B), which can be applied in the same form for implementations using other platforms and just by modifying the values of the relevant parameters. Thus, a Reader of the manuscript would like to know some more details specific to the chosen circuit-QED platform.
Finally, I must admit that the paper is relatively clearly and consistently written, so one can easily follow the presentation and understand the protocol even concerning many details.
In conclusion, I could recommend the publication of this work in SciPost if the manuscript was adequately revised according to at least some of the above-mentioned issues.