Review Reports

Round 1

Reviewer 1 Report

This work establishes link between SL2(C) character varieties with algebraic surfaces and topological quantum computing. This connection is proposed to be an alternative to the concept of anyons. The authors use Hopf link and links showing similar character variety for this work. Character variety of Bianchi groups and singular fibers are considered. The authors claim and highlight that the topological quantum computing aspects can be incorporated in quantum gravity. This is a proof-of-principal detailed study and is suitable for this journal.

I could not find much to criticize except that the introduction section can be improved by expanding and giving more general background related to the work. The writing is clear, and the observations are well described, hence I recommend this work for publication.

Author Response

Thank you for your positive report. We did slight improvements in the presentation and the English all over the document.

Reviewer 2 Report

Report on symmetry-1701571:

The authors investigated the character varieties and algebraic surfaces for the topology of quantum computing. They consider a few singular fibers in Kodaira’s classification of minimal elliptic surfaces. The authors make use of the Enriques-Kodaira classification of algebraic surfaces and the related topological tools to make such surfaces explicit. Qutrit and two-qubit magic state computing, derived from the trefoil knot in their previous work, may be seen as topological quantum computing from the Hopf link. Such results may have potential applications in topological quantum information processing devices.

The authors anticipate the approach of connecting knot/link theory, algebraic surfaces and topological quantum computing. A magic state is a non-stabilizer pure state (a non-eigenstate of a Pauli group gate) that adds to stabilizer operations (Clifford group unitaries, preparations, and measurements) in order to ensure the universality (the possibility of getting an arbitrary quantum gate). The manuscript reports a significant advance and offers incremental improvement to existing work in the references. The similar particle-like behavior of topological defects in linear wave packets in photonic graphene, and hierarchy of nonlinear entanglement dynamics for continuous variables [Phys. Rev. Lett. 122, 233905 (2019); Phys. Rev. Lett. 127, 150502 (2021)] are studied theoretically and experimentally. Moreover, the possible experimental realization techniques should be clarified and discussed in the revised manuscript ? In addition, the English throughout this manuscript should be further revised to meet the standard of Symmetry. The authors should give a brief discussion on this issue in the introduction section to highlight the novelty of the current work.

The author’s work of the character varieties and algebraic surfaces for the topology of quantum computing are useful. Such surfaces are candidates for a new type of topological quantum computing different from anyons. After above improvement, I could recommend publication of the manuscript in Symmetry.

Author Response

Thank you for your detailed report. We did slight improvements in the presentation and the English all over the document.

The experimental references you propose happen to be far from our purpose of introducing a new formal view of topological quantum computing. Hopefully we can reach some proposals for experiments in the future in the line of work done by T. Asselmeyer-Maluga in references [9]-[10].

Reviewer 3 Report

The authors study the representation theory of finitely presented groups and the connection of SL2(C) character varieties to topological quantum computing. Their study is based on the Hopf link and links showing Hopf link character variety. The main topics are the study of links in the Bianchi family and links for singular fibers in an elliptic fibration.

The manuscript is well organized and the results and discussions are presented in a very clear manner.

I do recommend its publication in your journal.

 

Author Response

Thank you for your positive report. We did slight improvements in the presentation and the English all over the document.