TQFTs and quantum computing (Q6573768)

Quantum computing is captured within the formalism of the monoidal subcategory of \(\boldsymbol{Vect}_{\mathbb{C}}\) generated by \(\mathbb{C}^{2}\), while topological quantum field theories à la \textit{M. Atiyah} [Publ. Math., Inst. Hautes Étud. Sci. 68, 175--186 (1988; Zbl 0692.53053)] are diagrams in \(\boldsymbol{Vect}_{\mathbb{C}}\) indexed by cobordisms. This paper, aiming to formalize the connection between the two theories, considers topological quantum field theories of thick tangled type, which can be regarded as a particular instance in a currently popular program of \textit{categorification} in mathematics. It is observed that the transport graphs are in fact quivers, giving rise to well-defined Nakajima quiver varieties in the sense of [\textit{H. Nakajima}, Duke Math. J. 76, No. 2, 365--416 (1994; Zbl 0826.17026)], which carry categorical actions of Lie algebras, both at the level of \(K\)-theory [\textit{H. Nakajima}, J. Am. Math. Soc. 14, No. 1, 145--238 (2001; Zbl 0981.17016)] and at a geometric level [\textit{S. Cautis} et al., Math. Ann. 357, No. 3, 805--854 (2013; Zbl 1284.14016)] being a potential source of gates. The authors argue that it suffices to consider double categories for obtaining a reasonable method for formulatiing quantum information in terms of topological field theories.\N\NThe authors equip cobordisms with machinery for producing linear maps by parallel transport along curves under a connection, assembling these structures into a higher category. The category \(\mathbb{F}\boldsymbol{Vect}_{\mathbb{C}}\) of finite-dimensional complex vector spaces and linear maps is given a suitable higher categorical structure. Quantum circuits are realized as images of cobordisms under monoidal functors from these modified cobordisms to \(\mathbb{F}\boldsymbol{Vect}_{\mathbb{C}}\), which are computed by taking parallel transports of vectors and then combining the results in a pattern encoded in this domain category.\N\NOne may ask how the constructions in this paper might be realized in a physical system of qubits, or whether there is a particular type of qubit that is better suited to these constructions than others. One candidate is provided by the synthetic hyperbolic lattices studied experimentally in [\textit{A. J. Kollár}, \textit{M. Fitzpatrick} and \textit{A. A. Houck}, ``Hyperbolic lattices in circuit quantum electrodynamics'', Nature 571, 45--50 (2019; \url{https://www.nature.com/articles/s41586-019-1348-3})] and theoretically in [\textit{J. Maciejko} and \textit{S. Rayan}, ``Hyperbolic band theory'', Sci. Adv. 7, No. 36 (2021; \url{https://www.science.org/doi/10.1126/sciadv.abe9170})]. The effective quantum electrodynamics of these materials take place on a Riemann surface of genus \(g\geq2\). Mathematically speaking, the curves arise from the quotient of the hyperbolic plane \(\mathbb{H}\) by the discrete Fuchsian group \(\Gamma \subset\mathrm{PSL}\left( 2\right) \) of translations of the lattice. These negatively-curved materials have a well-defined electronic band theory [loc. cit.] and a complete Bloch wave decomposition [\textit{J. Maciejko} and \textit{S. Rayan}, ``Automorphic Bloch theorems for hyperbolic lattices'', PNAS, Article ID 119 (9) e2116869119 (2022; \url{https://www.pnas.org/doi/10.1073/pnas.2116869119})] in terms of the representations of the fundamental group of the surface, which makes it possible to replace the eigenvalue problem for the Laplace-Beltrami operator and a periodic potential with genuinely topological data coming from the surface. Another recent article [\textit{E. Kienzle} and \textit{S. Rayan}, Adv. Math. 409, Part B, Article ID 108664, 53 p. (2022; Zbl 1535.81124)] presents a web of speculations and correspondences involving hyperbolic band theory and quantum field theories.