Citation: Ren, Yuanjie, Nathanan Tantivasadakarn, and Dominic J. Williamson. "Efficient Preparation of Solvable Anyons with Adaptive Quantum Circuits." arXiv preprint arXiv:2411.04985 (2024).
Web: https://arxiv.org/abs/2411.04985
Tags: Modular-tensor-categories, Kitaev-quantum-double, Abelian-anyons
This lovely paper shows that every solvable topological phase can be constructed using adaptive finite depth local unitary circuits. Here, solvable refers to the notion of solvable fusion category:
> Etingof, Pavel, Dmitri Nikshych, and Victor Ostrik. "Weakly group-theoretical and solvable fusion categories." Advances in Mathematics 226.1 (2011): 176-205.
Solvable fusion categories can always be obtained from the trivial category by a sequence of cyclic extensions. The authors argue that these cyclic extensions physically correspond to gauging cyclic symmetries. Thus, solvable phases can be obtained from the trivial phase by repeatedly gauging abelian symmetries. The authors then argue that gauging abelian symmetries can be implemented using generalized Krammers-Wannier type maps, which can be done using adaptive finite depth local unitary circuits.
By Mochon's work, non-solvable anyons can used to make a universal quantum computer. It doesn't feel like a coincidence that these non-solvable anyons are the only ones we can make using adaptive FDLU. The authors put it best: "An interesting unsolved challenge is to establish a firm connection between braiding universality and adaptive circuit depth lower bounds for anyon theories".
A fun aspect of this paper is that the algorithms are inherently tensor categorical. Suppose you want to implement the doubled SU(2)_4 phase. The first step of the algorithm in this paper is to search through the space of fusion categories to find a fusion category Morita equivalent to SU(2)_4 which is cyclically nilpotent. In this case, you can find a sequence of a Z3 extension and two Z2 extensions which makes a category Morita equivalent to SU(2)_4. Then, you use the gauging maps associated to this category to construct a string-net ground state. Then, you use the results of
> Lootens, Laurens, et al. "Mapping between Morita-equivalent string-net states with a constant depth quantum circuit." Physical Review B 105.8 (2022): 085130.
to get back to the desired SU(2)_4 string-net.
This sheds light on Nat's trick of implementing the D4 phase as a twisted Z2^3 quantum double. The twisted Z2^3 quantum double is Morita equivalent to D4, but it has smaller nilpotnency class.