Abstract
We compute an upper bound on the circuit complexity of quantum states in 3d Chern-Simons theory corresponding to certain classes of knots. Specifically, we deal with states in the torus Hilbert space of Chern-Simons that are the knot complements on the 3-sphere of arbitrary torus knots. These can be constructed from the unknot state by using the Hilbert space representation of the S and T modular transformations of the torus as fundamental gates. The upper bound is saturated in the semiclassical limit of Chern-Simons theory. The results are then generalized for a family of multi-component links that are obtained by “Hopf-linking” different torus knots. We also use the braid word presentation of knots to discuss states on the punctured sphere Hilbert space associated with 2-bridge knots and links. The calculations present interesting number theoretic features related with continued fraction representations of rational numbers. In particular, we show that the minimization procedure defining the complexity naturally leads to regular continued fractions, allowing a geometric interpretation of the results in the Farey tesselation of the upper-half plane. Finally, we relate our discussion to the framework of path integral optimization by generalizing the original argument to non-trivial topologies.
Highlights
In [2], Nielsen et al introduced a nice geometric approach to quantum complexity
These can be constructed from the unknot state by using the Hilbert space representation of the S and T modular transformations of the torus as fundamental gates
We propose a geometric interpretation of Cn,m in terms of geodesic paths on a graph connecting rational numbers, which has a natural representation in the upper-half plane
Summary
The Chern-Simons theory with gauge group G and level k, denoted by Gk, is defined on a (compact, connected, oriented) 3-manifold M by the action k. The level k is an integer in order to ensure gauge invariance of the path integral defining the quantum theory [33]. This action is topological in the sense that it is independent of the metric chosen in M. The expectation value of any product WR(L) ≡ i WRi(Ki) of Wilson loops computes a topological invariant of the link L = i Ki obtained by joining the (non-intersecting) knots Ki.3 This is calculated as usual by the path integral. We are interested in Chern-Simons theory defined on a topological 3-manifold M with a 2-torus as a boundary, ∂M = T 2 Any such M can be understood as the knot complement. The state |K contains all the Wilson loop knot invariants of the knot K at level k
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
