Abstract
We study notions of complexity for link complement states in Chern Simons theory with compact gauge group $G$. Such states are obtained by the Euclidean path integral on the complement of $n$-component links inside a 3-manifold $M_3$. For the Abelian theory at level $k$ we find that a natural set of fundamental gates exists and one can identify the complexity as differences of linking numbers modulo $k$. Such linking numbers can be viewed as coordinates which embeds all link complement states into $\mathbb{Z}_k ^{\otimes n(n-1)/2}$ and the complexity is identified as the distance with respect to a particular norm. For non-Abelian Chern Simons theories, the situation is much more complicated. We focus here on torus link states and show that the problem can be reduced to defining complexity for a single knot complement state. We suggest a systematic way to choose a set of minimal universal generators for single knot complement states and then evaluate the complexity using such generators. A detailed illustration is shown for $SU(2)_k$ Chern Simons theory and the results can be extended to general compact gauge group.
Highlights
Quantum information concepts play an important role in high energy and gravitational research
We review the result that such states have GHZ-like structure and show that by introducing controlled-NOT (CNOT) operators the problem is reduced to defining complexity of a single knot state
We have studied the computational complexity of link complement states in Chern-Simons theory
Summary
Quantum information concepts play an important role in high energy and gravitational research. [11] considered Gaussian states in free field theory, which can be generated by a finite set of generators They test κ norms for κ ∈ Rþ as cost functions and conclude that complexity defined by κ 1⁄4 1 has the cutoff dependence most similar to wormhole volume [6]. There is a natural way to define the fundamental gates for Uð1Þk Chern-Simons theory so the complexity of the link complement states is well defined. In this case the complexity is directly connected to the Gauss linking numbers between components of the link Ln. In this case the complexity is directly connected to the Gauss linking numbers between components of the link Ln This observation provides the first example that topological properties can manifest in complexity.
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