Quantum invariants of closed framed $3$-manifolds based on ideal triangulations
By S. Mihalache, Sakie Suzuki, Yuji Terashima. 2022
351 citations recorded by the source corpus.
We construct a new type of quantum invariant of closed framed $3$-manifolds with the vanishing first Betti number. The invariant is defined for any finite dimensional Hopf algebra, such as small quantum groups, and is based on ideal triangulations. We use the canonical element of the Heisenberg double, which satisfies a pentagon equation, and graphical representations of $3$-manifolds introduced by R. Benedetti and C. Petronio. The construction is simple and easy to be understood intuitively; the pentagon equation reflects the Pachner $(2,3)$ move of ideal triangulations and the non-involutiveness of the Hopf algebra reflects framings. For an involutory Hopf algebra, the invariant reduces to an invariant of closed combed $3$-manifolds. For an involutory unimodular counimodular Hopf algebra, the invariant reduces to the topological invariant of closed $3$-manifolds which is introduced in our previous paper. In this paper we formalize the construction using more generally a Hopf monoid in a symmetric pivotal category and use tensor networks for calculations.
Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/5bca393fb14214ef9f94621bfec855144c47af0b
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In Geometry and Topology
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- General parameter-shift rules for quantum gradients (2021, background): “First rising to prominence with the introduction of the variational quantum eigensolver (VQE) [7], they have evolved to cover topics such as optimization [8], quantum chemistry [9– 13], integer factorization [14], compilation [15], quantum control [16], matrix diagonalization [17, 18], and variation”
- Bounds on the capacity and power of quantum batteries (2018, background): “In this latter case, even though the bounds (9) are still respected, it will not always be possible to saturate them when the set of control operations is limited to unitary operations [18].”
- Quantum Stochastic Processes and Quantum non-Markovian Phenomena (2020, background): “For instance, the diamond norm is very convenient to work with when studying the statistical properties of quantum stochastic processes [284, 285].”
- The XZZX surface code (2020, background): “Fault-tolerant quantum computation by generalised lattice surgery [29, 47].”