A Tracy-Singh product of Yang-Baxter gates is implemented by swaps around the parallel application of the two gates, and the associated Turaev invariants factor in the non-swapped primitive case.
Kassel, Quantum groups , Graduate Texts in Mathematics 155 (1995), Springer Verlag, Berlin
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On the Realization of quantum gates coming from the Tracy-Singh product
A Tracy-Singh product of Yang-Baxter gates is implemented by swaps around the parallel application of the two gates, and the associated Turaev invariants factor in the non-swapped primitive case.