On the Realization of quantum gates coming from the Tracy-Singh product

Abstract The Tracy-Singh product of matrices (TS) is an efficient tool, for the construction of gates, which preserves properties such as unitary, primitive and entangling. Moreover, if c and $$c'$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>c</mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:math> are both Yang-Baxter gates, then $$c \boxtimes c'$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>c</mml:mi> <mml:mo>⊠</mml:mo> <mml:msup> <mml:mi>c</mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:mrow> </mml:math> , their TS product, is also a Yang-Baxter gate. A natural question arises whether the TS product preserves enhanced YBE pairs and the existing connections between primitive gates and knot invariants. Another question we consider is how to implement the gates $$c \boxtimes c'$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>c</mml:mi> <mml:mo>⊠</mml:mo> <mml:msup> <mml:mi>c</mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:mrow> </mml:math> .