Unitary and entangling solutions to the parametric Yang–Baxter equation in all dimensions

Abstract

We present a new class of solutions to the parameter-dependent Yang–Baxter equation across all dimensions, which includes a significant subclass of unitary and entangling solutions. In any dimension $d\ge 2$, we construct a well-structured matrix $R\left(u\right)$ that exhibits intriguing and useful symmetries. A key feature of $R\left(u\right)$ is its composition from four monomial-based layers: $R\left(u\right)=R^{a\left(u\right)}+R^{b\left(u\right)}+R^{x\left(u\right)}+R^{y\left(u\right)},$where $a\left(u\right)$, $b\left(u\right)$, $x\left(u\right)$ and $y\left(u\right)$ are four sets of monomial complex functions of the form, $a{u}^{n_a}$, $b{u}^{n_b}$, $x{u}^{n_x}$ and $y{u}^{n_y}$, respectively. This well-designed structure plays a crucial role in demonstrating that $R\left(u\right)$ is indeed a solution to the Yang–Baxter equation and in establishing the conditions for its unitarity and entangling properties. Additionally, it allows us to identify interesting non-trivial subfamilies with these properties.

As is widely recognized, unitary and entangling solutions to the Yang–Baxter equation serve as universal quantum logic gates for qudit quantum computing. Moreover, the search for new solutions to the Yang–Baxter equation in higher dimensions is a common endeavor in both mathematics and physics.