Let $H$ be a finite dimensional hilbert space. Let $L:H\otimes H\rightarrow H\otimes H$ be a unitary transformation. Then the equation
$$(L\otimes I)(I\otimes L)(L\otimes I)=(I\otimes L)(L\otimes I)(I\otimes L)$$
where $I:H\rightarrow H$ is the identity mapping is known as the Yang-Baxter equation.


We shall call a linear transformation $L:H\otimes H\rightarrow H\otimes H$ a universal quantum gate if there is some $n$ such that the unitary transformations of the form
$I^{\otimes i}\otimes L\otimes I^{\otimes (n-i-2)}$
along with the mappings $\lambda\cdot I^{\otimes n}$ where $|\lambda|=1$ generate a dense subgroup of the unitary group $U(2^{n})$.


Does there exist a universal quantum gate $L:H\otimes H\rightarrow H\otimes H$ which satisfies the Yang-Baxter equation? If so, then what is the least natural number $n$ so that there is a universal quantum gate $L:\mathbb{C}^{n}\otimes\mathbb{C}^{n}\rightarrow \mathbb{C}^{n}\otimes\mathbb{C}^{n}$ which satisfies the Yang-Baxter equation?


$\textbf{Motivation}$


If $L$ is a quantum gate which satisfies the Yang-Baxter equation, then for all $n$, the function $\phi:B\_{n}\rightarrow H^{\oplus n}$ defined by
$\phi(\sigma\_{i})=I^{\oplus(i-1)}\oplus L\oplus I^{\oplus(n-i-1)}$ is a group homomorphism where $B\_{n}$ denotes the $n$-strand braid group. 


If $L$ is also universal, then any quantum algorithm (called algorithm A) can be emulated by a unitary transformation $\phi(w)$ for some braid word $w$. However, if
$\sigma\_{i\_{1}}^{e\_{1}}...\sigma\_{i\_{n}}^{e\_{n}}$ is one of the normal forms for the braid for $w$, then the sequence of quantum gates
$(\phi(\sigma\_{i\_{1}})^{e\_{1}},...,\phi(\sigma\_{i\_{n}})^{e\_{n}})$ obfuscates the original quantum algorithm so that a person cannot figure out how algorithm A works simply by looking at the braid $\sigma\_{i\_{1}}^{e\_{1}}...\sigma\_{i\_{n}}^{e\_{n}}$.


The idea of using braid group representations to obfuscate quantum algorithms has been proposed in [this recent paper](https://arxiv.org/abs/1212.6458) where they have observed that the Fibonacci representation of braid groups could be used to obfuscate quantum algorithms. However, a universal unitary transformation that satisfies the Yang-Baxter transformation seems to be a more direct way of using braids to obfuscate quantum algorithms.



Are there any unitary matrices which satisfy the Yang-Baxter equation which are universal for quantum computation?
[Braiding Operators are Universal Quantum Gates](https://arxiv.org/abs/quant-ph/0401090), by Louis Kauffman and Samuel Lomonaco (2004)



> 
> In this paper, we prove that certain solutions to the Yang-Baxter equation together with local unitary two-dimensional operators form a universal set of quantum gates. In particular, we show that a single specific solution of the Yang-Baxter Equation, the Bell basis change matrix, is a universal gate.
> 
> 
> 


If we wish to do away with the local unitaries, and only use unitary solutions of the Yang-Baxter equation, then universal quantum computation is not known to be possible, because a classical simulation is available, see [Classical simulation of Yang-Baxter gates](https://arxiv.org/abs/1407.1361) (2004).


