# Is topological conventional computation possible?

A function $$f:X^{2}\rightarrow X^{2}$$ is said to satisfy the Yang-Baxter equation if $$(f\times\mathrm{Id}_{X})\circ(\mathrm{Id}_{X}\times f)\circ(f\times \mathrm{Id}_{X})=(\mathrm{Id}_{X}\times f)\circ (f\times \mathrm{Id}_{X})\circ(\mathrm{Id}_{X}\times f).$$

The positive braid monoid $$B_{n}^{+}$$ is the monoid presented with generators $$\sigma_{1},\dots,\sigma_{n-1}$$ and relations $$\sigma_{i}\sigma_{i+1}\sigma_{i}=\sigma_{i+1}\sigma_{i}\sigma_{i+1}$$ and $$\sigma_{i}\sigma_{j}=\sigma_{j}\sigma_{i}$$ whenever $$|i-j|>1$$. A positive braid $$b\in B_{n}^{+}$$ is said to be simple if it cannot be written in the form $$b=b_{0}\sigma_{i}^{2}b_{1}$$ for some $$i\in\{1,\dots,n-1\}$$.

If $$f$$ satisfies the Yang-Baxter equation, then define a monoid action $$\pi_{f,n}:X^{n}\times B_{n}\rightarrow X^{n}$$ by letting $$\pi_{f,n}((x_{1},\dots,x_{n}),\sigma_{i})=(x_{1},\dots,x_{i-1},f(x_{i},x_{i+1}),x_{i+2},\dots,x_{n}).$$

Does there exist

1. a function $$f:X^{2}\rightarrow X^{2}$$ with satisfying the Yang-Baxter equation where $$X$$ is finite,

2. some natural number $$c$$, and

3. for $$m,n$$ are natural numbers, functions $$i_{n,m}:\{0,1\}^{n}\rightarrow X^{m},j_{m,n}:X^{m}\rightarrow\{0,1\}^{n}$$ such that the maps $$(n,m,r)\mapsto i_{n,m}(r),(m,n,x)\mapsto j_{m,n}(x)$$ are computable in polynomial time such that

if $$C$$ is a circuit of width $$w$$ and depth $$d$$ that computes a function $$F_{C}:\{0,1\}^{m}\rightarrow\{0,1\}^{n}$$, then there is some $$v\leq w\cdot c$$ and positive braid $$b\in B_{v}$$ where $$F_{C}(x_{1},\dots,x_{m})=j_{v,n}(\pi_{f,n}(i_{m,v}(x_{1},\dots,x_{m}),b)))$$ for all inputs $$(x_{1},...,x_{m})\in\{0,1\}^{m}$$ such that $$b=b_{1}\dots b_{s}$$ for some simple braids $$b_{1},\dots,b_{s}$$ and some $$s\leq d\cdot c$$?