Straddling-gates problem in multipartite quantum systems

We study a variant of quantum circuit complexity, the binding complexity: Consider an $n$-qubit system divided into two sets ${k}_{1}, {k}_{2}$ qubits each (${k}_{1}\ensuremath{\le}{k}_{2}$) and gates within set are free; what is least cost two-qubit ``straddling'' for preparing arbitrary state, assuming no ancilla allowed? First, our work suggests that, without making assumptions on entanglement spectrum, $\mathrm{\ensuremath{\Theta}}({2}^{{k}_{1}})$ straddling always suffice. then prove any $\text{U}({2}^{n})$ unitary synthesis can be accomplished with $\mathrm{\ensuremath{\Theta}}({4}^{{k}_{1}})$ gates. Furthermore, we extend results to multipartite systems, show that $m$-partite Schmidt decomposable state has complexity linear in $m$, which hints its multiseparable property. This result not only resolves open problem posed by Vijay Balasubramanian, who was initially motivated = volume conjecture gravity, but also offers realistic applications distributed computation near future.