Straddling-gates problem in multipartite quantum systems

Abstract

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