Separation and approximate separation of multipartite quantum gates

The number of qubits of current quantum computers is one of the most dominating restrictions for applications. So it is naturally conceived to use two or more small capacity quantum computers to form a larger capacity quantum computing system by quantum parallel programming. To design the parallel program for quantum computers, the primary obstacle is to decompose quantum gates in the whole circuit to the tensor product of local gates. In the paper, we first devote to analyzing theoretically separability conditions of multipartite quantum gates on finite or infinite dimensional systems. Furthermore, we perform the separation experiments for $n$-qubit quantum gates on the IBM’s quantum computers by the software Q$|SI\rangle$. Not surprisedly, it is showed that there exist few separable ones among multipartite quantum gates. Therefore, we pay our attention to the approximate separation problems of multipartite gates, i.e., how a multipartite gate can be closed to separable ones.

💡 Research Summary

The paper addresses a fundamental obstacle in quantum parallel programming: the decomposition of a global multipartite quantum gate into a tensor product of local gates, which would enable the use of several small‑capacity quantum processors as a single larger system. After a brief motivation describing the scarcity of physical qubits on current devices and the emergence of quantum programming languages (QCL, Quipper, LIQUi|>, Q#, Q|SI⟩), the authors formulate the “separation problem” for a unitary operator U = exp(iH) acting on a composite Hilbert space ⊗{k=1}^n H_k. The Hamiltonian H is expressed as a finite sum of tensor‑product terms H = Σ{i=1}^{N_H} A_i^{(1)}⊗A_i^{(2)}⊗…⊗A_i^{(n)} with each A_i^{(j)} self‑adjoint.

The first major theoretical contribution is a complete necessary and sufficient condition for exact separability when H consists of a single tensor product term (N_H = 1). Theorem 2.2 shows that U can be written as C⊗D if and only if one of the factors A or B is proportional to the identity; the other factor must then be of the form exp