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We show that any unitary operator on the $d_A\times d_B$ system ($d_A\ge 2$)can be decomposed into the product of at most $4d_A-5$ controlled unitaryoperators. The number can be reduced to $2d_A-1$ when $d_A$ is a power of two.We also prove that three controlled unitaries can implement a bipartite complexpermutation operator, and discuss the connection to an analogous result onclassical reversible circuits. We further show that any $n$-partite unitary onthe space $\mathbb{C}^{d_1}\otimes...\otimes\mathbb{C}^{d_n}$ is the product ofat most $[2\prod^{n-1}_{j=1}(2d_j-2)-1]$ controlled unitary gates, each ofwhich is controlled from $n-1$ systems. The number can be further reduced for$n=4$. We also decompose any bipartite unitary into the product of a simpletype of bipartite gates and some local unitaries. We derivedimension-independent upper bounds for the CNOT-gate cost or entanglement costof bipartite permutation unitaries (with the help of ancillas of fixed size) asfunctions of the Schmidt rank of the unitary. It is shown that such costs undera simple protocol are related to the log-rank conjecture in communicationcomplexity theory via the link of nonnegative rank.

Decomposition of bipartite and multipartite unitary gates into the product of controlled unitary gates