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

TL;DR

This paper presents new bounds and methods for decomposing bipartite and multipartite unitary gates into controlled unitaries, improving understanding of their structure and resource costs in quantum circuits.

Contribution

It introduces tighter bounds for decomposing complex unitaries into controlled gates, including special cases and connections to classical reversible circuits and communication complexity.

Findings

Any bipartite unitary can be decomposed into at most 4d_A-5 controlled unitaries.

Three controlled unitaries suffice to implement bipartite complex permutation operators.

Upper bounds for CNOT and entanglement costs relate to the Schmidt rank and the log-rank conjecture.

Abstract

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 unitary operators. 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 complex permutation operator, and discuss the connection to an analogous result on classical reversible circuits. We further show that any $n$-partite unitary on the space $\mathbb{C}^{d_1}\otimes...\otimes\mathbb{C}^{d_n}$ is the product of at most $[2\prod^{n-1}_{j=1}(2d_j-2)-1]$ controlled unitary gates, each of which 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 simple type of bipartite gates and some local unitaries. We derive dimension-independent upper bounds for the CNOT-gate cost or entanglement cost of bipartite permutation unitaries (with the help of ancillas of fixed size) as functions of the Schmidt rank of the unitary. It is shown that such costs under a simple protocol are related to the log-rank conjecture in communication complexity theory via the link of nonnegative rank.