Techniques for fault-tolerant decomposition of a multicontrolled Toffoli gate

Abstract

Physical implementation of scalable quantum architectures faces an immense challenge in the form of fragile quantum states. To overcome it, quantum architectures with fault tolerance are desirable. These are achieved currently by using a surface code along with a transversal gate set. This indicates the need for decomposition of universal $n$-qubit multicontrolled Toffoli ($n$-MCT) gates using a transversal gate set. Additionally, the transversal non-Clifford phase gate incurs high latency, which makes it an important factor to consider during decomposition. Additionally, the decomposition of a large MCT gate without ancillas presents an additional hurdle. In this paper, we address both of these issues by introducing the $\mathrm{Clifford}+{Z}_{N}$ gate library. We present an ancilla-free decomposition of MCT gates with linear phase depth and quadratic phase count. Furthermore, we provide a technique for decomposition of MCT gates in unit phase depth using the $\mathrm{Clifford}+{Z}_{N}$ library, albeit at the cost of ancillary lines and exponential phase count.