Lower bound the T-count via unitary stabilizer nullity


We introduce magic measures for multi-qubit quantum gates and establish lower bounds on the non-Clifford resources for fault-tolerant quantum computation. First, we introduce the stabilizer nullity of an arbitrary multi-qubit unitary, which is based on the subgroup of the quotient Pauli group associated with the unitary. This unitary stabilizer nullity extends the state stabilizer nullity by Beverland et al. to a dynamic version. We in particular show this magic measure has desirable properties such as sub-additivity under composition and additivity under tensor product. Second, we prove that a given unitary’s stabilizer nullity is a lower bound for the T-count, utilizing the above properties in gate synthesis. Third, we compare the state and the unitary stabilizer nullity, proving that the lower bounds for the T-count obtained by the unitary stabilizer nullity are never less than the state stabilizer nullity. Moreover, we show an explicit $n$-qubit unitary family of unitary stabilizer nullity $2n$, which implies that its T-count is at least $2n$. This gives an example where the bounds derived by the unitary stabilizer nullity strictly outperform the state stabilizer nullity by a factor of $2$. We further connect the unitary stabilizer nullity and the state stabilizer nullity with auxiliary systems, showing that adding auxiliary systems and choosing proper stabilizer states can strictly improving the lower bound obtained by the state stabilizer nullity.

Physical Review Applied
Xin Wang
Xin Wang
Associate Professor

Prof. Xin Wang founded the QuAIR lab at HKUST(Guangzhou) in June 2023. His research primarily focuses on better understanding the limits of information processing with quantum systems and the power of quantum artificial intelligence. Prior to establishing the QuAIR lab, Prof. Wang was a Staff Researcher at the Institute for Quantum Computing at Baidu Research, where he concentrated on quantum computing research and the development of the Baidu Quantum Platform. Notably, he spearheaded the development of Paddle Quantum, a Python library designed for quantum machine learning. From 2018 to 2019, Prof. Wang held the position of Hartree Postdoctoral Fellow at the Joint Center for Quantum Information and Computer Science (QuICS) at the University of Maryland, College Park. He earned his doctorate in quantum information from the University of Technology Sydney in 2018, under the guidance of Prof. Runyao Duan and Prof. Andreas Winter. In 2014, Prof. Wang obtained his B.S. in mathematics (with Wu Yuzhang Honor) from Sichuan University.