In the stabilizer formalism of quantum computation, the stabilizer states and Clifford gates serve as classical objects since the Gottesman-Knill theorem asserts that any quantum circuit with these elements can be simulated efficiently by classical means. To achieve genuine quantum computation, the magic resource (i.e., nonstabilizer state or non-Clifford gate) is necessary. The quantum T gates are identified as fundamental operations in the paradigm Clifford+T. Among many quantifiers of the magic resource of quantum states, two basic and simple ones are the L1 norm of the characteristic function (Fourier transform) and Wigner negativity. When the magic resource is quantified by the L1 norm, the quantum T gate is optimal (for generating the magic resource from stabilizer states) among all diagonal gates in any prime dimensional system. When the magic resource is quantified by Wigner negativity, the qutrit T gate on C3 is optimal among all diagonal gates, and it remains open whether the quantum T gates are also optimal in other prime dimensional systems. Considering the operational significance of Wigner negativity in quantum computation, in this work, we investigate the optimal diagonal gates for generating Wigner negativity in the ququint system C5. In sharp contrast with the qutrit case C3, we demonstrate that the ququint T gate is suboptimal. Through theoretical analysis and numerical calculations, 500 distinct ququint diagonal gates are identified as optimal for generating Wigner negativity, which surpass the ququint T gate. All these gates are Clifford equivalent, exhibit special symmetries, and lie outside the Clifford hierarchy. We further explore the optimal gates in a family of diagonal gates lying in the Clifford hierarchy, revealing that, while the ququint T gate is not globally optimal, it can still generate considerable Wigner negativity.

Publication:

PHYSICAL REVIEW A

http://dx.doi.org/10.1103/d36z-tb7h

Author:

Lingxuan Feng

State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China

Shunlong Luo

School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China