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 $\text{Clifford}+T$. Among many quantifiers of the magic resource of quantum states, two basic and simple ones are the $L^1$ norm of the characteristic function (Fourier transform) and Wigner negativity. When the magic resource is quantified by the $L^1$ 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 ${\mathbb{C}}^3$ 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 ${\mathbb{C}}^5$. In sharp contrast with the qutrit case ${\mathbb{C}}^3$, 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.

Close

PrevNext

View figure in article

• Quantum information theory