# Optimized Unconventional Geometric Gates in Superconducting Circuits
## 1 Introduction  
Quantum computers can solve some complex problems that the classical ones cannot solve [1; 2], because of the coherence and entanglement of the quantum process. Constructing quantum computers requires a set of universal quantum gates, including a set of arbitrary one-qubit gates and a nontrivial two-qubit gate [3; 4]. However, the decoherence from the surrounding environment and the noise effects decrease the quantum gate fidelity. Thus, it is essential and challenging to construct quantum gates that have a high enough gate fidelity and a strong robustness against errors.  
In 1984, [PERSON] found the geometric phase in an adiabatic process, which was an element of the holonomy group defined by the connection in the parameters space [5]. Meanwhile, the geometric phase only depends on the geometric properties of the curve in the parameter space [6]. Hence, the quantum gates constructed by the geometric phase have a strong robustness against certain local errors [7; 8; 9]. However, the adiabatic condition requires the quantum system be exposed to the environment for a long time. In 1987, [PERSON] and [PERSON] generalized the Berry phase by relaxing the adiabatic condition [10]. The nonadiabatic geometric quantum computation (NGQC) needed less evolution time due to the nonadiabatic process. Since then, NGQC has been extensively studied theoretically [11; 12; 13; 14] and experimentally implemented in various quantum systems [15; 16; 17]. Furthermore, the Abelian NGQC can be promoted to the non-Abelian case [18], and has also been widely used to generate a quantum computation [19; 20; 21; 22; 23].  
The important task of geometric quantum computing is how to deal with dynamic phases that are not protected by geometric properties. There are two common approaches.  
The first is to design a special evolutionary path to completely eliminate the dynamic phase [24; 25; 26], which often requires a long evolutionary distance, thus exacerbating the decoherence effect caused by the external environment. For example, no matter how small the rotating gate is, the evolution path required in the orange-slice-shape-path (OSSP) scheme is formed by two lines of longitude.The second approach is unconventional geometric quantum computing, which manages to convert dynamical phases into geometric features. However, this method requires a complex system parameter design. Moreover, the ratio of the kinetic phase to the geometric phase is difficult to adjust in the latter scheme [27; 28].  
Now, we ask two questions. Inspired by general methods for producing nonadiabatic and holonomy quantum computations, whether we can also find a general approach to realize an unconventional geometric quantum computation [29; 30]. Here, the ratio of the dynamical phase to the geometric phase is arbitrarily adjusted. Assuming such a method exists, would different ratios affect the robustness of the quantum gates?  
In this work, we derive a general approach to implement quantum computing with unconventional geometries. This method can be understood as a mapping. As long as we input a set of pendant states and path parameters, we can get an accurate Hamiltonian and the ratio can be adjusted arbitrarily. Next, we optimized the ratio to improve the robustness of the quantum gates. Finally, we construct our ratio-optimized single-qubit T-gate, H-gate, and two-qubit controlled phase gate in superconducting circuits. We perform numerical simulations of quantum gates. The results show that the fidelities of the T-gate, H-gate and controlled phase gate reach 99.98%, 99.95%, and 99.85%, respectively. Therefore, our scheme provides a promising way to realize large-scale fault-tolerant quantum computation in superconducting circuits.