Recent Advances on Nonadiabatic Geometric Quantum Computation

TL;DR

This paper surveys nonadiabatic geometric quantum computation (NGQC) as a path to fast, robust quantum gates by exploiting geometric phases. It organizes NGQC within a unified path-design framework and examines multiple strategies to boost gate fidelity, including time-optimal control, short-path geometric gates, and dynamical decoupling, supplemented by numerical comparisons. It also analyzes robustness against local errors using composite pulses, optimal control, dynamical corrections, and doubly geometric control, highlighting trade-offs between speed, robustness, and decoherence. The review connects theory to experiment, outlines design principles for practical gate implementations, and points to future directions such as AI-assisted optimization and integration with error correction to enable scalable, fault-tolerant quantum computing.

Abstract

The geometric phase stands as a foundational concept in quantum physics, revealing deep connections between geometric structures and quantum dynamical evolution. Unlike dynamical phases, geometric phases exhibit intrinsic resilience to certain types of perturbation, making them particularly valuable for quantum information processing, where maintaining coherent quantum operations is essential. This article provides a review of geometric phases in the context of universal quantum gate construction, i.e., the geometric quantum computation (GQC), with special attention to recent progress in nonadiabatic implementations that enhance gate fidelity and/or operational robustness. We first review a unified theoretical framework that can encompass all existing nonadiabatic GQC approaches, then systematically examine the design principles of nonadiabatic geometric gates with a particular focus on how optimal control techniques can be leveraged to improve the accuracy and noise resistance. In addition, we conducted detailed numerical comparisons of various nonadiabatic GQC protocols, offering a quantitative assessment of their respective performance characteristics and practical limitations. Through this focused investigation, our aim is to provide researchers with both fundamental insights and practical guidance for advancing geometric approaches in quantum computing.

Figures (10)

• Figure 1: Illustration of evolution trajectories for a pair of evolution states $|\Psi_{1}(t)\rangle$ and $|\Psi_{2}(t)\rangle$ on Bloch sphere. $\mathcal{C}_1$ and $\mathcal{C}_2$ are the actual evolution path and the geodesic line that connects the initial point $\{\chi(0),\xi(0)\}$ and the final point $\{\chi(\tau),\xi(\tau)\}$ for $|\Psi_{1}(t)\rangle$, respectively. On the contrary, $\overline{\mathcal{C}}_1$ and $\overline{\mathcal{C}}_2$ are those for $|\Psi_{2}(t)\rangle$.
• Figure 2: Illustration of single-qubit geometric gates construction for orange-slice-shaped loop. (a) Evolution path of three segments forming an orange-slice-shaped loop for $|\Psi_1(t)\rangle$. (b) The energy structure for a driven transmon with a weak anharmonicity $\alpha$, where the lowest two energy levels are treated as the computational subspace.
• Figure 3: Comparative analysis between conventional NGQC and TOC-optimized schemes, illustrating (a) evolution trajectories and (b) gate duration differences. Reproduced from Ref. PhysRevApplied.14.064009
• Figure 4: Evolution trajectories for short-path NGQC schemes: (a) half-orange-slice loop li2021high, (b) triangular loop ding2021path, (c) circular loop ding2021nonadiabatic, and (d) unclosed loop ji2021noncyclic.
• Figure 5: Fidelity comparison of geometric schemes under decoherence rate $\Omega^m_1/5000$ and pulse shape $\Omega^m_1\sin^2(\pi t/\tau)$ ($\Omega^m_1=2\pi\times15$ MHz): conventional NGQC PhysRevA.96.052316PhysRevApplied.10.054051 (red), non-cyclic ji2021noncyclic (orange), triangular-path ding2021path (light-blue), circular-path ding2021nonadiabatic (blue), half-orange-slice li2021high (pink), TOC-optimized PhysRevApplied.14.064009 (black), and DD-protected PhysRevA.102.032627 (green, with 6 decoupling sequences).