Qutrit Clifford+T gates by two-body angular momentum couplings, rotations and one-axis-twistings

Abstract: We develop an angular momentum representation and implementation of the Clifford+T set of unitaries for qutrits. We show that local gates from this set can be realized by the sole use of suitable rotations and one-axis-twisting operations, which are at most quadratic in the angular momentum operators and thus can be experimentally realized in many quantum systems. Controlled rotations are shown to only require linear angular momentum couplings and, as a consequence, the full qutrit Clifford+T set is shown to be expressed solely in terms of two-body angular momentum couplings, rotations and one-axis-twisting operations. By employing the Jordan-Schwinger map, we show an analogous implementation in terms of bosonic modes, improving on the number of modes with regard to a previous scheme. Moreover, we employ the cross-Kerr interaction in order to obtain any qutrit Clifford+T gate for bosonic modes. We illustrate our findings with simple schemes for preparing entangled states of interest.

• The paper presents the effective realization of qutrit Clifford+T gates using two-body angular momentum couplings.
• It employs rotations and one-axis-twistings to achieve scalable, experimentally feasible gate operations.
• The integration of the Jordan-Schwinger map and Kerr interactions enhances mode efficiency and supports scalable quantum computing.

Qutrit Clifford+T Gates by Two-Body Angular Momentum Couplings, Rotations, and One-Axis-Twistings

Abstract

The paper "Qutrit Clifford+T gates by two-body angular momentum couplings, rotations, and one-axis-twistings" investigates the implementation and representation of qutrit Clifford+T gates using angular momentum operations. It demonstrates that the full Clifford+T gate set for qutrits can be effectively realized through rotations and one-axis-twisting operations. These gates require interactions that are at most quadratic in angular momentum operators, which have practical physical implementations. Moreover, the paper explores these gates' analogs in bosonic systems, emphasizing improvements in mode efficiency utilizing the Jordan-Schwinger map.

Introduction

Higher-dimensional systems, such as qutrits, present advantageous opportunities for quantum information processing over traditional qubit approaches. They demand fewer $N$-body interactions and serve as a superior testbed for theoretical quantum formulations. The paper outlines an approach using angular momentum coupling to realize their full gate set efficiently (2604.23007). This strategic use of angular momentum offers both theoretical clarity and experimental feasibility, reducing needed resources while sustaining operational precision.

Angular Momentum Realization

Local Gates

Angular momentum operators $J_x$, $J_y$, and $J_z$ facilitate the implementation of qutrit Clifford+T gates. For qutrit systems, $j=1$ representations cater to three-level encoding, offering multiple pathways to manipulate quantum states by rotations and one-axis-twisting operations. The paper details how these rotations represent typical interactions like spin and quadrupolar magnetic effects, which are prevalent in contemporary quantum systems.

Figure 1: Interferometric diagram for the preparation of a qutrit angular momentum graph state $|J_{GHZ}$.

Controlled Gates

Controlled gates such as CZ and CX are derived by leveraging two-body angular momentum couplings, typical in spin-orbit interactions. These gates integrate seamlessly into the experimental setups for quantum computations, illustrating resource sufficiency and scalability. The paper calls attention to experimental settings where controlled interactions can be convincingly reproduced.

Quantum Harmonic Oscillator Realizations

Jordan-Schwinger Map

The equivalence between angular momentum and bosonic systems, established through the Jordan-Schwinger map, facilitates a reduction in the operational complexity of implementing Clifford+T gates. Notably, qutrit systems only require two bosonic modes—an improvement over previous approaches needing $d$ modes, rendering experimental setups more compact and efficient.

Kerr Interactions

Implementations leveraging Kerr nonlinearities offer another viable pathway for gate realization. Kerr effects, including self-Kerr and cross-Kerr interactions, are pivotal in quantum optics and superconducting circuits, serving as the foundational mechanisms for intricate state manipulations within qutrit systems, as well as addressing scalability challenges.

Applications: Entangled States Preparation

Maximally Entangled States

The framework developed enables the creation of maximally entangled qutrit states, an advance from prior non-deterministic methods. Such states underscore the utility and application potential of qutrits in measuring quantum coherence and supporting quantum information tasks.

Qutrit Graph States

Expanding on simple gate implementations, the paper provides methodologies for constructing multipartite entangled states, primarily qutrit graph states. These states can be constructed using combinations of Clifford gates and are proficient in embodying complex entanglement patterns needed for quantum computation protocols.

Conclusions

The work effectively bridges theoretical and experimental quantum computing domains by demonstrating the qutrit Clifford+T gate set implementation through angular momentum and bosonic systems. It anticipates further research in optimizing angular momentum interactions and experimental setups, potentially catalyzing advancements in scalable quantum networks and many-body quantum systems. Future endeavors are poised to explore beyond the few-body implementations, harnessing the reduced physical resource requirements and interactions studied here. The applicability of these theoretical constructs to existing technologies provides promising directions for both immediate and long-term advances.