Transition-Aware Decomposition of Single-Qudit Gates

TL;DR

The paper tackles the problem of efficiently decomposing arbitrary single-qudit gates into sequences of physically allowed transitions under selection rules. It introduces Transition-Aware QR (TAQR), which achieves a worst-case of at most $d(d-1)/2$ transitions and provides static and adaptive strategies to minimize gate counts based on platform topology and matrix sparsity. The method demonstrates superior performance for trapped-ion qudits with arbitrary selection rules and competitive results for superconducting and photonic qudits, with potential to reduce single-qudit errors and simplify two-qudit gate construction. The approach promises a platform-agnostic, drop-in transpilation step for scalable qudit-based quantum computing, supported by available reference implementations.

Abstract

Quantum computation with $d$-level quantum systems, also known as qudits, benefits from the possibility to use a richer computational space compared to qubits. However, for an arbitrary qudit-based hardware platform, the issue is that a generic qudit operation has to be decomposed into the sequence of native operations $-$ pulses that are adjusted to the transitions between two levels in a qudit. Typically, not all levels in a qudit are simply connected to each other due to specific selection rules. Moreover, the number of pulses plays a significant role, since each pulse takes a certain execution time and may introduce error. In this paper, we propose a resource-efficient algorithm to decompose single-qudit operations into the sequence of pulses that are allowed by qudit selection rules. Using the developed algorithm, the number of pulses is at most $d(d{-}1)/2$ for an arbitrary single-qudit operation. For specific operations, the algorithm could produce even fewer pulses. We provide a comparison of qudit decompositions for several types of trapped ions, specifically $^{171}\text{Yb}^+$, $^{137}\text{Ba}^+$ and $^{40}\text{Ca}^+$ with different selection rules, and also decomposition for superconducting qudits. Although our approach deals with single-qudit operations, the proposed approach is important for realizing two-qudit operations since they can be implemented as a standard two-qubit gate that is surrounded by efficiently implemented single-qudit gates.

Figures (3)

• Figure 1: Row elimination schemes for a qudit unitary matrix with given allowed transitions. (a) Decomposition for a superconducting qudit with allowed transitions $\textsf{R}_{}^{n,{n \pm 1}}$. (b) Decomposition for a trapped-ion qudit with allowed transitions $\textsf{R}_{}^{0,n}$ and $\textsf{R}_{}^{n,0}$.
• Figure 2: Algorithm that determines indices for the qudit decomposition scheme. It utilizes the graph structure of selection rules and the breadth-first search algorithm.
• Figure 3: Decomposition of a matrix with zero non-diagonal elements: (a) Static and (b) Adaptive schemes.