Qudit Implementation of the Rodeo Algorithm for Quantum Spectral Filtering

TL;DR

This paper extends the Rodeo algorithm to qudits, introducing a spectral filter and a protocol for entropic estimation, demonstrating improved performance in quantum spectral analysis of multi-level systems.

Contribution

It formulates the Rodeo algorithm using qudits, introduces the Rodeo kernel as a spectral filter, and proposes a microcanonical protocol for entropic estimation.

Findings

Qudit implementation reduces fluctuations by 18% compared to qubits.

Numerical simulations on the Ising model validate the approach.

Qudits enhance spectral and thermodynamic analysis of multi-level systems.

Abstract

Qudits, the multi-level generalization of qubits, provide a natural extension of the binary paradigm in quantum computation and offer new opportunities to enhance algorithmic performance. Beyond their direct applicability to the simulation of multi-level quantum systems, higher-dimensional ancillae can improve sampling efficiency in quantum algorithms by enabling the simultaneous implementation of multiple control operations, thereby reducing circuit complexity. In this work, we pursue three main objectives. First, we present a formulation of the Rodeo algorithm employing a general $d$-level ancilla qudit. Second, we introduce the concept of the \emph{Rodeo kernel}, defined as a two-frequency interferometer, which acts as a spectral filter in the energy domain. Finally, we propose a microcanonical protocol for the Rodeo algorithm. This protocol enables the estimation of entropic quantities through a single energy sweep and admits a natural interpretation as a Gaussian convolution of the density of states. To support the theoretical analysis, we perform numerical evaluations of the corresponding quantum circuit using ancilla qudits of dimensions three, four, and five. The simulations are performed for the one-dimensional Ising model, considering both spin-$\frac{1}{2}$ and spin-$1$ particles. The ancilla qutrit implementation exhibits an $18\%$ reduction in fluctuations compared to the qubit implementation. Our results show that the qudits provide a framework for spectral analysis and thermodynamic characterization of multi-level quantum systems.