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Qudit Implementation of the Rodeo Algorithm for Quantum Spectral Filtering

Julio Rocha, Rodrigo Dias

Published Apr 23, 2026• 0 citations
Publisher

Abstract

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 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-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.

DOI
10.21203/rs.3.rs-9150631/v1

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APL Photonics•May 1, 2026•0 citations
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 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-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.
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 ddd-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-$\sfrac{1}{2}$ and spin-111 particles. The ancilla qutrit implementation exhibits an 18%18\%18% reduction in fluctuations compared to the qubit implementation. Our results show that qudits provide a framework for spectral analysis and thermodynamic characterization of multi-level quantum systems.
The density of states (DoS) encodes the thermodynamic and spectral properties of quantum many-body systems, yet its reconstruction becomes intractable for Hilbert spaces too large to diagonalize. Classically, the kernel polynomial method (KPM) addresses this by combining stochastic trace estimation with a smoothing kernel. Here we show that the Rodeo algorithm---one of the simplest eigenvalue-location protocols for near-term quantum hardware---provides a direct quantum analogue of this approach. Averaging the Rodeo response over Haar-random input states yields the DoS convolved with a spectral kernel fixed entirely by the distribution of evolution times: the random states play the role of stochastic trace estimation, and the temporal sampling distribution that of the damping kernel. The construction requires only the standard single-ancilla circuit, and quantum typicality suppresses the statistical error as the Hilbert-space dimension grows. We derive the estimator and its uncertainties, establish an explicit dictionary between signal-processing window functions and quantum reconstruction kernels, and validate the method on the one-dimensional transverse-field Ising and spin-1 models.
Computing non-equilibrium steady states of open quantum systems is a challenging task on conventional computers, motivating quantum algorithms for direct steady-state estimation. A natural route is to regard the steady state as the zero mode of the Liouvillian and to isolate this sector spectrally. We formulate this task as a known-zero-sector projection problem and implement the corresponding filter using the Rodeo algorithm, which performs stochastic spectral filtering through repeated controlled evolutions and measurement-conditioned filtering steps. In the steady-state setting, the filter can be centered directly at the known zero eigenvalue, avoiding the spectral search required in generic eigenstate preparation. Compared with a phase-estimation-based implementation of the same projection, the Rodeo approach enables restart on failure and reduces the target-error dependence of the filtering cost and controlled-evolution depth from power-law to logarithmic. This advantage becomes more pronounced as the spectral separation of the Hermitian Liouvillian embedding increases, allowing Rodeo filtering to outperform phase-estimation filtering already at modest controlled-evolution depths. Our results identify Rodeo filtering as a resource-efficient primitive for estimating steady-state observables in open quantum systems.
Quantum computation of trapped spectra for nuclear scattering requires resource-efficient circuits and reliable energy inputs. We present a cross-platform benchmark of ancilla-recycled RODEO filtering for a four-dimensional effective Hamiltonian describing a trapped neutron--proton model. Static circuits on IonQ Forte-1 and dynamic circuits on IBM Aachen implement the same ideal ten-cycle all-zero accepted filter using twelve and three active qubits, respectively. Mid-circuit measurement and reset reduce the active qubit count by 75\% while retaining all controlled evolutions. The interleaved IBM peak-center offset from the exact level is −0.017±0.507 keV-0.017 \pm 0.507~\mathrm{keV}−0.017±0.507 keV, compared with −0.568±0.694 keV-0.568 \pm 0.694~\mathrm{keV}−0.568±0.694 keV for IonQ (finite-shot bootstrap uncertainties). At ω=3.7 MeVω=3.7~\mathrm{MeV}ω=3.7 MeV, the modified effective range expansion maps both centers to finite-confinement scattering inputs consistent with the exact reference within propagated uncertainties. Three IBM batches exhibit center variations beyond finite-shot uncertainties (p=0.015p=0.015p=0.015), despite comparatively stable postselection amplitudes. Calibration-derived noisy simulations do not consistently reproduce the measured centers. Complete sampled support enables sample-based quantum diagonalization to recover the exact effective-space level, whereas larger-space simulator tests expose substantial errors from omitted configurations. The calculation validates one finite-confinement input, not a free-space phase shift. This benchmark provides a reusable circuit and validation protocol separating spectral precision, hardware repeatability, and sampled-subspace completeness for multilevel nuclear-scattering calculations.
The Rodeo Algorithm is a quantum computing method for computing the energy spectrum of a Hamiltonian and preparing its energy eigenstates. We discuss how to improve the performance of the rodeo algorithm for each of these two applications. In particular, we demonstrate that using a geometric series of time samples offers a near-optimal optimization space for a given total runtime by studying the Rodeo Algorithm performance on a model Hamiltonian representative of gapped many-body quantum systems. Analytics explain the performance of this time sampling and the conditions for it to maintain the established exponential performance of the Rodeo Algorithm. We finally demonstrate this sampling protocol on various physical Hamiltonians, showing its practical applicability. Our results suggest that geometric series of times provide a practical, near-optimal, and robust time-sampling strategy for quantum state preparation with the Rodeo Algorithm across varied Hamiltonians without requiring model-specific fine-tuning.
Many quantum algorithms require ancilla qubits and deep circuits, yet adding and interacting with them are difficult and resource-intensive. Moreover, the high-dimensional properties of quantum systems have been demonstrated to simplify quantum circuits while enhancing efficiency. We propose a novel “Dimension Expansion and Partial Measurement” scheme for quantum information processing based on qudits’ multi-level nature. Given a d-dimensional qudit (expandable to d × r dimensions) and local single-qudit operations in d and r dimensions, we can introduce r-dimensional equivalent ancilla qudits (logically introduced via dimension expansion) and enable quantum operations in deep circuits across d and r dimensions. With successfully generated photons, our scheme efficiently cascades two-qubit/qudit gates with unity theoretical success probability; existing schemes are either probabilistic or need additional particles. In particular, this scheme can be extended to multiple ancilla qudits with measurement feedforward by the “Partially Measure-Reexpand” procedure. This leads to a resource-efficient universal high-dimensional measurement-based quantum computation scheme. On a silicon-photonic chip, starting from 2 qubits (each expandable to a four-dimensional qudit), we implemented a measurement-based quantum computation system equivalent to 5 qubits and experimentally validated our scheme. The proposed scheme applies to simplifying and scaling up various quantum systems.