The rodeo algorithm has been proposed recently as an efficient method in quantum computing for projection of a given initial state onto a state of fixed energy for systems with discrete spectra. In the initial formulation of the rodeo algorithm these times were chosen randomly via a Gaussian distribution with fixed RMS times. In this paper it is shown that such a random approach for choosing times suffers from exponentially large fluctuations in the suppression of unwanted components: as the number of iterations gets large, the distribution of suppression factors obtained from random selection approaches a log-normal distribution leading to remarkably large fluctuations. We note that by choosing times intentionally rather than randomly such fluctuations can be avoided and strict upper bounds on the suppression can be obtained. Moreover, the average suppression using fixed computational cost can be reduced by many orders of magnitude relative to the random algorithm. A key to doing this is to choose times that vary over exponentially many times scales, starting from a modest maximum scale and going down to time scales exponentially smaller.The rodeo algorithm has been proposed recently as an efficient method in quantum computing for projection of a given initial state onto a state of fixed energy for systems with discrete spectra. In the initial formulation of the rodeo algorithm these times were chosen randomly via a Gaussian distribution with fixed RMS times. In this paper it is shown that such a random approach for choosing times suffers from exponentially large fluctuations in the suppression of unwanted components: as the number of iterations gets large, the distribution of suppression factors obtained from random selection approaches a log-normal distribution leading to remarkably large fluctuations. We note that by choosing times intentionally rather than randomly such fluctuations can be avoided and strict upper bounds on the suppression can be obtained. Moreover, the average suppression using fixed computational cost can be reduced by many orders of magnitude relative to the random algorithm. A key to doing this is to choose times that vary over exponentially many times scales, starting from a modest maximum scale and going down to time scales exponentially smaller.

arXiv:2305.19952May 31, 20230 citations

Rodeo Algorithm for Quantum Computing

Kenneth Choi, Dean Lee, Joey Bonitati +2 more

We present a stochastic quantum computing algorithm that can prepare any eigenvector of a quantum Hamiltonian within a selected energy interval $[E\ensuremath{-}\ensuremath{\epsilon},E+\ensuremath{\epsilon}]$. In order to reduce the spectral weight of all other eigenvectors by a suppression factor $\ensuremath{\delta}$, the required computational effort scales as $O[|\mathrm{log}\ensuremath{\delta}|/(p\ensuremath{\epsilon})]$, where $p$ is the squared overlap of the initial state with the target eigenvector. The method, which we call the rodeo algorithm, uses auxiliary qubits to control the time evolution of the Hamiltonian minus some tunable parameter $E$ . With each auxiliary qubit measurement, the amplitudes of the eigenvectors are multiplied by a stochastic factor that depends on the proximity of their energy to $E$ . In this manner, we converge to the target eigenvector with exponential accuracy in the number of measurements. In addition to preparing eigenvectors, the method can also compute the full spectrum of the Hamiltonian. We illustrate the performance with several examples. For energy eigenvalue determination with error $\ensuremath{\epsilon}$, the computational scaling is $O[(\mathrm{log}\ensuremath{\epsilon}{)}^{2}/(p\ensuremath{\epsilon})]$. For eigenstate preparation, the computational scaling is $O(\mathrm{log}\mathrm{\ensuremath{\Delta}}/p)$, where $\mathrm{\ensuremath{\Delta}}$ is the magnitude of the orthogonal component of the residual vector. The speed for eigenstate preparation is exponentially faster than that for phase estimation or adiabatic evolution.We present a stochastic quantum computing algorithm that can prepare any eigenvector of a quantum Hamiltonian within a selected energy interval $[E\ensuremath{-}\ensuremath{\epsilon},E+\ensuremath{\epsilon}]$. In order to reduce the spectral weight of all other eigenvectors by a suppression factor $\ensuremath{\delta}$, the required computational effort scales as $O[|\mathrm{log}\ensuremath{\delta}|/(p\ensuremath{\epsilon})]$, where $p$ is the squared overlap of the initial state with the target eigenvector. The method, which we call the rodeo algorithm, uses auxiliary qubits to control the time evolution of the Hamiltonian minus some tunable parameter $E$. With each auxiliary qubit measurement, the amplitudes of the eigenvectors are multiplied by a stochastic factor that depends on the proximity of their energy to $E$. In this manner, we converge to the target eigenvector with exponential accuracy in the number of measurements. In addition to preparing eigenvectors, the method can also compute the full spectrum of the Hamiltonian. We illustrate the performance with several examples. For energy eigenvalue determination with error $\ensuremath{\epsilon}$, the computational scaling is $O[(\mathrm{log}\ensuremath{\epsilon}{)}^{2}/(p\ensuremath{\epsilon})]$. For eigenstate preparation, the computational scaling is $O(\mathrm{log}\mathrm{\ensuremath{\Delta}}/p)$, where $\mathrm{\ensuremath{\Delta}}$ is the magnitude of the orthogonal component of the residual vector. The speed for eigenstate preparation is exponentially faster than that for phase estimation or adiabatic evolution.

arXiv:2009.04092Physical Review LettersJul 23, 202191 citations

Rodeo Filtering for Direct Steady-State Estimation in Open Quantum Systems

Hyeonjun Yeo, Jongin Jeong, Soyoung Shin +1 more

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.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.

quant-pharXiv:2606.29218Jun 30, 20260 citations

A Kernel-Based Density of States Estimator for Quantum Computing

Julio Cesar Siqueira Rocha

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.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.

quant-pharXiv:2607.24972Jul 29, 20260 citations

Hierarchical fusion method for scalable quantum eigenstate preparation

Matthew Patkowski, Onat Ayyildiz, Matjaž Kebrič +2 more

Robust and efficient eigenstate preparation is a central challenge in quantum simulation. The Rodeo Algorithm (RA) offers exponential convergence to a target eigenstate but suffers from poor performance when the initial state has low overlap with the desired eigenstate, hindering the applicability of the original algorithm to larger systems. In this work, we introduce a fusion method that preconditions the RA state by an adiabatic ramp to overcome this limitation. By incrementally building up large systems from exactly solvable subsystems and using adiabatic preconditioning to enhance intermediate state overlaps, we ensure that the RA retains its exponential convergence even in large-scale systems. We demonstrate this hybrid approach using numerical simulations of the spin- 1/2 XX model and find that the Rodeo Algorithm exhibits robust exponential convergence across system sizes. We benchmark against using only an adiabatic ramp as well as using the unmodified RA, finding that for state preparation precision at the level of $10^{-3}$ infidelity or better there a decisive computational cost advantage to the fusion method. These results together demonstrate the scalability and effectiveness of the fusion method for practical quantum simulations.Robust and efficient eigenstate preparation is a central challenge in quantum simulation. The Rodeo Algorithm (RA) offers exponential convergence to a target eigenstate but suffers from poor performance when the initial state has low overlap with the desired eigenstate, hindering the applicability of the original algorithm to larger systems. In this work, we introduce a fusion method that preconditions the RA state by an adiabatic ramp to overcome this limitation. By incrementally building up large systems from exactly solvable subsystems and using adiabatic preconditioning to enhance intermediate state overlaps, we ensure that the RA retains its exponential convergence even in large-scale systems. We demonstrate this hybrid approach using numerical simulations of the spin- 1/2 XX model and find that the Rodeo Algorithm exhibits robust exponential convergence across system sizes. We benchmark against using only an adiabatic ramp as well as using the unmodified RA, finding that for state preparation precision at the level of $10^{-3}$ infidelity or better there a decisive computational cost advantage to the fusion method. These results together demonstrate the scalability and effectiveness of the fusion method for practical quantum simulations.

arXiv:2510.19039Physical Review AApr 27, 20260 citations

Improved Rodeo Algorithm Performance for Spectral Functions and State Preparation

Matthew Patkowski, Onat Ayyildiz, Katherine Hunt +2 more

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.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.

quant-pharXiv:2602.05978Feb 6, 20260 citations

p

is the squared overlap of the initial state with the target eigenvector. The method, which we call the rodeo algorithm, uses auxiliary qubits to control the time evolution of the Hamiltonian minus some tunable parameter

E

. With each auxiliary qubit measurement, the amplitudes of the eigenvectors are multiplied by a stochastic factor that depends on the proximity of their energy to

E

. In this manner, we converge to the target eigenvector with exponential accuracy in the number of measurements. In addition to preparing eigenvectors, the method can also compute the full spectrum of the Hamiltonian. We illustrate the performance with several examples. For energy eigenvalue determination with error $\ensuremath{\epsilon}$, the computational scaling is $O[(\mathrm{log}\ensuremath{\epsilon}{)}^{2}/(p\ensuremath{\epsilon})]$. For eigenstate preparation, the computational scaling is $O(\mathrm{log}\mathrm{\ensuremath{\Delta}}/p)$, where $\mathrm{\ensuremath{\Delta}}$ is the magnitude of the orthogonal component of the residual vector. The speed for eigenstate preparation is exponentially faster than that for phase estimation or adiabatic evolution.

10^{-3}

infidelity or better there a decisive computational cost advantage to the fusion method. These results together demonstrate the scalability and effectiveness of the fusion method for practical quantum simulations.