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arXiv:2510.19039 · Physical Review A

Hierarchical fusion method for scalable quantum eigenstate preparation

Matthew Patkowski, Onat Ayyildiz, Matjaž Kebrič, Katharine L. C. Hunt, Dean Lee

Published Apr 27, 2026• 0 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/ycjd-5glj

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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 10−310^{-3}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.
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 ppp 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 EEE. 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 EEE. 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.
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.
We present a comparison of different quantum state preparation algorithms and their overall efficiency for the Schwinger model with a theta term. While adiabatic state preparation is proved to be effective, in practice it leads to large gate counts to prepare the ground state. The quantum approximate optimization algorithm (QAOA) provides excellent results while keeping the counts small by design, at the cost of an expensive classical minimization process. We introduce a “blocked” modification of the Schwinger Hamiltonian to be used in the QAOA that further decreases the length of the algorithms as the size of the problem is increased. The rodeo algorithm (RA) provides a powerful tool to efficiently prepare any eigenstate of the Hamiltonian, as long as its overlap with the initial guess is large enough. We obtain the best results when combining the blocked QAOA ansatz and the RA, as this provides an excellent initial state with a relatively short algorithm without the need to perform any classical steps for large problem sizes.
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.
Quantum simulation is expected to be a main application of quantum computers with realistic utility in quantum chemistry, materials science and beyond. However, preparing excited or general eigenstates is a central challenge, particularly when the desired eigenvalue is not known in advance, or when the overlap with the initial state is insufficient. We introduce the Dominant Eigenstate Filtering via Eigenprobability Amplification and Thresholding (DEFEAT) algorithm that identifies and filters the eigenstate with the largest overlap with the supplied initial state. Our key observation is that we do not need prior knowledge of the target eigenvalue, as we construct efficient twirling superoperators that map initial states to eigenprobability density operators ρρρ, diagonal in the Hamiltonian eigenbasis and encoding its spectral weights, alongside its block-encoding implementation. Our crucial innovation is the quadratic amplification of probabilities via the factorisation ρ=ρsqrt†ρsqrtρ=ρ_{\rm sqrt}^\daggerρ_{\rm sqrt}ρ=ρsqrt†​ρsqrt​, analogous to the recently introduced sum-of-squares spectral amplification (SOSSA), which we use here to amplify the separation between dominant and subdominant components. Thresholding then yields the dominant-eigenstate projector. Compared with conventional phase estimation, DEFEAT improves the dependence on the overlap with the initial state while requiring substantially fewer ancillary qubits. We prove that the query complexity of the filtering step is optimal up to logarithmic factors and establish a complementary lower bound for eigenstate preparation under purified query access to ρρρ. We validate in numerical simulations that the convergence rate of DEFEAT matches our theoretical results. Our results provide a general eigenvalue-agnostic primitive for dominant eigenstate filtering and preparation, and for estimating properties of dominant eigenstates.
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, evolution times were chosen randomly via a Gaussian distribution with a fixed standard deviation. 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 timescales, starting from a modest maximum scale and going down to timescales exponentially smaller.