Rodeo Algorithm: Quantum Eigenstate Preparation

• Rodeo Algorithm is a measurement-driven quantum method that filters states to isolate a desired energy eigenstate with exponential suppression outside a target energy window.
• It employs cycles of controlled time evolution and phase rotations, using ancilla qubits to iteratively reduce unwanted spectral contributions with resource scaling of O(|log δ|/(pε)).
• Demonstrations on quantum hardware and hybrid strategies like variational preconditioning and fusion methods highlight RA's efficiency in eigenstate preparation and spectral analysis.

The Rodeo Algorithm (RA) is a stochastic, measurement-driven quantum algorithm for both eigenstate preparation and spectral estimation of quantum Hamiltonians. It operates by applying cycles of controlled time evolution and phase rotation, using auxiliary (ancilla) qubits to exponentially suppress undesired eigenstate components outside a target energy window. Unlike adiabatic or phase estimation methods, RA directly “filters” quantum states to amplify the desired energy eigenvector. Its computational effort for eigenstate preparation with final suppression factor $\delta$ outside an energy interval $[E-\epsilon, E+\epsilon]$ scales as $O[|\log \delta|/(p\epsilon)]$, with $p$ the squared overlap of the initial state with the target eigenvector. This exponential convergence in measurement depth and resource efficiency has been validated across quantum simulation domains and on real quantum hardware.

1. Fundamental Principles and Circuit Operation

RA prepares an eigenvector of a Hamiltonian $H_{\text{obj}}$ by iteratively removing all contributions from states distant from a target energy $E$. Each cycle involves an object register (system qubits) and ancilla qubits initialized in $|1\rangle$. The ancilla qubit sequence per cycle consists of:

• Hadamard ($H$) rotation,
• Controlled time evolution, implementing $e^{-i H_{\text{obj}} t_n}$ controlled by the ancilla,
• Phase rotation $P(E t_n)$: adds a phase $e^{i E t_n}$ to the ancilla,
• Second Hadamard,
• Measurement of the ancilla.

Upon measurement, amplitudes of each $H_{\text{obj}}$ eigenvector $|E_\text{obj}\rangle$ are multiplied by $\cos^2 [(E_\text{obj} - E)t_n / 2]$, so when $E_\text{obj} \approx E$ the factor approaches unity, but for non-target energies this factor leads to exponential suppression over cycles. Letting $t_n$ be chosen stochastically (typically Gaussian-distributed), after $N$ cycles, the overall success probability for measuring the ancilla in $|1\rangle$ is: $$P_N = \prod_{n=1}^N \cos^2\left[\frac{1}{2}(E_\text{obj} - E)t_n\right]$$ States within the filter window $[E-\epsilon, E+\epsilon]$ are retained, all others are exponentially suppressed.

2. Computational Scaling, Suppression Metrics, and Efficiency

The required resources for state purification and eigenvalue estimation in RA are governed by overlap $p = |\langle E_\text{target} | \psi_I \rangle|^2$ and precision parameters:

Task	Scaling Formula	Primary Parameters
Eigenvector preparation	$O(|\log \delta|/(p\epsilon))$	$\delta$ (suppression), $p$, $\epsilon$
Eigenvalue estimation	$O((\log\epsilon)^2/(p\epsilon))$	$\epsilon$ (energy error), $p$
Residual suppression	$O(\log \Delta/p)$	$\Delta$ (residual amplitude), $p$

For a suppression factor per cycle (geometric mean) $\sim 1/4$, after $N = |\log \delta|$ cycles the residual is $4^{-N}$. For eigenvalue scans, the procedure involves repeated log-bisection over energy intervals with cumulative cost $(\log\epsilon)^2/(p\epsilon)$. Compared to phase estimation ($O(1/(p\epsilon))$) and adiabatic evolution ($O(1/\Delta)$), RA achieves exponential speedup especially in the limit of small residual error $\Delta$.

3. Spectral Filtering and Spectrum Determination

RA supports both eigenstate preparation and spectrum computation:

• Eigenstate Preparation: For a single target energy, the iterative projection leaves the final wavefunction nearly a pure eigenvector in the energy interval.
• Spectral Function Extraction: By scanning the target energy $E$ over a range, RA constructs the spectral function $S(E)$, with peaks indicating eigenstates with $|E_\text{obj} - E| \lesssim \epsilon$. This is realized efficiently for models such as the ten-site Heisenberg chain and the one-dimensional Anderson model, where the spectral function and corresponding errors (e.g., $\log \Delta$) match exact diagonalization.

In practice, for moderate $N < 10$, RA reconstructs spectra with high fidelity and low error, requiring only short gate depth and moderate measurement counts, as demonstrated in the referenced quantum simulation studies.

4. Algorithmic Improvements: Deterministic Scheduling and Super-Iteration

Optimal suppression in RA requires careful control of time evolution intervals. (Cohen et al., 2023) identifies that random scheduling (random times) can lead to log-normal distributions of suppression factors, resulting in large fluctuations and unreliable bounds for residual unwanted amplitudes. The deterministic "super iteration" approach sets time intervals as exponentially decreasing sequences ($T_0$, $T_0/2$, $T_0/4$, etc.), ensuring that suppression regions are spaced so that problematic resonance points (where suppression is poor) are nullified by subsequent sub-iterations.

The overall suppression factor after super iteration is: $$s_\text{sup} = j_0^2(\pi \zeta_\text{sup}) = \left( \frac{\sin(\pi\zeta_\text{sup})}{\pi\zeta_\text{sup}} \right)^2$$ For lowest excited states ($E = \Delta$), $s_\text{sup} = 0$ for $T_0 = 2\pi\hbar/\Delta$. Thus, with intentional multi-scale time selection, strict upper bounds on suppression are obtained (e.g., $<10^{-14}$ for $T_\text{tot}/T_0 \approx 6.4$). This approach both reduces computational cost and guarantees reliability compared to the random algorithm.

5. Extensions: Variational Preconditioning, Fusion Methods, and Multi-Qubit Systems

As the system size grows, the initial state overlap $p$ becomes the limiting factor for RA efficiency. Several strategies have been developed:

• Variational Rodeo Algorithm (VRA) (Bonitati, 19 Dec 2024): Embeds RA into a variational optimization loop, tuning circuit parameters (e.g., QAOA ansatz) to maximize overlap before RA cycles.
• Hybrid Blocked QAOA + RA (Bazavov et al., 31 Oct 2024): Uses blocked QAOA ansatz to produce a high-overlap initial state, then applies RA for final purification, significantly reducing required Trotter steps and gate counts for models such as the Schwinger Hamiltonian.
• Fusion Method (Patkowski et al., 21 Oct 2025): Constructs large systems by fusing exactly- or near-exactly-prepared subsystems, with adiabatic preconditioning (ramp of Hamiltonian couplings) to enhance intermediate state overlaps. RA is then applied post-fusion to reach near-unit fidelity. The computational cost $\kappa_H$ for the fusion procedure is:

$\kappa_H = p^{-1}(t_A + t_R)$

where $t_A$ is the adiabatic ramp time, $t_R$ RA evolution time, and $p$ the initial overlap. Numerical benchmarks (e.g., spin-1/2 XX model) confirm a decisive advantage vs. pure adiabatic or unmodified RA for achieving $\lesssim 10^{-3}$ infidelity.

• Scalability: These hybrid strategies retain RA's exponential suppression of non-target states, surmounting challenges of the orthogonality catastrophe and low initial overlaps as $N_\text{qubit}$, system complexity, and spectral density increase.

6. Practical Implementations and Experimental Demonstrations

Proof-of-principle demonstrations have been performed on multiple quantum platforms:

• IBM Q Casablanca: Energy eigenvalues for random one-qubit Hamiltonians prepared by RA with relative error $0.08\%$; observable expectation values via Hellmann–Feynman theorem with $0.7\%$ error; no error mitigation required (Qian et al., 2021).
• PennyLane/Xanadu simulators and IBM Q Experience: Spectral peaks and eigenstate projections for the Zeeman model, including multi-qubit, degenerate/entangled scenarios (Gomes et al., 16 Jul 2024). Real device runs confirm that parameter optimization and repetition strategies are critical to mitigate noise and gate errors.

Key circuit depth findings:

• Three RA cycles typically require $\sim$6 CNOT gates per experiment on IBM Q.
• For multi-qubit models, Trotterization and controlled operations are necessary; efficient decomposition strategies (variational, blocked, fusion) minimize resource overhead.

7. Applications and Limitations

RA and its variants contribute across domains:

• Quantum many-body simulation, spectrum analysis, and eigenstate preparation for condensed matter (Heisenberg, Anderson, Schwinger, Ising, XX models).
• Thermodynamic property extraction via number of states function (Rocha et al., 2023), with less than $1\%$ error in specific heat calculations.
• Nuclear structure, quantum chemistry, and quantum statistical mechanics simulations.

Limitations arise primarily when initial overlap $p$ is small, increasing post-selection cost and/or requiring hybrid preconditioning (variational or adiabatic). Tuning stochastic evolution parameters (e.g., $t_{\text{RMS}}$) is crucial and remains an implementation-dependent consideration. The method's strength is maximal for NISQ-era devices with short coherence windows due to low gate-depth per cycle; for large systems, scalability is now addressed via fusion and hybrid strategies.

• “Rodeo Algorithm for Quantum Computing” (Choi et al., 2020)
• “Demonstration of the Rodeo Algorithm on a Quantum Computer” (Qian et al., 2021)
• “Optimizing rodeo projection” (Cohen et al., 2023)
• “Estimating the Number of States via the Rodeo Algorithm for Quantum Computation” (Rocha et al., 2023)
• “Unraveling Rodeo Algorithm Through the Zeeman Model” (Gomes et al., 16 Jul 2024)
• “Efficient State Preparation for the Schwinger Model with a Theta Term” (Bazavov et al., 31 Oct 2024)
• “Eigenstate Preparation on Quantum Computers” (Bonitati, 19 Dec 2024)
• “High-Fidelity Scalable Quantum State Preparation via the Fusion Method” (Patkowski et al., 21 Oct 2025)