Unraveling Rodeo Algorithm Through the Zeeman Model

1. Problem Statement

The paper addresses the challenge of characterizing quantum systems by determining the eigenstates and eigenvalue spectrum of an arbitrary Hamiltonian ($H|\psi\rangle = E|\psi\rangle$).

• Limitations of Existing Methods: Traditional approaches like Phase Estimation and Adiabatic Evolution often require significant resources or prior knowledge.
• Limitations of the Original Rodeo Algorithm: While the Rodeo Algorithm (proposed by Choi et al.) offers an exponentially faster timescale for finding eigenvalues, the original formulation has critical constraints:
  1. It relies heavily on the overlap between the initial state ($|\psi_I\rangle$) and the target eigenstates. If the overlap is low, detection fails.
  2. It requires prior knowledge of the eigenstates to define the input effectively.
  3. It does not provide a detailed procedure for constructing the full set of eigenstates after measurement, particularly in cases of degeneracy or entanglement.
• Goal: To develop a methodology that allows the Rodeo Algorithm to determine the full spectrum (eigenvalues and eigenstates) using arbitrary initial states without prior knowledge of the system's eigenstates, while optimizing performance for Noisy Intermediate-Scale Quantum (NISQ) devices.

2. Methodology

The authors propose a novel framework based on the Zeeman Model (spin interactions in a magnetic field) to test and refine the Rodeo Algorithm.

A. Theoretical Framework: The "Bull Operator" and "Rider State"

The authors formalize the algorithm mathematically to generalize its application:

• Rider State ($|\Psi_0\rangle$): The global state composed of $N$ ancillary qubits (initialized in $|-\rangle$) and $M$ target qubits (in an arbitrary state $|\psi_I\rangle$).
• Bull Operator ($B$): A composite operator representing the sequence of controlled time evolution ($C O_n$) and phase shift ($P_n$) applied sequentially to the ancillas.
  • The operator effectively "twists" the states based on the difference between the trial energy $E$ and the true eigenvalues $E_x$.
• Measurement Strategy: Instead of relying on a single shot, the authors calculate the expectation value of the Pauli-Z operator ($\sigma_z$) for the ancillas. They define a function $h(E, \psi_I)$ which represents the average measurement outcome over random time intervals $t_k$ drawn from a Gaussian probability density function (PDF).
  • Key Insight: When the trial energy $E$ matches an eigenvalue $E_x$, the function $-h(E, \psi_I)$ peaks. The amplitude of this peak corresponds to the probability $P(x) = |\langle x|\psi_I\rangle|^2$ of the initial state projecting onto that eigenstate.

B. Simulation and Experimental Platforms

The study utilizes two platforms to validate the method:

1. Pennylane (Xanadu): Used for high-fidelity simulations of single and two-qubit systems to optimize parameters and analyze noise-free scenarios.
2. IBM Q Experience (Real Device): Specifically the ibmq_lima superconducting device, used to test the algorithm's robustness against real-world noise, decoherence, and gate errors.

C. Optimization Strategies

To overcome NISQ limitations and the dependency on initial state overlap, the authors introduce four strategies:

1. Measurement Repetition: Repeating the experiment ($N_{rounds}$) with different random time intervals to reduce statistical dispersion (standard deviation).
2. Increasing Ancillary Qubits: Using more ancillas ($N > 1$) to parallelize the filtering process, though this increases circuit depth and noise susceptibility.
3. PDF Parameter Tuning: Optimizing the standard deviation ($d$) and mean ($\tau$) of the Gaussian distribution used for time intervals to sharpen peaks and reduce oscillations.
4. Initial State Tuning: Systematically varying the Bloch sphere angles ($\theta, \phi$) of the input state to maximize the overlap with specific eigenstates, effectively "scanning" the Hilbert space.

3. Key Contributions

1. Generalization to Arbitrary States: The paper demonstrates that the Rodeo Algorithm can successfully identify eigenvalues and eigenstates even when the initial state has low overlap with the target, provided the data is analyzed via the proposed $h(E, \psi_I)$ function and optimization strategies.
2. Handling Degeneracy and Entanglement:
   • Degeneracy: The method correctly identifies degenerate eigenvalues (e.g., $E=0$ for two spins) where multiple eigenstates share the same energy.
   • Entanglement: The algorithm successfully processes maximally entangled initial states (Bell states) and partially entangled states, correctly mapping the resulting probability distributions to the underlying Hamiltonian spectrum.
3. Noise Resilience Analysis: The study quantifies how real-device noise (decoherence, gate errors) affects the peak height (preventing it from reaching exactly 1.0) and proposes that increasing measurement repetitions and tuning PDF parameters can mitigate these effects.
4. Open Data and Code: The authors made the datasets and computational codes available on Zenodo to foster collaborative research.

4. Results

• Single-Spin System:
  • The algorithm successfully identified eigenvalues $E = \pm B$ for arbitrary initial states.
  • When the initial state was an eigenstate, the peak reached $-h \approx 1$.
  • For superposition states (e.g., $|+\rangle$), two peaks appeared at $\pm B$ with amplitudes proportional to the probability amplitudes of the superposition (e.g., 0.5 each).
  • Increasing $N_{rounds}$ from 1 to 50 significantly reduced fluctuations, making the eigenvalues clearly distinguishable.
• Two-Spin System (Bipartite):
  • Non-interacting Spins: The method resolved the spectrum $E = \{-2B, 0, 0, 2B\}$, correctly identifying the degeneracy at $E=0$.
  • Entangled States:
    • For Bell states $|\Phi^\pm\rangle$ (superposition of $|00\rangle$ and $|11\rangle$), peaks appeared at $\pm 2B$.
    • For Bell states $|\Psi^\pm\rangle$ (superposition of $|01\rangle$ and $|10\rangle$), a single peak appeared at $E=0$ due to degeneracy.
    • Partially entangled states produced peaks with amplitudes reflecting the specific superposition coefficients.
• Real Device (IBM ibmq_lima):
  • The results mirrored the simulator data but showed reduced peak heights (max $\approx 0.8$) due to noise.
  • Despite the noise, the locations of the peaks (eigenvalues) remained accurate, confirming the algorithm's viability on current NISQ hardware.

5. Significance

This work is significant for several reasons:

• Practicality for NISQ: It provides a concrete roadmap for using the Rodeo Algorithm on current, noisy quantum computers by introducing parameter tuning and repetition strategies that do not require error correction.
• Removal of Prior Knowledge Barrier: By showing that arbitrary initial states can be used to reconstruct the full spectrum, the paper removes a major bottleneck in quantum eigenvalue problems, making the algorithm more accessible for unknown systems.
• Scalability: The successful application to two-qubit systems with entanglement and degeneracy suggests the method is scalable to larger, more complex many-body systems.
• Benchmarking: It serves as a robust benchmark for comparing simulator performance against real hardware, highlighting specific noise sources (like SWAP gate overhead) that affect algorithmic efficiency.

In conclusion, the authors successfully "unraveled" the Rodeo Algorithm, transforming it from a method requiring specific initial conditions into a versatile tool for spectral analysis of quantum systems, validated through both simulation and real quantum hardware.