Rodeo Algorithm for Quantum Computing.

Area of Science:

• Quantum Computing
• Quantum Algorithms
• Computational Physics

Background:

• Accurate preparation and characterization of quantum states are crucial for quantum computation and simulation.
• Existing methods for eigenvector preparation, such as phase estimation and adiabatic evolution, face limitations in speed and efficiency.

Purpose of the Study:

• To introduce a novel stochastic quantum computing algorithm for preparing specific quantum eigenvectors within a given energy range.
• To demonstrate the algorithm's capability in computing the full spectrum of a quantum Hamiltonian.
• To analyze the computational scaling and efficiency of the proposed method.

Main Methods:

• The rodeo algorithm utilizes auxiliary qubits to control time evolution, stochastically adjusting eigenvector amplitudes based on energy proximity to a tunable parameter E.
• Measurements on auxiliary qubits introduce stochastic factors, driving convergence towards the target eigenvector.
• The algorithm's performance is analyzed through its computational scaling with respect to desired accuracy and initial state overlap.

Main Results:

• The algorithm achieves exponential accuracy in target eigenvector preparation with a computational effort scaling of O[|logδ|/(pε)] for spectral weight suppression.
• Energy eigenvalue determination with error ε scales as O[(logε)^{2}/(pε)].
• Eigenstate preparation demonstrates a computational scaling of O(logΔ/p), significantly outperforming existing methods.

Conclusions:

• The rodeo algorithm provides an efficient and accurate method for preparing quantum eigenvectors and computing energy spectra.
• Its exponential speedup in eigenstate preparation offers a significant advancement over phase estimation and adiabatic evolution.
• The stochastic nature of the algorithm allows for high precision with a reduced computational burden.