Eigenpath traversal by Poisson-distributed phase randomisation
<|reference_start|>Eigenpath traversal by Poisson-distributed phase randomisation: We present a framework for quantum computation, similar to Adiabatic Quantum Computation (AQC), that is based on the quantum Zeno effect. By performing randomised dephasing operations at intervals determined by a Poisson process, we are able to track the eigenspace associated to a particular eigenvalue.   We derive a simple differential equation for the fidelity, leading to general theorems bounding the time complexity of a whole class of algorithms. We also use eigenstate filtering to optimise the scaling of the complexity in the error tolerance $\epsilon$.   In many cases the bounds given by our general theorems are optimal, giving a time complexity of $O(1/\Delta_m)$ with $\Delta_m$ the minimum of the gap. This allows us to prove optimal results using very general features of problems, minimising the problem-specific insight necessary.   As two applications of our framework, we obtain optimal scaling for the Grover problem (i.e.\ $O(\sqrt{N})$ where $N$ is the database size) and the Quantum Linear System Problem (i.e.\ $O(\kappa\log(1/\epsilon))$ where $\kappa$ is the condition number and $\epsilon$ the error tolerance) by direct applications of our theorems.<|reference_end|>
@article{cunningham2024eigenpath,
    title={Eigenpath traversal by Poisson-distributed phase randomisation},
    author={Joseph Cunningham and J'er'emie Roland},
    journal={arXiv preprint arXiv:2406.03972},
    year={2024},
    archivePrefix={arXiv},
    eprint={2406.03972},
    primaryClass={quant-ph cs.DS}
}