Beyond asymptotic reasoning: the practicalities of a quantum ground state projector based on the wall-Chebyshev expansion

TL;DR

This paper explores a quantum algorithm for ground-state preparation using wall-Chebyshev expansion, analyzing its practical performance, success probabilities, and potential advantages over existing methods in quantum computing.

Contribution

It provides a detailed analysis of the wall-Chebyshev projector’s asymptotic behavior, success probabilities, and practical benchmarks, highlighting its robustness and resource trade-offs.

Findings

Requires fewer Hamiltonian oracle applications for a given fidelity.

Success probability decays exponentially, limiting practical use.

Maintains non-trivial success probability in certain regimes.

Abstract

We consider a quantum algorithm for ground-state preparation based on a Chebyshev series approximation to the wall function. In a classical setting, this approach is appealing as it guarantees rapid convergence. We analyze the asymptotic scaling and success probabilities of different quantum implementations and provide numerical benchmarks, comparing the performance of the wall-Chebyshev projectors with current state-of-the-art approaches. We find that this approach requires fewer serial applications of the Hamiltonian oracle to achieve a given ground state fidelity, but is severely limited by exponentially decaying success probability. However, we find that some implementations maintain non-trivial success probability in regimes where wall-Chebyshev projection leads to a fidelity improvement over other approaches. As the wall-Chebyshev projector is highly robust to loose known upper bounds on the true ground state energy, it offers a potential resource trade-off, particulary in the early fault-tolerant regime of quantum computation.