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Ground-state energy estimation with a guiding state

Given a unitary UU with largest eigenphase θmax⁡\theta_{\max} and a guiding state ∣ψ⟩|\psi\rangle having overlap at least γ\gamma with the eigenspace corresponding to θmax⁡\theta_{\max}, estimate θmax⁡\theta_{\max} to prescribed accuracy using queries to UU. The problem asks whether this can be done with query complexity matching the known lower bound in the parameters γ\gamma and the estimation accuracy, eliminating the previously known multiplicative factor of log⁡(1/γ)\log(1/\gamma). Equivalently, for Hamiltonian ground-state estimation, given a Hamiltonian and a guiding state with overlap at least γ\gamma with the ground space, estimate the smallest eigenvalue at query complexity matching the corresponding lower bound.

Equivalent formulations 2Other wordings

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. Guided largest-eigenphase estimation

    Given a unitary UU and a guiding state with overlap at least γ\gamma with the eigenspace of the largest eigenphase, estimate the largest eigenphase of UU with query complexity matching the known lower bound, rather than one larger by a factor of log⁡(1/γ)\log(1/\gamma).

    source: Optimal Quantum Algorithm for Ground-State Energy Estimation with a Guiding State

  2. Guided Hamiltonian ground-state energy estimation

    Given a Hamiltonian and a guiding state with overlap at least γ\gamma with its ground space, estimate the smallest eigenvalue with query complexity matching the known lower bound.

    source: Optimal Quantum Algorithm for Ground-State Energy Estimation with a Guiding State

References

Primary source

arXiv

Additional references

quantum-algorithmsquantum-complexity
Posted

Progress summary

Refreshed
Claimed solved

A new preprint claims an optimal quantum method for estimating ground-state energy with a guiding state, but the claim has not been independently verified.

The problem concerns guided quantum ground-state energy estimation and asks whether the query complexity can reach the known lower bound. The reported method is intended to answer an open question of Mande and de Wolf.

Known results

  • Mande and de Wolf (2023) established tight lower bounds for quantum phase estimation with an advice state, up to logarithmic factors, including limits governed by the overlap parameter γ\gamma.
  • Chia et al. (2022) established strong complexity-theoretic hardness results for guided local Hamiltonian problems.
  • Waite, Lin, Elman, and Bremner (2025) characterized broader families of guiding states and reported BQP-completeness results in a canonical setting.

August 2026 claimed optimal algorithm

An August 25, 2026 report describes the preprint Optimal Quantum Algorithm for Ground-State Energy Estimation with a Guiding State, which claims to improve the previous query complexity by a factor of log⁡(1/γ)\log(1/\gamma) and attain the query lower bound. This is a claimed complete resolution, but the source identifies the work as an unrefereed preprint and no independent verification was found.

Current status (as of August 2026): The preprint claims the optimal query complexity and resolution of the open question, but no independently verified proof is recorded.

Sources

Solutions 0

No solutions have been posted yet.