• Abstract Determining the energy gap in a quantum many-body system is critical to understanding its behavior and is important in quantum chemistry and condensed matter physics. • The challenge of determining the energy gap requires identifying both the excited and ground states of a system. • In this work, we consider preparing the prior distribution and circuits for the eigenspectrum of structured, time-independent Hamiltonians (e.g., local, sparse, or symmetry-constrained), which can benefit both classical and quantum algorithms for solving eigenvalue problems. • The proposed algorithm unfolds in three strategic steps: Hamiltonian transformation, parameter representation, and classical clustering. • These steps are underpinned by two key insights: the use of quantum circuits to approximate the ground state of transformed Hamiltonians and the analysis of parameter representation to distinguish between eigenvectors. • The algorithm is showcased through applications to the 1D Heisenberg system and the LiH molecular system, highlighting its potential for both near-term quantum devices and fault-tolerant quantum devices, restricted to structured Hamiltonians where approximate state preparation is feasible.

Article Summaries:

  • Hybrid quantum‑classical clustering offers a new route to estimate the eigenspectrum of structured, time‑independent Hamiltonians. The method proceeds in three stages: (1) a unitary transformation of the Hamiltonian, (2) a parameter‑representation that captures key features of the eigenstates, and (3) classical clustering to group parameter vectors into distinct eigenvalue bands. Quantum circuits are used to approximate the ground state of the transformed Hamiltonian, while classical analysis separates ground and excited states. The algorithm is benchmarked on a 1‑D Heisenberg chain and the LiH molecule, showing promise for both near‑term and fault‑tolerant quantum devices. The study highlights scalability and resource efficiency for eigenvalue problems in quantum chemistry and condensed‑matter physics.
  • Hybrid quantum‑classical clustering is proposed to generate a prior distribution over the eigenspectrum of structured, time‑independent Hamiltonians, such as local or symmetry‑constrained systems. The algorithm proceeds in three stages: (1) a Hamiltonian transformation that simplifies the spectrum, (2) a parameter‑representation step that encodes eigenvectors into quantum circuit parameters, and (3) classical clustering that groups similar parameter sets to distinguish ground and excited states. Quantum circuits are used to approximate the ground state of the transformed Hamiltonian, while classical analysis separates eigenvectors. Demonstrations on a 1‑D Heisenberg chain and LiH molecule show the method’s applicability to both near‑term and fault‑tolerant quantum devices, offering a resource‑efficient route to accurate eigenvalue estimation.

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