Quantum Eigensolvers

Purpose

This file is the category index and routing guide for eigensolver tasks.

Use this folder when the user wants to compute eigenvalues or eigenstates of a quantum operator, compare exact and variational eigensolver strategies, or implement an eigensolver workflow in Qiskit.

Routing Rules

Choose the leaf skill by problem type:

• If the user wants exact eigenvalues, full diagonalization behavior, sparse or dense matrix backends, or auxiliary operator evaluation on exact eigenstates:
  • Read ./numyeigensolver/SKILL.md
• If the user wants a variational workflow, low-lying excited states, ansatz-based optimization, fidelity penalties, or hybrid primitive-based execution:
  • Read ./vqd/SKILL.md

Selection Guide

1. NumPy Eigensolver

See reference: ./numyeigensolver/SKILL.md

Use this path when:

• the operator is small enough for exact classical treatment,
• the user needs deterministic eigenpairs,
• the task involves dense or sparse matrix eigendecomposition,
• auxiliary operators should be evaluated exactly on returned eigenstates.

2. Variational Quantum Deflation

See reference: ./vqd/SKILL.md

Use this path when:

• the user needs the ground state plus excited states,
• the workflow depends on a parameterized ansatz,
• the solution should use Qiskit primitives and a classical optimizer,
• overlap penalties or orthogonality constraints are part of the algorithm design.

Quick Intent Router

• "Compute the lowest few exact eigenvalues" -> ./numyeigensolver/SKILL.md
• "Diagonalize a SparsePauliOp exactly" -> ./numyeigensolver/SKILL.md
• "Evaluate auxiliary observables on exact eigenstates" -> ./numyeigensolver/SKILL.md
• "Find excited states with a variational ansatz" -> ./vqd/SKILL.md
• "Implement VQD with estimator and fidelity primitives" -> ./vqd/SKILL.md
• "Compare exact and variational eigensolvers" -> read this file first, then both leaf skills.

Engineering Notes

• NumPyEigensolver is exact but scales poorly with qubit count because the operator dimension grows as $2^n$.
• VQD is more scalable in workflow design, but its result quality depends on the ansatz, optimizer, fidelity evaluation, and penalty tuning.
• Use NumPyEigensolver as a baseline when validating a VQD setup on small instances.

Response Contract

When handling eigensolver requests:

1. Decide whether the task is exact classical diagonalization or variational excited-state estimation.
2. Read the corresponding leaf skill before producing code.
3. Keep category-level guidance here and leave implementation detail to the leaf skills.