Quantum

Circuit Synthesis for Variational Quantum Algorithms: The Depth of Algebra and Geometry

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Briefings

This article analyzes Hamiltonian Variational Ansatz centers using dynamical Lie algebras to determine exact state reachability and control depth. It establishes that nondegenerate eigenstates of control Hamiltonians have closed projective orbits, guaranteeing finite-depth ground state preparation for diagonal problem Hamiltonians. The study bounds native control exponentials via commutator filtration growth and constructs global frameworks on invariant sectors, explicitly applying these to Pauli-generated systems and constrained maximum independent set multi-angle QAOA circuits.

Key points

• Nondegenerate eigenstates of a control Hamiltonian possess a closed projective orbit, which guarantees exact, finite-depth ground state preparation for architectures using diagonal problem Hamiltonians and irreducible stoquastic mixers.
• Native control costs for local unitary synthesis and state preparation are bounded by the commutator filtration depth and the reachable orbit dimensions, respectively.

Context

The research addresses three fundamental questions in Hamiltonian Variational Ansatz centers: exact state reachability, native control depth sufficiency, and bounded circuit extensions for cost reduction. By leveraging dynamical Lie algebras and geometry of the initial state's orbit, the authors provide new theoretical bounds for quantum circuit optimization.

Limitations

Results apply specifically to architectures with diagonal problem Hamiltonians and irreducible stoquastic mixers. Global frameworks are constructed on invariant sectors with full or special unitary dynamical Lie algebras, but explicit applications to Pauli-generated systems and constrained maximum independent set multi-angle QAOA circuits are demonstrated. The study relies on preprint arXiv:2610.22689v1, and the analysis of Hamiltonian Variational Ansatz centers remains an active research area with significant uncertainty regarding the general applicability of these bounds.

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