Circuit Synthesis for Variational Quantum Algorithms: The Depth of Algebra and Geometry
Abstract
The analysis of Hamiltonian Variational Ansätze centers on three fundamental questions: which states are exactly reachable from a given initial state, what native control depth suffices for local and global synthesis, and whether a bounded circuit extension can expose a direction of decreasing cost at a suboptimal output. We address these challenges by leveraging the dynamical Lie algebras (DLAs) associated with the circuits and the geometry of the initial state's orbit under the action of the corresponding dynamical Lie group G. We show that an initial state has a closed projective orbit whenever it is a nondegenerate eigenstate of a control Hamiltonian. Combined with asymptotic QAOA convergence, this guarantees exact, finite depth ground state preparation for architectures employing diagonal problem Hamiltonians and irreducible stoquastic mixers. We bound the number of native control exponentials required for local unitary synthesis and state preparation using DLA commutator filtration growth alongside reachable orbit dimensions. Complementing these local results, we construct global frameworks on invariant sectors equipped with full unitary or special unitary dynamical Lie algebras. These constructions yield explicit applications to Pauli-generated systems and constrained maximum independent set multi-angle QAOA circuits; specifically, we show that their native control costs are directly determined by commutator filtration depth and independent-set toggle-group geometry, respectively. Finally, if the dynamical Lie group is compact, the observable HP generates an admissible evolution (iHP ∈ g, where g is the DLA), and the projective orbit of the initial state contains a ground state ray of HP, then a bounded depth, identity-initialized circuit extension exhibits either first-order descent or negative curvature at every suboptimal circuit output.
Related
- Optimization Geometry of QAOA and Variational Quantum Algorithms
- Accelerating noisy variational quantum algorithms with physics-informed denoising networks
- Scaling law of variational quantum algorithms
- Towards Reproducible Evaluation of Distributed Quantum Circuit Partitioning Algorithms
- Circuit-depth optimization of quantum partial-search algorithms
BibTeX
@misc{tsvelikhovskiy2026circuit,
title = {{Circuit Synthesis for Variational Quantum Algorithms: The Depth of Algebra and Geometry}},
author = {Boris Tsvelikhovskiy},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.22689},
eprint = {2610.22689},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.22689},
}
RIS
TY - UNPB
TI - Circuit Synthesis for Variational Quantum Algorithms: The Depth of Algebra and Geometry
AU - Boris Tsvelikhovskiy
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.22689
UR - https://arxiv.org/abs/2610.22689
AB - The analysis of Hamiltonian Variational Ansätze centers on three fundamental questions: which states are exactly reachable from a given initial state, what native control depth suffices for local and global synthesis, and whether a bounded circuit extension can expose a direction of decreasing cost at a suboptimal output. We address these challenges by leveraging the dynamical Lie algebras (DLAs) associated with the circuits and the geometry of the initial state's orbit under the action of the corresponding dynamical Lie group G. We show that an initial state has a closed projective orbit whenever it is a nondegenerate eigenstate of a control Hamiltonian. Combined with asymptotic QAOA convergence, this guarantees exact, finite depth ground state preparation for architectures employing diagonal problem Hamiltonians and irreducible stoquastic mixers. We bound the number of native control exponentials required for local unitary synthesis and state preparation using DLA commutator filtration growth alongside reachable orbit dimensions. Complementing these local results, we construct global frameworks on invariant sectors equipped with full unitary or special unitary dynamical Lie algebras. These constructions yield explicit applications to Pauli-generated systems and constrained maximum independent set multi-angle QAOA circuits; specifically, we show that their native control costs are directly determined by commutator filtration depth and independent-set toggle-group geometry, respectively. Finally, if the dynamical Lie group is compact, the observable H_P generates an admissible evolution (iH_P ∈ g, where g is the DLA), and the projective orbit of the initial state contains a ground state ray of H_P, then a bounded depth, identity-initialized circuit extension exhibits either first-order descent or negative curvature at every suboptimal circuit output.
ER -