Effective-Hamiltonian Quantum Solvers for Differential Equations: Alternative Constructions and Function Encodings
Abstract
Differential equations can be encoded as ground-state problems by constructing a positive-semidefinite effective Hamiltonian whose minimum-energy state represents the solution. We extend this framework in several directions. First, we show how multiple nonzero initial, boundary, and data conditions can be incorporated into the homogeneous formulation A | f⟩=0 using a known nonzero reference condition. We then analyse an alternative formulation based on A |f⟩ = | b⟩, with Hamiltonian Hb=A†(I−|b⟩⟨ b|)A, which incorporates source terms and nonzero constraints through an augmented system. For nonlinear differential equations, we examine the unphysical ground-state degeneracy introduced by tensor-product representations and consider both Hamiltonian constraints and ansatz-level restrictions for targeting physically valid product states. Finally, we develop grid-value amplitude-encoded versions of both Hamiltonian constructions and compare them with spectral coefficient encoding. Through linear and nonlinear examples, we assess how the choice of Hamiltonian formulation and function representation affects solution recovery, spectral gap, degeneracy, and readout. These results broaden the applicability of ground-state-based quantum differential-equation solvers while clarifying their principal practical trade-offs.
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BibTeX
@misc{paine2026effectivehamiltonian,
title = {{Effective-Hamiltonian Quantum Solvers for Differential Equations: Alternative Constructions and Function Encodings}},
author = {Annie E. Paine},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.26330},
eprint = {2609.26330},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26330},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Effective-Hamiltonian Quantum Solvers for Differential Equations: Alternative Constructions and Function Encodings
AU - Annie E. Paine
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.26330
UR - https://arxiv.org/abs/2609.26330
AB - Differential equations can be encoded as ground-state problems by constructing a positive-semidefinite effective Hamiltonian whose minimum-energy state represents the solution. We extend this framework in several directions. First, we show how multiple nonzero initial, boundary, and data conditions can be incorporated into the homogeneous formulation A | f⟩=0 using a known nonzero reference condition. We then analyse an alternative formulation based on A |f⟩ = | b⟩, with Hamiltonian H_b=A^†(I−|b⟩⟨ b|)A, which incorporates source terms and nonzero constraints through an augmented system. For nonlinear differential equations, we examine the unphysical ground-state degeneracy introduced by tensor-product representations and consider both Hamiltonian constraints and ansatz-level restrictions for targeting physically valid product states. Finally, we develop grid-value amplitude-encoded versions of both Hamiltonian constructions and compare them with spectral coefficient encoding. Through linear and nonlinear examples, we assess how the choice of Hamiltonian formulation and function representation affects solution recovery, spectral gap, degeneracy, and readout. These results broaden the applicability of ground-state-based quantum differential-equation solvers while clarifying their principal practical trade-offs.
ER -