Reducing the arithmetic cost of quantum derivative pricing with language-model circuit design

Quantum amplitude estimation can estimate expected discounted payoffs with quadratically fewer oracle calls than Monte Carlo, but each call requires coherent computation. We use language-model-assisted circuit design to reduce reversible payoff arithmetic for a three-asset worst-of autocallable and a target accrual redemption forward (TARF). The constructions retain compact integer representations of market paths alongside contract history, reconstructing quantities temporarily for contract decisions and payoff-angle evaluation. Relative to matched internal references, the factor-sum autocallable and windowed TARF circuits reduce the product of Toffoli count and peak logical qubits $(N_{Tof} Q_{max})$ by factors of 2.39 and 4.84, respectively. At a second autocallable setting, fresh search recovers a 1.35-fold saving after the tested fixed adaptation loses its advantage. For the factor-sum circuit, we prove exact coherent execution of its integer specification for every input, incoming accumulator and measurement record, with clean workspace and phase cancellation. A separate lattice calculation gives a numerical angle-tolerance-violation probability of 6.248e-8 and evaluates pricing bias under the declared loading law, with bounded omission but no certified numerical enclosure. The results connect compact representations to lower-cost payoff arithmetic and separately assessed logical and financial correctness. Their resource scope excludes loading, payoff rotation and estimation.