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This research establishes rigorous finite-shot success guarantees for constrained quantum optimization by decoupling mixer-driven exploration from spectral selection via Fejér filtering. By deriving an explicit lower bound on sampling probability, the work provides a tractable framework for compensating limited phase separation or weak envelope mass through increased filter orders and additional circuit shots. These analytical tools offer critical insights into resource management for near-term quantum algorithms targeting feasible solutions.
Exposing Finite-Depth, Finite-Shot Guarantees for Constrained Quantum Optimization via Fej\'er Filtering
arXiv:2603.01809v3 Announce Type: replace Abstract: Constrained quantum optimization algorithms need quantitative guarantees that connect circuit resources to the probability of actually sampling feasible or optimal solutions in finitely many shots. We establish such a connection by exposing a positive sampling law in which mixer-driven exploration and spectral selection can be controlled separately. We show that after removing interference between distinct cost eigenspaces as an analytic device, the measurement distribution becomes the normalized product of a mixer-induced exploration envelope and a Fej\'er spectral weight, with the former describing how the mixer spreads probability over the encoded manifold and the latter enhancing the target cost phase while suppressing spectrally separated nontarget phases. In this model, finite-shot success becomes a tractable competition between target weight and off-target leakage, yielding an explicit lower bound on the probability of sampling an optimum. For the primary bound, we rescale the cost Hamiltonian to an integer-valued spectrum, placing the wrapped cost phases on a controlled lattice for Fej\'er filtering. We then define $\delta$ as the minimum circular separation between the optimal phase and every nontarget phase. The single-shot success probability $q_0$ satisfies \[ q_0 \ge \frac{x}{1+x}, \qquad x = (p+1)^2 \sin^2\!\left(\frac{\delta}{2}\right) C_{\beta}, \] where $p$ is the filter order and $C_{\beta}$ is the mixer-envelope mass on the optimal set, exposing a finite-resource compensation law in which weaker phase separation or smaller envelope mass can be compensated by increased filter order and additional shots. The same filtering principle exposes a feasibility guarantee when applied to penalty phases. We further prove analogous bounds for nonlattice spectra through off-target suppression, extending our results beyond exact lattice normalization.
Abstract:Constrained quantum optimization algorithms need quantitative guarantees that connect circuit resources to the probability of actually sampling feasible or optimal solutions in finitely many shots. We establish such a connection by exposing a positive sampling law in which mixer-driven exploration and spectral selection can be controlled separately. We show that after removing interference between distinct cost eigenspaces as an analytic device, the measurement distribution becomes the normalized product of a mixer-induced exploration envelope and a Fejér spectral weight, with the former describing how the mixer spreads probability over the encoded manifold and the latter enhancing the target cost phase while suppressing spectrally separated nontarget phases. In this model, finite-shot success becomes a tractable competition between target weight and off-target leakage, yielding an explicit lower bound on the probability of sampling an optimum. For the primary bound, we rescale the cost Hamiltonian to an integer-valued spectrum, placing the wrapped cost phases on a controlled lattice for Fejér filtering. We then define $\delta$ as the minimum circular separation between the optimal phase and every nontarget phase. The single-shot success probability $q_0$ satisfies \[ q_0 \ge \frac{x}{1+x}, \qquad x = (p+1)^2 \sin^2\!\left(\frac{\delta}{2}\right) C_{\beta}, \] where $p$ is the filter order and $C_{\beta}$ is the mixer-envelope mass on the optimal set, exposing a finite-resource compensation law in which weaker phase separation or smaller envelope mass can be compensated by increased filter order and additional shots. The same filtering principle exposes a feasibility guarantee when applied to penalty phases. We further prove analogous bounds for nonlattice spectra through off-target suppression, extending our results beyond exact lattice normalization.
Substantially revised version. Clarified the positive Fejér-reweighted sampling law and its interpretation, refined the finite-depth/finite-shot guarantees and phase-normalization analysis, and strengthened the resource-tradeoff discussion to include inverse-polynomial success-probability regimes. Title and exposition revised
Quantum Physics (quant-ph); Information Theory (cs.IT); Mathematical Physics (math-ph); Numerical Analysis (math.NA)
arXiv:2603.01809 [quant-ph]
(or arXiv:2603.01809v3 [quant-ph] for this version)
Full text under the publisher's open licence · 315 words
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