Superspace Concentration and Adversarial Robustness in Quantum Algorithms

2026-06-09

AI Summary

This work investigates how to quantify the capacity of quantum systems to concentrate information in an extended degree-of-freedom space to enhance adversarial robustness. It introduces a focus measure \( F(\rho) \) and establishes, for the first time, a complete resource theory of hyperspace concentration, rigorously distinguishing it operationally from \( U(d_S) \)-asymmetry. The study further uncovers a direct link between \( F(\rho) \) and the success probability of the marked state in Grover’s algorithm. Through GPU-accelerated simulations across six system configurations and over ten thousand random states, \( F(\rho) \) demonstrates strict monotonicity under various noise channels with no violations observed; analytical decoherence predictions achieve an accuracy of \( 1.11 \times 10^{-16} \). Experiments show that focused states maintain \( F > 0.9 \) even under attack strength \( \varepsilon = 0.302 \), substantially outperforming conventional fidelity-based metrics (\( \varepsilon = 0.174 \)), with the focus capacity gap \( \Delta F \) obeying a \( \log_2(d_S) \) scaling law.

Abstract

We study superspace concentration as a quantum resource, formalized through the focus measure F(\r{ho}) = λ_max(\r{ho}_super) - the largest eigenvalue of the reduced superspace state - which quantifies the capacity of a quantum system to concentrate informational weight into a preferred subspace of an extended degree-of-freedom space. We develop a complete resource-theoretic framework around this measure and validate its properties through GPU-accelerated numerical simulation. Analytic decoherence predictions are confirmed to machine precision (1.11 x 10^{-16}) for superspace dimensions dS in {2,4,8,16,32}. Focus monotonicity holds across 10,000 random states with zero violations under four focus-non-generating channels across six system configurations. Focused quantum states resist coherent unitary attacks with significantly greater resilience than standard fidelity predicts, with focus remaining above 0.9 at attack strength ε = 0.302 versus ε = 0.174 for fidelity. We further demonstrate that the focus measure and the U(dS)-asymmetry measure are operationally distinct: asymmetry remains near zero and provides no robustness signal under coherent and targeted attacks while focus tracks spectral concentration and remains robust until ε > 0.3. The connection between Grover's algorithm and superspace concentration is made explicit via the identity F(|ψ_k><ψ_k|) = P(marked), providing a resource-theoretic interpretation of oracle query complexity. Finally, we provide the first numerical characterization of the focus capacity gap ΔF, identifying a log_2(dS) scaling law confirmed for both product and correlated noise channels.

Problem

• superspace concentration
• adversarial robustness
• quantum resource
• focus measure
• quantum algorithms

Innovation

• superspace concentration
• focus measure
• adversarial robustness
• quantum resource theory
• Grover's algorithm