Treffer: Quantum Risk Modeling: Theoretical Frameworks for Financial Uncertainty Quantification

We develop a comprehensive theoretical framework for quantum-enhanced risk modeling in financial systems, establishing mathematical foundations for representing and computing risk factors using quantum states and operations. The theory begins by formulating portfolio risk as quantum observables, where correlations between assets are naturally encoded in entangled quantum states, enabling exponentially compact representations of high-dimensional covariance structures. We derive theoretical bounds on the accuracy of quantum risk estimates and prove convergence theorems for quantum algorithms that compute value-at-risk, conditional value-atrisk, and higher-moment risk measures. Our framework incorporates counterdiabatic evolution techniques to navigate complex risk landscapes efficiently, establishing conditions under which quantum annealing approaches converge to optimal portfolio configurations with provable performance guarantees. We analyze the computational complexity of quantum versus classical risk calculation, identifying scenarios where quantum advantage emerges based on portfolio size, correlation structure complexity, and required solution precision. The theory extends to dynamic risk assessment under changing market conditions, proving that quantum algorithms can adapt to evolving risk profiles with polynomial overhead. We establish fundamental limits on quantum speedups for risk computation and identify problem structures that maximize quantum advantage. This theoretical foundation bridges quantum computing and quantitative finance, offering rigorous principles for designing quantum algorithms that address computational bottlenecks in modern risk management.