Stabilized Maximum-Likelihood Iterative Quantum Amplitude Estimation for Structural CVaR under Correlated Random Fields

TL;DR

This work tackles tail-risk quantification in stochastic structural mechanics by casting CVaR estimation as a bounded expectation and implementing it with a stabilized quantum iterative amplitude-estimation framework (ML-IQAE). It combines a Nyström low-rank representation for spatially correlated random fields with a state-preparation oracle and adaptive Grover amplification, enabling finite-sample confidence guarantees and variance reduction. The key contributions include likelihood-constrained interval tracking, multi-hypothesis management, adaptive Grover-depth and shot allocation, and periodic disambiguation, all bounded by an explicit restart-on-empty mechanism. Numerical benchmarks on a 1D axial bar, a 2D cantilever, and a stress-concentration L-bracket demonstrate substantial oracle-efficiency gains over classical Monte Carlo while maintaining statistical reliability under correlated material uncertainty. The framework provides a practically robust pathway for quantum-enhanced tail-risk evaluation in high-fidelity structural simulations and can integrate with existing FE workflows as quantum hardware matures.

Abstract

Conditional Value-at-Risk (CVaR) is a central tail-risk measure in stochastic structural mechanics, yet its accurate evaluation under high-dimensional, spatially correlated material uncertainty remains computationally prohibitive for classical Monte Carlo methods. Leveraging bounded-expectation reformulations of CVaR compatible with quantum amplitude estimation, we develop a quantum-enhanced inference framework that casts CVaR evaluation as a statistically consistent, confidence-constrained maximum-likelihood amplitude estimation problem. The proposed method extends iterative quantum amplitude estimation (IQAE) by embedding explicit maximum-likelihood inference within a rigorously controlled interval-tracking architecture. To ensure global correctness under finite-shot noise and the non-injective oscillatory response induced by Grover amplification, we introduce a stabilized inference scheme incorporating multi-hypothesis feasibility tracking, periodic low-depth disambiguation, and a bounded restart mechanism governed by an explicit failure-probability budget. This formulation preserves the quadratic oracle-complexity advantage of amplitude estimation while providing finite-sample confidence guarantees and reduced estimator variance. The framework is demonstrated on benchmark problems with spatially correlated lognormal Young's modulus fields generated using a Nystrom low-rank Gaussian kernel model. Numerical results show that the proposed estimator achieves substantially lower oracle complexity than classical Monte Carlo CVaR estimation at comparable confidence levels, while maintaining rigorous statistical reliability. This work establishes a practically robust and theoretically grounded quantum-enhanced methodology for tail-risk quantification in stochastic continuum mechanics.

Figures (11)

• Figure 1: Oracle-$A$ circuit used in this work to encode scenario-dependent values $g_i$ into ancilla amplitudes via scenario-controlled $R_y$ rotations. Pattern conversions required for controls-on-$|0\rangle$ are implemented using surrounding $X$ gates, yielding a compact "lookup-table" implementation.
• Figure 2: Circuit implementation of the reflection $S_0 = I - 2|000\rangle\langle 000|$ using Pauli-$X$ conjugation of a multi-controlled $Z$ gate.
• Figure 3: Quantum circuit for one Grover amplification step $G = -A S_0 A^\dagger S_\chi$. The circuit consists of the success reflection $S_\chi$, the inverse oracle $A^\dagger$, the zero-state reflection $S_0$ implemented with a multi-controlled $Z$ gate, and the oracle $A$.
• Figure 4: 1D axial bar benchmark. Left end fixed ($u(0)=0$) and point load $P$ applied at $x=L$. The bar is discretized into linear elements with constant properties.
• Figure 5: Absolute error in $\mathrm{CVaR}_{\alpha}(Q)$ for compliance $Q=C$ of the bar benchmark versus requested compute budget.