The analysis of credit risk is crucial for the efficient operation of financial institutions. Quantum Amplitude Estimation (QAE) offers the potential for a quadratic speedup over classical methods used to estimate metrics such as Value at Risk (VaR) and Conditional Value at Risk (CVaR). However, numerous limitations remain in efficiently scaling the implementation of quantum circuits that solve these estimation problems. One of the main challenges is the use of costly and restrictive arithmetic that must be implemented within the quantum circuit. In this paper, we propose using Quantum Singular Value Transformation (QSVT) to significantly reduce the cost of implementing the state preparation operator, which underlies QAE for credit risk analysis. We also present an end-to-end code implementation and the results of a simulation study to validate the proposed approach and demonstrate its benefits.
The analysis of credit risk is crucial for the efficient operation of financial institutions. Quantum Amplitude Estimation (QAE) offers the potential for a quadratic speedup over classical methods used to estimate metrics such as Value at Risk (VaR) and Conditional Value at Risk (CVaR). However, numerous limitations remain in efficiently scaling the implementation of quantum circuits that solve these estimation problems. One of the main challenges is the use of costly and restrictive arithmetic that must be implemented within the quantum circuit. In this paper, we propose using Quantum Singular Value Transformation (QSVT) to significantly reduce the cost of implementing the state preparation operator, which underlies QAE for credit risk analysis. We also present an end-to-end code implementation and the results of a simulation study to validate the proposed approach and demonstrate its benefits.
Monte Carlo (MC) simulations are widely used in financial risk management, from estimating value-at-risk (VaR) to pricing over-the-counter derivatives. However, they come at a significant computational cost due to the number of scenarios required for convergence. If a probability distribution is available, Quantum Amplitude Estimation (QAE) algorithms can provide a quadratic speed-up in measuring its properties as compared to their classical counterparts. Recent studies have explored the calculation of common risk measures and the optimisation of QAE algorithms by initialising the input quantum states with pre-computed probability distributions. If such distributions are not available in closed form, however, they need to be generated numerically, and the associated computational cost may limit the quantum advantage. In this paper, we bypass this challenge by incorporating scenario generation – i.e. simulation of the risk factor evolution over time to generate probability distributions – into the quantum computation; we refer to this process as Quantum MC (QMC) simulations. Specifically, we assemble quantum circuits that implement stochastic models for equity (geometric Brownian motion), interest rate (mean-reversion models), and credit (structural, reduced-form, and rating migration credit models) risk factors. We then integrate these models with QAE to provide end-to-end examples for both market and credit risk use cases.
Abstract Problem Statement Accurate estimation of extreme financial risks, such as Conditional Value-at-Risk (CVaR) at high confidence levels ( α ≥ 0.95), poses significant computational challenges for classical Monte Carlo methods, which require O (1/ ϵ 2 ) samples and struggle to scale under rare-event scenarios. Methodology To address this, we propose a hybrid variational quantum–spintronic framework integrating Variational Quantum State Preparation (VQA), a threshold comparator oracle, and Maximum-Likelihood Amplitude Estimation (MLAE) to enable quantum-accelerated CVaR estimation suitable for NISQ devices. Results Using only 6 qubits and K = 6 amplification levels, our approach achieves CVaR ^ 0.95 = 0.281 ± 0.012 {\widehat {{\rm{CVaR}}}_{0.95}} = 0.281 \pm 0.012 (error < 1.1% compared to classical benchmarks), with tail probability p ^ = 0.0478 ± 0.0009 \hat p = 0.0478 \pm 0.0009 and state fidelity 0.967 on the ibm_brisbane backend. MLAE reduces circuit depth by 3.3 × relative to canonical QAE, and a conceptual spintronic implementation achieves ∼2.0 fJ/gate, corresponding to an 80% energy reduction versus CMOS. Contributions This work introduces the first NISQ-compatible, energy-efficient quantum–spintronic pipeline for regulatory-compliant financial risk analytics, providing a quadratic sampling speedup, accurate tail-risk estimation, and a pathway toward sustainable quantum finance.
Financial modeling and risk analysis involve computationally intensive tasks -- Monte Carlo simulation for derivative pricing, portfolio optimisation over high-dimensional asset spaces, and Value-at-Risk estimation from complex multivariate distributions -- where quantum computing offers potential polynomial-to-quadratic speedups. This paper proposes the Quantum Financial Computation Framework (QFCF), a systematic evaluation of quantum algorithms for four core financial computing tasks: option pricing (quantum amplitude estimation for Monte Carlo), portfolio optimisation (QAOA and VQE on quadratic unconstrained binary optimisation formulations), credit risk assessment (quantum Bayesian inference), and Value-at-Risk estimation (quantum arithmetic circuits). QFCF is evaluated on 16 financial benchmark problems spanning European and Asian option pricing, 50-500 asset portfolio optimisation, and credit portfolio loss distributions on near-term (NISQ) and fault-tolerant quantum hardware models. Key results: quantum amplitude estimation achieves quadratic speedup over classical Monte Carlo for option pricing with circuit depth scaling as O(1/ε) versus classical O(1/ε2); QAOA achieves portfolio optimisation solution quality within 4.8% of classical branch-and-bound for 50-asset problems; quantum Bayesian credit scoring reduces inference time by 38x for 200-variable credit networks. The framework introduces the Quantum Financial Utility Index (QFUI) and quantitative quantum advantage thresholds for each financial task.
Risk measures are important key figures to measure the adequacy of the reserves of a company. The most common risk measures in practice are Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR). Recently, quantum-based algorithms are introduced to calculate them. These procedures are based on the so-called quantum amplitude estimation algorithm, which leads to a quadratic speed-up compared to classical Monte-Carlo based methods. Based on these ideas, we construct quantum-based algorithms to calculate alternatives for VaR and CVaR, namely the Expectile Value-at-Risk (EVaR) and the Range Value-at-Risk (RVaR). These algorithms are also based on quantum amplitude estimation. In two case studies, we compare their performance with the quantum-based algorithms for VaR and CVaR. We find that all of the algorithms perform sufficiently well on a quantum simulator. Further, the calculations of EVaR and VaR are robust against noise on a real quantum device. This is not the case for CVaR and RVaR.
Abstract In recent years a CRA (Credit Risk Analysis) quantum algorithm with a quadratic speedup over classical analogous methods has been introduced [1]. We propose a new variant of this quantum algorithm with the intent of overcoming some of the most significant limitations (according to business domain experts) of this approach. In particular, we describe a method to implement a more realistic and complex risk model for the default probability of each portfolio’s asset, capable of taking into account multiple systemic risk factors. In addition, we present a solution to increase the flexibility of one of the model’s inputs, the Loss Given Default, removing the constraint to use integer values. This specific improvement addresses the need to use real data coming from the financial sector in order to establish fair benchmarking protocols. Although these enhancements come at a cost in terms of circuit depth and width, they nevertheless show a path towards a more realistic software solution. Recent progress in quantum technology shows that eventually, the increase in the number and reliability of qubits will allow for useful results and meaningful scales for the financial sector, also on real quantum hardware, paving the way for a concrete quantum advantage in the field. The paper also describes experiments conducted on simulators to test the circuit proposed and contains an assessment of the scalability of the approach presented.
We present a rigorously derived Quantum Risk Management Framework (QRMF) integrating quantum amplitude estimation for tail risk metrics, entanglement-based systemic risk analytics, and variational quantum risk optimization. Our comprehensive approach encompasses three novel modules: (i) Quantum Risk Amplitude Estimation (QRAE) achieving quadratic sample complexity improvements over classical Monte Carlo for Value-at-Risk (VaR) and Conditional VaR (CVaR) computation through binary search with amplitude estimation subroutines; (ii) Quantum Systemic Risk Analysis (QSRA) encoding financial exposure networks into entangled bipartite states with purity-based early warning observables and QAOA partitioning for cascade mitigation; and (iii) Variational Quantum Risk Optimizer (VQRO) implementing constraint-aware portfolio optimization using risk-adjusted Hamiltonians with feasible-state ansätze and penalty-based formulations. We establish formal complexity theorems under explicit QRAM/QROM cost models, prove convergence guarantees for variational components under convex surrogate conditions, and demonstrate theoretical correspondence between entanglement purity responses and classical contagion path-sum sensitivities in linearized network models. The hybrid framework combines gate-model quantum circuits for amplitude estimation and entanglement encoding with variational quantum eigensolvers for optimization, validated through comprehensive simulations on synthetic financial networks and real market data. Extensive ablation studies on circuit depth versus accuracy trade-offs, noise resilience analysis, and resource scaling complement theoretical results, with all quantum amplitude estimation and entanglement simulations performed using Qiskit on classical hardware while baseline classical Monte Carlo and network centrality methods provide benchmarking baselines. This distinction ensures accurate interpretation of quantum advantages across the integrated risk management pipeline.