Monte Carlo simulations underpin computational quantitative finance, where they are used for derivative pricing, risk quantification and scenario analysis whenever closed-form solutions are unavailable. Their main limitation is the slow O(N^-1/2) convergence of classical sampling. Quantum amplitude estimation promises a quadratic improvement in query complexity over classical Monte Carlo, reducing the sample requirement from O(eps^-2) to O(eps^-1) under idealised conditions. This thesis measures the extent to which the quadratic improvement in query complexity survives a concrete risk task under gate noise when compared to classical Monte Carlo. A four-obligor correlated portfolio, whose loss distribution can be computed exactly, is encoded into quantum circuits, and its value at risk and expected shortfall are estimated by iterative amplitude estimation under exactly simulated depolarising noise on every two-qubit gate. Six configurations of 11 to 15 qubits vary the portfolio, the factor grid and the arithmetic construction. At matched loss-model evaluation budgets and without noise, amplitude estimation reduces the RMSE of the tail probability against classical Monte Carlo by a factor of 13.6 at the largest budget tested. The error budget is dominated by the circuit encoding, not the estimator. As a result, the value at risk at the 95% level picks the wrong loss level in none of the noiseless repetitions, while the encoding understates expected shortfall by 0.177 loss units. Under noise, the signal decays with every amplification round. The number of rounds that still help falls from 64 at an error rate of 3x10^-6 to none at 3x10^-2. This work concludes that amplification benefits only while the product of gate count and error rate stays small against one. Because every obligor added enlarges the circuit, the error rate at which amplification remains useful falls as the portfolio grows, and advantage starts depending on circuit compression instead of estimator design.
Monte Carlo simulations underpin computational quantitative finance, where they are used for derivative pricing, risk quantification and scenario analysis whenever closed-form solutions are unavailable. Their main limitation is the slow O(N^-1/2) convergence of classical sampling. Quantum amplitude estimation promises a quadratic improvement in query complexity over classical Monte Carlo, reducing the sample requirement from O(eps^-2) to O(eps^-1) under idealised conditions. This thesis measures the extent to which the quadratic improvement in query complexity survives a concrete risk task under gate noise when compared to classical Monte Carlo. A four-obligor correlated portfolio, whose loss distribution can be computed exactly, is encoded into quantum circuits, and its value at risk and expected shortfall are estimated by iterative amplitude estimation under exactly simulated depolarising noise on every two-qubit gate. Six configurations of 11 to 15 qubits vary the portfolio, the factor grid and the arithmetic construction. At matched loss-model evaluation budgets and without noise, amplitude estimation reduces the RMSE of the tail probability against classical Monte Carlo by a factor of 13.6 at the largest budget tested. The error budget is dominated by the circuit encoding, not the estimator. As a result, the value at risk at the 95% level picks the wrong loss level in none of the noiseless repetitions, while the encoding understates expected shortfall by 0.177 loss units. Under noise, the signal decays with every amplification round. The number of rounds that still help falls from 64 at an error rate of 3x10^-6 to none at 3x10^-2. This work concludes that amplification benefits only while the product of gate count and error rate stays small against one. Because every obligor added enlarges the circuit, the error rate at which amplification remains useful falls as the portfolio grows, and advantage starts depending on circuit compression instead of estimator design.
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.
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.
Abstract We present a quantum algorithm that analyzes risk more efficiently than Monte Carlo simulations traditionally used on classical computers. We employ quantum amplitude estimation to price securities and evaluate risk measures such as Value at Risk and Conditional Value at Risk on a gate-based quantum computer. Additionally, we show how to implement this algorithm and how to trade-off the convergence rate of the algorithm and the circuit depth. The shortest possible circuit depth—growing polynomially in the number of qubits representing the uncertainty—leads to a convergence rate of O ( M −2/3 ), where M is the number of samples. This is already faster than classical Monte Carlo simulations which converge at a rate of O ( M −1/2 ). If we allow the circuit depth to grow faster, but still polynomially, the convergence rate quickly approaches the optimum of O ( M −1 ). Thus, for slowly increasing circuit depths our algorithm provides a near quadratic speed-up compared to Monte Carlo methods. We demonstrate our algorithm using two toy models. In the first model we use real hardware, such as the IBM Q Experience, to price a Treasury-bill (T-bill) faced by a possible interest rate increase. In the second model, we simulate our algorithm to illustrate how a quantum computer can determine financial risk for a two-asset portfolio made up of government debt with different maturity dates. Both models confirm the improved convergence rate over Monte Carlo methods. Using simulations, we also evaluate the impact of cross-talk and energy relaxation errors.
Monte Carlo integration using quantum computers has been widely investigated, including applications to concrete problems. It is known that quantum algorithms based on quantum amplitude estimation (QAE) can compute an integral with a smaller number of iterative calls of the quantum circuit which calculates the integrand, than classical methods call the integrand subroutine. However, the issues about the iterative operations in the integrand circuit have not been discussed so much. That is, in the high-dimensional integration, many random numbers are used for calculation of the integrand and in some cases similar calculations are repeated to obtain one sample value of the integrand. In this paper, we point out that we can reduce the number of such repeated operations by a combination of the nested QAE and the use of pseudorandom numbers (PRNs), if the integrand has the separable form with respect to contributions from distinct random numbers. The use of PRNs, which the authors originally proposed in the context of the quantum algorithm for Monte Carlo, is the key factor also in this paper, since it enables parallel computation of the separable terms in the integrand. Furthermore, we pick up one use case of this method in finance, the credit portfolio risk measurement, and estimate to what extent the complexity is reduced.
Numerical integration with Monte Carlo methods is a central computational task in many scientific and industrial applications, including financial derivative pricing and risk management. Classical Monte Carlo algorithms are computationally demanding: achieving an accuracy ε typically requires a number of function evaluations scaling as O(1/ε2). Quantum-accelerated Monte Carlo methods based on quantum amplitude estimation can in principle quadratically improve this dependence. However, \textit{quasi}-Monte Carlo methods have not been explored in the quantum context. In this work, we introduce a quantum quasi-Monte Carlo algorithm that combines low-discrepancy nets with quantum amplitude estimation. The proposed method prepares the quasi-random point set coherently in superposition. The method does not yield an asymptotic improvement over classical quasi-Monte Carlo, since the total error separates into a discretization error, determined by the finite net, and a quantum estimation error. Instead, we explore a pre-asymptotic advantage window: for a target accuracy that would classically require 2q low discrepancy points, one can prepare a higher-resolution net of size 2Q, with Q>q, in superposition and reach the same accuracy using significantly fewer function queries. This window can be controlled by tuning the circuit resolution and amplitude-estimation parameters, making the approach relevant for practical regimes where the number of queries is finite rather than asymptotically large.
Classical Monte Carlo methods for pricing catastrophe insurance tail risk converge at order reciprocal root N, requiring large simulation budgets to resolve upper-tail percentiles of the loss distribution. This sample-sparsity problem can lead to AI models trained on impoverished tail data, producing poorly calibrated risk estimates where insolvency risk is greatest. Quantum Amplitude Estimation (QAE), following Montanaro, achieves convergence approaching order reciprocal N in oracle queries - a quadratic speedup that, at scale, would enable high-resolution tail estimation within practical budgets. We validate this advantage empirically using a Qiskit Aer simulator with genuine Grover amplification. A complete pipeline encodes fitted lognormal catastrophe distributions into quantum oracles via amplitude encoding, producing small readout probabilities that enable safe Grover amplification with up to k=16 iterations. Seven experiments on synthetic and real (NOAA Storm Events, 58,028 records) data yield three main findings: an oracle-model advantage, that strong classical baselines win when analytical access is available, and that discretisation, not estimation, is the current bottleneck.