We introduce a quantum algorithm to compute the market risk of financial derivatives. Previous work has shown that quantum amplitude estimation can accelerate derivative pricing quadratically in the target error and we extend this to a quadratic error scaling advantage in market risk computation. We show that employing quantum gradient estimation algorithms can deliver a further quadratic advantage in the number of the associated market sensitivities, usually called greeks . By numerically simulating the quantum gradient estimation algorithms on financial derivatives of practical interest, we demonstrate that not only can we successfully estimate the greeks in the examples studied, but that the resource requirements can be significantly lower in practice than what is expected by theoretical complexity bounds. This additional advantage in the computation of financial market risk lowers the estimated logical clock rate required for financial quantum advantage from Chakrabarti et al. [Quantum 5, 463 (2021)] by a factor of ~7, from 50MHz to 7MHz, even for a modest number of greeks by industry standards (four). Moreover, we show that if we have access to enough resources, the quantum algorithm can be parallelized across 60 QPUs, in which case the logical clock rate of each device required to achieve the same overall runtime as the serial execution would be ~100kHz. Throughout this work, we summarize and compare several different combinations of quantum and classical approaches that could be used for computing the market risk of financial derivatives.
We introduce a quantum algorithm to compute the market risk of financial derivatives. Previous work has shown that quantum amplitude estimation can accelerate derivative pricing quadratically in the target error and we extend this to a quadratic error scaling advantage in market risk computation. We show that employing quantum gradient estimation algorithms can deliver a further quadratic advantage in the number of the associated market sensitivities, usually called greeks . By numerically simulating the quantum gradient estimation algorithms on financial derivatives of practical interest, we demonstrate that not only can we successfully estimate the greeks in the examples studied, but that the resource requirements can be significantly lower in practice than what is expected by theoretical complexity bounds. This additional advantage in the computation of financial market risk lowers the estimated logical clock rate required for financial quantum advantage from Chakrabarti et al. [Quantum 5, 463 (2021)] by a factor of ~7, from 50MHz to 7MHz, even for a modest number of greeks by industry standards (four). Moreover, we show that if we have access to enough resources, the quantum algorithm can be parallelized across 60 QPUs, in which case the logical clock rate of each device required to achieve the same overall runtime as the serial execution would be ~100kHz. Throughout this work, we summarize and compare several different combinations of quantum and classical approaches that could be used for computing the market risk of financial derivatives.
We introduce two quantum algorithms to compute the Value at Risk (VaR) and Conditional Value at Risk (CVaR) of financial derivatives using quantum computers: the first by applying existing ideas from quantum risk analysis to derivative pricing, and the second based on a novel approach using Quantum Signal Processing (QSP). Previous work in the literature has shown that quantum advantage is possible in the context of individual derivative pricing and that advantage can be leveraged in a straightforward manner in the estimation of the VaR and CVaR. The algorithms we introduce in this work aim to provide an additional advantage by encoding the derivative price over multiple market scenarios in superposition and computing the desired values by applying appropriate transformations to the quantum system. We perform complexity and error analysis of both algorithms, and show that while the two algorithms have the same asymptotic scaling the QSP-based approach requires significantly fewer quantum resources for the same target accuracy. Additionally, by numerically simulating both quantum and classical VaR algorithms, we demonstrate that the quantum algorithm can extract additional advantage from a quantum computer compared to individual derivative pricing. Specifically, we show that under certain conditions VaR estimation can lower the latest published estimates of the logical clock rate required for quantum advantage in derivative pricing by up to ∼30x. In light of these results, we are encouraged that our formulation of derivative pricing in the QSP framework may be further leveraged for quantum advantage in other relevant financial applications, and that quantum computers could be harnessed more efficiently by considering problems in the financial sector at a higher level.
Pricing financial derivatives on quantum computers typically includes quantum arithmetic components which contribute heavily to the quantum resources required by the corresponding circuits. In this manuscript, we introduce a method based on Quantum Signal Processing (QSP) to encode financial derivative payoffs directly into quantum amplitudes, alleviating the quantum circuits from the burden of costly quantum arithmetic. Compared to current state-of-the-art approaches in the literature, we find that for derivative contracts of practical interest, the application of QSP significantly reduces the required resources across all metrics considered, most notably the total number of T-gates by ∼ 16 x and the number of logical qubits by ∼ 4 x. Additionally, we estimate that the logical clock rate needed for quantum advantage is also reduced by a factor of ∼ 5 x. Overall, we find that quantum advantage will require 4.7 k logical qubits, and quantum devices that can execute 10 9 T-gates at a rate of 45 MHz. While in this work we focus specifically on the payoff component of the derivative pricing process where the method we present is most readily applicable, similar techniques can be employed to further reduce the resources in other applications, such as state preparation.
Quantum computing has emerged as a transformative computational paradigm, offering novel capabilities to tackle complex problems in finance—most notably in derivative pricing, risk management, and portfolio optimization. This review synthesizes recent advances in quantum algorithms and hybrid quantum–classical frameworks as applied to financial services, examining how technologies such as quantum-annealing, quantum amplitude estimation, and variational quantum circuits can improve efficiency, accuracy, and scalability over classical methods. The study provides a systematic assessment of key use-cases: (1) derivative pricing and greeks computation via quantum Monte Carlo and amplitude estimation; (2) risk management enhancements, including Value-at-Risk (VaR) and conditional VaR through quantum gradient estimation; and (3) portfolio optimization via QUBO (quadratic unconstrained binary optimization) mapping, the Quantum Approximate Optimization Algorithm (QAOA) and quantum annealing approaches. In addition to reviewing technological potential, the paper addresses critical barriers—hardware noise, coherence limitations, data integration, regulatory considerations—and outlines future research directions, including quantum-resilient cryptography in finance and industry coupling. By bringing together algorithmic, financial and regulatory perspectives, this review seeks to guide academics and practitioners toward the next frontier in financial computing.
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.
The continuous growth of financial markets dictates the need for its participants to seek new approaches to financial analysis to gain competitive advantages, including through the use of new approaches in the field of computing. Quantum computing can be used as a tool for obtaining these advantages over competitors. In particular, Monte Carlo modeling, although widely used in financial risk management, requires significant computing resources due to the large number of scenarios required to obtain an accurate result. To optimize this approach, quantum amplitude estimation algorithms are used, which accelerate this process if pre-calculated probability distributions are used to initialize input quantum states. However, in the absence of these distributions in existing approaches to this topic, they are generated numerically using classical computing, which completely negates the advantage of the quantum approach. This article proposes a solution to this problem by using quantum computing, including for the generation of probability distributions. The article discusses the creation of quantum circuits for modeling the evolution of risk factors over time for capital flows, interest rates, and credit risks, and presents the combination of these models with quantum amplitude estimation algorithms as an example of using the obtained algorithms for credit risk management. In conclusion, the article analyzes the possibility of using the obtained circuits in financial analysis.
Quantum algorithms have been proposed for most computational tasks in trading, but the resulting literature is not a single body of work with a single standard of evidence. This survey partitions it into strands with distinct foundations and assesses each against the strongest available evidence. Pricing and market-risk computation rest on quantum amplitude estimation and carry a proven quadratic advantage in inverse precision; circuit-level resource estimates place the requirement at roughly 8000 logical qubits and a logical clock rate of tens of megahertz, several orders of magnitude beyond demonstrated hardware, with joint estimation of prices and sensitivities the main lever that lowers it. Combinatorial formulations of portfolio selection, multi-period trading trajectories and arbitrage detection map onto quadratic unconstrained binary optimization and have the field’s most mature hardware demonstrations, but an extensive benchmark on 250 real-stock instances finds classical mixed-integer programming and tailored heuristics dominant. Quantum machine learning for return prediction has been substantially dequantized, and a controlled evaluation on cross-sectional equity returns attributes reported advantages to evaluation protocol rather than to the model. A fourth strand exploits entanglement for decision coordination under latency constraints, where the advantage follows from Bell’s theorem rather than a complexity assumption. We identify the classical-quantum interface, superquadratic structure, resource-estimate methodology, nonlocal-game design and adversarial benchmarking as the productive open directions.