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arXiv:2105.12087 · New Journal of Physics

Quantum algorithm for credit valuation adjustments

Javier A. Alcazar, Andrea Cadarso, Amara Katabarwa, Marta Mauri, Borja Peropadre, Guoming Wang, Yudong Cao

Published Jan 28, 2022• 18 citations
arXivHTMLPDF

Abstract

Abstract Quantum mechanics is well known to accelerate statistical sampling processes over classical techniques. In quantitative finance, statistical samplings arise broadly in many use cases. Here we focus on a particular one of such use cases, credit valuation adjustment (CVA), and identify opportunities and challenges towards quantum advantage for practical instances. To build a NISQ-friendly quantum circuit able to solve such problem, we draw on various heuristics that indicate the potential for significant improvement over well-known techniques such as reversible logical circuit synthesis. In minimizing the resource requirements for amplitude amplification while maximizing the speedup gained from the quantum coherence of a noisy device, we adopt a recently developed Bayesian variant of quantum amplitude estimation using engineered likelihood functions. We perform numerical analyses to characterize the prospect of quantum speedup in concrete CVA instances over classical Monte Carlo simulations.

DOI
10.1088/1367-2630/ac5003

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Abstract Quantum mechanics is well known to accelerate statistical sampling processes over classical techniques. In quantitative finance, statistical samplings arise broadly in many use cases. Here we focus on a particular one of such use cases, credit valuation adjustment (CVA), and identify opportunities and challenges towards quantum advantage for practical instances. To build a NISQ-friendly quantum circuit able to solve such problem, we draw on various heuristics that indicate the potential for significant improvement over well-known techniques such as reversible logical circuit synthesis. In minimizing the resource requirements for amplitude amplification while maximizing the speedup gained from the quantum coherence of a noisy device, we adopt a recently developed Bayesian variant of quantum amplitude estimation using engineered likelihood functions. We perform numerical analyses to characterize the prospect of quantum speedup in concrete CVA instances over classical Monte Carlo simulations.
Credit Valuation Adjustment (CVA) requires repeated risk-neutral expectation estimation, making it a natural test bed for quantum amplitude estimation, whose coherent amplification can in principle reduce Monte Carlo sampling cost. Whether this advantage survives realistic financial encoding and noisy hardware remains open. We develop an end-to-end, noise-aware quantum workflow for CVA, covering market calibration, discretisation, oracle construction, hardware execution and error-budget analysis. The model combines a correlated two-asset exposure with discount and default factors, encoded through a QCBM-based joint time-market distribution and controlled payoff rotations. We introduce contrast-aware Bayesian iterative quantum amplitude estimation (CABIQAE), which incorporates experimentally calibrated Grover-contrast loss into Bayesian inference and circuit-depth selection. Hardware-calibrated experiments show that CABIQAE exploits the limited amplification available on current devices more effectively than noise-agnostic alternatives and achieves a much lower classical post-processing runtime than the noise-aware BAE baseline. The analysis further decomposes the total CVA error into statistical, encoding, discretisation and hardware contributions. The full CVA oracle remains limited by circuit depth and discretisation resolution.
A critical problem in the financial world deals with the management of risk, from regulatory risk to portfolio risk. Many such problems involve the analysis of securities modelled by complex dynamics that cannot be captured analytically, and hence rely on numerical techniques that simulate the stochastic nature of the underlying variables. These techniques may be computationally difficult or demanding. Hence, improving these methods offers a variety of opportunities for quantum algorithms. In this work, we study the problem of Credit Valuation Adjustments (CVAs) which has significant importance in the valuation of derivative portfolios. As a variant, we also consider the problem of pricing a portfolio of many different financial options. We propose quantum algorithms that accelerate statistical sampling processes to approximate the price of the multi-option portfolio and the CVA under different measures of dispersion. Technically, our algorithms are based on enhancing the quantum Monte Carlo (QMC) algorithms by Montanaro with an unbiased version of quantum amplitude estimation. We analyse the conditions under which we may employ these techniques and demonstrate the application of QMC techniques on CVA approximation when particular bounds for the variance of CVA are known.
The approximation of an integral with numerical integration methods is a relevant problem in many fields, from finance to chemistry. Monte Carlo methods are a family of methods widely used for numerical integration but which are not always applicable to industrial tasks. Instead, there are quantum computing methods such as Quantum Amplitude Estimation able to offer a speed-up compared to the Monte Carlo methods for numerical integration which promise a significant impact, especially in finance. The Quantum Amplitude Estimation algorithm is a fundamental quantum algorithm with the potential to achieve a quadratic speed-up. Unfortunately, being competitive over classical Monte Carlo methods would require hardware performances currently not available. Therefore, alternative methods have been developed in order to exploit fewer resources while maintaining an advantageous theoretical scaling. We present a comparison between the standard QAE algorithm and two NISQ-friendly versions of the QAE on a numerical integration task. The Monte Carlo technique of Metropolis-Hastings instead is used as a classical benchmark. The scaling of the different algorithms are evaluated in terms of estimation error as a function of the number of samples, computational time, and length of the quantum circuits required by the quantum solutions. The conclusion highlights that an alternative approach is preferable with respect to employng the phase estimation routine. Indeed, the maximum likelihood estimation guarantees the best trade-off between the length of the quantum circuits and the precision in the integral estimation.
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.
Quantum computers are expected to have substantial impact on the finance industry, as they will be able to solve certain problems considerably faster than the best known classical algorithms. In this article we describe such potential applications of quantum computing to finance, starting with the state-of-the-art and focusing in particular on recent works by the QC Ware team. We consider quantum speedups for Monte Carlo methods, portfolio optimization, and machine learning. For each application we describe the extent of quantum speedup possible and estimate the quantum resources required to achieve a practical speedup. The near-term relevance of these quantum finance algorithms varies widely across applications - some of them are heuristic algorithms designed to be amenable to near-term prototype quantum computers, while others are proven speedups which require larger-scale quantum computers to implement. We also describe powerful ways to bring these speedups closer to experimental feasibility - in particular describing lower depth algorithms for Monte Carlo methods and quantum machine learning, as well as quantum annealing heuristics for portfolio optimization. This article is targeted at financial professionals and no particular background in quantum computation is assumed.
The Quantum Amplitude Estimation (QAE) algorithm is a major quantum algorithm designed to achieve a quadratic speed-up. Until fault-tolerant quantum computing is achieved, being competitive over classical Monte Carlo (MC) remains elusive. Alternative methods have been developed so as to require fewer resources while maintaining an advantageous theoretical scaling. We compared the standard QAE algorithm with two Noisy Intermediate-Scale Quantum (NISQ)-friendly versions of QAE on a numerical integration task, with the Monte Carlo technique of Metropolis–Hastings as a classical benchmark. The algorithms were evaluated in terms of the estimation error as a function of the number of samples, computational time, and length of the quantum circuits required by the solutions, respectively. The effectiveness of the two QAE alternatives was tested on an 11-qubit trapped-ion quantum computer in order to verify which solution can first provide a speed-up in the integral estimation problems. We concluded that an alternative approach is preferable with respect to employing the phase estimation routine. Indeed, the Maximum Likelihood estimation guaranteed the best trade-off between the length of the quantum circuits and the precision in the integral estimation, as well as greater resistance to noise.