论文标题
量子实施与风险分析相关的Copulas
Quantum Implementation of Risk Analysis-relevant Copulas
论文作者
论文摘要
现代的定量风险管理依赖于对尾巴依赖性的适当建模，并且可能准确地量化风险度量，例如风险的价值（VAR），在2000年的100分之一或什至1个。参见（Woerner and Egger，2018年），以及有关更广泛的观点（Orús等，2018）。风险分析工具箱的一个重要要素是Copula，请参见有关财务应用的（Jouanin等，2004）。但是，据作者的最佳知识，到目前为止，还没有以明确的形式从风险建模相关的副总统进行取样的量子计算实现。我们在这里的重点是实施简单而强大的Copula模型，能够令人满意地捕获建模风险因素的关节尾行行为。本文介绍了一些简单的Copula家族，其中包括（Milek，2014年）中介绍的多元B11（MB11）Copula家族。我们将证明这个副群体家族适合风险聚集，因为它非常能够再现尾部依赖性结构。有关最终可行的双变量尾巴依赖性结构的相关基准，请参见（Emberechts等，2016），以及必要的和充分的条件。事实证明，可以使用量子计算中存在的简单构造来表达这种离散的副物：二进制分数扩展格式，共酮/独立随机变量，受控门和凸组合，因此适用于量子计算机实现。本文在IBM量子计算机上介绍了量子实现电路，数值和符号模拟结果以及实验验证背后的设计。本文也提出了一种通用方法，用于实施任何离散的副总裁。
Modern quantitative risk management relies on an adequate modeling of the tail dependence and a possibly accurate quantification of risk measures, like Value at Risk (VaR), at high confidence levels like 1 in 100 or even 1 in 2000. Quantum computing makes such a quantification quadratically more efficient than the Monte Carlo method; see (Woerner and Egger, 2018) and, for a broader perspective, (Orús et al., 2018). An important element of the risk analysis toolbox is copula, see (Jouanin et al., 2004) regarding financial applications. However, to the best knowledge of the author, no quantum computing implementation for sampling from a risk modeling-relevant copula in explicit form has been published so far. Our focus here is implementation of simple yet powerful copula models, capable of a satisfactory capturing the joint tail behaviour of the modelled risk factors. This paper deals with a few simple copula families, including Multivariate B11 (MB11) copula family, presented in (Milek, 2014). We will show that this copula family is suitable for the risk aggregation as it is exceptionally able to reproduce tail dependence structures; see (Embrechts et al., 2016) for a relevant benchmark as well as necessary and sufficient conditions regarding the ultimate feasible bivariate tail dependence structures. It turns out that such a discretized copula can be expressed using simple constructs present in the quantum computing: binary fraction expansion format, comonotone/independent random variables, controlled gates, and convex combinations, and is therefore suitable for a quantum computer implementation. This paper presents design behind the quantum implementation circuits, numerical and symbolic simulation results, and experimental validation on IBM quantum computer. The paper proposes also a generic method for quantum implementation of any discretized copula.
