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.

中文翻译：

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使用量子梯度算法在金融市场风险中获得量子优势

我们引入了一种量子算法来计算金融衍生品的市场风险。以前的工作表明，量子幅度估计可以在目标误差中二次加速衍生品定价，我们将其扩展到市场风险计算中的二次误差缩放优势。我们表明，采用量子梯度估计算法可以在相关市场敏感性的数量上提供进一步的二次优势，通常称为 $greeks$。通过对实际感兴趣的金融衍生品上的量子梯度估计算法进行数值模拟，我们证明了我们不仅可以成功估计所研究示例中的希腊人，而且在实践中资源需求可以显着低于理论复杂性界限的预期. 计算金融市场风险的这一额外优势降低了 Chakrabarti 等人的金融量子优势所需的估计逻辑时钟速率。[Quantum 5, 463 (2021)] 约 7 倍，从 50MHz 到 7MHz，即使按行业标准 (4) 对少量希腊人来说也是如此。此外，我们表明，如果我们能够访问足够的资源，量子算法可以在 60 个 QPU 上并行化，在这种情况下，实现与串行执行相同的整体运行时间所需的每个设备的逻辑时钟速率约为 100kHz。在整个工作中，我们总结和比较了几种不同的量子和经典方法组合，可用于计算金融衍生品的市场风险。