We describe a general-purpose framework to design quantum algorithms. This framework relies on two pillars: a basic data structure called quantum matrix and a modular structure based on three quasi-independent modules. These latter include a loading module, a tool-kit of basic quantum arithmetic operations and a read-out module. We briefly discuss the loading and read-out modules, while the arithmetic module is analyzed in more depth. Eventually, we give explicit examples regarding the manipulation of generic oracles and hint at possible applications.

We review the state of the art and recent advances in quantum computing applied to derivative pricing and the computation of risk estimators like Value at Risk. After a brief description of the financial derivatives, we first review the main models and numerical techniques employed to assess their value and risk on classical computers. We then describe some of the most popular quantum algorithms for pricing and VaR. Finally, we discuss the main remaining challenges for the quantum algorithms to achieve their potential advantages.

We introduce the Real Quantum Amplitude Estimation (RQAE) algorithm, an extension of Quantum Amplitude Estimation (QAE) which is sensitive to the sign of the amplitude. RQAE is an iterative algorithm which offers explicit control over the amplification policy through an adjustable parameter. We provide a rigorous analysis of the RQAE performance and prove that it achieves a quadratic speedup, modulo logarithmic corrections, with respect to unamplified sampling. Besides, we corroborate the theoretical analysis with a set of numerical experiments.

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