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.

Parametrized quantum circuits (PQC) are quantum circuits which consist of both fixed and parametrized gates. In recent approaches to quantum machine learning (QML), PQCs are essentially ubiquitous and play the role analogous to classical neural networks. They are used to learn various types of data, with an underlying expectation that if the PQC is made sufficiently deep, and the data plentiful, the generalization error will vanish, and the model will capture the essential features of the distribution. While there exist results proving the approximability of square-integrable functions by PQCs under the L2 distance, the approximation for other function spaces and under other distances has been less explored. In this work we show that PQCs can approximate the space of continuous functions, p-integrable functions and the Hk Sobolev spaces under specific distances. Moreover, we develop generalization bounds that connect different function spaces and distances. These results provide a theoretical basis for different applications of PQCs, for example for solving differential equations. Furthermore, they provide us with new insight on the role of the data normalization in PQCs and of loss functions which better suit the specific needs of the users.

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.

In this work we present an alternative methodology to the standard Quantum Accelerated Monte Carlo (QAMC) applied to derivatives pricing. Our pipeline benefits from the combination of a new encoding protocol, referred to as the direct encoding, and an amplitude estimation algorithm, the modified Real Quantum Amplitude Estimation (mRQAE) algorithm. On the one hand, the direct encoding prepares a quantum state which contains the information about the sign of the expected payoff. On the other hand, the mRQAE is able to read all the information contained in the quantum state. Although the procedure we describe is different from the standard one, the main building blocks are almost the same. Thus, all the extensive research that has been performed is still applicable. Moreover, we experimentally compare the performance of the proposed methodology against the standard QAMC employing a quantum emulator and show that we retain the speedups.

We describe a general-purpose framework to implement quantum algorithms relying upon an efficient handling of arrays. The cornerstone of the framework is the direct embedding of information into quantum amplitudes, thus avoiding hampering square roots. We discuss the entire pipeline, from data loading to information extraction. Particular attention is devoted to the definition of an efficient toolkit of basic quantum operations on arrays. We comment on strong and weak points of the proposed quantum manipulations, especially in relation to an effective exploitation of quantum parallelism. We describe in detail some general-purpose routines as well as their embedding in full algorithms. Their efficiency is critically discussed both locally, at the level of the routine, and globally, at the level of the full algorithm. Finally, we comment on some applications in the quantitative finance domain.