### Quantum Amplitude Estimation for Binomial Tree Model of zero-coupon bonds

In this article, we introduce an application of the **Quantum Amplitude Estimation (QAE**) algorithm for the calculation of the expected value of the price of a **zero-coupon bond** using the **binomial tree model**.

For its part, the **QAE** algorithm is one of the most popular algorithms in quantum computing. Among its various applications is the calculation of the expected value of a probability distribution^{1}. From a theoretical point of view, it offers a computational advantage over its traditional counterparts.

The **binomial tree model** (Figure 1), on the other hand, is a model used to assign a price to a financial product in the future. It consists of discretizing the time from the present moment to the expiration date and, at each of the time steps, assigning a certain probability for the increase in the price and a probability for the decrease in the price. By applying the method iteratively, we will obtain multiple values for the final price, each with a certain probability associated with it. The value assigned to the financial product will be the expected value of this distribution. This is the calculation we can make using **QAE**.

**Figure 1**

One of these financial products where we can implement the **binomial tree model** is the **zero-coupon bond**. There are simple works of one-step implementations of this model^{2 }in a quantum computer. However, despite the theoretical superiority of QAE, there are physical limitations to its implementation, among which are the number of qubits or the circuit depth and noise. In this regard, a paper has recently been published in which the performance of this model, on different physical platforms (superconducting circuits and trapped ions), has been comparatively studied^{3}.

In conclusion, it should be noted that hardware improvements will be necessary for this method to be used in more realistic situations with a practical component.

^{1} Abrams, D. S., & Williams, C. P. (1999). Fast quantum algorithms for numerical integrals and stochastic processes. *arXiv preprint quant-ph/9908083*.

^{2} Woerner, S., & Egger, D. J. (2019). Quantum risk analysis. *npj Quantum Information*, *5*(1), 15.

^{3} Heo, J., & Lee, M. (2024). Comparative Study of Quantum-Circuit Scalability in a Financial Problem. *arXiv preprint arXiv:2404.04911*.