Toward pricing financial derivatives with an IBM quantum computer

Pricing interest-rate financial derivatives is a major problem in finance, in which it is crucial to accurately reproduce the time evolution of interest rates. Several stochastic dynamics have been proposed in the literature to model either the instantaneous interest rate or the instantaneous forward rate. A successful approach to model the latter is the celebrated Heath-Jarrow-Morton framework, in which its dynamics is entirely specified by volatility factors. In its multifactor version, this model considers several noisy components to capture at best the dynamics of several time-maturing forward rates. However, as no general analytical solution is available, there is a trade-off between the number of noisy factors considered and the computational time to perform a numerical simulation. Here, we employ the quantum principal component analysis to reduce the number of noisy factors required to accurately simulate the time evolution of several time-maturing forward rates. The principal components are experimentally estimated with the five-qubit IBMQX2 quantum computer for $2\times 2$ and $3\times 3$ cross-correlation matrices, which are based on historical data for two and three time-maturing forward rates. This paper is a step towards the design of a general quantum algorithm to fully simulate on quantum computers the Heath-Jarrow-Morton model for pricing interest-rate financial derivatives. It shows indeed that practical applications of quantum computers in finance will be achievable in the near future.

Figure 1

Quantum circuit implementation for $n+logN$ qubits. The first $n$ qubits are dedicated to the binary codification of the maximum eigenvalue of the matrix ${\sigma }_N$ and they are initialized in the site $|0\rangle$. The rest of the qubits, a total of $logN$, encode the estimation of the corresponding eigenvector and are initialized on a random state $|b\rangle$. The single qubit gate $H$ corresponds to the Hadamard gate. The rest of the gates are controlled operations. The controlled $U_{\rho }^{2^k}$ gate applies the matrix $U=e^{it{\sigma }_N}$ $2^k$ times on the last set of qubits. The controlled $R_k^{†}$ gate applies the matrix $\left(\begin{array}{cc}1& 0\\ 0& e^{2\pi i/2^k}\end{array}\right)$ to each target qubit. After performing all the operations, if the initial $|b\rangle$ is appropriate, one gets the final state $|b_1{b}_2\cdots b_n\rangle \otimes |u_{\max }\rangle$, where $|b_1{b}_2\cdots b_n\rangle$ is the $n$-bit estimation of the eigenvalue ${\lambda }_j^{\left(n\right)}$ and $|u_{\max }\rangle$ is the best estimation of the exact eigenvector of ${\sigma }_N$.

Figure 2

(a) Quantum circuit implementation for the $2\times 2$ matrix. The first two qubits encode the 2-bit estimation of the greatest eigenvalue of ${\rho }_2$ and are initialized on the state $|0\rangle$. The last qubit is dedicated to the estimation of the corresponding eigenvector. It is initialized on a random state $|b\rangle$. For the first iteration, we initialize it on the state $|+\rangle$ by applying a Hadamard gate. The single-qubit gates represented by the letter $H$ refer to the Hadamard gate. The controlled $U$ gates represent the unitary controlled operations called $U_{\rho }^{2^k}$ in Fig. 1. The last two-qubit gate is a controlled $S^{†}$ gate, controlling the first qubit and acting on the second. The final state of the system after running the circuit and taking measures is $|11\rangle \otimes (0.719|0\rangle +0.659|1\rangle )$. (b) Populations for each iteration. The graphic shows the experimental probabilities of finding the three qubits in each state and its corresponding errors for the four iterations of the algorithm. We have considered both statistical and experimental errors, assuming for the latter an error of $8\%$ for each two-qubit gate.

Figure 3

(a) Quantum phase estimation circuit implementation for the 3-bit estimation of the greatest eigenvalue, ${\mathrm{\Lambda }}^{\left(3\right)}$. After estimating the eigenvector $|u_{\max }\rangle$, it is used to improve the estimation of the corresponding eigenvalue $\lambda$ by applying the quantum phase estimation algorithm. In this case we dedicate three qubits to the binary codification of $\lambda$. The fourth qubit is initialized on the estimated eigenvector $|u_{\max }\rangle$. The fifth two-qubit gate represents a controlled $T^{†}$ gate, controlling the first qubit and acting on the third one. The rest of the gates are the same that have been applied on the previous part of the algorithm. Finally, one takes measures and gets the final state of the system: $|{\mathrm{\Lambda }}^{\left(3\right)}\rangle \otimes |u_{\max }\rangle =|111\rangle \otimes \left[0.8150|0\rangle +0.5794|1\rangle \right]$. (b) Populations of the 3-bit eigenvalue estimation. This chart shows the probabilities of each state in the qiskit simulator (green bars) and in the real quantum processor (yellow bars) for the quantum phase estimation algorithm taking the previously obtained eigenvector given in Eq. (C1). The first qubit refers to the subspace of the eigenvector estimation. The next three qubits refers to the subspace of the binary estimation of the eigenvalue. The quantum circuit comprises at least six entangling gates, which leads the system to an almost total decoherence, as reflected in the homogeneous distribution of probabilities in the real chip.

Figure 4

Quantum circuit implementation for the $4\times 4$ matrix. The first qubit is the only one dedicated for the binary codification of the greatest eigenvalue $\lambda$ of the matrix ${\rho }_4$. It is initialized on the state $|0\rangle$. The last two qubits encode the estimation of the corresponding eigenvector and are initialized on a random state $|b\rangle$. The final state of the system after measuring is $|1\rangle \otimes |u_{\max }\rangle$.