Efficient Quantum State Preparation for the Cauchy Distribution Based on Piecewise Arithmetic

The benefits of the quantum Monte Carlo algorithm heavily rely on the efficiency of the superposition state preparation. So far, most reported Monte Carlo algorithms use the Grover–Rudolph state preparation method, which is suitable for efficiently integrable distribution functions. Consequently, most reported works are based on log-concave distributions, such as normal distributions. However, non-log-concave distributions still have many uses, such as in financial modeling. Recently, a new method was proposed that does not need integration to calculate the rotation angle for state preparation. However, performing efficient state preparation is still difficult due to the high cost associated with high precision and low error in the calculation for the rotation angle. Many methods of quantum state preparation use polynomial Taylor approximations to reduce the computation cost. However, Taylor approximations do not work well with heavy-tailed distribution functions that are not bounded exponentially. In this article, we present a method of efficient state preparation for heavy-tailed distribution functions. Specifically, we present a quantum gate-level algorithm to prepare quantum superposition states based on the Cauchy distribution, which is a non-log-concave heavy-tailed distribution. Our procedure relies on a piecewise polynomial function instead of a single Taylor approximation to reduce computational cost and increase accuracy. The Cauchy distribution is an even function, so the proposed piecewise polynomial contains only a quadratic term and a constant term to maintain the simplest approximation of an even function. Numerical analysis shows that the required number of subdomains increases linearly as the approximation error decreases exponentially. Furthermore, the computation complexity of the proposed algorithm is independent of the number of subdomains in the quantum implementation of the piecewise function due to quantum parallelism. An example of the proposed algorithm based on a simulation conducted in Qiskit is presented to demonstrate its capability to perform state preparation based on the Cauchy distribution.