Efficient Cauchy distribution based quantum state preparation by using the comparison algorithm

The quantum Monte Carlo algorithm can provide significant speedup compared to its classical counterpart. So far, most reported works have utilized Grover’s state preparation algorithm. However, this algorithm relies on costly controlled Y rotations to apply the correct amplitudes onto the superposition states. Recently, a comparison-based state preparation method was proposed to reduce computational complexity by avoiding rotation operations. One critical aspect of this method is the generation of the comparison threshold associated with the amplitude of the quantum superposition states. The direct computation of the comparison threshold is often very costly. An alternative is to estimate the threshold with a Taylor approximation. However, Taylor approximations do not work well with heavy-tailed distribution functions such as the Cauchy distribution, which is widely used in applications such as financial modeling. Therefore, a new state preparation method needs to be developed. In this study, an efficient comparison-based state preparation method is proposed for the heavy-tailed Cauchy distribution. Instead of a single Taylor approximation for the entire function domain, this study uses quantum piecewise arithmetic to increase accuracy and reduce computational cost. The proposed piecewise function is in the simplest form to estimate the comparison threshold associated with the amplitudes. Numerical analysis shows that the number of required subdomains increases linearly as the maximum tolerated approximation error decreases exponentially. 197 subdomains are required to keep the error below 18192 of the maximum amplitude. Quantum parallelism ensures that the computational complexity of estimating the amplitudes is independent from the number of subdomains.