Abstract
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 Grover's state preparation algorithm 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 work, 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.
Highlights
Q UANTUM computers have the ability to outperform their classical counterpart in many ways [1] [2] [3]
Instead of applying the controlled Y rotations on the flag qubit with quantum circuits based on a piecewise polynomial, the rotation angles are calculated through arithmetic and stored in their register where Y rotations are performedbased on the values in the register
Because the purpose of this example case is to demonstrate the feasibility of using a piecewise polynomial approximation in state preparation for the Cauchy distribution, a built-in piecewise polynomial pauli rotation function is used
Summary
Q UANTUM computers have the ability to outperform their classical counterpart in many ways [1] [2] [3]. There is a detailed circuit-based method proposed in [31] to prepare states when the rotation angle is approximated with a polynomial function. The most common method to compute the rotation angle for each quantum superposition state is to use a Taylor polynomial series [30]. This method is difficult to implement for the Cauchy distribution due to its heavy-tailed nature. Instead of applying the controlled Y rotations on the flag qubit with quantum circuits based on a piecewise polynomial, the rotation angles are calculated through arithmetic and stored in their register where Y rotations are performedbased on the values in the register. The computation complexity is kept low by limiting the arithmetic operation to its simplest form of a quadratic term and a constant term for all subdomains
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