In this article, I will explain a new method of Quantum Phase Estimation, which was published this year. This method is called Statistical Quantum Phase Estimation. It is a method that aims to set the evaluation function, including the inner product, to 1 by iterative calculation. This is a way to make effective use of coefficients that are normally discarded in quantum phase estimation. The evaluation function is given as $\mid \nu_k \rangle$ of the target state (eigenstate)

$C(\mid \Phi\rangle, \theta_R)=\sum_{k=0}^{d_t-1}\mid \langle \nu_k\mid \Phi\rangle \mid^2\mid \sum_{n=0}^{d_c-1}e^{2\pi n i(\theta_k-\theta _R)}\mid$ -(1)

where $d_t$ and $d_c$ are the number of target bits and control bits, respectively. The number of control bits in this algorithm is 1, while $d_c$ is a value that varies depending on the required accuracy. Also, $\theta_k$ is the eigenphase of the eigenstate $\mid \nu_k \rangle$ . The objective of this algorithm is to update $\theta_R$ and converge the input initial state set $\mid \Phi\rangle\in\{\mid B_k \rangle\}$ to the exact solution.

The processing steps of this algorithm are as follows

1, Determine an orthogonal initial set in a $2d_t-1$ dimensional basis containing ｜Φ〉.

2, Evaluate equation (1) after turning the circuit in Figure 1, and choose the θ_R at the value of equation (1) that is the smallest.

3, Update $2d_t-1$ dimensional |Φ〉 according to the following equation.

$\mid \Phi ' \rangle = \mid \Phi \rangle + za(1-C(\mid \Phi \rangle, \theta_R))\mid B_{m~mod~d_t}\rangle$ -(2)

$z=i ~if ~m\geq d_t, z=1~ otherwise.$

4, If |Φ and $\theta_R$ are not updated in 3, set a to a/2 and iterate 3.

5, Extract the result when convergence is reached or after the number of iterations.

Figure 1 Quantum circuit for statistical quantum phase estimation and its processing schematic.

In the paper, the results of the calculation with water molecules were posted after this, and the simulation was also successful with high accuracy. I will try to implement it when I get a chance.

https://arxiv.org/abs/2104.10285

Paper Introduction: Statistical Quantum Phase Estimation.

Hikaru Wakaura