Paper Introduction: Statistical Quantum Phase Estimation.


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 νk\mid \nu_k \rangle of the target state (eigenstate)


C(Φ,θR)=k=0dt1νkΦ2n=0dc1e2πni(θkθR)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 dtd_t and dcd_c are the number of target bits and control bits, respectively. The number of control bits in this algorithm is 1, while dcd_c is a value that varies depending on the required accuracy. Also, θk\theta_k is the eigenphase of the eigenstate νk\mid \nu_k \rangle . The objective of this algorithm is to update θR\theta_R and converge the input initial state set Φ{Bk}\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 2dt12d_t-1 dimensional basis containing |Φ〉.


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


3, Update 2dt12d_t-1  dimensional |Φ〉 according to the following equation.


Φ=Φ+za(1C(Φ,θR))Bm mod dt\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 mdt,z=1 otherwise.z=i ~if ~m\geq d_t, z=1~ otherwise.


4, If |Φ and θR\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



Hikaru Wakaura
個人研究者の若浦 光です。量子アルゴリズムの実装結果や論文の紹介などを載せていきます。 mail: hikaruwakaura@gmail.com
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Hikaru Wakaura
個人研究者の若浦 光です。量子アルゴリズムの実装結果や論文の紹介などを載せていきます。 mail: hikaruwakaura@gmail.com
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