# Bayesian quantum phase estimation.

In this article, I will explain the Bayesian quantum phase estimation method, which uses Bayesian phase estimation to simultaneously calculate the phase, eigenvalues, and eigenstates in multiple states. This algorithm is quite different from the usual quantum phase estimation method. This algorithm uses the circuit shown in Figure 1. This circuit consists of a group of target bits corresponding to d this state and a control operation space of d+1 dimensions. Since it is structurally safe for a quantum computer to handle a 2^{n}-dimensional space, it is better to set d+1=2^{n}. Here, H^{d+1} is not the usual Hadamard gate, but a gate that satisfies $\langle m \mid H^{d+1} \mid n \rangle=e^{-2\pi i nm/(d+1)}/\sqrt{d+1}$ , and S_{φ}^{d+1} is a diagonal term with $\langle n \mid S_\phi^{d+1}\mid n \rangle=e^{-i\phi_n}$ . φn is a randomly determined phase. Also, UC is a gate to multiply each state in the target bit group by U only if the control bit group is $\mid d\rangle$ . With this circuit in action, and with θ_{n} as the final intrinsic phase of state n, the existence probability for phase φ and M in state o is

$P(o\mid \theta,\phi,M)=\frac{1}{(d+1)^2}(d+1+2\sum_n cos[M\theta_n+\beta_n]+2\sum_{n<m}cos[M(\theta_n-\theta_m)+\gamma_{nm}])$

$\beta_n=\phi_n-\phi_0+\frac{2\pi no}{(d+1)}, \gamma_{nm}=\phi_n-\phi_m+\frac{2\pi(n-m)o}{(d+1)}$

Then From there, we have the Bayesian probability distribution

$P(\zeta\mid o, \phi, M)=\frac{P(o\mid \theta,\phi,M)P(\zeta)}{\int d^d\zeta P(o\mid \zeta, \phi, M)P(\zeta)}$

is calculated. The entire flow including them is shown in Figure 2.

The study of quantum phase estimation, including this algorithm, has developed rapidly in recent years. This has led to the development of several derivatives of the algorithm, and the discussion of error probability has also begun. This field is still in its infancy.

Figure 1: Circuit of Bayesian quantum phase estimation.

Figure 2: Flowchart of Bayesian quantum phase estimation.

[2010.09075] Bayesian Quantum Multiphase Estimation Algorithm (arxiv.org)