qutip：计算门保真度

问题描述

已知实际门矩阵（$U_{meas}$）和理想门矩阵（$U_{ideal}$），求实际门的保真度。

思路

用QPT(quantum process tomography)将U矩阵转化为$\chi$矩阵，然后求保真度。

具体的计算可直接调用qutip的相关接口。

实例

单比特门

理想门为$R_x(\frac{\pi}{2})$，实际门为$R_x(\frac{\pi}{2}+\delta)$
矩阵形式：
$U_{ideal}=e^{-\frac{i}{2}\sigma_x*\frac{\pi}{2}}$
$U_{meas}=e^{-\frac{i}{2}\sigma_x*(\frac{\pi}{2}+\delta)}$

计算过程：

import numpy as np
from scipy.linalg import expm 
import qutip as qt

# 已知U_ideal和U_meas
U_ideal = expm(-1j/2*sigma_x*PI/2)
delta = 5/180*PI
U_meas = expm(-1j/2*sigma_x*(PI/2+delta))

# 将U矩阵转化为chi矩阵
U_psi_ideal = qt.Qobj(U_ideal)
U_rho_ideal = qt.spre(U_psi_ideal) * qt.spost(U_psi_ideal.dag())
U_psi_meas = qt.Qobj(U_meas)
U_rho_meas = qt.spre(U_psi_meas) * qt.spost(U_psi_meas.dag())
op_basis = [[qt.qeye(2), qt.sigmax(), qt.sigmay(), qt.sigmaz()]]
op_label = [["i", "x", "y", "z"]]
chi_ideal = qt.qpt(U_rho_ideal, op_basis)
chi_meas = qt.qpt(U_rho_meas, op_basis)

# 由chi矩阵计算门保真度
fidelity = qt.process_fidelity(qt.Qobj(chi_ideal),qt.Qobj(chi_meas))
print(f'fidelity:{fidelity:.4f}')

结果

fidelity:0.9981

注：有了$\chi$矩阵，也可以直接可视化看一下QPT的结果。

# plot process tomography
fig = plt.figure(figsize=(14, 6))
ax1 = fig.add_subplot(121, projection='3d')
ax2 = fig.add_subplot(122, projection='3d')
qt.qpt_plot_combined(chi_ideal, op_label, ax=ax1)
qt.qpt_plot_combined(chi_meas, op_label, ax=ax2)
ax1.set_title(r'$U_{ideal}$')
ax2.set_title(r'$U_{meas}$')

两比特门

todo