Real-time quantum state estimation in circuit QED via the Bayesian approach

Using a superconducting circuit QED device, we present a theoretical study of real-time quantum state estimation via the quantum Bayesian approach. Suitable conditions under which the quantum Bayesian approach can accurately update the density matrix of the qubit are analyzed. We also consider the correlation between some basic and physically meaningful parameters of the circuit QED and the performance of the Bayesian approach. Our results advance the understanding of the quantum Bayesian approach.

Figure 1

(a) Measured signal and (b) filtered signal of a single measurement in circuit QED with $t_m=0.01\mu \mathrm{s},{\omega }_q=0$ Hz, ${\mathrm{\Delta }}_r=0$ Hz, $\chi =1$ MHz, ${\varepsilon }_d=15$ MHz, $\kappa =20$ MHz, ${\alpha }_e={\alpha }_g=0,{\gamma }_1={\gamma }_{\psi }=0$, and $\varphi =\pi$.

Figure 2

Real-time quantum state estimation via the quantum Bayesian approach. (a)–(c) depict in detail the estimation of the three Bloch components $\langle {\widehat{\sigma }}_i\rangle$ with the quantum Bayesian approach (solid blue curves) and the original evolution of $\langle {\sigma }_i\rangle$ (dashed red curves) under the exact SME (6). (d) depicts the trajectories in the Bloch sphere, where the blue curve is the estimated trajectory and the red curve is the evolution one under the exact SME (6). To make it easier to observe, we only intercept the first $0.75\mu \mathrm{s}$ of the trajectory. Here, $\eta =0.1$ and the rest of the parameters are the same as those in Fig. 1.

Figure 3

Evolution of the error functions $E_{{\gamma }_1},E{z}_{{\gamma }_1},E{y}_{{\gamma }_1}$, and $E{x}_{{\gamma }_1}$ for a total time of $10\mu \text{s}$ and with 5000 trajectories, as functions of the parameter ${\gamma }_1$. In (a), the blue solid curve, the red dashed curve, the yellow dashed-dotted curve, and the purple dotted curve represent $E_{{\gamma }_1},E{z}_{{\gamma }_1},E{y}_{{\gamma }_1}$, and $E{x}_{{\gamma }_1}$, respectively. The solid red dot is the critical point of ${\gamma }_1$ at which $E_{{\gamma }_1}$ reaches 0.1. For clarity, the time evolution of the estimated $\langle {\widehat{\sigma }}_z\rangle$ (solid blue curve) and the evolution of $\langle {\sigma }_z\rangle$ (dashed red curve) at this point are plotted in (b).

Figure 4

Evolutions of the error functions $E_a,E{x}_a,E{y}_a$, and $E{z}_a$ for a total time of $10\mu \text{s}$ and with 5000 trajectories, as functions of the parameters $\chi ,{\mathrm{\Delta }}_r,{\varepsilon }_d,\eta ,{\gamma }_{\phi }$, and $\kappa$ in (a)–(f), respectively. The blue solid curves, the red dashed lines, the yellow dashed-dotted curves, and the purple dotted curves represent $E_a,E{z}_a,E{y}_a$, and $E{x}_a$, respectively. The solid red dot in each subfigure represents the point at which the overall error $E_a$ reaches 0.1.