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Quantum parameter estimation via dispersive measurement in circuit QED

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Abstract

We investigate the quantum parameter estimation in circuit quantum electrodynamics via dispersive measurement. Based on the Metropolis–Hastings algorithm and the Markov chain Monte Carlo (MCMC) integration, a new algorithm is proposed to calculate the Fisher information by the stochastic master equation. The Fisher information is expressed in the form of log-likelihood functions and further approximated by the MCMC integration. Numerical results show that the evolution of the Fisher information can approach the quantum Fisher information in a short time interval. These results demonstrate the effectiveness of the proposed algorithm. Finally, based on the proposed algorithm, we consider the effects of the measurement operator and the measurement efficiency on the Fisher information.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China under Grant 61873317 and by the Fundamental Research Funds for the Central Universities.

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Correspondence to Wei Cui.

Appendices

Appendix A: The lemma of the multi-dimensional Itô formula

In the multi-dimensional Itô formula, it is worth noting that if x(t) were continuously differentiable with respect to time t, then the term \(\frac{1}{2}\hbox {d}{x^T}(t){V_{xx}}({x(t),t})\hbox {d}x(t)\) would not appear owing to the classical calculus formula for total derivatives. For example, if \(V( {{x_1},{x_2}})\) is continuously differentiable with respect to t, e.g., \(V( {{x_1},{x_2}}) = {x_1}( t){x_2}(t)\), then its derivation should be \(\hbox {d}V( {{x_1},{x_2}}) = {x_1}\hbox {d}{x_2} + {x_2}\hbox {d}{x_1} + \hbox {d}{x_1}\hbox {d}{x_2}\).

Appendix B: Metropolis–Hastings algorithm

In Markov chains, suppose we generate a sequence of random variables \({X_1},{X_2},\ldots ,{X_{n }}\) with Markov property, namely the probability of moving to the next state depends only on the present state instead of the previous state:

$$\begin{aligned} { Pr} \left\{ {{X_{n + 1}} = x\left| {{X_1} = {x_1},\ldots ,{X_n} = {x_n}} \right. } \right\} = { Pr} \left\{ {{X_{n + 1}} = x\left| {{X_n} = {x_n}} \right. } \right\} . \end{aligned}$$
(18)

Then, for a given state \(X_t\), the next state \(X_{t+1}\) does not depend further on the hist of the chain \({X_1},{X_2},\ldots ,{X_{t - 1}}\), but comes from a distribution which only depends on the current state of the chain \(X_t\). For any time instant t, if the next state is the first sample reference point Y obeying distribution \(q\left( { \cdot \left| {{X _t}} \right. } \right) \) which is called the transition kernel of the chain, then obviously it depends on the current state \(X_t\). In generally, \(q\left( { \cdot \left| {{X _t}} \right. } \right) \) may be a multi-dimensional normal distribution with mean X, so the candidate point Y is accepted with probability \(\alpha \left( {{X _t},Y} \right) \) where

$$\begin{aligned} \alpha \left( {X ,Y} \right) = \min \left( {1,\frac{{\pi \left( Y \right) q\left( {X \left| Y \right. } \right) }}{{\pi \left( X \right) q\left( {Y\left| X \right. } \right) }}} \right) . \end{aligned}$$
(19)

Here, \(\pi \left( A \right) \) stands for a function only depends on A. If the candidate point is accepted, the next state becomes \(X_{t+1}=Y\). If the candidate point is rejected, it means that the chain does not move, and the next state will be \(X_{t+1}=X_t\). We illustrate this sampling process with a simple example (see Fig. 6). Here, the initial value is \(X(1) = -10\). Figure 6a represents the stationary distribution N(0, 0.1). In Fig. 6b, we plot 500 iterations from Metropolis–Hastings algorithm [36] with the stationary distribution N(0, 1) and proposal distribution N(0, 0.1). Obviously, sampling data selecting from the latter part would be better.

Fig. 6
figure 6

Illustration of the Metropolis– Hastings algorithm with the initial value \(X(1) = -10\). a Stationary distribution N(0, 0.1), and in b, 500 iterations from Metropolis–Hastings algorithm with the stationary distribution N(0, 1) and proposal distribution N(0, 0.1) are plotted

Appendix C: Makov Chain Monte Carlo integration

In Markov chain, the Monte Carlo integration [38] can be used to evaluate E[f(X)] by drawing samples \(\{ {X _1},\ldots {X _n} \}\) from the Metropolis–Hastings algorithm. Here,

$$\begin{aligned} E\left[ {f\left( X \right) } \right] \approx \frac{1}{n}\sum \limits _{i = 1}^n {f\left( {{X _i}} \right) } \end{aligned}$$
(20)

means that the population mean of \(f\left( X \right) \) is approximated by the sample mean. When the sample \({X_t}\) is independent, the law of large numbers ensures that the approximation can be made as accurate as desired by increasing the sample. Note that here n is not the total amount of samples by Metropolis–Hastings algorithm but the length of drawing samples.

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Gong, B., Yang, Y. & Cui, W. Quantum parameter estimation via dispersive measurement in circuit QED. Quantum Inf Process 17, 301 (2018). https://doi.org/10.1007/s11128-018-2078-4

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