CDJ-Pontryagin Optimal Control for General Continuously Monitored Quantum Systems

Abstract

The Chantasri-Dressel-Jordan (CDJ) stochastic path integral formalism (Chantasri et al. 2013 and 2015) characterizes the statistics of the readouts and the most likely conditional evolution of continuously monitored quantum systems. In our work, we generalize the CDJ formalism to arbitrary continuously monitored systems by introducing a costate operator. We then prescribe a generalized Pontryagin's maximum principle for quantum systems undergoing arbitrary evolution and find conditions on optimal control protocols. We show that the CDJ formalism's most likely path can be cast as a quantum Pontryagin's maximum principle, where the cost function is the readout probabilities along a quantum trajectory. This insight allows us to derive general optimal control equations for arbitrary control parameters. We apply our results to a monitored oscillator in the presence of a parametric quadratic potential and variable quadrature measurements. We find the optimal potential strength and quadrature angle for fixed-end point problems. The optimal parametric potential is analytically shown to have a "bang-bang" form. We apply our protocol to three quantum oscillator examples relevant to Bosonic quantum computing. The first example considers a binomial codeword preparation from an error word, the second example looks into cooling to the ground state from an even cat state, and the third example investigates a cat state to cat state evolution. We compare the statistics of the fidelities of the final state with respect to the target state for trajectories generated under the optimal control with those generated under a sample control. Compared to the latter case, we see a 40-196% increase in the number of trajectories reaching more than 95% fidelities under the optimal control. Our work provides a systematic prescription for finding quantum optimal control for continuously monitored systems.

Figures (9)

• Figure 1: We sketch the schematic of a quantum system coupled to a detector that monitors the system. The readout $r$ obtained due to continuous measurement is noisy. In general, the system is controlled through a parameter $\chi_1$ that changes the system Hamiltonian (unitary control) and another parameter $\chi_2$ that modifies the measurements (dissipative control) performed on the system.
• Figure 2: The plots show the optimal path for Gaussian states under position measurements (i.e. $\theta=0$ in Eq. \ref{['l_theta']}). The top two panels show the time evolution of the expectation values of position and momentum. The next three panels show the time evolution of the position variance, position-momentum covariance, and the momentum variance. The final panel shows the most likely readout obtained as a function of time. All the observables are scaled to be dimensionless (see Sec. \ref{['sho']}). The time and the collapse timescale ($\tau=15.0$) are in units of the inverse of the oscillator frequency. In the plots, the green curve shows evolution obtained from Ref. PRXQuantum.3.010327, with a Gaussian state assumption. The red dashed lines show the most likely path obtained from the general formalism presented in Sec. \ref{['sho_op_strato']}. The blue dots in the first five panes denote the initial (a squeezed vacuum state), and the red crosses denote the final state (a squeezed coherent state). The squeezing is determined by the $\tau$ (see Appendix \ref{['xmeasurement']}). We see that the expectation values match very well, while the deviations in the variances and the covariance signify the failure of the steady-state assumption.
• Figure 3: Optimal control for $\ket{\psi(t=0)}=\ket{\psi_i}=\frac{\ket{0}-\ket{4}}{\sqrt{2}}$ and $\ket{\psi(t=3.0)}=\ket{\psi_f}=\frac{\ket{0}+\ket{4}}{\sqrt{2}}$ with collapse timescale $\tau=15.0$. Time and the collapse timescale are in units of the inverse of the oscillator frequency. The left panels from top to bottom show the time evolution of the position and momentum expectation values, position variance, and the covariance of position and momentum, respectively, under optimal control. The top panel on the right-hand side shows the evolution of the momentum variance. All the observables are scaled to be dimensionless (see Sec. \ref{['sho']}). The blue dots show the initial state, and the red '$\times$' show the final state. The second panel (from the top) on the right-hand side shows the most likely readout under optimal control. The third and fourth panels on the right-hand side show the optimal control parameters $\theta_1^\star$ and $\lambda_1^\star$. In other words, they show the optimal measurement quadrature and parametric potential strength. As explained in the text, the potential strength, $\lambda_1^{\star}$, is of "bang-bang" form. The final state here is reached with 95.46% fidelity (see Appendix \ref{['numerics_OC']}).
• Figure 4: From top to bottom, the panels show the time evolution of the position expectation value, momentum expectation value, the variance of position, the covariance of position and momentum, the variance of momentum, and the readout under the control in Figure \ref{['fig:optimal_control_binomial_code']}. The green dashed line shows the most likely path under optimal control shown in Figure \ref{['fig:optimal_control_binomial_code']}. The blue curve shows a sample simulated stochastic trajectory under the optimal control shown in the bottom two right panels of Figure \ref{['fig:optimal_control_binomial_code']}, starting from the initial state $\ket{\psi_i}=\frac{\ket{0}-\ket{4}}{\sqrt{2}}$. The collapse timescale is $\tau=15.0$ (time and the collapse timescale are in units of the inverse of the oscillator frequency). The quadratures and the readout are dimensionless (see Sec. \ref{['sho']}). We see that the expectation values along the trajectory jitter around the most likely path. Additionally, the trajectory closely follows the evolution of the variances and the covariance. The expectation values for the most likely path and the optimal readout appear flat because the corresponding scales involved with the stochastic trajectory are much larger (see Appendix \ref{['numerics_trajectory']} for the numerical method adopted to simulate trajectories).
• Figure 5: In panel (a), the top (bottom) plot shows sample measurement quadrature (parametric potential strength) as a function of time for collapse timescales $\tau=15.0$ (in units of $1/$oscillator frequency) such that trajectories starting from $\ket{\psi_i}=\frac{\ket{0}-\ket{4}}{\sqrt{2}}$ have a significant probability of reaching $\ket{\psi_f}=\frac{\ket{0}+\ket{4}}{\sqrt{2}}$ at $t=3.0$. Note, $\lambda_1(t)$ has a structure almost identical to $\lambda_1^\star(t)$ shown in Figure \ref{['fig:optimal_control_binomial_code']}. However, $\theta(t)$ here is very different from $\theta^\star(t)$. Panel (b) shows the histogram of final state fidelities (with respect to the target state $\ket{\psi_f}=\frac{\ket{0}+\ket{4}}{\sqrt{2}}$) of 10,000 simulated trajectories under the optimal control in Figure \ref{['fig:optimal_control_binomial_code']} (blue, vertical hatched) and the sample control presented in panel (a) (orange, diagonal hatched). We see that a significantly higher number of trajectories can achieve very high fidelities under optimal control.