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Optimal Controls for Open Quantum Systems Using Stochastic Path Integrals

Wirawat Kokaew, Thiparat Chotibut, Areeya Chantasri

Published Jan 1, 2020• 0 citations
Publisher

Abstract

We propose a new optimal control protocol for quantum state preparation in the presence of environmental noises based on the Most Likely Path (MLP) approach. The standard method for dealing with unknown noises in open quantum systems is via Lindblad master equation, which describes the Mean Path (MP) of noisy quantum trajectories. However, the mean path does not always faithfully represent the ensemble of trajectories, especially when the trajectory distribution is multimodal. The most likely path of quantum trajectories is extracted from stochastic path integral formulation, constructed from the joint probability of quantum statetrajectories and the noises. We study a quantum control problem of a single-qubit state preparation, controlling the qubit from an arbitrary initial state to the desired target state in the presence of a dephasing noise, where the control is the Rabi oscillation. This approach yields the analytical solution for the optimal control protocol for the Rabi drive, which is not possible using the standard MP approach. We also investigate the benefits of optimal Rabi drive by looking at the distribution of final states in the Bloch sphere; we find that the distribution of final state is concentrated around the desired target state and does not spread out over the qubit.

DOI
10.58837/chula.sp.2020.63

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We propose a new optimal control protocol for quantum state preparation in the presence of environmental noises based on the Most Likely Path (MLP) approach. The standard method for dealing with unknown noises in open quantum systems is via Lindblad master equation, which describes the Mean Path (MP) of noisy quantum trajectories. However, the mean path does not always faithfully represent the ensemble of trajectories, especially when the trajectory distribution is multimodal. The most likely path of quantum trajectories is extracted from stochastic path integral formulation, constructed from the joint probability of quantum statetrajectories and the noises. We study a quantum control problem of a single-qubit state preparation, controlling the qubit from an arbitrary initial state to the desired target state in the presence of a dephasing noise, where the control is the Rabi oscillation. This approach yields the analytical solution for the optimal control protocol for the Rabi drive, which is not possible using the standard MP approach. We also investigate the benefits of optimal Rabi drive by looking at the distribution of final states in the Bloch sphere; we find that the distribution of final state is concentrated around the desired target state and does not spread out over the qubit.
We investigate open-loop quantum state preparation for a class of open quantum systems whose dynamics follow a Gorini-Kossakowski-Lindblad-Sudarshan (GKLS) master equation that admits a trajectory-based stochastic representation. The deterministic control objective is reformulated as a stochastic optimal control problem -- interpreting stochasticity as a methodological tool akin to stochastic Schrödinger equation unravelings -- which situates the problem within the path integral control framework. For the class of GKLS generators under consideration, this reformulation leads to an explicit expression for the optimal control as a weighted average over stochastic quantum trajectories, thereby eliminating the need for gradient evaluations. Building on this theoretical result, we derive a control update rule for piecewise-constant control pulses and demonstrate that adaptive importance sampling progressively enhances the control estimator during optimization, culminating in the algorithm we term Path integral Quantum Control (PiQC). We further introduce an annealed variant of PiQC, wherein a synthetic noise schedule gradually steers open-system trajectories toward closed-system dynamics, enabling high-fidelity unitary state preparation. Numerical studies on a dissipative single-qubit system and a multi-qubit Nuclear Magnetic Resonance model verify that PiQC yields precise open-loop controls and displays robustness to Hamiltonian perturbations. We propose PiQC as a trajectory-based alternative to gradient-based approaches, which might offer a viable solution in quantum control problems where gradient computation is infeasible or computationally demanding.
Creating robust quantum operations is a major challenge in the current noisy intermediate-scale quantum computing era. Recently, the importance of noise-resilient control methods has become more pronounced in the field. Ordinarily, noisy quantum systems are described by the Lindblad equation. However, minimizing noise susceptibility using this equation has proven challenging because of its irreversibility. In this study, we propose a method for creating robust pulses based on the stochastic Schr\"odinger equation. This equation describes individual noise realizations under any colored noise process, contrary to the Lindblad equation, which describes mean system behavior under white noise. Using stochastic optimal control techniques, our method, fidelity-enhanced variational quantum optimal control, is able to construct higher-fidelity paths than its nonstochastic counterpart. By accounting for both environmental noise sources as well as noise sources inherent to the control system, highly significant increases in fidelity are noted for both single- and multiple-qubit state preparations.
Path integral control is a stochastic optimal control framework that computes optimal policies as weighted averages over state trajectories. It was recently shown that, combined with stochastic quantum trajectories, the path integral control approach constitutes a promising alternative for quantum optimal control. In this work, we propose an extension of path integral control to the complex plane using circular complex Wiener processes. The change of measure is defined on the driving noise space under an exact control-noise matching condition, which also accommodates rank-deficient state diffusions. Applied to stochastic Schrödinger equations, the resulting formalism gives a compact treatment of paired non-Hermitian jump channels with unequal positive dissipation rates and yields cross-entropy-optimal feedback and open-loop control formulas.
We generalize and extend the stochastic path-integral formalism and action principle for continuous quantum measurement introduced in Chantasri, Dressel, and Jordan [Phys. Rev. A 88, 042110 (2013)], where the optimal dynamics, such as the most likely paths, is obtained by extremizing the action of the stochastic path integral. In this work, we apply exact functional methods as well as develop a perturbative approach to investigate the statistical behavior of continuous quantum measurement. Examples are given for the qubit case. For qubit measurement with zero-qubit Hamiltonian, we find analytic solutions for average trajectories and their variances while conditioning on fixed initial and final states. For qubit measurement with unitary evolution, we use the perturbation method to compute expectation values, variances, and multi-time-correlation functions of qubit trajectories in the short-time regime. Moreover, we consider continuous qubit measurement with feedback control, using the action principle to investigate the global dynamics of its most likely paths, and finding that in an ideal case, qubit state stabilization at any desired pure state is possible with linear feedback. We also illustrate the power of the functional method by computing correlation functions for the qubit trajectories with a feedback loop to stabilize the qubit Rabi frequency.
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 involving a few qubits or quantum harmonic oscillators in Gaussian states. 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.
We consider the quantum state control of a multi-state system which evolves an initial state into a target state. We explicitly demonstrate the control method in an interesting case involving the transfer and rotation of a Schr\"{o}dinger cat state through a coupled harmonic oscillator chain at a predetermined time TTT. We use the gradient-based Krotov's method to design the time-dependent parameters of the coupled chain to find an optimal control shape that will evolve the system into a target state. We show that the prescribed quantum state control can be achieved with high fidelity, and the robustness of the control against generic environment noises is explored. Our findings will be of interest for the optimal control of a many-body open quantum system in the presence of environmental noise.