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CDJ-Pontryagin Optimal Control for General Continuously Monitored Quantum Systems

Quantum Journal

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Researchers Tathagata Karmakar and Andrew N. Jordan have developed a generalized framework for quantum optimal control in continuously monitored systems, extending the CDJ stochastic path integral formalism to arbitrary quantum systems. The team introduced a costate operator and adapted Pontryagin’s maximum principle to quantum systems, enabling derivation of optimal control equations for any control parameters. This bridges classical and quantum control theory. Applied to a monitored quantum oscillator, the optimal control protocol demonstrated "bang-bang" behavior, where parameters switch abruptly between extreme values for maximum efficiency. Three bosonic quantum computing examples—binomial codeword preparation, ground-state cooling, and cat-state evolution—showed 40–196% higher fidelity under optimal control compared to standard protocols. This work provides a systematic method for designing state preparation and error correction protocols in real-time monitored quantum systems, advancing fault-tolerant quantum computing.

CDJ-Pontryagin Optimal Control for General Continuously Monitored Quantum Systems

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AbstractThe 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.Featured image: 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.Popular summaryTime continuous measurements enable real-time monitoring of quantum systems. These measurements describe the evolution of a quantum system conditioned on a sequence of readouts through stochastic quantum trajectories. On the other hand, quantum optimal control protocols identify cost-effective methods for preparing and stabilizing quantum states. In our work, we present a framework for determining the most likely evolution-based optimal control for general monitored quantum systems. The Chantasri-Dressel-Jordan (CDJ) stochastic path integral formulation lets us find the most likely evolution under continuous monitoring. Developed in 2013, the CDJ formalism expresses the probability densities of the quantum trajectories in terms of a stochastic action. Recent investigations have proposed CDJ-based optimal control protocols for single and two-qubit systems. For these systems, a connection between the CDJ approach and Pontryagin's maximum principle (PMP) has been established. In classical control theory, the PMP provides the necessary conditions for finding optimal controls in dynamical systems. Our work prescribes a general PMP for quantum systems undergoing arbitrary evolution. We then describe the CDJ most likely paths as a special case of such PMP. We apply the analysis to a monitored oscillator under a parametric control potential. For this example, we show the optimality of “bang-bang'' controls. We compare the performance of optimal protocols with non-optimal sample controls. The former sees a 40-196% increase in the final state fidelities. Our results are useful for designing state preparation protocols in continuously monitored systems and inspire future investigations into problems such as cooling, continuous error correction, etc.► BibTeX data@article{Karmakar2026cdjpontryagin, doi = {10.22331/q-2026-03-24-2043}, url = {https://doi.org/10.22331/q-2026-03-24-2043}, title = {{CDJ}-{P}ontryagin {O}ptimal {C}ontrol for {G}eneral {C}ontinuously {M}onitored {Q}uantum {S}ystems}, author = {Karmakar, Tathagata and Jordan, Andrew N.}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2043}, month = mar, year = {2026} }► References [1] John Preskill ``Quantum Computing in the NISQ era and beyond'' Quantum 2, 79 (2018). https://doi.org/10.22331/q-2018-08-06-79 [2] Steven M. Girvin ``Introduction to quantum error correction and fault tolerance'' SciPost Phys. Lect. 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Could not fetch ADS cited-by data during last attempt 2026-03-24 13:42:36: No response from ADS or unable to decode the received json data when getting the list of citing works.This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions. AbstractThe 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.Featured image: 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.Popular summaryTime continuous measurements enable real-time monitoring of quantum systems. These measurements describe the evolution of a quantum system conditioned on a sequence of readouts through stochastic quantum trajectories. On the other hand, quantum optimal control protocols identify cost-effective methods for preparing and stabilizing quantum states. In our work, we present a framework for determining the most likely evolution-based optimal control for general monitored quantum systems. The Chantasri-Dressel-Jordan (CDJ) stochastic path integral formulation lets us find the most likely evolution under continuous monitoring. Developed in 2013, the CDJ formalism expresses the probability densities of the quantum trajectories in terms of a stochastic action. Recent investigations have proposed CDJ-based optimal control protocols for single and two-qubit systems. For these systems, a connection between the CDJ approach and Pontryagin's maximum principle (PMP) has been established. In classical control theory, the PMP provides the necessary conditions for finding optimal controls in dynamical systems. Our work prescribes a general PMP for quantum systems undergoing arbitrary evolution. We then describe the CDJ most likely paths as a special case of such PMP. We apply the analysis to a monitored oscillator under a parametric control potential. For this example, we show the optimality of “bang-bang'' controls. We compare the performance of optimal protocols with non-optimal sample controls. The former sees a 40-196% increase in the final state fidelities. Our results are useful for designing state preparation protocols in continuously monitored systems and inspire future investigations into problems such as cooling, continuous error correction, etc.► BibTeX data@article{Karmakar2026cdjpontryagin, doi = {10.22331/q-2026-03-24-2043}, url = {https://doi.org/10.22331/q-2026-03-24-2043}, title = {{CDJ}-{P}ontryagin {O}ptimal {C}ontrol for {G}eneral {C}ontinuously {M}onitored {Q}uantum {S}ystems}, author = {Karmakar, Tathagata and Jordan, Andrew N.}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2043}, month = mar, year = {2026} }► References [1] John Preskill ``Quantum Computing in the NISQ era and beyond'' Quantum 2, 79 (2018). https://doi.org/10.22331/q-2018-08-06-79 [2] Steven M. Girvin ``Introduction to quantum error correction and fault tolerance'' SciPost Phys. Lect. 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CDJ-Pontryagin Optimal Control for General Continuously Monitored Quantum Systems

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