Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Mathematical Physics

Mayer problem for quantum stochastic control

M. O. Ogundiran, E. O. Ayoola

Published Feb 1, 2010• 7 citations
Publisher

Abstract

This paper is concerned with the value function for optimal quantum stochastic control. We established the Lipschitz continuity of the value function, which is inherent from the Lipschitz property of the set-valued map describing the quantum stochastic control. By using a feedback multivalued map, we also prove a characterization of optimal solutions relating to a noncommutative generalization of Mayer problem.

DOI
10.1063/1.3300332

Related papers

Optimal control equation for quantum stochastic differential equations

Javad Sharifi, Hamidreza Momeni

Interaction of open quantum systems with fundamental noncommutative quantum noises can be described by quantum stochastic differential equations (QSDE). These equations have a key role in quantum network analysis and design, especially for quantum information processing. Hence, in this paper, we derive a Hamilton-Jacobi-Bellman equation for quantum stochastic differential equations. The Bellman optimality principle is developed for open quantum systems. The cost functional of quantum observable to be minimized is considered to be general noncommutative polynomial of quantum operator. Since the method directly deals with QSDE, then it is a useful tool for optimal control of quantum optical networks. In addition, we will exhibit some electro-optical and all-optical feedback control schematics for implementation of quantum control based on QSDEs.

IEEE Conference on Decision and Control (CDC)•Dec 1, 2010•2 citations

On the separation principle of quantum control

Luc Bouten, Ramon van Handel

It is well known that quantum continuous observations and nonlinear filtering can be developed within the framework of the quantum stochastic calculus of Hudson-Parthasarathy. The addition of real-time feedback control has been discussed by many authors, but the foundations of the theory still appear to be relatively undeveloped. Here we introduce the notion of a controlled quantum flow, where feedback is taken into account by allowing the coefficients of the quantum stochastic differential equation to be adapted processes in the observation algebra. We then prove a separation theorem for quantum control: the admissible control that minimizes a given cost function is a memoryless function of the filter, provided that the associated Bellman equation has a sufficiently regular solution. Along the way we obtain results on existence and uniqueness of the solutions of controlled quantum filtering equations and on the innovations problem in the quantum setting.

arXiv:math-ph/0511021•Nov 5, 2005•32 citations

Hamilton–Jacobi–Bellman equations for quantum optimal feedback control

John Edward Gough, V. P. Belavkin, Oleg Georgievich Smolyanov

We exploit the separation of the filtering and control aspects of quantum feedback control to consider the optimal control as a classical stochastic problem on the space of quantum states. We derive the corresponding Hamilton-Jacobi-Bellman equations using the elementary arguments of classical control theory and show that this is equivalent, in the Stratonovich calculus, to a stochastic Hamilton-Pontryagin setup. We show that, for cost functionals that are linear in the state, the theory yields the traditional Bellman equations treated so far in quantum feedback.

arXiv:quant-ph/0502155•Journal of Optics B Quantum and Semiclassical Optics•Sep 14, 2005•62 citations

A DISCRETE INVITATION TO QUANTUM FILTERING AND FEEDBACK CONTROL ∗

Luc Bouten, Ramon van Handel, Matthew +1 more

Abstract. The engineering and control of devices at the quantum-mechanical level—such as those consisting of small numbers of atoms and photons—is a delicate business. The fundamental uncertainty that is inherently present at this scale manifests itself in the unavoidable presence of noise, making this a novel field of application for stochastic estimation and control theory. In this expository paper we demonstrate estimation and feedback control of quantum mechanical systems in what is essentially a noncommutative version of the binomial model that is popular in mathematical finance. The model is extremely rich and allows a full development of the theory, while remaining completely within the setting of finite-dimensional Hilbert spaces (thus avoiding the technical complications of the continuous theory). We introduce discretized models of an atom in interaction with the electromagnetic field, obtain filtering equations for photon counting and homodyne detection, and solve a stochastic control problem using dynamic programming and Lyapunov function methods.

•Jan 1, 2006•71 citations

Quantum speedups for stochastic optimization

Aaron Sidford, Chenyi Zhang

We consider the problem of minimizing a continuous function given quantum access to a stochastic gradient oracle. We provide two new methods for the special case of minimizing a Lipschitz convex function. Each method obtains a dimension versus accuracy trade-off which is provably unachievable classically and we prove that one method is asymptotically optimal in low-dimensional settings. Additionally, we provide quantum algorithms for computing a critical point of a smooth non-convex function at rates not known to be achievable classically. To obtain these results we build upon the quantum multivariate mean estimation result of Cornelissen et al. 2022 and provide a general quantum-variance reduction technique of independent interest.

arXiv:2308.01582•neural information processing systems•Jan 1, 2023•1 citation

A Discrete Invitation to Quantum Filtering and Feedback Control

Luc Bouten, Ramon van Handel, M. R. James

The engineering and control of devices at the quantum mechanical level—such as those consisting of small numbers of atoms and photons—is a delicate business. The fundamental uncertainty that is inherently present at this scale manifests itself in the unavoidable presence of noise, making this a novel field of application for stochastic estimation and control theory. In this expository paper we demonstrate estimation and feedback control of quantum mechanical systems in what is essentially a noncommutative version of the binomial model that is popular in mathematical finance. The model is extremely rich and allows a full development of the theory while remaining completely within the setting of finite-dimensional Hilbert spaces (thus avoiding the technical complications of the continuous theory). We introduce discretized models of an atom in interaction with the electromagnetic field, obtain filtering equations for photon counting and homodyne detection, and solve a stochastic control problem using dynamic programming and Lyapunov function methods.

arXiv:math/0606118•SIAM Review•May 1, 2009•89 citations
This paper is concerned with the value function for optimal quantum stochastic control. We established the Lipschitz continuity of the value function, which is inherent from the Lipschitz property of the set-valued map describing the quantum stochastic control. By using a feedback multivalued map, we also prove a characterization of optimal solutions relating to a noncommutative generalization of Mayer problem.
Interaction of open quantum systems with fundamental noncommutative quantum noises can be described by quantum stochastic differential equations (QSDE). These equations have a key role in quantum network analysis and design, especially for quantum information processing. Hence, in this paper, we derive a Hamilton-Jacobi-Bellman equation for quantum stochastic differential equations. The Bellman optimality principle is developed for open quantum systems. The cost functional of quantum observable to be minimized is considered to be general noncommutative polynomial of quantum operator. Since the method directly deals with QSDE, then it is a useful tool for optimal control of quantum optical networks. In addition, we will exhibit some electro-optical and all-optical feedback control schematics for implementation of quantum control based on QSDEs.
It is well known that quantum continuous observations and nonlinear filtering can be developed within the framework of the quantum stochastic calculus of Hudson-Parthasarathy. The addition of real-time feedback control has been discussed by many authors, but the foundations of the theory still appear to be relatively undeveloped. Here we introduce the notion of a controlled quantum flow, where feedback is taken into account by allowing the coefficients of the quantum stochastic differential equation to be adapted processes in the observation algebra. We then prove a separation theorem for quantum control: the admissible control that minimizes a given cost function is a memoryless function of the filter, provided that the associated Bellman equation has a sufficiently regular solution. Along the way we obtain results on existence and uniqueness of the solutions of controlled quantum filtering equations and on the innovations problem in the quantum setting.
We exploit the separation of the filtering and control aspects of quantum feedback control to consider the optimal control as a classical stochastic problem on the space of quantum states. We derive the corresponding Hamilton-Jacobi-Bellman equations using the elementary arguments of classical control theory and show that this is equivalent, in the Stratonovich calculus, to a stochastic Hamilton-Pontryagin setup. We show that, for cost functionals that are linear in the state, the theory yields the traditional Bellman equations treated so far in quantum feedback.
Abstract. The engineering and control of devices at the quantum-mechanical level—such as those consisting of small numbers of atoms and photons—is a delicate business. The fundamental uncertainty that is inherently present at this scale manifests itself in the unavoidable presence of noise, making this a novel field of application for stochastic estimation and control theory. In this expository paper we demonstrate estimation and feedback control of quantum mechanical systems in what is essentially a noncommutative version of the binomial model that is popular in mathematical finance. The model is extremely rich and allows a full development of the theory, while remaining completely within the setting of finite-dimensional Hilbert spaces (thus avoiding the technical complications of the continuous theory). We introduce discretized models of an atom in interaction with the electromagnetic field, obtain filtering equations for photon counting and homodyne detection, and solve a stochastic control problem using dynamic programming and Lyapunov function methods.
We consider the problem of minimizing a continuous function given quantum access to a stochastic gradient oracle. We provide two new methods for the special case of minimizing a Lipschitz convex function. Each method obtains a dimension versus accuracy trade-off which is provably unachievable classically and we prove that one method is asymptotically optimal in low-dimensional settings. Additionally, we provide quantum algorithms for computing a critical point of a smooth non-convex function at rates not known to be achievable classically. To obtain these results we build upon the quantum multivariate mean estimation result of Cornelissen et al. 2022 and provide a general quantum-variance reduction technique of independent interest.
The engineering and control of devices at the quantum mechanical level—such as those consisting of small numbers of atoms and photons—is a delicate business. The fundamental uncertainty that is inherently present at this scale manifests itself in the unavoidable presence of noise, making this a novel field of application for stochastic estimation and control theory. In this expository paper we demonstrate estimation and feedback control of quantum mechanical systems in what is essentially a noncommutative version of the binomial model that is popular in mathematical finance. The model is extremely rich and allows a full development of the theory while remaining completely within the setting of finite-dimensional Hilbert spaces (thus avoiding the technical complications of the continuous theory). We introduce discretized models of an atom in interaction with the electromagnetic field, obtain filtering equations for photon counting and homodyne detection, and solve a stochastic control problem using dynamic programming and Lyapunov function methods.