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

arXiv:0708.3513

Computational complexity of quantum optimal control landscapes

Raj Chakrabarti, Re-Bing Wu, HERSCHEL A. RABITZ

Published Aug 27, 2007• 2 citations
arXivHTMLPDF

Abstract

We study the Hamiltonian-independent contribution to the complexity of quantum optimal control problems. The optimization of controls that steer quantum systems to desired objectives can itself be considered a classical dynamical system that executes an analog computation. The system-independent component of the equations of motion of this dynamical system can be integrated analytically for various classes of discrete quantum control problems. For the maximization of observable expectation values from an initial pure state and the maximization of the fidelity of quantum gates, the time complexity of the corresponding computation belongs to the class continuous log (CLOG), the lowest analog complexity class, equivalent to the discrete complexity class NC. The simple scaling of the Hamiltonian-independent contribution to these problems with quantum system dimension indicates that with appropriately designed search algorithms, quantum optimal control can be rendered efficient even for large systems.

DOI
10.48550/arxiv.0708.3513

Related papers

Search complexity and resource scaling for the quantum optimal control of unitary transformations

Katharine W. Moore, Raj Chakrabarti, Gregory Riviello +1 more

The optimal control of unitary transformations is a fundamental problem in quantum control theory and quantum information processing. The feasibility of performing such optimizations is determined by the computational and control resources required, particularly for systems with large Hilbert spaces. Prior work on unitary transformation control indicates that (i) for controllable systems, local extrema in the search landscape for optimal control of quantum gates have null measure, facilitating the convergence of local search algorithms, but (ii) the required time for convergence to optimal controls can scale exponentially with the Hilbert space dimension. Depending on the control-system Hamiltonian, the landscape structure and scaling may vary. This work introduces methods for quantifying Hamiltonian-dependent and kinematic effects on control optimization dynamics in order to classify quantum systems according to the search effort and control resources required to implement arbitrary unitary transformations.

arXiv:1006.1829•Physical Review A•Jan 31, 2011•58 citations

Quantum control landscapes

Raj Chakrabarti, HERSCHEL A. RABITZ

Numerous lines of experimental, numerical and analytical evidence indicate that it is surprisingly easy to locate optimal controls steering quantum dynamical systems to desired objectives. This has enabled the control of complex quantum systems despite the expense of solving the Schrödinger equation in simulations and the complicating effects of environmental decoherence in the laboratory. Recent work indicates that this simplicity originates in universal properties of the solution sets to quantum control problems that are fundamentally different from their classical counterparts. Here, we review studies that aim to systematically characterize these properties, enabling the classification of quantum control mechanisms and the design of globally efficient quantum control algorithms.

arXiv:0710.0684•International Reviews in Physical Chemistry•Oct 1, 2007•179 citations

Efficient Optimal Control of Open Quantum Systems

Wenhao He, Tongyang Li, Xiantao Li +3 more

The optimal control problem for open quantum systems can be formulated as a time-dependent Lindbladian that is parameterized by a number of time-dependent control variables. Given an observable and an initial state, the goal is to tune the control variables so that the expected value of some observable with respect to the final state is maximized. In this paper, we present algorithms for solving this optimal control problem efficiently, i.e., having a poly-logarithmic dependency on the system dimension, which is exponentially faster than best-known classical algorithms. Our algorithms are hybrid, consisting of both quantum and classical components. The quantum procedure simulates time-dependent Lindblad evolution that drives the initial state to the final state, and it also provides access to the gradients of the objective function via quantum gradient estimation. The classical procedure uses the gradient information to update the control variables. At the technical level, we provide the first (to the best of our knowledge) simulation algorithm for time-dependent Lindbladians with an $\ell_1$-norm dependence. As an alternative, we also present a simulation algorithm in the interaction picture to improve the algorithm for the cases where the time-independent component of a Lindbladian dominates the time-dependent part. On the classical side, we heavily adapt the state-of-the-art classical optimization analysis to interface with the quantum part of our algorithms. Both the quantum simulation techniques and the classical optimization analyses might be of independent interest.

arXiv:2405.19245•Leibniz international proceedings in informatics•Jan 1, 2024•1 citation

Optimal control theory for continuous-variable quantum gates

Re-Bing Wu, Raj Chakrabarti, HERSCHEL A. RABITZ

The methodology of optimal control theory is applied to the problem of implementing quantum gates in continuous-variable (CV) systems with quadratic Hamiltonians. We demonstrate that it is possible to define a fidelity measure for CV gate optimization that is devoid of traps, such that the search for optimal control fields using local algorithms will not be hindered. The optimal control of several quantum computing gates, as well as that of algorithms composed of these primitives, is investigated using several typical physical models and compared for discrete-variable and continuous-variable quantum systems. Numerical simulations indicate that the optimization of generic CV quantum gates is inherently more expensive than that of generic discrete variable quantum gates, but can be routinely achieved for all the major classes of computing primitives. The exact-time controllability of CV systems, hitherto largely ignored in the design of information processing models, is shown to play an important role in determining the maximal achievable gate fidelity. Moreover, the ability to control interactions between qunits can be exploited to delimit the total control fluence. Future experimental model systems should carefully tune these parameters so as to enable the implementation of CV quantum information processing with optimal fidelity.

arXiv:0708.2118•Physical Review A•May 1, 2008•34 citations

Quantum algorithms for optimal control: large-scale finite-horizon linear-quadratic regulation

Xiajie Huang, Cristina Álvarez Yuste, Nikita Guseynov +2 more

We develop quantum algorithms for optimal control over a prescribed finite time interval (finite horizon), focusing on continuous-time linear-quadratic regulation (LQR). This foundational control problem often requires solving a nonlinear matrix Riccati equation, which can be costly for large systems and long control durations on conventional classical solvers. We instead work directly with the state-costate optimality equations, reformulating the LQR problem as a multiple-shooting linear system. Importantly, we prove a condition-number bound for the system that is independent of the horizon, under stabilizability and detectability assumptions. By incorporating the initial and final conditions into a single linear system, we can solve the subsequent linear system more efficiently with a quantum linear systems subroutine. For a given initial state, our quantum algorithms return classical estimates of the optimal cost and the optimal control inputs at a selected time, without constructing a Riccati solution operator. Given sparse-matrix block-encodings, an initial-state preparation oracle and a fixed number of control inputs, the query and gate complexities scale polylogarithmically with system size and control duration, provided that sparsity, relevant norms, and stability parameters are bounded. The dependence on the target error is inverse-linear up to logarithmic factors. We establish the required bounds uniformly in system size and duration for a family of damped mass-spring chains with two endpoint controls. Numerical experiments support the analysis.

quant-pharXiv:2610.26843•Oct 20, 2026•0 citations

Exploring quantum control landscapes: Topology, features, and optimization scaling

Katharine W. Moore, HERSCHEL A. RABITZ

Quantum optimal control experiments and simulations have successfully manipulated the dynamics of systems ranging from atoms to biomolecules. Surprisingly, these collective works indicate that the effort (i.e., the number of algorithmic iterations) required to find an optimal control field appears to be essentially invariant to the complexity of the system. The present work explores this matter in a series of systematic optimizations of the state-to-state transition probability on model quantum systems with the number of states $N$ ranging from 5 through 100. The optimizations occur over a landscape defined by the transition probability as a function of the control field. Previous theoretical studies on the topology of quantum control landscapes established that they should be free of suboptimal traps under reasonable physical conditions. The simulations in this work include nearly 5000 individual optimization test cases, all of which confirm this prediction by fully achieving optimal population transfer of at least 99.9$%$ on careful attention to numerical procedures to ensure that the controls are free of constraints. Collectively, the simulation results additionally show invariance of required search effort to system dimension $N$. This behavior is rationalized in terms of the structural features of the underlying control landscape. The very attractive observed scaling with system complexity may be understood by considering the distance traveled on the control landscape during a search and the magnitude of the control landscape slope. Exceptions to this favorable scaling behavior can arise when the initial control field fluence is too large or when the target final state recedes from the initial state as $N$ increases.

arXiv:1006.3702•Physical Review A•Jul 12, 2011•54 citations
We study the Hamiltonian-independent contribution to the complexity of quantum optimal control problems. The optimization of controls that steer quantum systems to desired objectives can itself be considered a classical dynamical system that executes an analog computation. The system-independent component of the equations of motion of this dynamical system can be integrated analytically for various classes of discrete quantum control problems. For the maximization of observable expectation values from an initial pure state and the maximization of the fidelity of quantum gates, the time complexity of the corresponding computation belongs to the class continuous log (CLOG), the lowest analog complexity class, equivalent to the discrete complexity class NC. The simple scaling of the Hamiltonian-independent contribution to these problems with quantum system dimension indicates that with appropriately designed search algorithms, quantum optimal control can be rendered efficient even for large systems.
The optimal control of unitary transformations is a fundamental problem in quantum control theory and quantum information processing. The feasibility of performing such optimizations is determined by the computational and control resources required, particularly for systems with large Hilbert spaces. Prior work on unitary transformation control indicates that (i) for controllable systems, local extrema in the search landscape for optimal control of quantum gates have null measure, facilitating the convergence of local search algorithms, but (ii) the required time for convergence to optimal controls can scale exponentially with the Hilbert space dimension. Depending on the control-system Hamiltonian, the landscape structure and scaling may vary. This work introduces methods for quantifying Hamiltonian-dependent and kinematic effects on control optimization dynamics in order to classify quantum systems according to the search effort and control resources required to implement arbitrary unitary transformations.
Numerous lines of experimental, numerical and analytical evidence indicate that it is surprisingly easy to locate optimal controls steering quantum dynamical systems to desired objectives. This has enabled the control of complex quantum systems despite the expense of solving the Schrödinger equation in simulations and the complicating effects of environmental decoherence in the laboratory. Recent work indicates that this simplicity originates in universal properties of the solution sets to quantum control problems that are fundamentally different from their classical counterparts. Here, we review studies that aim to systematically characterize these properties, enabling the classification of quantum control mechanisms and the design of globally efficient quantum control algorithms.
The optimal control problem for open quantum systems can be formulated as a time-dependent Lindbladian that is parameterized by a number of time-dependent control variables. Given an observable and an initial state, the goal is to tune the control variables so that the expected value of some observable with respect to the final state is maximized. In this paper, we present algorithms for solving this optimal control problem efficiently, i.e., having a poly-logarithmic dependency on the system dimension, which is exponentially faster than best-known classical algorithms. Our algorithms are hybrid, consisting of both quantum and classical components. The quantum procedure simulates time-dependent Lindblad evolution that drives the initial state to the final state, and it also provides access to the gradients of the objective function via quantum gradient estimation. The classical procedure uses the gradient information to update the control variables. At the technical level, we provide the first (to the best of our knowledge) simulation algorithm for time-dependent Lindbladians with an ℓ1\ell_1ℓ1​-norm dependence. As an alternative, we also present a simulation algorithm in the interaction picture to improve the algorithm for the cases where the time-independent component of a Lindbladian dominates the time-dependent part. On the classical side, we heavily adapt the state-of-the-art classical optimization analysis to interface with the quantum part of our algorithms. Both the quantum simulation techniques and the classical optimization analyses might be of independent interest.
The methodology of optimal control theory is applied to the problem of implementing quantum gates in continuous-variable (CV) systems with quadratic Hamiltonians. We demonstrate that it is possible to define a fidelity measure for CV gate optimization that is devoid of traps, such that the search for optimal control fields using local algorithms will not be hindered. The optimal control of several quantum computing gates, as well as that of algorithms composed of these primitives, is investigated using several typical physical models and compared for discrete-variable and continuous-variable quantum systems. Numerical simulations indicate that the optimization of generic CV quantum gates is inherently more expensive than that of generic discrete variable quantum gates, but can be routinely achieved for all the major classes of computing primitives. The exact-time controllability of CV systems, hitherto largely ignored in the design of information processing models, is shown to play an important role in determining the maximal achievable gate fidelity. Moreover, the ability to control interactions between qunits can be exploited to delimit the total control fluence. Future experimental model systems should carefully tune these parameters so as to enable the implementation of CV quantum information processing with optimal fidelity.
We develop quantum algorithms for optimal control over a prescribed finite time interval (finite horizon), focusing on continuous-time linear-quadratic regulation (LQR). This foundational control problem often requires solving a nonlinear matrix Riccati equation, which can be costly for large systems and long control durations on conventional classical solvers. We instead work directly with the state-costate optimality equations, reformulating the LQR problem as a multiple-shooting linear system. Importantly, we prove a condition-number bound for the system that is independent of the horizon, under stabilizability and detectability assumptions. By incorporating the initial and final conditions into a single linear system, we can solve the subsequent linear system more efficiently with a quantum linear systems subroutine. For a given initial state, our quantum algorithms return classical estimates of the optimal cost and the optimal control inputs at a selected time, without constructing a Riccati solution operator. Given sparse-matrix block-encodings, an initial-state preparation oracle and a fixed number of control inputs, the query and gate complexities scale polylogarithmically with system size and control duration, provided that sparsity, relevant norms, and stability parameters are bounded. The dependence on the target error is inverse-linear up to logarithmic factors. We establish the required bounds uniformly in system size and duration for a family of damped mass-spring chains with two endpoint controls. Numerical experiments support the analysis.
Quantum optimal control experiments and simulations have successfully manipulated the dynamics of systems ranging from atoms to biomolecules. Surprisingly, these collective works indicate that the effort (i.e., the number of algorithmic iterations) required to find an optimal control field appears to be essentially invariant to the complexity of the system. The present work explores this matter in a series of systematic optimizations of the state-to-state transition probability on model quantum systems with the number of states NNN ranging from 5 through 100. The optimizations occur over a landscape defined by the transition probability as a function of the control field. Previous theoretical studies on the topology of quantum control landscapes established that they should be free of suboptimal traps under reasonable physical conditions. The simulations in this work include nearly 5000 individual optimization test cases, all of which confirm this prediction by fully achieving optimal population transfer of at least 99.9% on careful attention to numerical procedures to ensure that the controls are free of constraints. Collectively, the simulation results additionally show invariance of required search effort to system dimension NNN. This behavior is rationalized in terms of the structural features of the underlying control landscape. The very attractive observed scaling with system complexity may be understood by considering the distance traveled on the control landscape during a search and the magnitude of the control landscape slope. Exceptions to this favorable scaling behavior can arise when the initial control field fluence is too large or when the target final state recedes from the initial state as NNN increases.