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Controlling open quantum systems: Tools, achievements, and limitations
Christiane P. Koch
TL;DR
Quantum technologies require control that preserves coherence and entanglement despite environmental decoherence. This review surveys optimal-control methods for open quantum systems, including strategies that suppress dissipation and exploit environmental memory, and concludes that non-Markovian controllability remains insufficiently understood.
Problem
Open-system control must preserve nonclassical quantum features despite decoherence caused by interaction with the environment.
Method
The review synthesizes controllability results and optimal-control strategies for avoiding decoherence, optimizing relaxation, and exploiting non-Markovian environmental dynamics.
Results
The reviewed examples show that optimal control can suppress detrimental dissipation, accelerate cooling, and exploit suitably coupled environmental modes to enhance controllability.
Takeaways & Limitations
Environmental interactions can be either obstacles or resources for quantum control, depending on their dynamical properties and how they are incorporated into control design.
Takeaways & Limitations
The role of non-Markovian effects and the features that can be exploited for quantum control remain insufficiently understood.
Abstract
from arXiv · showhide
The advent of quantum devices, which exploit the two essential elements of quantum physics, coherence and entanglement, has sparked renewed interest in the control of open quantum systems. Successful implementations face the challenge to preserve the relevant nonclassical features at the level of device operation. A major obstacle is decoherence which is caused by interaction with the environment. Optimal control theory is a tool that can be used to identify control strategies in the presence of decoherence. We review here recent advances in optimal control methodology that allow for tackling typical tasks in device operation for open quantum systems and discuss examples of relaxation-optimized dynamics. Optimal control theory is also a useful tool to exploit the environment for control. We discuss examples and point out possible future extensions.
I. INTRODUCTION
Quantum optimal control applies classical fields to steer quantum dynamics, while open-system control must balance useful operations against environmental noise and loss of coherence. The review surveys strategies for preserving quantum features, avoiding decoherence, and exploiting environmental interactions.
- Quantum optimal control uses classical controls, such as RF amplitudes or laser electric fields, to induce desired quantum dynamics.
- Optimal-control methods were extended from NMR and matter-wave dynamics to quantum-information tasks including gates, entanglement, measurement, state preparation, transport, and storage.
- In photoinduced chemical reactions, calculated time-shaped pulses may be experimentally incompatible with spectral shaping and inaccurate models can optimize the wrong dynamics.
- QIP is a promising setting because quantum systems are typically well characterized and slow enough for electronic pulse shaping.
- Open-system control must balance control-induced noise with sufficient isolation of coherence and other desired quantum features.
- Dissipation challenges quantum control but can also enable control, motivating strategies that either suppress environmental effects or exploit them.
II. OPEN QUANTUM SYSTEMS
Open quantum systems are modeled through dynamics that may be memoryless or memory-bearing, with non-Markovianity characterized by several competing measures. Their analysis remains computationally and experimentally difficult, although memory effects may provide control resources.
- An open quantum system is represented by a density operator whose dynamics must be calculated and whose control success must be quantified.
- Markovian dynamics are memoryless because the dynamical map is divisible into completely positive and trace-preserving maps.
- Markovian evolution can be solved with arbitrarily high precision, but the required resources scale exponentially with Hilbert- and Liouville-space size.
- Non-Markovian dynamics lack a unified Lindblad-type framework, so methods with different assumptions and applicability ranges are used.
- Non-Markovian simulations may replace the environment with a surrogate of effective modes, enabling longer propagation than exact system-environment dynamics permits.
- Memory effects can be quantified through deviations from divisibility, recoherence, information backflow, or state distinguishability.
- Measuring non-Markovianity in condensed-phase experiments remains difficult because limited control restricts access to measurable quantities.
- Non-Markovianity may revive coherence and correlations, suggesting greater control power than in the Markovian regime.
B. Measuring success of control in open quantum systems
Control success in open quantum systems is assessed with figures of merit that uniquely identify the target while remaining computationally feasible. The review describes state, gate, and flexible-equivalence objectives alongside reductions in the number of required test states.
- A suitable control figure of merit must attain its optimum if and only if the target is reached and must be computable.
- For state transfer, success is measured by the projection of the final system state onto the desired target state.
- Unitary-gate control requires simultaneous state-to-state transitions for every state in the logical basis.
- The Liouville-space gate objective evaluates how closely the actual evolution implements the target operation on a d-dimensional logical subspace.
- The d-scaling of the full Liouville-space objective restricts its use to systems with very few qubits.
- Three states suffice to define a figure of merit for a target unitary, independent of system size, reducing resources relative to process tomography.
- A d+1-state variant can converge faster than the three-state formulation while remaining more efficient than the d2-state objective.
- Locally equivalent gates can replace a specific gate target when numerical searches benefit from a more flexible objective.
III. CONTROL OF OPEN QUANTUM SYSTEMS
Controlling open quantum systems involves first determining whether a desired task is reachable and then synthesizing controls that realize it. The review distinguishes controllability analysis from control synthesis and emphasizes their practical dependence on available resources.
- Open-system control separates controllability analysis from control synthesis, also called motion planning in classical automatic control.
- Checking whether a task is possible should precede searching for controls that reach the target.
- The review surveys controllability, environment-based synthesis strategies, and optimal control theory as complementary parts of open-system control.
A. Controllability of open quantum systems
Controllability analysis asks whether target states are reachable, but open-system dissipation can prevent or enable reachability beyond what Hamiltonian analysis captures. Rigorous reduced-dynamics results remain limited, especially for non-Markovian systems.
- Controllability limitations: Hamiltonian Lie-rank controllability is insufficient for open systems because dissipation can prevent or enable access to target states.Fast decoherence can turn pure states into mixed states, inhibiting pure-state transitions even when the Hamiltonian satisfies the full-rank condition.
- Controllability limitations: Numerical searches are often required, but their local character cannot rigorously establish non-reachability with a prescribed error.This limitation is particularly relevant for systems with nearly unitary dynamics.
- Dissipative control: Dissipation can also enable reachability, as cooling transforms mixed states into pure ones beyond the scope of Hamiltonian-only Lie-rank analysis.Purity changes are not captured by controllability criteria based solely on unitary Hamiltonian dynamics.
- Open problems: Dynamic controllability remains largely uncharted because it must account for available controls, environmental couplings, and measurements rather than merely map existence.Kinematic controllability can establish that a dynamical map exists but generally provides no practical dynamical information.
- Analysis routes: Two routes analyze open-system controllability: model the complete system-environment dynamics or use a reduced system description.The complete description permits unitary controllability tools, while reduced dynamics require methods suited to dissipation and memory.
- Open problems: Reduced-dynamics controllability has rigorous analyses mainly for Markovian systems; no rigorous non-Markovian analysis had been performed, and its effect on controllability remained unclear.Markovian results include reachable-state characterizations and Lie-wedge conditions, whereas non-Markovian evidence was numerical.
B. Control strategies for open quantum systems
Open-system control strategies either exploit symmetries to protect states from decoherence or use timescale separation and spectral engineering to suppress environmental effects. These approaches have practical boundaries, including unavailable symmetries and finite control-field durations.
- Symmetry-based protection: Symmetries in the system-bath interaction can create decoherence-free or noiseless subsystems that protect logical information from noise.A preserved quantum number defines the protected subsystem.
- Symmetry-based protection: The main limitation of symmetry-based protection is that a suitable system-bath symmetry may not exist.The protection depends on an interaction symmetry such as indistinguishable environmental coupling.
- Timescale-based control: When decoherence is slow, operations can be performed faster than decoherence or its effects can be reduced through average Hamiltonian and spin-echo techniques.These methods rely on assumptions about the timescale of system-bath interaction.
- Dynamical decoupling: Dynamical decoupling has been extended to pulse imperfections and platforms including ions and superconducting circuits.Its principal limitation is that control fields have finite duration and cannot always be made sufficiently short.
- Spectral control: Transfer-filter-function methods connect time- and frequency-domain strategies, predict fidelities for weak noise, and support spectral separation from the environment.They can cancel system-bath interactions at least to second order.
- Spectral control: Noise spectroscopy can characterize noise spectral density and help derive microscopic system-environment models across physical platforms.This provides a starting point for more thorough understanding of system-environment interactions.
IV. OPTIMAL CONTROL OF OPEN QUANTUM SYSTEMS
Quantum optimal control synthesizes external fields from control targets, constraints, and system dynamics. Because most open-system problems lack closed-form solutions and are difficult computationally, numerical gradient-based optimization is the prevailing approach.
- Framework: Quantum optimal control methods synthesize external control fields using the target, constraints, and modeled quantum time evolution.The target is formulated as a functional of the controls and evaluated using the system dynamics.
- Computational approach: Closed-form geometric solutions exist only for exceptional systems, so most open-system control problems require numerical optimization.Examples of exceptional cases include one or two spins, harmonic oscillators, and decaying Λ-systems.
- Computational approach: Open-system control is difficult enough that algorithmic efficiency is important, with gradient-based methods predominating in existing work.The review presents these methods before discussing numerical strategies for avoiding decoherence and exploiting the environment.
A. Optimal control theory applied to open quantum systems
The reviewed optimization framework drives states or operations toward specified targets by iteratively updating controls using forward states and backward co-states. It extends to dissipative dynamics, unitary operations, flexible constraints, and spectral penalties, while scaling and nonstandard targets remain challenging.
- State-to-state control: State-to-state control finds a field that drives a known initial density operator into a target state at final time T with prescribed error ǫ.The target functional depends implicitly on the controls through the open-system dynamical map.
- Dissipative dynamics: For Markovian dynamics, the framework models coherent control through a Hamiltonian and dissipation through Lindblad operators representing decay channels.In laser cooling, the control can be an electric field and the Lindblad operators can describe spontaneous decay rates.
- Iterative optimization: Krotov-style optimization seeks an extremum of the target functional while providing monotonic convergence through sequential control updates.The update uses a shape function and an algorithm parameter that determines the control step size.
- Iterative optimization: The update depends on forward states and backward co-states obtained from coupled equations solved iteratively from an initial control guess.For linear control coupling, the explicit dependence of the update on the new control vanishes.
- Unitary operations: Unitary-operation optimization treats a gate as simultaneous state-to-state transfers driven by the same control field.The associated state and co-state equations are solved concurrently for multiple basis states.
- Unitary operations: M can be reduced from d^2 to 3 for unitary-operation optimization, using suitable initial states, while co-state weights can accelerate convergence.The reduced-state result changes the number of propagated states required by the algorithm.
- Flexible targets: Flexible targets such as arbitrary perfect entanglers require modified target-function derivatives, and open-system applications remain under exploration.Storage of all propagated states can become limiting as system size grows.
- Control constraints: Additional constraints can enforce desired control properties, including spectral constraints implemented through a Fredholm equation and efficiently solved with Fourier transforms.Multiple constraints can be combined with different weights to emphasize their relative importance.
B. Fighting and avoiding decoherence
The review presents strategies for reducing decoherence by optimizing known controls, identifying decoherence-resistant subspaces, or operating near the quantum speed limit.
- Known control strategies: Optimized dynamical decoupling can target previously unaccounted-for noise features and complement numerically optimized pulses.These approaches include gradient-ascent techniques and genetic algorithms, with combined use demonstrated for entanglement generation and distribution in NV centers.
- Decoherence-free strategies: Quantum optimal control can dynamically identify decoherence-free subspaces and noiseless subsystems when direct identification is difficult.Direct searches may be hindered by numerous traps in the search space.
- Quantum speed limits: When decoherence affects Hilbert-space regions similarly, the fastest possible operation can mitigate its impact, subject to the quantum speed limit.For complex systems, optimal control can identify both the speed limit and the control that achieves it.
C. Cooling and quantum reservoir engineering
The review describes cooling as control that deliberately uses dissipation and extends this framework toward quantum reservoir engineering, while highlighting unresolved numerical and condensed-phase challenges.
- Cooling: Laser cooling of molecular vibrations is possible even when molecular structure favors heating rather than cooling.Without control constraints, one state with moderate-probability spontaneous emission into the cooling target is sufficient.
- Cooling: An optimized laser pulse substantially reduces cooling cycles relative to an unshaped pulse when molecular structure favors vibrational cooling.Similar speedups have been reported for an optomechanical resonator and trapped, quasi-condensed cold atoms.
- Quantum reservoir engineering: The cooling framework can generalize to quantum reservoir engineering, in which a desired state becomes the ground state of a driven dissipative system.The framework is especially relevant when coherent and dissipative timescales differ, as in optical pumping.
- Quantum reservoir engineering: Quantum reservoir engineering is challenging in condensed-phase settings because desired and undesired dissipative channels coexist and non-Markovian effects may occur.Efficient numerical implementation remains open because the search space is larger than in standard quantum control.
- Quantum reservoir engineering: Control synthesis may combine environmental population distributions, measurements, tailored coherent excitation, or optimization toward a desired steady state.The review identifies exploration under undesired dissipation and non-Markovian dynamics as an open computational challenge.
D. Exploiting non-Markovianity for quantum control
The review presents non-Markovianity as a possible resource for quantum control, while emphasizing that its control-relevant features and broader potential remain insufficiently understood.
- Potential benefits: Non-Markovianity may improve control through information backflow, system-environment correlations, and cooperative effects of dissipation and driving.Reported possibilities include improved gate fidelities, entropy export for cooling, and enhanced quantum information processing and communication.
- Controllability enhancement: In an anharmonic ladder system, a sufficiently isolated and strongly coupled two-level environmental defect extended possible operations from SO(N) to SU(N).The review notes that such conditions occur in some superconducting circuits and systems with small natural spin baths.
- Open challenges: Only a limited number of optimal-control studies have examined open systems with non-Markovian dynamics, leaving their full control potential largely uncharted.Open questions include how memory buildup and spectral-density features can be exploited.
V. CONCLUSIONS
The conclusions formulate tentative rules for controlling open quantum systems: suppress harmful Markovian decoherence for purity-preserving tasks, use dissipation for purity-changing targets, and exploit non-Markovianity selectively.
- Conclusions: The review spans applications in quantum thermodynamics, biological chromophore complexes, and other fields beyond quantum technologies.These examples include optimizing noisy heat-engine efficiency and maximizing exciton transfer.
- Conclusions: For purity-preserving operations, Markovian dynamics are unwanted, so optimal control can seek the shortest operation and identify decoherence-resistant subspaces.Neglecting dissipation simplifies optimization, while explicitly including it costs more numerically but can reveal protected subspaces.
- Conclusions: For operations that change state purity, the environment is necessary, and quantum reservoir engineering can make the target a fixed state of the Liouvillian.External control fields may also select among multiple fixed points.
- Conclusions: Non-Markovian dynamics can help or hinder control; benefits include improved controllability and an improved quantum speed limit under suitable environmental conditions.Improved controllability requires a few strongly coupled, sufficiently isolated environmental modes that can act as ancillas.
- Conclusions: The review concludes that understanding controllability and exploitable non-Markovian features remains limited despite growing successful-control examples.It proposes broader model studies and systematic non-Markovianity analysis to clarify these questions.