Advances in Quantum Control - Techniques, Applications, and Challenges

We show how to apply the Pontryagin Maximum Principle (PMP) in Quantum Optimal control. We consider low dimensional quantum systems for which the optimal solutions can be derived analytically. After a pedagogical introduction to these techniques [2,3], different examples will be described. We discuss how the PMP is modified to optimize parameterized functions. We also show how to derive robust or selective optimal controls in a two-level quantum system. The limitations of this approach and its link with numerical optimization techniques will be also explored. [1]- Quantum optimal control in quantum technologies. Strategic report on current status, visions ans goals for research in Europe C. P. Koch, U. Boscain, T. Calarco, G. Dirr, S. Filipp, S. Glaser, R. Kosloff, S. Montangero, T. Schulte-Herbruggen, D. Sugny and F. K. Wilhelm EPJ Quantum Technol. 9, 19 (2022) [2]- Introduction to the Pontryagin Maximum Principle for Quantum Optimal Control U. Boscain, M. Sigalotti, and D. Sugny PRX Quantum 2, 030203 (2021) [3]- Introduction to the theoretical and experimental aspects of quantum optimal control Q. Ansel, E. Dionis, F. Arrouas, B. Peaudecerf, S. Guérin, D. Guéry-Odelin and D. Sugny J. Phys. B 57, 133001 (2024)

The joint state of a continuously monitored quantum system and the classical filtered measurement record has recently been shown to be described by a quantum Fokker-Planck master equation. We present a numerically efficient approach to compute the steady state of the system and detector in the absence of feedback. We use this to extract detailed statistics of the filtered signal, including all moments, the full probability distribution, the mutual information, correlation functions, as well as the Fisher information for parameter estimation. Building on this, we demonstrate how perturbative corrections allow one to efficiently determine the steady state solution for weak feedback driving.

Variational quantum algorithms (VQAs) are a significant focus of the quantum computing community. These algorithms operate by wrapping a classical optimization loop around a parameterized quantum circuit, and iteratively searching for the parameter configuration that produces the best solution to the problem under consideration. A critical challenge in VQAs is the difficulty of this classical optimization problem, which can become intractable as the number of quantum circuit parameters increases. I will introduce feedback-based quantum algorithms (FQAs) as an alternative paradigm that is optimization-free and applicable to a broad range of applications. Within this paradigm, quantum circuit parameter values are assigned in a layer-wise manner using a deterministic, measurement-based feedback law derived from quantum Lyapunov control principles. The use of feedback in this manner guarantees a monotonic improvement in solution quality with respect to the depth of the quantum circuit. I will overview quantum Lyapunov control theory as a motivation for this framework and go on to discuss concrete formulations of FQAs for applications including quantum simulation and combinatorial optimization. I will conclude by presenting results from a hardware implementation and numerical investigations of convergence, scalability, and robustness. Sandia National Labs is managed and operated by NTESS under DOE NNSA contract DENA0003525. SAND2022-10773 A.

Nanoscale devices that convert energy into useful work are becoming increasingly prevalent. A critical challenge in this area is managing energy transduction at the nanoscale. Quantum measurement and the associated acquisition of information can be harnessed to guide and enhance work output through feedback control. In this context, we explore a quantum information engine (QIE) as a prototype energy-transducing nanodevice controlled by measurement. This engine leverages the information exchange between a working medium, represented by a two-level system, and a meter, modeled as a quantum harmonic oscillator. However, this information transfer is not instantaneous; it relies on the measurement time, which is the duration needed to establish a correlation between the quantum system and the meter. This measurement time sets a lower limit on the cycle time of the QIE, making time-dependent information acquisition a crucial resource for the process. We examine the relationship between the energetic cost of quantum measurement, the associated acquisition of information, and the potential extractable work during finite-time operations. To achieve this, we investigate performance quantifiers, including the conversion of information to extractable work, the efficiency of converting input energy to net extracted work, and power, defined as the rate of net work extraction during the cycle operation. Our findings indicate that all these factors are influenced by measurement time in various ways. For example, achieving high efficiency necessitates a long measurement time, but this, in turn, reduces power. To identify an optimal balance among these performance indicators, we employ Pareto optimization, a multi-objective optimization method, to determine ideal measurement times.

In this talk, I will introduce a new quantum control method for on-demand noise filtering that integrates transition filter functions directly into continuous, time-dependent Hamiltonian pulses. This approach offers a powerful and flexible alternative to conventional $\pi$-pulse-based dynamical decoupling techniques. Our method enables precise control over signal transfer at selected frequencies, bandwidths, and phases, providing a high degree of tunability to match specific noise environments. It is inherently robust and significantly extends the coherence time of quantum systems. We will explore the principles behind the method, its practical advantages, and demonstrate its effectiveness through experimental results in nitrogen-vacancy (NV) centers in diamond, showing excellent agreement between theory and implementation.

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Controlling quantum systems requires performing work on them, but at the quantum scale, work is inherently stochastic and must be described by a probability distribution. Existing methods for computing these distributions rely on approximate master equations, such as the Redfield, polaron, or reaction coordinate approaches. While effective within their regimes of validity, these methods break down when applied beyond those limits. In this work, we present a novel approach that combines the two-point measurement protocol with process tensor theory, enabling the calculation of quantum work statistics to arbitrary precision—without approximations or assumptions about the underlying physics. This new development is general due to its reliance on process tensors and can be easily applied to study the work statistics for any time dependent system Hamiltonian within the spin-boson model. We show this generality by calculating the work statistics for two interesting problems, being Landauer erasure and the Landau-Zener transition. Our results establish a general and exact method for studying stochastic work in driven quantum systems, providing new insights into quantum thermodynamics and control.

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The achievement of control of noisy quantum systems has become pivotal in recent years. This holds especially true for NISQ quantum devices [1]. Quantum optimal control techniques rely on sophisticated numerical approaches [2]. However, no conclusive solution has been devised so far in the case of noisy quantum systems, i.e., quantum systems evolving under the influence of their surroundings [3]. Consequently, alternative approaches are desirable to reduce the complexity of the control techniques and increase the level of understanding of the physics involved. We propose to control prototypical instances of driven open quantum systems [4], i.e., two level systems and quantum harmonic oscillators, by means of a reverse engineering approach of the control Hamiltonians, which is based on the theory of dynamical invariants [5]. In contrast to previous theoretical approaches, in our work the influence of the environment explicitly depends on the control fields, i.e., dissipative effects change as functions of the control protocols [6]. We show that open-loop control protocols working in a finite amount of time can be devised to be robust against the effects of decoherence and dissipation. Our work is open to extensions to complex systems involving many interacting degrees of freedom, e.g., qubits in a programmable quantum processor or a quantum simulator [7]. [1] J. Preskill, Quantum 2, 79 (2018). [2] Koch, C.P., Boscain, U., Calarco, T. et al., EPJ Quantum Technol. 9, 19 (2022). [3] Koch, C.P., J. Phys.: Condens. Matter 28 213001 (2016). [4] Cangemi, L.M., Espinós, H., Puebla, R., Torrontegui, E. and Levy. A. arXiv:2311.13164v2 (2023). [5] Levy, A. et al, New J. Phys. 20 025006 (2018). [6] Dann, R., Levy, A., and Kosloff, R., Phys. Rev. A 98, 052129 (2018). [7] Espinós, H., Cangemi, L.M., Levy, A., Puebla, R. and Torrontegui, E., arXiv:2309.05469v2 (2023).

Thouless pumping is a transport phenomenon, wherein charge is transferred without dissipation by varying a many-body Hamiltonian periodically with time. In the thermodynamic limit, the charge pumped per driving period is topologically quantized, rendering the transport robust even in the presence of interactions and disorder. In its original formulation, Thouless pumping requires adiabatic (slow) driving, which maintains the system in its many-body ground state. I will describe how shortcuts to adiabaticity can be used to accelerate Thouless pumping, thereby removing the requirement of adiabaticity. Remarkably, the pumped charge remains topologically (hence robustly) quantized in this accelerated regime. I will show how this strategy applies to the Rice-Mele model, a simple and illustrative example of Thouless pumping.

Quantum many-body systems are emerging as key elements in the quest for quantum-based technologies and in the study of fundamental physics. Here I will address the challenge of achieving fast and high-fidelity evolutions across quantum phase transitions, a crucial requirement for practical applications. This is achieved by a control technique based on dynamical invariants tailored to ensure adiabatic-like evolution within the lowest-energy subspace of many-body systems [1], whose suitability and performance is illustrated in the paradigmatic transverse-field Ising and long-range Kitaev models [1]. By resorting to the derived analytical control, the dynamics of the many-body system results in high-fidelity adiabatic evolutions operating close to the speed limit. Remarkably, this approach leads to the breakdown of Kibble-Zurek scaling laws, offering tunable and significantly improved time scaling behavior. Detailed numerical simulations demonstrate the scalability with the system size and robustness against noisy controls and disorder, as well as the prospect of applying this technique to a non-integrable system. [1] Invariant-based control of quantum many-body systems across critical points H. Espinós, L. M. Cangemi, A. Levy, R. Puebla, E. Torrontegui, arXiv:2309.05469

The control and stabilization of many-body quantum systems whose classical counterparts exhibit highly chaotic motion is a challenging problem. The presence of many-body quantum chaotic dynamics is often conceptualized as the ultimate enemy of quantum device control as it leads rapidly to thermalization, and is certainly a fundamental hindrance to controlling quantum computation. However, what if chaos could be harnessed instead as a resource for quantum control just as has been shown for classical systems? One of the principal goals of controlling classically chaotic dynamical systems is known as targeting, which is the very weakly perturbative process of using the system's extreme sensitivity to initial conditions in order to arrive at a predetermined target state. It relies on a kind of "inverse butterfly effect": fast exponential convergence. In this talk we develop a many-body quantum control technique inspired by classical targeting. Starting from an initial quantum state in a quantum chaotic system: “how can one transfer the chaotic many-body system to a predetermined remote target state most efficiently?’’

We introduce a novel quantum algorithm solving Boolean satisfiability (k-SAT) problems using only k-local generalized measurement with adjustable strength. The algorithm operates by measuring each clause term of the k-SAT problem while gradually changing the measurement basis. We develop a filtering technique to detect clause failures when employing weak measurement. While the algorithm can be implemented autonomously with Lindbladian dissipation, benchmarking reveals that heralding on clause measurement outcomes with subsequent filtering significantly improve the algorithm's performance. We then clarify the role of measurement strength. While the weak continuous limit is seen to be advantageous in terms of the optimal time-to-solution (TTS) value, the scaling of the optimal TTS with qubit number is similar in both continuous (weak) and discrete (stronger) measurement settings. We show that the quantum system converges to the solution state deterministically in the long time limit and analyze the tradeoffs in the overall TTS as we approach that limit. Finally, we investigate the use of feedback to further speedup the algorithm.

We utilize a recent approach for constructing approximate shortcuts-to-adiabaticity (STA) algorithms [1] and apply it to solve combinatorial optimization problems including k-SAT and maximum independent set (MIS). While adiabatic algorithms necessitate slowly varying con- trols to maintain the system in its ground state, shortcut-to-adiabaticity (STA) methods engineer counter-diabatic forces that suppress diabatic transitions regardless of the control variation rate [2]. Despite naive expectations that STA protocols can solve hard problems at incredi- ble speeds, the computation resources required to engineer STA Hamiltonians and physical resources required to realize them can be huge. Luckily, approximations to STA Hamiltonians can have better performance metrics. Ref. [1] suggests expanding the STA Hamiltonian using odd orders of nested commutators [3], with expansion coefficients found by fitting the function 1/x with Chebyshev polynomials in x. We apply this approach for solving k-SAT and MIS. We analyze how the degeneracy of the solution state, the decay of high-frequency tails in the spectral function, and the expansion order affect the fidelity. [1] J. R. Finzgar, S. Notarnicola, M. Cain, M. D. L. Mikhail, and D. Sels, arXiv preprint arXiv:2503.01958 (2025). [2] D. Guéry-Odelin, A. Ruschhaupt, A. Kiely, E. Torrontegui, S. Martínez-Garaot, and J. G. Muga, Reviews of Modern Physics 91, 045001 (2019). [3] P. W. Claeys, M. Pandey, D. Sels, and A. Polkovnikov, Physical review letters 123, 090602 (2019).

Quantum optimal control (QOC) supports the advance of quantum technologies by tackling its problems at the pulse level: numerical approaches iteratively work toward a given target by parametrizing the applied time-dependent fields with a finite set of variables. The effectiveness of the resulting optimization depends on the complexity of the problem and the number of variables. We consider different parametrizations in terms of basis functions, asking whether the choice of the applied basis affects the quality of the optimization. Furthermore, we consider strategies to choose the most suitable basis. For comparison, we test three different randomizable bases—introducing the sinc and sigmoid bases as alternatives to the Fourier basis—on QOC problems of varying complexity. For each problem, the basis-specific convergence rates result in a unique ranking. Especially for expensive evaluations, e.g., in closed loop, a potential speedup by a factor of up to 10 may be crucial for the optimization's feasibility. We conclude that a problem-dependent basis choice is an influential factor for QOC efficiency and provide advice for its approach.

Quantum simulation uses a precisely controlled quantum system to model and understand the behavior of more complex quantum systems. For effective simulation, two key elements are crucial: the ability to prepare the initial state of interest and the precise shaping of the environment or Hamiltonian. This talk will demonstrate how quantum control tools directly address both of these requirements, significantly expanding the capabilities of quantum simulation. We'll present two of our recent experimental results, leveraging Bose-Einstein condensates in time-dependent optical lattices. First, we'll explore weak and strong localization in disordered media. We'll show how quantum state control helps identify new signatures of strong localization and discuss the influence of symmetries on these signatures. Second, we'll shift our focus to open quantum systems. Next, we'll illustrate how strategically shaping a Hamiltonian's parameters allows us to use a shaken isolated system to accurately emulate the behavior of an open system.

Quantum imaginary time evolution (ITE) is a powerful method for investigating the spectral properties of a many-body Hamiltonian. In this talk we present quantum algorithms that implement ITE by adaptively growing a quantum circuit based on feedback from previous steps with performance guarantees. This is achieved by first identifying ITE as a Riemannian gradient flow that can be created through a sequence of unitary transformations arbitrarily well. We derive an upper bound for the error between the two evolutions and present a stochastic Riemannian gradient descent algorithm in which each step is efficiently implementable on a quantum computer and that simulates ITE with performance guarantees on average. We prove that for a sufficiently small step size, the stochastic evolution concentrates around the imaginary time evolution with high probability, cooling the system by moving into random tangent space directions.

We will derive Floquet theory from the adiabatic theorem. We demonstrate how concepts from counterdiabatic driving allow us to acquire a new understanding of periodically driven systems.

Perfectly entangling gates (PE) are crucial for various applications in quantum information. One method to realize these gates is with the help of an external control field, whose concrete shape is found using optimal control theory. In- stead of optimizing the shape that realizes a specific gate, the optimization target can be extended to the full set of PE. This increases the flexibility of optimization and allows to find the best PE from the set of all PE. First, we show that it is possible to construct the unitary part of an unknown coherent evolution by propagating specifically talilored density matrices. We then extend this construction method to approximate the unitary part of a non-unitary evolution. Lastly, we employ this method to superconducting qubits, where we numerically find opti- mized control fields that generate maximally entangled states for a desired gate duration, even if dissipation is present in the system

We propose a quantum optimal control framework based on the Pontryagin Maximum Principle to design energy- and time-efficient pulses for open quantum systems. By formulating the Langevin equation of a dissipa- tive LC circuit as a linear control problem, we derive optimized pulses with exponential scaling in energy cost, outperforming conventional shortcut-to-adiabaticity methods such as counter-diabatic driving. When applied to a resonator dispersively coupled to a qubit, these optimized pulses achieve an excellent signal-to-noise ratio comparable to longitudinal coupling schemes across varying critical photon numbers. Our results provide a sig- nificant step toward efficient control in dissipative open systems and improved qubit readout in circuit quantum electrodynamics.

Quantum technology is a fast-emerging field with both scientific and technological importance. The performance relies on unique features like superposition and entanglement and depends on sophisticated mechanisms of control to perform the desired tasks. Quantum Optimal Control (QOC) has proven to be a powerful tool to accomplish this task. I will give a brief overview on the use of QOC for quantum technology application with NV centers in diamond [1], the CRAB algorithm for Optimal Control [2], and the optimal-control software QuOCS [3], focussing on recent demonstrations of closed-loop control [4,5,6]. [1] P. Rembold et al., AVS Quantum Sci. 2, 024701 (2020) [2] M. M. Müller et al., Rep. Prog. Phys. 85 076001 (2022) [3] M. Rossignolo et al. Comp. Phys. Comm. 291, 108782 (2023) [4] N. Oshnik et al., Phys. Rev. A 106, 013107 (2022) [5] A. Marshall et al., Phys. Rev. Res. 4, 043179 (2022) [6] P. Vetter et al., npj Quantum Information 10 (1), 96 (2024)

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Shortcuts to Adiabaticity (STA) are quantum control protocols motivated by adiabatic processes and mainly derived using analytical techniques. However, there are still quantum systems where these STA methods cannot be applied. Therefore, I will then present a new technique for such scenarios, called “Enhanced Shortcuts to Adiabaticity” (eSTA) [1], which provides an easy to calculate analytical correction to existing STA protocols. After giving some introductory overview about STA, I start with a presentation of the formalism of eSTA. I present then some applications of eSTA [2, 3], for example, also spin squeezing in internal bosonic Josephson junctions [4, 7]. Finally, I will also give an outlook towards Shortcut-Enhanced Quantum Thermodynamics. [1] C. Whitty, A. Kiely and A. Ruschhaupt, Phys. Rev. Research 2, 023360 (2020) [2] C. Whitty, A. Kiely and A. Ruschhaupt, Phys. Rev.A 105, 013311 (2022) [3] C. Whitty, A. Kiely and A. Ruschhaupt, J. Phys. B: At. Mol. Opt. Phys. 55, 194003 (2022) [4] M. Odelli, V. M. Stojanović and A. Ruschhaupt, Phys. Rev. Applied 20, 054038 (2023) [5] J. Li, E. Ya Sherman and A. Ruschhaupt, Phys. Rev. A 106, L030201 (2022) [6] J. Li, E. Ya Sherman and A. Ruschhaupt, New J. Phys. 26, 053031 (2024) [7] M. Odelli, A. Ruschhaupt and V. M. Stojanović, Phys. Rev. A 110, 022610 (2024)

While the fundamental limits of single-parameter estimation are well established, practical applications—such as quantum sensing and quantum imaging—often demand the simultaneous estimation of multiple parameters, where precision limits remain poorly understood. This challenge stems from the inherent incompatibility of optimal estimation strategies for different parameters. In this talk, I will examine these incompatibilities in both measurement and control, presenting fundamental trade-off relations that constrain the achievable precision for estimating multiple parameters.

In quantum metrology, the aim is to estimate parameters with a precision that outperforms classical strategies, typically quantified in terms of the (quantum) Fisher information. In this work, we consider the problem of designing a feedback control law to enhance the Fisher information along individual stochastic trajectories of a continuously monitored quantum system. As a prototypical example, we consider parameter estimation in a single bosonic mode controlled via a squeezing Hamiltonian while subject to continuous homodyne detection. Since such a system admits a linear Gaussian representation, it allows for tractable numerical algorithms. To solve the problem, we propose a receding horizon approach, where a small optimization problem is solved at each control step based on the latest estimate of the state and parameter. As a benchmark, we formulate the stochastic optimal control problem for maximizing the Fisher information and subsequently solve it via a numerical solution to the corresponding Hamilton-Jacobi-Bellman (HJB) equation. Numerical simulations show that the receding horizon control significantly improves the Fisher information along individual trajectories compared to open-loop strategies, although the performance is still worse than the benchmark HJB solution.

In this talk, I will demonstrate how control theory is valuable in quantum computing. I will begin by discussing gradient flows and the algorithms derived from them. As a practical example, I will use the problem of finding the ground state. While several variational methods exist to address this problem, our approach specifically focuses on feedback control theory. Among the most studied algorithms in this context is the so-called I-FALQON algorithm, which utilizes Lyapunov theory in combination with the invariance principle. Additionally, alternative approaches, such as gradient flow methods, will be explored during the talk. These methods involve considering gradients on complex projective spaces or unitary Lie groups, each resulting in distinct flow lines and therefore different algorithmic implementations. An optimal algorithm is characterized by its minimal use of quantum gates, particularly controlled gates. Each algorithm thus represents a distinct path or homotopy on a manifold from an initial to a terminal state. The central issue we address is determining the path requiring the fewest gates. This challenge directly relates to the well-known controllability problem within control theory.

Universal quantum computing requires the ability to perform every unitary operation, i.e., evolution operator controllability. In view of developing resource-efficient quantum processing units (QPUs), it is important to determine how many local controls and qubit-qubit couplings are required for controllability. Unfortunately, assessing the controllability of large qubit arrays is a difficult task, due to the exponential scaling of Hilbert space dimension. This work shows that it is sufficient to connect two qubit arrays that are operator controllable by a single entangling two-qubit gate in order to obtain a composite qubit array that is operator controllable. The proof provides a template to build up modular QPUs from smaller building blocks with reduced numbers of local controls and couplings.

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