QOC — Control landscapes

The Quantum Sphere: A Physically Realizable Optimization Benchmark with Provable Linear Convergence in White- and Black-Box Settings

2026-09-30T00:00:00Z

We revisit an established Quantum Control objective, characterize its exact local metric geometry, and reposition it as a rigorous benchmark for Search and Optimization. The trap-free topology of the landscape, established two decades ago, guarantees unhindered optimal pathways, but global topology alone does not govern convergence speed. Although the so-called Quantum Sphere is defined by the fourth power of the control field, we show that its yield gap admits an exact representation as a nonnegative linear combination of $\sin^2$ terms. From this single identity, we derive a two-sided quadratic sandwich with ratio $4/π^2$ on a certified, worst-case quarter-wave region, proving the optimal peak nondegenerate modulo its affine gauge and well-conditioned. For white-box optimization, on a halved (eighth-wave) region, the objective satisfies a Polyak-Łojasiewicz inequality with an explicit constant, granting Gradient Descent a linear rate of $1 - 16/(π^4κ_r)$, where $κ_r$ is the condition number of the relevant Hessian block. For black-box optimization, we prove that the elitist single-parent Evolution Strategy converges linearly, almost surely and in expected hitting time, from a certified sublevel set, despite the landscape's continuous gauge symmetry and its lattice of periodic optima; the certified level is explicit whenever the landscape is trap-free, and near the optimum the strategy provably sees only the quadratic tangent of the peak. Systematic numerical simulations corroborate the theory. Finally, we show that additive input (actuator) noise reweights each term of the representation by a Debye-Waller factor without disrupting the landscape's structure, and outline open challenges: output (signal) noise, far-field plateau escape, angular-synchronization certificates, and another physically scalable benchmark (SHG with Glass).

Engineered Dissipation for Thermal-State Tracking Across Quantum Criticality

2026-09-30T00:00:00Z

Engineering dissipation has emerged as a powerful strategy for controlling quantum many-body dynamics, complementing purely coherent control for state preparation. However, finite-time driving across a quantum critical point generates nonadiabatic excitations and coherences, and existing dissipation-engineering approaches have largely characterized how a fixed environment modifies critical dynamics rather than actively steering a system toward a prescribed thermal trajectory. Here we develop a Markovian dissipative framework for finite-time thermal-state tracking based on engineered population currents with non-negative transition rates, and apply it to the transverse-field Ising model driven across its quantum critical point. Compared with unitary dynamics and a natural thermal bath, the engineered dynamics strongly suppresses thermal excitation deviations and provides additional control over coherence damping through a free common current $Q_k$, while exhibiting a substantially steeper residual scaling, $δn_{\mathrm{ex}}^{\mathrm{th}}\sim(v/v_{\mathrm{ch}})^2$, than the Kibble--Zurek behavior recovered for unitary evolution. These results point toward engineered dissipation as a route for controlling thermal-state preparation in driven quantum critical systems.

Qudit-ADAPT-VQE: an adaptive variational algorithm with counterdiabatic-inspired improvements for qudits

2026-08-15T00:00:00Z

Variational quantum algorithms based on qudits have attracted significant attention in recent years. However, as in their qubit-based counterparts, challenges such as barren plateaus and the design of efficient ansatz remain major obstacles. In this work, we propose to address these issues through a qudit implementation of the ADAPT-VQE algorithm, which constructs the ansatz iteratively. Specifically, we introduce an operator pool inspired by adiabatic evolution enhanced with counterdiabatic driving for ansatz construction and employ it to solve Max 3-Cut. We show that the warm-start strategy inherent to ADAPT-VQE, together with an ansatz construction based on counterdiabatic operators, achieves higher accuracy and lower native gates implementation than approaches on fixed ansatz. Furthermore, we show that, in qudit-based quantum computing, ADAPT-VQE with a counterdiabatic operator pool can navigate rough optimization landscapes with local traps through the burrowing mechanism, suggesting robustness against barren plateau effects and providing a scalable framework for variational quantum algorithms with qudits.

Pulse engineering via projection of response functions at infinite nonlinear order

2026-07-27T00:00:00Z

Optimal control problems arise in a wide range of scientific disciplines, but the corresponding optimization algorithms often display a strong dependence on hyperparameters that significantly influence performance and convergence. For the optimal implementation of quantum algorithms, these challenges are further amplified by high-dimensional control landscapes and the need for high-fidelity operations. Here, we propose an algorithm for optimal control problems in quantum computing to efficiently generate high-fidelity control protocols for multi-qubit systems in a hyperparameter and gradient free manner. The method, referred to as Pulse Engineering via Projection of response functions at infinite nonlinear order (PEPRino), leverages the framework of response theory to navigate the control landscape to find high-fidelity implementations. This is achieved by determining the control landscape via response functions to infinite order, efficiently evaluated by resummation in terms of the first and second order response function. To demonstrate the approach, we apply it to quantum systems consisting of two and three qubits for the optimal implementation of the Quantum Fourier Transform (QFT). We benchmark the proposed algorithm against the Chopped Random Basis (CRAB) algorithm utilizing the Nelder-Mead method, focusing on the 2-qubit scenario. The results indicate faster convergence regarding iteration steps and computational time, highlighting the advantages of our approach.

Pulse Quality Optimisation in Quantum Optimal Control

2026-04-28T00:00:00Z

Quantum optimal control methods are widely used to design experimental control pulses such as laser amplitudes, phases, or detunings, that implement a target unitary evolution. In practice, what makes a pulse "good" depends not only on its fidelity, but also on the experimental setting and the relevant hardware constraints. Here, we introduce geometric quantum control with kernel optimisation (GECKO), a model-agnostic method for improving control pulses after a high-fidelity solution has been found. GECKO uses the Riemannian geometry of the special unitary group to identify directions in pulse space that leave the implemented unitary unchanged to first order, allowing one to traverse level sets of the control landscape while optimising a chosen differentiable pulse-quality function. We demonstrate GECKO on a transverse-field Ising Hamiltonian implementing CZ and CNOT gates, optimising pulse properties including spectral filtering, smoothness, robustness to parameter deviations, and pulse duration. In all cases, GECKO finds substantially improved pulse solutions.

Adaptive Tensor Network Sampling for Quantum Optimal Control

2026-04-27T00:00:00Z

Quantum optimal control (QOC) provides a systematic framework for achieving high-fidelity operations in quantum systems and plays a central role in tasks such as gate synthesis, state transfer, and pulse design. Existing QOC methods broadly fall into two categories: gradient-based and gradient-free algorithms. The associated optimization landscape is often high-dimensional, non-convex, and populated by numerous local minima, making efficient gradient-free search strategies essential. To address this, we introduce a gradient-free matrix product state/tensor train (MPS/TT) sampling heuristic for discrete quantum optimal control. In our approach, the MPS defines a score function over the space of discrete control parameters, which in turn induces a sampling distribution over candidate control sequences. This distribution is iteratively refined through selection of better performing sequences and local tensor updates to bias the search toward high-performing sequences. We evaluate the method on a range of benchmark problems, including single-qubit state transfer, Bell-pair preparation, qutrit gate implementation, and open-system population transfer. Across these tasks, the method exhibits stable convergence behavior and competitive empirical performance relative to established gradient-free baselines. These results suggest that tensor network sampling offers a viable heuristic framework for discrete quantum control.

Classical feature map surrogates and metrics for quantum control landscapes

2025-09-30T00:00:00Z

We derive and analyze three feature map representations of parametrized quantum dynamics, which generalize variational quantum circuits. These are (i) a Lie-Fourier partial sum, (ii) a Taylor expansion, and (iii) a finite-dimensional sinc kernel regression representation. The Lie-Fourier representation is shown to have a dense spectrum with discrete peaks, that reflects control Hamiltonian properties, but that is also compressible in typically found symmetric systems. We prove boundedness in the spectrum and the cost function derivatives, and discrete symmetries of the coefficients, with implications for learning and simulation. We further show the landscape is Lipschitz continuous, linking global variation bounds to local Taylor approximation error - key for step size selection, convergence estimates, and stopping criteria in optimization. This also provides a new form of polynomial barren plateaux originating from the Lie-Fourier structure of the quantum dynamics. These results may find application in local and general surrogate model learning, which we benchmark numerically, in characterizations of hardness in the problem instances, and for meta-parameter heuristics in quantum optimizers.

Exploring Quantum Control Landscape and Solution Space Complexity through Dimensionality Reduction & Optimization Algorithms

2025-02-17T00:00:00Z

Understanding the quantum control landscape (QCL) is important for designing effective quantum control strategies. In this study, we analyze the QCL for a single two-level quantum system (qubit) using various control strategies. We employ Principal Component Analysis (PCA), to visualize and analyze the QCL for higher dimensional control parameters. Our results indicate that dimensionality reduction techniques such as PCA, can play an important role in understanding the complex nature of quantum control in higher dimensions. Evaluations of traditional control techniques and machine learning algorithms reveal that Genetic Algorithms (GA) outperform Stochastic Gradient Descent (SGD), while Q-learning (QL) shows great promise compared to Deep Q-Networks (DQN) and Proximal Policy Optimization (PPO). Additionally, our experiments highlight the importance of reward function design in DQN and PPO demonstrating that using immediate reward results in improved performance rather than delayed rewards for systems with short time steps. A study of solution space complexity was conducted by using Cluster Density Index (CDI) as a key metric for analyzing the density of optimal solutions in the landscape. The CDI reflects cluster quality and helps determine whether a given algorithm generates regions of high fidelity or not. Our results provide insights into effective quantum control strategies, emphasizing the significance of parameter selection and algorithm optimization.

Topological Phase Transitions in a Constrained Two-Qubit Quantum Control Landscape

2024-11-13T00:00:00Z

In optimal quantum control, control landscape phase transitions (CLPTs) indicate sharp changes occurring in the set of optimal protocols, as a physical model parameter is varied. Here, we demonstrate the existence of a new class of CLPTs, associated with changes in the topological properties of the optimal level set in a two-qubit state-preparation problem. In particular, the distance distribution of control protocols sampled through stochastic homotopic dynamics reveals discontinuous changes in the number of connected components in the optimal level set, as a function of the protocol duration. We demonstrate how topological CLPTs can be detected in modern-day experiments.

The Top Manifold Connectedness of Quantum Control Landscapes

2024-09-23T00:00:00Z

The control of quantum systems has been proven to possess trap-free optimization landscapes under the satisfaction of proper assumptions. However, many details of the landscape geometry and their influence on search efficiency still need to be fully understood. This paper numerically explores the path-connectedness of globally optimal control solutions forming the top manifold of the landscape. We randomly sample a plurality of optimal controls in the top manifold to assess the existence of a continuous path at the top of the landscape that connects two arbitrary optimal solutions. It is shown that for different quantum control objectives including state-to-state transition probabilities, observable expectation values and unitary transformations, such a continuous path can be readily found, implying that these top manifolds are fundamentally path-connected. The significance of the latter conjecture lies in seeking locations in the top manifold where an ancillary objective can also be optimized while maintaining the full optimality of the original objective that defined the landscape.

Machine learning for predicting control landscape maps of quantum molecular dynamics: Laser-induced three-dimensional alignment of asymmetric top molecules

2024-08-30T00:00:00Z

Machine learning for predicting control landscape maps of full quantum molecular dynamics is examined through a case study of the laser-induced three-dimensional (3D) alignment of asymmetric top molecules, an essential technique for observing and/or manipulating molecular dynamics in a molecule-fixed frame. We consider the "prolate-type" asymmetryic top molecules with the asymmetry parameters $-1 < κ< 0$ and the C2v symmetry in the low-temperature limiting case, which are aligned by using mutually orthogonal linearly polarized double laser pulses. The landscape map for each molecule consists of 6000 pixels, each pixel of which represents the maximum degree of alignment achieved by each set of control parameters. After examining ways to deal with the markedly different molecular parameters in a unified manner for suitably training a convolutional neural network (CNN) model, we train the CNN model by using 55 training sample molecules to predict the control landscape maps of 35 test sample molecules with reasonably high accuracy. As the predicted landscape maps provide a big picture of the alignment control, we show, for example, that the double pulse control scheme is especially effective for a molecule having a polarizability component that is much larger in value than the other two components.

Circumventing Traps in Analog Quantum Machine Learning Algorithms Through Co-Design

2024-08-26T00:00:00Z

Quantum machine learning QML algorithms promise to deliver near-term, applicable quantum computation on noisy, intermediate-scale systems. While most of these algorithms leverage quantum circuits for generic applications, a recent set of proposals, called analog quantum machine learning (AQML) algorithms, breaks away from circuit-based abstractions and favors leveraging the natural dynamics of quantum systems for computation, promising to be noise-resilient and suited for specific applications such as quantum simulation. Recent AQML studies have called for determining best ansatz selection practices and whether AQML algorithms have trap-free landscapes based on theory from quantum optimal control (QOC). We address this call by systematically studying AQML landscapes on two models: those admitting black-boxed expressivity and those tailored to simulating a specific unitary evolution. Numerically, the first kind exhibits local traps in their landscapes, while the second kind is trap-free. However, both kinds violate QOC theory's key assumptions for guaranteeing trap-free landscapes. We propose a methodology to co-design AQML algorithms for unitary evolution simulation using the ansatz's Magnus expansion. We show favorable convergence in simulating dynamics with applications to metrology and quantum chemistry. We conclude that such co-design is necessary to ensure the applicability of AQML algorithms.

Toward a Theory of Phase Transitions in Quantum Control Landscapes

2024-08-20T00:00:00Z

Control landscape phase transitions (CLPTs) occur as abrupt changes in the cost function landscape upon varying a control parameter, and can be revealed by non-analytic points in statistical order parameters. A prime example are quantum speed limits (QSL) which mark the onset of controllability as the protocol duration is increased. Here we lay the foundations of an analytical theory for CLPTs by developing Dyson, Magnus, and cumulant expansions for the cost function that capture the behavior of CLPTs with a controlled precision. Using linear and quadratic stability analysis, we reveal that CLPTs can be associated with different types of instabilities of the optimal protocol. This allows us to explicitly relate CLPTs to critical structural rearrangements in the extrema of the control landscape: utilizing path integral methods from statistical field theory, we trace back the critical scaling of the order parameter at the QSL to the topological and geometric properties of the set of optimal protocols, such as the number of connected components and its dimensionality. We verify our predictions by introducing a numerical sampling algorithm designed to explore this optimal set via a homotopic stochastic update rule. We apply this new toolbox explicitly to analyze CLPTs in the single- and two-qubit control problems whose landscapes are analytically tractable, and compare the landscapes for bang-bang and continuous protocols. Our work provides the first steps towards a systematic theory of CLPTs and paves the way for utilizing statistical field theory techniques for generic complex control landscapes.

Control landscapes for high-fidelity generation of C-NOT and C-PHASE gates with coherent and environmental driving

2024-05-23T00:00:00Z

High fidelity generation of two-qubit gates is important for quantum computation, since such gates are components of popular universal sets of gates. Here we consider the problem of high fidelity generation of two-qubit C-NOT and C-PHASE (with a detailed study of C-Z) gates in presence of the environment. We consider the general situation when qubits are manipulated by coherent and incoherent controls; the latter is used to induce generally time-dependent decoherence rates. For estimating efficiency of optimization methods for high fidelity generation of these gates, we study quantum control landscapes which describe the behaviour of the fidelity as a function of the controls. For this, we generate and analyze the statistical distributions of best objective values obtained by incoherent GRadient Ascent Pulse Engineering (inGRAPE) approach. We also apply inGRAPE and stochastic zero-order method to numerically estimate minimal infidelity values. The results are different from the case of single-qubit gates and indicate a smooth trap-free behaviour of the fidelity.

On the Role of Controllability in Pulse-based Quantum Machine Learning Models

2024-05-15T00:00:00Z

Pulse-based quantum machine learning (QML) models possess full expressivity when they are ensemble controllable. However, it has also been shown that barren plateaus emerge in such models, rendering training intractable for systems with large dimension. In this paper, we show that the trade-off is closely related to the controllability of the underlying pulse-based models. We first apply the Fliess-series expansion to pulse-based QML models to investigate the effect of control system structure on model expressivity, which leads to a universal criterion for assessing the expressivity of generic QML models. Guided by this criterion, we then demonstrate how designing pulse-based models on low-dimensional manifolds can balance expressivity and trainability. Finally, numerical experiments are carried out to verify the proposed criterion and our analysis, which futher demonstrate that increasing dimensionality enhances expressivity but avoids barren plateaus if the model is designed with limited controllability on a submanifold. Our approach provides a promising path for designing pulse-based QML models that are both highly expressive and trainable.

Barren Plateaus in Variational Quantum Computing

2024-05-01T00:00:00Z

Variational quantum computing offers a flexible computational paradigm with applications in diverse areas. However, a key obstacle to realizing their potential is the Barren Plateau (BP) phenomenon. When a model exhibits a BP, its parameter optimization landscape becomes exponentially flat and featureless as the problem size increases. Importantly, all the moving pieces of an algorithm -- choices of ansatz, initial state, observable, loss function and hardware noise -- can lead to BPs when ill-suited. Due to the significant impact of BPs on trainability, researchers have dedicated considerable effort to develop theoretical and heuristic methods to understand and mitigate their effects. As a result, the study of BPs has become a thriving area of research, influencing and cross-fertilizing other fields such as quantum optimal control, tensor networks, and learning theory. This article provides a comprehensive review of the current understanding of the BP phenomenon.

Phenomenon of a stronger trapping behaviour in $Λ$-type quantum systems with symmetry

2024-04-10T00:00:00Z

$Λ$, $V$, $Ξ$ (ladder), and other three-level quantum systems with one forbidden transition (referred here as $Λ$-type systems) play an important role in quantum physics. Various applications require manipulation of such systems using as control shaped laser field. In this work, we study how degeneracy of energy states or of Bohr frequencies in these systems affects the efficiency or difficulty of finding optimal shape of the control field. For this, we adopt the notion of higher order traps, which was introduced in [A.N. Pechen and D.J. Tannor, Are there traps in quantum control landscapes? Phys. Rev. Lett. {\bf 106}, 120402 (2011)], where second/third order traps were discovered for $Λ$-atom with one forbidden transition and with non-degenerate energy levels. We theoretically study control of such systems with and without degeneracy in their eigenstates and Bohr frequencies, and investigate numerically using GRAPE and l-BFGS algorithms how this degeneracy influences on the efficiency of optimizing the control laser field. We find that the degeneracy of the Bohr frequencies in the $Ξ$ system, which makes the system energy levels symmetrically distributed, leads to the appearance of a seventh order trap with a more significant attracting domain resulting in a more difficult optimization, while the degeneracy of energy states in generic $Λ$-type systems does not lead to an increase of the order of the zero control trap compared to the non-degenerate case. We also find that when not only the Bohr frequencies are degenerate in the system $Ξ$, but also the dipole moments for the two allowed transitions coincide (in this case $Ξ$ system is not controllable), then true traps arise in the quantum control landscape. In particular, the constant zero control becomes a trap.

Shaping the Laser Control Landscape of a Hydrogen Transfer Reaction by Vibrational Strong Coupling. A Direct Optimal Control Approach

2024-01-02T00:00:00Z

Controlling molecular reactivity by shaped laser pulses is a long-standing goal in chemistry. Here we suggest a direct optimal control approach which combines external pulse optimization with other control parameters arising in the upcoming field of vibro-polaritonic chemistry, for enhanced controllability The direct optimal control approach is characterized by a simultaneous simulation and optimization paradigm, meaning that the equations of motion are discretized and converted into a set of holonomic constraints for a nonlinear optimization problem given by the control functional. Compared with indirect optimal control this procedure offers great flexibility such as final time or Hamiltonian parameter optimization. Simultaneous direct optimal control (SimDOC) theory will be applied to a model system describing H-atom transfer in a lossy Fabry-Pérot cavity under vibrational strong coupling conditions. Specifically, optimization of the cavity coupling strength and thus of the control landscape will be demonstrated.

Quantum control landscape for generation of $H$ and $T$ gates in an open qubit with both coherent and environmental drive

2023-09-05T00:00:00Z

An important problem in quantum computation is generation of single-qubit quantum gates such as Hadamard ($H$) and $π/8$ ($T$) gates which are components of a universal set of gates. Qubits in experimental realizations of quantum computing devices are interacting with their environment. While the environment is often considered as an obstacle leading to decrease of the gate fidelity, in some cases it can be used as a resource. Here we consider the problem of optimal generation of $H$ and $T$ gates using coherent control and the environment as a resource acting on the qubit via incoherent control. For this problem, we study quantum control landscape which represents the behaviour of the infidelity as a functional of the controls. We consider three landscapes, with infidelities defined by steering between two, three (via Goerz-Reich-Koch approach), and four matrices in the qubit Hilbert space. We observe that for the $H$ gate, which is Clifford gate, for all three infidelities the distributions of minimal values obtained with gradient search have a simple form with just one peak. However, for $T$ gate which is a non-Clifford gate, the situation is surprisingly different - this distribution for the infidelity defined by two matrices also has one peak, whereas distributions for the infidelities defined by three and four matrices have two peaks, that might indicate possible existence of two isolated minima in the control landscape. Important is that among these three infidelities only those defined with three and four matrices guarantee closeness of generated gate to a target and can be used as a good measure of closeness. We study sets of optimized solutions for this most general and not treated before case of coherent and incoherent controls acting together, and discover that they form submanifolds in the control space, and unexpected, in some cases two isolated submanifolds.

Orthogonality catastrophe and quantum speed limit for dynamical quantum phase transition

2023-08-09T00:00:00Z

We investigate the orthogonality catastrophe and quantum speed limit in the Creutz model for dynamical quantum phase transitions. We demonstrate that exact zeros of the Loschmidt echo can exist in finite-size systems for specific discrete values. We highlight the role of the zero-energy mode when analyzing quench dynamics near the critical point. We also examine the behavior of the time for the first exact zeros of the Loschmidt echo and the corresponding quantum speed limit time as the system size increases. While the bound is not tight, it can be attributed to the scaling properties of the band gap and energy variance with respect to system size. As such, we establish a relation between the orthogonality catastrophe and quantum speed limit by referencing the full form of the Loschmidt echo. Significantly, we find the possibility of using the quantum speed limit to detect the critical point of a static quantum phase transition, along with a decrease in the amplitude of noise induced quantum speed limit.

Control landscape of measurement-assisted transition probability for a three-level quantum system with dynamical symmetry

2023-07-14T00:00:00Z

Quantum systems with dynamical symmetries have conserved quantities which are preserved under coherent controls. Therefore such systems can not be completely controlled by means of only coherent control. In particular, for such systems maximal transition probability between some pair of states over all coherent controls can be less than one. However, incoherent control can break this dynamical symmetry and increase the maximal attainable transition probability. Simplest example of such situation occurs in a three-level quantum system with dynamical symmetry, for which maximal probability of transition between the ground and the intermediate state by only coherent control is $1/2$, and by coherent control assisted by incoherent control implemented by non-selective measurement of the ground state is about $0.687$, as was previously analytically computed. In this work we study and completely characterize all critical points of the kinematic quantum control landscape for this measurement-assisted transition probability, which is considered as a function of the kinematic control parameters (Euler angles). This used in this work measurement-driven control is different both from quantum feedback and Zeno-type control. We show that all critical points are global maxima, global minima, saddle points and second order traps. For comparison, we study the transition probability between the ground and highest excited state, as well as the case when both these transition probabilities are assisted by incoherent control implemented by measurement of the intermediate state.

Spin-echo and quantum versus classical critical fluctuations in TmVO$_4$

2023-06-22T00:00:00Z

Using spin-echo Nuclear Magnetic Resonance in the model Transverse-Field Ising system TmVO$_4$, we show that low frequency quantum fluctuations at the quantum critical point have a very different effect on $^{51}$V nuclear-spins than classical low-frequency noise or fluctuations that arise at a finite temperature critical point. Spin-echos filter out the low frequency classical noise but not the quantum fluctuations. This allows us to directly visualize the quantum critical fan and demonstrate the persistence of quantum fluctuations at the critical coupling strength in TmVO$_4$ to high temperatures in an experiment that remains transparent to finite temperature classical phase transitions. These results show that while dynamical decoupling schemes can be quite effective in eliminating classical noise in a qubit, a quantum critical environment may lead to rapid entanglement and decoherence.

Laziness, Barren Plateau, and Noise in Machine Learning

2022-06-19T00:00:00Z

We define \emph{laziness} to describe a large suppression of variational parameter updates for neural networks, classical or quantum. In the quantum case, the suppression is exponential in the number of qubits for randomized variational quantum circuits. We discuss the difference between laziness and \emph{barren plateau} in quantum machine learning created by quantum physicists in \cite{mcclean2018barren} for the flatness of the loss function landscape during gradient descent. We address a novel theoretical understanding of those two phenomena in light of the theory of neural tangent kernels. For noiseless quantum circuits, without the measurement noise, the loss function landscape is complicated in the overparametrized regime with a large number of trainable variational angles. Instead, around a random starting point in optimization, there are large numbers of local minima that are good enough and could minimize the mean square loss function, where we still have quantum laziness, but we do not have barren plateaus. However, the complicated landscape is not visible within a limited number of iterations, and low precision in quantum control and quantum sensing. Moreover, we look at the effect of noises during optimization by assuming intuitive noise models, and show that variational quantum algorithms are noise-resilient in the overparametrization regime. Our work precisely reformulates the quantum barren plateau statement towards a precision statement and justifies the statement in certain noise models, injects new hope toward near-term variational quantum algorithms, and provides theoretical connections toward classical machine learning. Our paper provides conceptual perspectives about quantum barren plateaus, together with discussions about the gradient descent dynamics in \cite{together}.

On the detailed structure of quantum control landscape for fast single qubit phase-shift gate generation

2022-04-28T00:00:00Z

In this work, we study the detailed structure of quantum control landscape for the problem of single-qubit phase shift gate generation on the fast time scale. In previous works, the absence of traps for this problem was proven on various time scales. A special critical point which was known to exist in quantum control landscapes was shown to be either a saddle or a global extremum, depending on the parameters of the control system. However, in the case of saddle the numbers of negative and positive eigenvalues of Hessian at this point and their magnitudes have not been studied. At the same time, these numbers and magnitudes determine the relative ease or difficulty for practical optimization in a vicinity of the critical point. In this work, we compute the numbers of negative and positive eigenvalues of Hessian at this saddle point and moreover, give estimates on magnitude of these eigenvalues. We also significantly simplify our previous proof of the theorem about this saddle point of the Hessian [Theorem~3 in B.O.~Volkov, O.V.~Morzhin, A.N.~Pechen, J.~Phys.~A: Math. Theor. {\bf 54}, 215303 (2021)].

Kibble-Zurek scaling due to environment temperature quench in the transverse field Ising model

2022-03-08T00:00:00Z

The Kibble-Zurek mechanism describes defect production due to non-adiabatic passage through a critical point. Here we study its variant from ramping the environment temperature to a critical point. We find that the defect density scales as $τ^{-dν}$ or $τ^{-d/z}$ for thermal or quantum critical points, respectively, in terms of the usual critical exponents and $1/τ$ the speed of the drive. Both scalings describe reduced defect density compared to conventional Kibble-Zurek mechanism, which stems from the enhanced relaxation due to bath-system interaction. Ramping to the quantum critical point is investigated by studying the Lindblad equation for the transverse field Ising chain in the presence of thermalizing bath, with couplings to environment obeying detailed balance, confirming the predicted scaling. The von-Neumann or the system-bath entanglement entropy follows the same scaling. Our results are generalized to a large class of dissipative systems with power-law energy dependent bath spectral densities as well.

The Optimization Landscape of Hybrid Quantum-Classical Algorithms: from Quantum Control to NISQ Applications

2022-01-19T00:00:00Z

This review investigates the landscapes of prevalent hybrid quantum-classical optimization algorithms in many rapidly developing quantum technologies, where the objective function is either computed by a natural quantum system or a quantum ansatz that is engineered, but the optimizer is classical. In any particular case, the nature of the underlying control landscape is fundamentally important for systematic optimization of the objective. In early studies on the optimal control of few-body dynamics, the optimizer could take full control of the quantum systems to be manipulated whose Hilbert space dimension is relatively small. Stepping into the noisy intermediate-scale quantum (NISQ) era, the experimentally growing computational power of the ansatz expressed as quantum hardware may bring quantum advantage over classical computers, but the classical optimizer is often limited by the available control resources. Across these different scales, we will show that the landscape's geometry experiences morphological changes from favorable trap-free landscapes to easily trapping rugged landscapes, and eventually to barren-plateau landscapes on which the optimizer can hardly move. This unified view provides the basis for understanding classes of systems that may be readily controlled out to those with special consideration, including the difficulties and potential advantages of NISQ technologies, as well as seeking possible ways to escape traps or plateaus, in particular circumstances.

Mitigated barren plateaus in the time-nonlocal optimization of analog quantum-algorithm protocols

2021-11-15T00:00:00Z

Quantum machine learning has emerged as a promising utilization of near-term quantum computation devices. However, algorithmic classes such as variational quantum algorithms have been shown to suffer from barren plateaus due to vanishing gradients in their parameters spaces. We present an approach to quantum algorithm optimization that is based on trainable Fourier coefficients of Hamiltonian system parameters. Our ansatz is exclusive to the extension of discrete quantum variational algorithms to analog quantum optimal control schemes and is non-local in time. We demonstrate the viability of our ansatz on the objectives of compiling the quantum Fourier transform and preparing ground states of random problem Hamiltonians. In comparison to the temporally local discretization ansätze in quantum optimal control and parameterized circuits, our ansatz exhibits faster and more consistent convergence. We uniformly sample objective gradients across the parameter space and find that in our ansatz the variance decays at a non-exponential rate with the number of qubits, while it decays at an exponential rate in the temporally local benchmark ansatz. This indicates the mitigation of barren plateaus in our ansatz. We propose our ansatz as a viable candidate for near-term quantum machine learning.

Diagnosing Barren Plateaus with Tools from Quantum Optimal Control

2021-05-29T00:00:00Z

Variational Quantum Algorithms (VQAs) have received considerable attention due to their potential for achieving near-term quantum advantage. However, more work is needed to understand their scalability. One known scaling result for VQAs is barren plateaus, where certain circumstances lead to exponentially vanishing gradients. It is common folklore that problem-inspired ansatzes avoid barren plateaus, but in fact, very little is known about their gradient scaling. In this work we employ tools from quantum optimal control to develop a framework that can diagnose the presence or absence of barren plateaus for problem-inspired ansatzes. Such ansatzes include the Quantum Alternating Operator Ansatz (QAOA), the Hamiltonian Variational Ansatz (HVA), and others. With our framework, we prove that avoiding barren plateaus for these ansatzes is not always guaranteed. Specifically, we show that the gradient scaling of the VQA depends on the degree of controllability of the system, and hence can be diagnosed through the dynamical Lie algebra $\mathfrak{g}$ obtained from the generators of the ansatz. We analyze the existence of barren plateaus in QAOA and HVA ansatzes, and we highlight the role of the input state, as different initial states can lead to the presence or absence of barren plateaus. Taken together, our results provide a framework for trainability-aware ansatz design strategies that do not come at the cost of extra quantum resources. Moreover, we prove no-go results for obtaining ground states with variational ansatzes for controllable system such as spin glasses. Our work establishes a link between the existence of barren plateaus and the scaling of the dimension of $\mathfrak{g}$.

Optimal training of variational quantum algorithms without barren plateaus

2021-04-29T00:00:00Z

Variational quantum algorithms (VQAs) promise efficient use of near-term quantum computers. However, training VQAs often requires an extensive amount of time and suffers from the barren plateau problem where the magnitude of the gradients vanishes with increasing number of qubits. Here, we show how to optimally train VQAs for learning quantum states. Parameterized quantum circuits can form Gaussian kernels, which we use to derive adaptive learning rates for gradient ascent. We introduce the generalized quantum natural gradient that features stability and optimized movement in parameter space. Both methods together outperform other optimization routines in training VQAs. Our methods also excel at numerically optimizing driving protocols for quantum control problems. The gradients of the VQA do not vanish when the fidelity between the initial state and the state to be learned is bounded from below. We identify a VQA for quantum simulation with such a constraint that thus can be trained free of barren plateaus. Finally, we propose the application of Gaussian kernels for quantum machine learning.

Quantum control landscape for ultrafast generation of single-qubit phase shift quantum gates

2021-04-26T00:00:00Z

In this work, we consider the problem of ultrafast controlled generation of single-qubit phase shift quantum gates. Globally optimal control is a control which realizes the gate with maximal possible fidelity. Trap is a control which is optimal only locally but not globally. It was shown before that traps do not exist for controlled generation of arbitrary single-qubit quantum gates for sufficiently long times, as well as for fast control of quantum gates other than phase shift gates. Ultrafast generation of phase-shift gates was missed in the previous analysis. In this work we show, combining analytical and numerical optimization methods such as Gradient Ascent Pulse Engineering (GRAPE), differential evolution, and dual annealing, that control landscape for ultrafast generation of phase shift gates is also free of traps. Mathematical analysis of quantum control landscapes, which aims to prove either absence or existence of traps for quantum control objective functionals, is an important topic in quantum control. In this work, we provide a rigorous analysis of quantum control landscapes for ultrafast generation of single-qubit quantum gates and show, combining analytical methods based on a sophisticated analysis of spectrum of the Hessian, and numerical optimization methods such as Gradient Ascent Pulse Engineering (GRAPE), differential evolution, and dual annealing, that control landscape for ultrafast generation of phase shift gates is free of traps.