Department of Mathematical Methods for Quantum Technologies
Steklov Mathematical Institute, Moscow, Russia
- Tracked papers
- 52 (2004–2026)
Topics
The most frequent field-map topics among the PI' tracked papers.
- Open systems 20
- Landscapes 15
- Controllability 10
- GRAPE 9
- Gates 8
- Problem formulation 7
- State preparation 6
- Feedback 5
Recent papers
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A complete classification of operator controllability for closed three-level quantum systems driven by a single control
Abstract
In this work, we study operator controllability of finite-dimensional closed quantum systems driven by a single coherent control. Two general Lie-algebraic results on controllability of such systems are proved. The first gives sufficient conditions under which controllability of the system is preserved when controlled energy-level shifts are added. The second gives a sufficient basis-element condition under which the dynamical Lie algebra contains $\mathfrak{su}(N)$ and the system is therefore operator controllable. These results, together with known controllability conditions, are applied to obtain a complete classification of operator controllability of all three-level quantum systems in terms of their free and interaction Hamiltonian matrix elements. This classification covers all combinations of energy-level and level-spacing degeneracies, two or three allowed transitions, as well as the presence or absence of controlled energy-level shifts, and gives necessary and sufficient conditions for operator controllability in each case.
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Fundamental limits on state preparation for an open qubit
Abstract
We analytically determine the ultimate limits of state preparation in two-level open quantum systems driven by coherent control. For a dissipative qubit governed by a GKSL master equation, we give an exact characterization of the reachable set in the Bloch ball. Dissipation excludes a region of states in the Bloch ball which cannot be approached even under arbitrarily strong coherent driving, and we prove that this region has a nontrivial geometry whose boundary is a surface of revolution around the $x$-axis which is analytic except for two conical singularities. We derive a closed-form control protocol for moving on this boundary, and construct an explicit protocol that steers the system arbitrarily close to any prescribed boundary state. These results provide a complete geometric constructive description of reachable qubit states in the standard dissipative environment, establishing fundamental bounds on controllability and state-preparation fidelity for open two-level quantum systems.
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Diverse efficiency of observable optimization for four-level quantum systems with higher-order traps
Abstract
In this work, we perform an analytical and numerical analysis of quantum landscapes for controlling special four-level quantum systems for which we prove that the null control is a five-order trap: a $V-V$ system and an anharmonic system. As a control goal, an observable optimization is considered. The rigorous theoretical analysis is followed by the numerical experiments based on the GRadient Ascent Pulse Engineering (GRAPE) algorithm and Gradient Projection Method (GPM), performed to investigate the behavior of the efficiency of optimization for unconstrained (using GRAPE) and constrained (using GPM) controls. As the main result, we observe an interesting phenomenon with a diverse behavior of the optimization efficiency depending on the system Hamiltonian -- sharp increase of the optimization efficiency up to 100% at certain distance from the null control for a V-V system, while much slower and less significant increase (and even small decrease) for a system with the chain interaction. This sharp difference might be related with the fine structure of the subspace of controls where second derivative of the objective functional is zero.
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Gradient projection method for constrained quantum control
Abstract
In this work, we adopt the Gradient Projection Method (GPM) to problems of quantum control. For general $N$-level closed and open quantum systems, we derive the corresponding adjoint systems and gradients of the objective functionals, and provide the projection versions of the Pontryagin maximum principle and the GPM, all directly in terms of quantum objects such as evolution operator, Hamiltonians, density matrices, etc. Various forms of the GPM, including one- and two-step, are provided and compared. We formulate the GPM both for closed and open quantum systems, latter for the general case with simultaneous coherent and incoherent controls. The GPM is designed to perform local gradient based optimization in the case when bounds are imposed on the controls. The main advantage of the method is that it allows to exactly satisfy the bounds, in difference to other approaches such as adding constraints as weight to objective. We apply the GPM to several examples including generation of one- and two-qubit gates and two-qubit Bell and Werner states for models of superconducting qubits under the constraint when controls are zero at the initial and final times, and steering an open quantum system state to a target density matrix for simulating action of the Werner-Holevo channel, etc.
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Quantum channels, complex Stiefel manifolds, and optimization
Abstract
Most general dynamics of an open quantum system is commonly represented by a quantum channel, which is a completely positive trace-preserving map (CPTP or Kraus map). Well-known representations of quantum channels are described by Choi matrices and by Kraus operator-sum representation (OSR). As was shown before, one can use Kraus OSR to parameterize quantum channels by points of a suitable quotient of some complex Stiefel manifold by the action of the unitary group. In this work, we establish a homeomorphism between the topological space of quantum channels and the quotient of the complex Stiefel manifold. This homeomorphism can be applied to various quantum optimization problems. As an example, we apply it to the analysis of extrema points for a wide variety of quantum control objective functionals defined on the complex Stiefel manifolds, including mean value, generation of quantum gates, thermodynamic quantities involving entropy, etc. Finally, a metric on the space of quantum channels induced by the Riemannian metric on the Stiefel manifold is defined, and we show that it is a generalization of the Bures angle between density matrices.
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Control landscapes for high-fidelity generation of C-NOT and C-PHASE gates with coherent and environmental driving
Abstract
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.
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Control of the von Neumann Entropy for an Open Two-Qubit System Using Coherent and Incoherent Drives
Abstract
This article is devoted to developing an approach for manipulating the von Neumann entropy $S(ρ(t))$ of an open two-qubit system with coherent control and incoherent control inducing time-dependent decoherence rates. The following goals are considered: (a) minimizing or maximizing the final entropy $S(ρ(T))$; (b) steering $S(ρ(T))$ to a given target value; (c) steering $S(ρ(T))$ to a target value and satisfying the pointwise state constraint $S(ρ(t)) \leq \overline{S}$ for a given $\overline{S}$; (d) keeping $S(ρ(t))$ constant at a given time interval. Under the Markovian dynamics determined by a Gorini--Kossakowski--Sudarshan--Lindblad type master equation, which contains coherent and incoherent controls, one- and two-step gradient projection methods and genetic algorithm have been adapted, taking into account the specifics of the objective functionals. The corresponding numerical results are provided and discussed.
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Phenomenon of a stronger trapping behaviour in $Λ$-type quantum systems with symmetry
Abstract
$Λ$, $V$, $Ξ$ (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.
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Incoherent GRAPE (inGRAPE) for optimization of quantum systems with environmentally assisted control
Abstract
In this work, we review several results on development and application of incoherent version of GRAPE (Gradient Ascent Pulse Engineering) approach, inGRAPE, to optimization for open quantum systems driven by both coherent and incoherent controls. In the incoherent control approach, the environment serves as a control together with coherent field, and decoherence rates become generally time-dependent. For a qubit, explicit analytic expressions for evolution of the density matrix were obtained by solving a cubic equation via Cardano method. We discuss applications of incoherent GRAPE method to high fidelity gate generation for open one- and two-qubit systems and surprising properties of the underlying control landscapes, forming two groups - smooth single peak landscapes for Hadamard, C-NOT and C-Z gates, and more complicated with two peaks for T (or $π/8$) gate. For a qutrit, a formulation of the environment-assisted incoherent control with time-dependent decoherence rates is provided.
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Energy control in a quantum oscillator using coherent control and engineered environment
Abstract
We develop and analyze a new method for manipulation of energy in a quantum harmonic oscillator using coherent, e.g., electromagnetic, field and incoherent control. Coherent control is typically implemented by shaped laser pulse or tailored electromagnetic field. Incoherent control is implemented by engineered environment, whose mean number of excitations at the frequency of the oscillator is used as a control variable. An approach to coherent and incoherent controls design based on the speed gradient algorithms in general, finite and differential forms is proposed. It is proved that the differential form is able to completely manipulate the energy of the oscillator: an arbitrary energy can be achieved starting from any initial state of the oscillator. The key instrument which allows for complete energy manipulation in this case is the use of the engineered environment. A robustified speed-gradient control algorithm in differential form is also proposed. It is shown that the proposed robustified control algorithm ensures exponential stability of the closed loop system which is preserved for sampled-data control.