arXiv subjects

Alexander Pechen

Publications and source records attributed to Alexander Pechen.

At least 37 records · Page 2Linked to original sources

arXiv2023

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

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.

quant-ph↗

arXiv2023

Krotov Type Optimization of Coherent and Incoherent Controls for Open Two-Qubit Systems

This work considers two-qubit open quantum systems driven by coherent and incoherent controls. Incoherent control induces time-dependent decoherence rates via time-dependent spectral density of the environment which is used as a resource for controlling the system. The system evolves according to the Gorini-Kossakowski-Sudarshan-Lindblad master equation with time-dependent coefficients. For two types of interaction with coherent control, three types of objectives are considered: 1) maximizing the Hilbert-Schmidt overlap between the final and target density matrices; 2) minimizing the Hilbert-Schmidt distance between these matrices; 3) steering the overlap to a given value. For the first problem, we develop the Krotov type methods directly in terms of density matrices with or without regularization for piecewise continuous constrained controls and find the cases where the methods produce (either exactly or with some precision) zero controls which satisfy the Pontryagin maximum principle and produce the overlap's values close to their upper estimates. For the problems 2) and 3), we find cases when the dual annealing method steers the objectives close to zero and produces a non-zero control.

quant-ph↗

arXiv2023

Optimization of Time-Dependent Decoherence Rates and Coherent Control for a Qutrit System

The work considers an open qutrit system whose density matrix $ρ(t)$ evolution is governed by the Gorini-Kossakowski-Sudarshan-Lindblad master equation with simultaneous coherent (in the Hamiltonian) and incoherent (in the superoperator of dissipation) controls. Incoherent control makes the decoherence rates depending on time in a specific controlled manner and within clear physical mechanics. We consider the problem of maximizing the Hilbert-Schmidt overlap between the system's final state $ρ(T)$ and a given target state $ρ_{\rm target}$ and the problem of minimizing the squared Hilbert-Schmidt distance between these states. For the both problems, we perform their realifications, derive the corresponding Pontryagin function, adjount system (with the two cases of transversality conditions in view of the two terminal objectives), and gradients of the objectives, adapt the one-, two-, three-step gradient projection methods. For the problem of maximizing the overlap, we also adapt the regularized first-order Krotov method. In the numerical experiments, we analyze, first, the methods' operation and, second, the obtained control processes, in respect to considering the environment as a resource via incoherent control.

quant-ph↗

arXiv2023

GRAPE optimization for open quantum systems with time-dependent decoherence rates driven by coherent and incoherent controls

The GRadient Ascent Pulse Engineering (GRAPE) method is widely used for optimization in quantum control. GRAPE is gradient search method based on exact expressions for gradient of the control objective. It has been applied to coherently controlled closed and open quantum systems. In this work, we adopt GRAPE method for optimizing objective functionals for open quantum systems driven by both coherent and incoherent controls. In our case, the tailored or engineered environment acts on the system as control via it time-dependent decoherence rates $γ_k(t)$ or, equivalently, via it spectral density of the environment $n_ω(t)$. To develop GRAPE approach for this problem, we compute gradient of various objectives for general N-level open quantum systems both for piecewise class of control. The case of a single qubit is considered in details and solved analytically. For this case, an explicit analytical expression for evolution and objective gradient is obtained via diagonalization of a $3\times 3$ matrix determining the system's dynamics in the Bloch ball. The diagonalization is obtained by solving a cubic equation via Cardano's method. The efficiency of the algorithm is demonstrated through numerical simulations for the state-to-state transition problem and its complexity is estimated.

quant-ph↗

arXiv2023

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

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.

quant-ph↗

arXiv2023

Optimal State Manipulation for a Two-Qubit System Driven by Coherent and Incoherent Controls

Optimal control of two-qubit quantum systems attracts high interest due to applications ranging from two-qubit gate generation to optimization of receiver for transferring coherence matrices along spin chains. State preparation and manipulation is among important tasks to study for such systems. Typically coherent control, e.g. a shaped laser pulse, is used to manipulate two-qubit systems. However, the environment can also be used $\unicode{x2013}$ as an incoherent control resource. In this article, we consider optimal state manipulation for a two-qubit system whose dynamics is governed by the Gorini-Kossakowski-Sudarshan-Lindblad master equation, where coherent control enters into the Hamiltonian and incoherent control into both the Hamiltonian (via Lamb shift) and the superoperator of dissipation. We exploit two physically different classes of interaction with coherent control and optimize the Hilbert-Schmidt overlap between final and target density matrices, including optimization of its steering to a given value. We find the conditions when zero coherent and incoherent controls satisfy the Pontryagin maximum principle, and in addition, when they form a stationary point of the objective functional. Moreover, we find a case when this stationary point provides the globally minimal value of the overlap. Using upper and lower bounds for the overlap, we develop one- and two-step gradient projection methods operating with functional controls.

quant-ph↗

arXiv2023

Quantum Gate Generation in Two-Level Open Quantum Systems by Coherent and Incoherent Photons Found with Gradient Search

In this work, we consider an environment formed by incoherent photons as a resource for controlling open quantum systems via an incoherent control. We exploit a coherent control in the Hamiltonian and an incoherent control in the dissipator which induces the time-dependent decoherence rates $γ_k(t)$ (via time-dependent spectral density of incoherent photons) for generation of single-qubit gates for a two-level open quantum system which evolves according to the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) master equation with time-dependent coefficients determined by these coherent and incoherent controls. The control problem is formulated as minimization of the objective functional, which is the sum of Hilbert-Schmidt norms between four fixed basis states evolved under the GKSL master equation with controls and the same four states evolved under the ideal gate transformation. The exact expression for the gradient of the objective functional with respect to piecewise constant controls is obtained. Subsequent optimization is performed using a gradient type algorithm with an adaptive step size that leads to oscillating behaviour of the gradient norm vs iterations. Optimal trajectories in the Bloch ball for various initial states are computed. A relation of quantum gate generation with optimization on complex Stiefel manifolds is discussed. We develop methodology and apply it here for unitary gates as a testing example. The next step is to apply the method for generation of non-unitary processes and to multi-level quantum systems.

quant-ph↗

arXiv2022

Optimal control for state preparation in two-qubit open quantum systems driven by coherent and incoherent controls via GRAPE approach

In this work, we consider a model of two qubits driven by coherent and incoherent time-dependent controls. The dynamics of the system is governed by a Gorini-Kossakowski-Sudarshan-Lindblad master equation, where coherent control enters into the Hamiltonian and incoherent control enters into both the Hamiltonian (via Lamb shift) and the dissipative superoperator. We consider two physically different classes of interaction with coherent control and study the optimal control problem of state preparation formulated as minimization of the Hilbert-Schmidt distance's square between the final density matrix and a given target density matrix at some fixed target time. Taking into account that incoherent control by its physical meaning is a non-negative function of time, we derive an analytical expression for the gradient of the objective and develop optimization approaches based on adaptation for this problem of GRadient Ascent Pulse Engineering (GRAPE). We study evolution of the von Neumann entropy, purity, and one-qubit reduced density matrices under optimized controls and observe a significantly different behavior of GRAPE optimization for the two classes of interaction with coherent control in the Hamiltonian.

quant-ph↗

arXiv2021

Reachable sets for two-level open quantum systems driven by coherent and incoherent controls

In this work, we study controllability in the set of all density matrices for a two-level open quantum system driven by coherent and incoherent controls. In [A. Pechen, Phys. Rev. A 84, 042106 (2011)] an approximate controllability, i.e., controllability with some precision, was shown for generic $N$-level open quantum systems driven by coherent and incoherent controls. However, the explicit formulation of this property, including the behavior of this precision as a function of transition frequencies and decoherence rates of the system, was not known. The present work provides a rigorous analytical study of reachable sets for two-level open quantum systems. First, it is shown that for $N=2$ the presence of incoherent control does not affect the reachable set (while incoherent control may affect the time necessary to reach particular state). Second, the reachable set in the Bloch ball is described and it is shown that already just for one coherent control any point in the Bloch ball can be achieved with precision $δ\sim γ/ω$, where $γ$ is the decoherence rate and $ω$ is the transition frequency. Typical values are $δ\lesssim10^{-3}$ that implies high accuracy of achieving any density matrix. Moreover, we show that most points in the Bloch ball can be exactly reached, except of two lacunae of size $\simδ$. For two coherent controls, the system is shown to be completely controllable in the set of all density matrices. Third, the reachable set as a function of the final time is found and shown to exhibit a non-trivial structure.

quant-ph↗

arXiv2019

On extrema of the objective functional for short-time generation of single qubit quantum gates

In the present work the extrema of the objective functional for the problem of generation of quantum gates (logical elements for quantum computations) for two--level systems are investigated for short duration of the control. The problem of existence of local but not global extrema, the so called traps, is considered. In prior works the absence of traps was proved for a sufficiently long control duration. In this paper we prove that for almost all target unitary operators and system Hamiltonians traps are absent for an arbitrarily small control duration. For the remainder target unitary operators and Hamiltonians we obtain a new estimate for the lower boundary of the control duration which guarantees the absence of traps.

quant-ph↗

arXiv2019

Control landscape for ultrafast manipulation by a qubit

In this work we study extrema of objective functionals for ultrafast manipulation by a qubit. Traps are extrema of the objective functionals which are optimal for manipulation by quantum systems only locally but not globally. Much effort in prior works was devoted to the analysis of traps for quantum systems controlled by long enough laser pulses and, for example, manipulation by a qubit with long control pulses was shown to be trap-free. Ultrafast femtosecond and attosecond control becomes now widely applicable that motivates the necessity for the analysis of traps on the ultrafast time scale. We do such analysis for a qubit and show that ultrafast state transfer in a qubit remains trap-free for a wide range of the initial and final states of the qubit. We prove that for this range the probability of transition between the initial and the final states has a saddle but has no traps.

quant-ph↗

arXiv2019

Entropy non-conservation and boundary conditions for Hamiltonian dynamical systems

Applying the theory of self-adjoint extensions of Hermitian operators to Koopman von Neumann classical mechanics, the most general set of probability distributions is found for which entropy is conserved by Hamiltonian evolution. A new dynamical phase associated with such a construction is identified. By choosing distributions not belonging to this class, we produce explicit examples of both free particles and harmonic systems evolving in a bounded phase-space in such a way that entropy is nonconserved. While these nonconserving states are classically forbidden, they may be interpreted as states of a quantum system tunneling through a potential barrier boundary. In this case, the allowed boundary conditions are the only distinction between classical and quantum systems. We show that the boundary conditions for a tunneling quantum system become the criteria for entropy preservation in the classical limit. These findings highlight how boundary effects drastically change the nature of a system.

cond-mat.stat-mech↗

arXiv2019

Krotov Method for Optimal Control in Closed Quantum Systems

Mathematical problems of optimal control in quantum systems attract high interest in connection with fundamental questions and existing and prospective applications. An important problem is the development of methods for constructing controls for quantum systems. One of the commonly used methods is the Krotov method initially proposed beyond quantum control in the articles by V.F.~Krotov and I.N.~Feldman (1978, 1983). The method was used to develop a novel approach for finding optimal controls for quantum systems in [D.J. Tannor, V. Kazakov, V. Orlov, In: Time-Dependent Quantum Molecular Dynamics, Boston, Springer, 347--360 (1992)] and [J.~Somlói, V.A.~Kazakov, D.J.~Tannor, Chem. Phys., 172:1, 85--98 (1993)], and in many works of various scientists, as described in details in this review. The review discusses mathematical aspects of this method for optimal control of closed quantum systems. It outlines various modifications with respect to defining the improvement function (which in most cases is linear or linear-quadratic), constraints on control spectrum and on the states of a quantum system, regularizers, etc. The review describes applications of the Krotov method to control of molecular dynamics, manipulation of Bose-Einstein condensate, quantum gate generation. We also discuss comparison with GRAPE (GRadient Ascent Pulse Engineering), CRAB (Chopped Random-Basis), the Zhu---Rabitz and the Maday---Turinici methods.

quant-ph↗

arXiv2015

Quantum multipole noise

Quantum multipole noise is defined as a family of creation and annihilation operators with commutation relations proportional to derivatives of delta function of difference of the times, $[c^-_n(t),c^+_n(τ)]\approx δ^{(n)}(τ-t)$. In this paper an explicit operator representation of the quantum multipole noise is constructed in a suitable pseudo-Hilbert space (i.e., in a Hilbert space with indefinite metric). For making this representation, we introduce a class of Hilbert spaces obtained as completion of the Schwartz space in specific norms. Using this representation, we obtain an asymptotic expansion as a series in quantum multipole noise for multitime correlation functions which describe the dynamics of open quantum systems weakly interacting with a reservoir.

math-ph↗

arXiv2015

On critical points of the objective functional for maximization of qubit observables

We study unconstrained control of a two-level quantum system and analyse critical points of the objective functional which represents quantum average of system observable at some final time $T$. In Proc. Steklov Inst. Math. 285, 233-240 (2014) it was shown that all maxima and minima of the objective functional are global if $T\ge π$ (in suitable units). In the present work we show that all maxima and minima are global as soon as $T\ge π/2$. Hence we reduce by the factor of two the minimal time for which traps, i.e., local but not global maxima or minima of the objective functional, do not exist.

quant-ph↗

arXiv2015

Quantum measurements as a control resource

We discuss the use of back-action of quantum measurements as a resource for controlling quantum systems and review its application to optimal approximation of quantum anti-Zeno effect.

quant-ph↗

arXiv2015

Quantum control landscape for a $Λ$-atom in the vicinity of second order traps

We show that the second order traps in the control landscape for a three-level $Λ$-system found in our previous work {\it Phys. Rev. Lett.} {\bf 106}, 120402 (2011) are not local maxima: there exist directions in the space of controls in which the objective grows. The growth of the objective is slow --- at best 4th order for weak variations of the control. This implies that simple gradient methods would be problematic in the vicinity of second order traps, where more sophisticated algorithms that exploit the higher order derivative information are necessary to climb up the control landscape efficiently. The theory is supported by a numerical investigation of the landscape in the vicinity of the $\varepsilon(t)=0$ second order trap, performed using the GRAPE and BFGS algorithms.

quant-ph↗

arXiv2014

Coherent control of a qubit is trap-free

There is a strong interest in optimal manipulating of quantum systems by external controls. Traps are controls which are optimal only locally but not globally. If they exist, they can be serious obstacles to the search of globally optimal controls in numerical and laboratory experiments, and for this reason the analysis of traps attracts considerable attention. In this paper we prove that for a wide range of control problems for two-level quantum systems all locally optimal controls are also globally optimal. Hence we conclude that two-level systems in general are trap-free. In particular, manipulating qubits---two-level quantum systems forming a basic building block for quantum computation---is free of traps for fundamental problems such as the state preparation and gate generation.

quant-ph↗