We present a monotonic numerical algorithm including time optimization for generating quantum gates for open systems. Such systems are assumed to be governed by Lindblad master equations for the density operators on a large Hilbert-space whereas the quantum gates are relative to a sub-space of small dimension. Starting from an initial seed of the control input, this algorithm consists in the repetition of the following two steps producing a new control input: (A) backwards integration of adjoint Lindblad-Master equations (in the Heisenberg-picture) from a set of final conditions encoding the quantum gate to generate; (B) forward integration of Lindblad-Master equations in closed-loop where a Lyapunov based control produced the new control input. The numerical stability is ensured by the stability of both the open-loop adjoint backward system and the forward closed-loop system. A clock-control input can be added to the usual control input. The obtained monotonic algorithm allows then to optimise not only the shape of the control input, but also the gate time. Preliminary numerical implementations indicate that this algorithm is well suited for cat-qubit gates, where Hilbert-space dimensions (2 for the Z-gate and 4 for the CNOT-gate) are much smaller than the dimension of the physical Hilbert-space involving mainly Fock-states (typically 20 or larger for a single cat-qubit). This monotonic algorithm, based on Lyapunov control techniques, is shown to have a straightforward interpretation in terms of optimal control: its stationary conditions coincides with the first-order optimality conditions for a cost depending linearly on the final values of the quantum states.
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 γk(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.
RIGA (Reference Input Generation Algorithm) is a monotonic numerical method for generating quantum gates for closed systems that are described by Schrödinger equations. In a previous paper, the authors have presented a monotonic quantum gate generation algorithm, called here by L-RIGA (Lindblad-RIGA), that is able to consider open quantum systems described by Lindblad master equations. The authors have claimed in that paper, without proof, that L-RIGA was originally obtained from a version of RIGA. In this paper we present this version of RIGA, called here F-RIGA (Fock-RIGA) that can consider open quantum systems after converting them to the Fock-Liouville descripton. This conversion is based on the Fock-Map, that is, the map F that sends an × n Hermitian matrix into an 2-vector of a real Euclidean space. The contribution of this paper is to show that that L-RIGA and F-RIGA are equivalent in the sense that, for each step ℓ, the data that is obtained by F-RIGA is transformed by the inverse of the Fock-Map into the data that is obtained in the same step of L-RIGA, letting the corresponding Lyapunov functions invariant. Furthermore, thanks to the great similarity between L-RIGA and a version of the Krotov method, a subproduct of this work is also establish a strong connection between that version of the Krotov method and the family of algorithms that are called by RIGA.
The problem of encoded quantum gate generation is studied in this paper. The idea is to consider a quantum system of higher dimension
n than the dimension
nˉ of the quantum gate to be synthesized. Given two orthonormal subsets
E={e1,e2,…,enˉ} and
F={f1,f2,…,fnˉ} of
Cn, the problem of encoded quantum gate generation consists in obtaining an open loop control law defined in an interval
[0,Tf] in a way that all initial states
ei are steered to
exp(ϕ)fi,i=1,2,…,nˉ up to some desired precision and to some global phase
ϕ∈R. This problem includes the classical (full) quantum gate generation problem, when
nˉ=n, the state preparation problem, when
nˉ=1, and finally the encoded gate generation when $ 1 < \bar n < n$. Hence, three problems are unified here within a unique common approach. The \emph{Reference Input Generation Algorithm (RIGA)} is generalized in this work for considering the encoded gate generation problem for closed quantum systems. A suitable Lyapunov function is derived from the orthogonal projector on the support of the encoded gate. Three case-studies of physical interest indicate the potential interest of such numerical algorithm: two coupled transmon-qubits, a cavity mode coupled to a transmon-qubit, and a chain of
N qubits, including a large dimensional case for which
N=10.
Quandary numerically simulates and optimizes the time-evolution of open quantum systems. The underlying dynamics are modelled by Lindblad's master equation, a linear ordinary differential equation (ODE) describing quantum systems interacting with the environment. Quandary solves this ODE numerically by applying a time-stepping integration scheme, and utilizes a gradient-based optimization approach to determine optimal control pulses that drive the quantum system to a desired target state. Two optimization objectives are considered: (a) Unitary gate optimization that finds controls to realize a unitary gate transformation, and (b) optimal reset that aims to drive the quantum system to the ground states. Gradient-based optimization schemes utilizing Petsc's Tao optimization package are applied to generate control pulses that minimize the respective measure. To evaluate the gradient of the objective function, the discrete adjoint method is used while leveraging techniques from Algorithmic Differentiation to produce exact and consistent gradients. To mitigate excessive execution run times, the software can be build together with the XBraid software library which provides a parallelization strategy to distribute the time-evolution of the underlying dynamics onto multiple processor.
This paper considers some control problems for closed and open two-level quantum systems. The closed system's dynamics is governed by the Schrödinger equation with coherent control. The open system dynamics is governed by the Gorini-Kossakowski-Sudarshan-Lindblad master equation whose Hamiltonian depends on coherent control and superoperator of dissipation depends on incoherent control. For the closed system, we consider the problem of generation of the phase shift gate for some values of phases and final times for which we numerically show that zero coherent control, which is a stationary point of the objective functional, is not optimal; it gives an example of subtle point for practical solving quantum control problems. The two-stage method, which was developed in [48] for generic
N-level open quantum systems for approximate generation of a given target density matrix, is modified here for the case of two-level systems. We modify the first ("incoherent") stage by numerically optimizing piecewise constant incoherent control instead of using constant incoherent control analytically computed using eigenvalues of the target density matrix. Exact analytical formulas are derived for the system state evolution, the objective functions and their gradients for the modified first stage. These formulas are then applied in the two-step gradient projection method. The numerical simulations show that the modified first stage duration can be significantly less than the unmodified first stage duration, but at the cost of performing optimization in the class of piecewise constant controls.
This article considers some control problems for closed and open two-level quantum systems. The closed system's dynamics is governed by the Schrödinger equation with coherent control. The open system's dynamics is governed by the Gorini-Kossakowski-Sudarshan-Lindblad master equation whose Hamiltonian depends on coherent control and superoperator of dissipation depends on incoherent control. For the closed system, we consider the problem for generation of the phase shift gate for some values of phases and final times for which we numerically show that zero coherent control, which is a stationary point of the objective functional, is not optimal; it gives an example of subtle point for practical solving quantum control problems. The two-stage method, which was developed in [A. Pechen, Phys. Rev. A., 84, 042106 (2011)] for generic
N-level open quantum systems for approximate generation of a given target density matrix, is modified here for the case of two-level systems. We modify the first ("incoherent") stage by numerically optimizing piecewise constant incoherent control instead of using constant incoherent control analytically computed using eigenvalues of the target density matrix. Exact analytical formulas are derived for the system's state evolution, the objective functions and their gradients for the modified first stage. These formulas are then applied in the two-step gradient projection method. The numerical simulations show that the modified first stage's duration can be significantly less than the unmodified first stage's duration, but at the cost of performing optimization in the class of piecewise constant controls. Bibliography: 72 titles.