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Cybernetics and Physics

Speed gradient control over qubit states

Sergey Borisenok, Elena Gogoleva

Published Nov 30, 2024• 1 citation

Abstract

We discuss the model of a quantum bit driven by an external classical field without decay in the rotating wave approximation. In such a model, the whole evolution of the qubit states takes place on the Bloch sphere. We reformulate the model as a unitless set of real ordinary differential equations and use the normalized external field as a feedback control parameter. The closed-loop algorithm is designed in the form of the speed gradient, driving the dynamical system towards minimizing a given nonnegative goal function expressed via the qubit variables. We investigate the achievability of the control goal, and focus on the most important features of the speed gradient algorithm applied to a quantum system in comparison with classical systems. Our approach is valid for the control over the ground and excited population levels, and over the qubit phase variables. The paper was presented at PhysCon2024.

DOI: 10.35470/2226-4116-2024-13-3-193-196

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We discuss the model of a quantum bit driven by an external classical field without decay in the rotating wave approximation. In such a model, the whole evolution of the qubit states takes place on the Bloch sphere. We reformulate the model as a unitless set of real ordinary differential equations and use the normalized external field as a feedback control parameter. The closed-loop algorithm is designed in the form of the speed gradient, driving the dynamical system towards minimizing a given nonnegative goal function expressed via the qubit variables. We investigate the achievability of the control goal, and focus on the most important features of the speed gradient algorithm applied to a quantum system in comparison with classical systems. Our approach is valid for the control over the ground and excited population levels, and over the qubit phase variables. The paper was presented at PhysCon2024.

Quantum state on Bloch sphere for superconducting charge qubit, phase qubit and flux qubit for all time in absence of external drive is stable to initial state. By driving the qubits, approximation of charge and flux Hamiltonian lead to quantum state rotation in Bloch sphere around an axis completely differ from rotation vector of exact Hamiltonian. The trajectory of quantum state for phase qubit for approximated and exact Hamiltonian is the same but the expectation of quantum observable has considerable errors as two other qubits. microwave drive control is designed for approximated Hamiltonian and exerted on actual systems and shows completely different trajectory with respect to desired trajectory. Finally a nonlinear control with external

μ

V voltage control and nA current control is designed for general qubit which completely stabilizes quantum state toward a desired state.

Abstract Quantum state on Bloch sphere for superconducting charge qubit, phase qubit and flux qubit for all time in absence of external drive is stable to initial state. By driving the qubits, approximation of charge and flux Hamiltonian lead to quantum state rotation in Bloch sphere around an axis completely differ from rotation vector of exact Hamiltonian. The trajectory of quantum state for phase qubit for approximated and exact Hamiltonian is the same but the expectation of quantum observable has considerable errors as two other qubits. microwave drive control is designed for approximated Hamiltonian and exerted on actual systems and shows completely different trajectory with respect to desired trajectory. Finally a nonlinear control with external µV voltage control and nA current control is designed for general qubit which completely stabilizes quantum state toward a desired state.

In this paper we investigate the application of optimal control of a single qubit coupled to an ohmic heat bath. For the weak bath coupling regime, we derive a Bloch-Redfield master equation describing the evolution of the qubit state parametrized by vectors in the Bloch sphere. By use of the optimal control methodology we determine the field that generates an arbitrary single qubit rotation in minimum time. In particular we develop an efficient method for solving the time-optimal control problem, namely the implementation of

X

gate, which consists of a rotation about the $\stackrel{P\vec}{ox}$ axis through angle $\ensuremath{\pi}$. The time-optimal control problem requires a bounded control. If upper and lower bounds are imposed for the external control, the optimal solution is of bang-bang type and switches from the upper to the lower values of the control bounds. We compare our results using the techniques of automatic differentiation to compute the gradient for the cost functional with some known results using Pontryagin's minimum principle. We use an alternative numerical approach for the optimization strategy.

The paper proposes a gradient method for feedback control and stabilization of quantum systems using Schröodinger’s and Lindblad’s descriptions. The eigenstates of the quantum system are defined by the spin model. First, a gradient-based control law is computed using Schröodinger’s description. Next, an estimate of state of the quantum system is obtained using Lindblad’s differential equation. In the latter case, by applying Lyapunov’s stability theory and LaSalle’s invariance principle one can compute a gradient control law which assures that the quantum system’s state will track the desirable state within acceptable accuracy levels. The performance of the control loop is studied through simulation experiments for the case of a two-qubit quantum system.