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Time-optimal state transfer for an open qubit

L. V. Lokutsievskiy, A. N. Pechen, M. I. Zelikin

Abstract

Finding minimal time and establishing the structure of the corresponding optimal controls which can transfer a given initial state of a quantum system into a given target state is a key problem of quantum control. In this work, this problem is solved for a basic component of various quantum technology processes -- a qubit interacting with the environment and experiencing an arbitrary time-dependent coherent driving. We rigorously derive both upper and lower estimates for the minimal steering time. Surprisingly, we discover that the optimal controls have a very special form -- they consist of two impulses, at the beginning and at the end of the control period, which can be assisted by a smooth time-dependent control in between. Moreover, an important for practical applications explicit almost optimal state transfer protocol is provided which only consists of four impulses and gives an almost optimal time of motion. The results can be directly applied to a variety of experimental situations for estimation of the ultimate limits of state control for quantum technologies.

Time-optimal state transfer for an open qubit

Abstract

Finding minimal time and establishing the structure of the corresponding optimal controls which can transfer a given initial state of a quantum system into a given target state is a key problem of quantum control. In this work, this problem is solved for a basic component of various quantum technology processes -- a qubit interacting with the environment and experiencing an arbitrary time-dependent coherent driving. We rigorously derive both upper and lower estimates for the minimal steering time. Surprisingly, we discover that the optimal controls have a very special form -- they consist of two impulses, at the beginning and at the end of the control period, which can be assisted by a smooth time-dependent control in between. Moreover, an important for practical applications explicit almost optimal state transfer protocol is provided which only consists of four impulses and gives an almost optimal time of motion. The results can be directly applied to a variety of experimental situations for estimation of the ultimate limits of state control for quantum technologies.
Paper Structure (10 sections, 9 theorems, 63 equations, 1 figure)

This paper contains 10 sections, 9 theorems, 63 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Controlled qubit interacting with the environment
  3. Auxiliary 2D control problem in cylindrical coordinates
  4. Obtaining a desired level of purity
  5. Continuity and analyticity of the optimal control in the auxiliary problem (\ref{['eq:auxiliary_system']})
  6. Absence of Impulses
  7. Reconstruction of the Optimal Control from the Auxiliary Problem
  8. Estimation of the minimum motion time in the state-to-state problem
  9. Proof of Proposition \ref{['prop:auxiliary_existence']}
  10. Proof of Proposition \ref{['prop:pureness']}

Key Result

Proposition 1

Put $\sigma=\text{sgn}(P_1-P_0)=\pm1$. If $P_1<1$, then there exists an optimal solution to problem problem:pureness for system eq:auxiliary_system. On any optimal solution, $R(t)$ does not change its sign. Without loss of generality, we assume $R(t)\ge 0$, and in this case, where If $P_1=1$ and $P_0<1$, then there is no optimal solution because there is no control that can bring the system to t

Figures (1)

  • Figure 1: This figure shows lower and upper estimates for the minimal time as a function of purity of the initial and final states. The upper estimate is plotted neglecting term $\pi/\omega$ which is typically several orders of magnitude smaller than term $e/\gamma$. Time is in the units of inverse $\gamma$.

Theorems & Definitions (21)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Proposition 1
  • Proposition 2
  • Corollary 1
  • proof : Proof of Corollary \ref{['cor:R_zeros']}
  • Theorem 1
  • ...and 11 more