Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Quantum Optimal Control of a Lambda System in the Density Matrix Formulation

Julia Cen, Domenico D'Alessandro

Abstract

In various physical implementations of quantum information processing, qubits are realized in a Lambda type system configuration as two stable lower energy levels coupled indirectly via an unstable higher energy level, that is, in comparison, a lot more susceptible to decoherence. We consider the quantum control problem of optimal state transfer between two isospectral density matrices, over an arbitrary finite time horizon, for the quantum Lambda system. The cost considered is a compromise between the energy of the control field and the average occupancy in the highest energy level. We apply a geometric approach that combines the use of the Pontryagin Maximum Principle, a symmetry reduction technique to reduce the number of parameters in the resulting optimization problem, and several auxiliary techniques to bound the parameter space in the search for the optimal solution. We prove several properties of the optimal control and trajectories for this problem, including their normality and smoothness. We obtain a system of differential equations that must be satisfied by the optimal pair of control and trajectory we treat in detail, with numerical simulations, and solve a case study involving a Hadamard-like transformation. Our techniques can be adapted to other contexts and promise to push to a more consequential level, the application of geometric control in quantum systems.

Quantum Optimal Control of a Lambda System in the Density Matrix Formulation

Abstract

In various physical implementations of quantum information processing, qubits are realized in a Lambda type system configuration as two stable lower energy levels coupled indirectly via an unstable higher energy level, that is, in comparison, a lot more susceptible to decoherence. We consider the quantum control problem of optimal state transfer between two isospectral density matrices, over an arbitrary finite time horizon, for the quantum Lambda system. The cost considered is a compromise between the energy of the control field and the average occupancy in the highest energy level. We apply a geometric approach that combines the use of the Pontryagin Maximum Principle, a symmetry reduction technique to reduce the number of parameters in the resulting optimization problem, and several auxiliary techniques to bound the parameter space in the search for the optimal solution. We prove several properties of the optimal control and trajectories for this problem, including their normality and smoothness. We obtain a system of differential equations that must be satisfied by the optimal pair of control and trajectory we treat in detail, with numerical simulations, and solve a case study involving a Hadamard-like transformation. Our techniques can be adapted to other contexts and promise to push to a more consequential level, the application of geometric control in quantum systems.

Paper Structure

This paper contains 16 sections, 8 theorems, 110 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. The model and statement of the problem
  3. Background in geometric optimal control theory; The Pontryagin Maximum Principle
  4. Differential equations for the optimal candidates
  5. Reduction of the number of parameters and auxiliary results
  6. Dependence of the optimal cost on $\gamma_0$
  7. Bounds on the norm of the control $\|P\|$
  8. Symmetries and elimination of redundancy
  9. Bounds on the values of $h_9$
  10. A case study: An Hadamard-like transformation on the density matrix
  11. Conclusions
  12. Proof of the bound (\ref{['UB3']})
  13. The zero occupancy case for the case study of Section \ref{['CS']}; Proof of Theorem \ref{['Final']}
  14. Case 1: $e^{-A+P}$ scalar
  15. Case 2: $e^{-A+P}$ possibly nonscalar
  16. ...and 1 more sections

Key Result

Theorem 1

Assume $(x^*,u^*)$ is an optimal pair for the above problem. Then there exists a function $\lambda=\lambda(t)$ with values in the cotangent bundle of $M$, $T^*M$, and a constant $\mu_0 \leq 0$ not both (identically) zero such that, defined the function,Notice the abuse of notation here. Since $\lamb we have

Figures (4)

  • Figure 1: Minimum distance square ($\|\rho(1)-\rho_1\|^2$) as a function of $|\tilde{h}_9|$ for $|\tilde{h}_9|\in [0,80]$.
  • Figure 2: Minimum distance square ($\|\rho(1)-\rho_1\|^2$) as a function of $\|P\|$ for $|\tilde{h}_9|\in [3.18,3.3]$.
  • Figure 3: Time dependence for the optimal controls for the problem with $\gamma_0=1$ and state transfer (\ref{['statetransf']})
  • Figure 4: Time dependence for the optimal density matrix for the problem with $\gamma_0=1$ and state transfer (\ref{['statetransf']})

Theorems & Definitions (11)

  • Theorem 1
  • Theorem 2
  • proof
  • Theorem 3
  • Proposition 4
  • Proposition 5
  • proof
  • Theorem 6
  • Lemma 7
  • proof
  • ...and 1 more