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Pointwise and dynamic programming control synthesis for finite-level open quantum memory systems

Igor G. Vladimirov, Ian R. Petersen, Guodong Shi

Abstract

This paper is concerned with finite-level quantum memory systems for retaining initial dynamic variables in the presence of external quantum noise. The system variables have an algebraic structure, similar to that of the Pauli matrices, and their Heisenberg picture evolution is governed by a quasilinear quantum stochastic differential equation. The latter involves a Hamiltonian whose parameters depend affinely on a classical control signal in the form of a deterministic function of time. The memory performance is quantified by a mean-square deviation of quantum system variables of interest from their initial conditions. We relate this functional to a matrix-valued state of an auxiliary classical control-affine dynamical system. This leads to a pointwise control design where the control signal minimises the time-derivative of the mean-square deviation with an additional quadratic penalty on the control. In an alternative finite-horizon setting with a terminal-integral cost functional, we apply dynamic programming and obtain a quadratically nonlinear Hamilton-Jacobi-Bellman equation, for which a solution is outlined in the form of a recursively computed asymptotic expansion.

Pointwise and dynamic programming control synthesis for finite-level open quantum memory systems

Abstract

This paper is concerned with finite-level quantum memory systems for retaining initial dynamic variables in the presence of external quantum noise. The system variables have an algebraic structure, similar to that of the Pauli matrices, and their Heisenberg picture evolution is governed by a quasilinear quantum stochastic differential equation. The latter involves a Hamiltonian whose parameters depend affinely on a classical control signal in the form of a deterministic function of time. The memory performance is quantified by a mean-square deviation of quantum system variables of interest from their initial conditions. We relate this functional to a matrix-valued state of an auxiliary classical control-affine dynamical system. This leads to a pointwise control design where the control signal minimises the time-derivative of the mean-square deviation with an additional quadratic penalty on the control. In an alternative finite-horizon setting with a terminal-integral cost functional, we apply dynamic programming and obtain a quadratically nonlinear Hamilton-Jacobi-Bellman equation, for which a solution is outlined in the form of a recursively computed asymptotic expansion.

Paper Structure

This paper contains 8 sections, 5 theorems, 91 equations, 1 figure.

Table of Contents

  1. INTRODUCTION
  2. FINITE-LEVEL OPEN QUANTUM MEMORY SYSTEM
  3. EVOLUTION OF SYSTEM VARIABLES AND THEIR MOMENTS
  4. MEAN-SQUARE DEVIATION FROM INITIAL CONDITIONS
  5. AUXILIARY CONTROL-AFFINE DYNAMICAL SYSTEM
  6. POINTWISE CONTROL OPTIMISATION
  7. FINITE-HORIZON OPTIMAL MEMORY CONTROL
  8. CONCLUSION

Key Result

Lemma 1

At any time $t\geqslant 0$, the quantum system variables in (dX) satisfy where $\alpha$, $\beta$ are the structure constants from (Xalg), the function $\psi$ is associated by (psi) with the fundamental matrix $G$ of the ODE (EEts), and $\mu(t)$ is the mean vector from (mu). $\square$

Figures (1)

  • Figure 3: An open quantum memory system governed by (\ref{['dX']}), which is driven by the external quantum noise $W$ and affected by the control signal $U$.

Theorems & Definitions (10)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof