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Partially Unitary Learning

Mikhail Gennadievich Belov, Vladislav Gennadievich Malyshkin

TL;DR

An iterative algorithm for finding the global maximum of this optimization problem is developed, and its application to a number of problems is demonstrated.

Abstract

The problem of an optimal mapping between Hilbert spaces $IN$ of $\left|ψ\right\rangle$ and $OUT$ of $\left|φ\right\rangle$ based on a set of wavefunction measurements (within a phase) $ψ_l \to φ_l$, $l=1\dots M$, is formulated as an optimization problem maximizing the total fidelity $\sum_{l=1}^{M} ω^{(l)} \left|\langleφ_l|\mathcal{U}|ψ_l\rangle\right|^2$ subject to probability preservation constraints on $\mathcal{U}$ (partial unitarity). The constructed operator $\mathcal{U}$ can be considered as an $IN$ to $OUT$ quantum channel; it is a partially unitary rectangular matrix (an isometry) of dimension $\dim(OUT) \times \dim(IN)$ transforming operators as $A^{OUT}=\mathcal{U} A^{IN} \mathcal{U}^{\dagger}$. An iterative algorithm for finding the global maximum of this optimization problem is developed, and its application to a number of problems is demonstrated. A software product implementing the algorithm is available from the authors.

Partially Unitary Learning

TL;DR

An iterative algorithm for finding the global maximum of this optimization problem is developed, and its application to a number of problems is demonstrated.

Abstract

The problem of an optimal mapping between Hilbert spaces of and of based on a set of wavefunction measurements (within a phase) , , is formulated as an optimization problem maximizing the total fidelity subject to probability preservation constraints on (partial unitarity). The constructed operator can be considered as an to quantum channel; it is a partially unitary rectangular matrix (an isometry) of dimension transforming operators as . An iterative algorithm for finding the global maximum of this optimization problem is developed, and its application to a number of problems is demonstrated. A software product implementing the algorithm is available from the authors.
Paper Structure (18 sections, 69 equations, 2 figures, 1 table)

This paper contains 18 sections, 69 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Formulation of the Problem
  3. A Quantum System Time Evolution
  4. A Classical System $\mathbf{x}\to \mathbf{f}$ Vector Mapping
  5. Learning Unitary Dynamics
  6. Traditional Unitary Learning
  7. Variational quantum algorithms
  8. Algebraic Structure of the Optimization Problem
  9. Numerical Solution
  10. A Demonstration Of Recovering Unitary Dynamics From Phase-Stripped Data
  11. A Demonstration Of Polynomial Mapping Recovery
  12. A Demonstration of Partially Unitary Behavior $D<n$
  13. A Demonstration Of Function Interpolation
  14. Conclusion
  15. On Lagrange Multipliers Calculation With Selected States Variation
  16. ...and 3 more sections

Figures (2)

  • Figure 1: For $D\le n=20$, a sample $\psi_l\to\phi_l$ is constructed with a known $u_{jk}$, the "original" one. The result is compared with the fidelity-maximizing solution of optimization problem (\ref{['allProjUKxfAppendix']}). For $D<n$ (partial unitarity), the fidelity of the optimization problem solution is always greater than the original fidelity. For $D=n$ (unitarity), the original and optimization problem solutions match exactly. A deviation of $\mathcal{F}^{orig}/M$ from $1$ indicates that the transform does not preserve trace --- it is a \ref{['operatorTransform']} only for $D=n$.
  • Figure 2: Scalar function $f=x^2$ (red) interpolation with: green: Radon-Nikodym (\ref{['approxRN']}), blue: $f$ corresponding to the maximal probability (\ref{['probFXUpExpanded']}), the dependence has discontinuities; some numerical instability presents even in this $D=n=6$ case. Pink: probabilities corresponding to (\ref{['probFXUpExpanded']}) and (\ref{['f_Pmax']}).