Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Optimal approximation to unitary quantum operators with linear optics

Juan Carlos Garcia-Escartin, Vicent Gimeno, Julio José Moyano-Fernández

TL;DR

The paper tackles the problem of approximating arbitrary unitary evolutions $U \in U(M)$—as generated by $n$ photons in $m$ modes—by evolutions $\widetilde{U}$ that can be realized with linear optics. It introduces a differential-geometric framework with a bi-invariant metric on $U(M)$ and uses Toponogov's comparison theorem to drive a geodesic-based projection of $U$ onto the image subgroup $\mathrm{im}(\varphi)$, yielding a convergent, locally optimal iterative algorithm. The method decomposes the principal logarithm $v=\log U$ into tangential and normal components relative to $\mathrm{im}(\varphi)$, with $U_a=\exp(v_T) \in \mathrm{im}(\varphi)$ and a bound $\|U-U_a\| \le \|v_N\|$ guiding each iteration. A Quantum Fourier Transform example demonstrates multiple local optima and shows that the approximants can be realized by simple optical elements such as balanced beam splitters and phase shifters, highlighting practical paths to implement approximate quantum operations with linear optics.

Abstract

Linear optical systems acting on photon number states produce many interesting evolutions, but cannot give all the allowed quantum operations on the input state. Using Toponogov's theorem from differential geometry, we propose an iterative method that, for any arbitrary quantum operator $U$ acting on $n$ photons in $m$ modes, returns an operator $\widetilde{U}$ which can be implemented with linear optics. The approximation method is locally optimal and converges. The resulting operator $\widetilde{U}$ can be translated into an experimental optical setup using previous results.

Optimal approximation to unitary quantum operators with linear optics

TL;DR

The paper tackles the problem of approximating arbitrary unitary evolutions —as generated by photons in modes—by evolutions that can be realized with linear optics. It introduces a differential-geometric framework with a bi-invariant metric on and uses Toponogov's comparison theorem to drive a geodesic-based projection of onto the image subgroup , yielding a convergent, locally optimal iterative algorithm. The method decomposes the principal logarithm into tangential and normal components relative to , with and a bound guiding each iteration. A Quantum Fourier Transform example demonstrates multiple local optima and shows that the approximants can be realized by simple optical elements such as balanced beam splitters and phase shifters, highlighting practical paths to implement approximate quantum operations with linear optics.

Abstract

Linear optical systems acting on photon number states produce many interesting evolutions, but cannot give all the allowed quantum operations on the input state. Using Toponogov's theorem from differential geometry, we propose an iterative method that, for any arbitrary quantum operator acting on photons in modes, returns an operator which can be implemented with linear optics. The approximation method is locally optimal and converges. The resulting operator can be translated into an experimental optical setup using previous results.

Paper Structure

This paper contains 11 sections, 9 theorems, 55 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The image algebra and its orthogonal complement
  3. Prerrequisites and Notation
  4. A bi-invariant Riemannian metric
  5. An application of Toponogov's comparison theorem
  6. Random unitary matrices and a basis for the image
  7. An iterative process for the approximation
  8. Convergence
  9. Application example
  10. Random initial matrices
  11. Summary, recommendations and future improvements

Key Result

Theorem 2.1

For $U\in U(M)$ such that $U\not\in {\rm im}\,\varphi$, let $v\in \mathfrak{u}(M)$ be the principal logarithm of $U$. Let be the orthogonal decomposition of $v$, with a tangent component $v_T \in {\rm im}\, d\varphi$ and a normal component $v_N \in {\rm im}\, (d\varphi)^\perp$. Then,

Figures (2)

  • Figure 1: Geodesic triangle $\Delta(p_1p_2p_3)$.
  • Figure 2: Submanifold $\mathrm{im}(\varphi)$ in $U(M)$, matrix $U$ and approximation $U_a$.

Theorems & Definitions (14)

  • Theorem 2.1
  • Theorem 3.1: Hopf-Rinow
  • Corollary 3.2
  • Theorem 4.1: Milnor Mil76
  • Lemma 4.2
  • proof : Proof of Lemma \ref{['principalminimal']}
  • Definition 5.1
  • Theorem 5.2: Toponogov
  • Proposition 7.1
  • proof
  • ...and 4 more