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Advancing quantum imaging through learning theory

Yunkai Wang, Changhun Oh, Junyu Liu, Liang Jiang, Sisi Zhou

TL;DR

The orthogonalized SPADE method is introduced-a nontrivial generalization of existing superresolution techniques-that achieves superior performance when multiple compact sources are closely spaced, marking an important step toward developing more general and practically applicable approaches.

Abstract

We study quantum imaging by applying the resolvable expressive capacity (REC) formalism developed for physical neural networks (PNNs). In this paradigm of quantum learning, the imaging system functions as a physical learning device that maps input parameters to measurable features, while complex practical tasks are handled by training only the output weights, enabled by the systematic identification of well-estimated features (eigentasks) and their corresponding sample thresholds. Using this framework, we analyze both direct imaging and superresolution strategies for compact sources, defined as sources with sizes bounded below the Rayleigh limit. In particular, we introduce the orthogonalized SPADE method-a nontrivial generalization of existing superresolution techniques-that achieves superior performance when multiple compact sources are closely spaced. This method relaxes the earlier superresolution studies' strong assumption that the entire source must lie within the Rayleigh limit, marking an important step toward developing more general and practically applicable approaches. Using the example of face recognition, which involve complex structured sources, we demonstrate the superior performance of our orthogonalized SPADE method and highlight key advantages of the quantum learning approach-its ability to tackle complex imaging tasks and enhance performance by selectively extracting well-estimated features.

Advancing quantum imaging through learning theory

TL;DR

The orthogonalized SPADE method is introduced-a nontrivial generalization of existing superresolution techniques-that achieves superior performance when multiple compact sources are closely spaced, marking an important step toward developing more general and practically applicable approaches.

Abstract

We study quantum imaging by applying the resolvable expressive capacity (REC) formalism developed for physical neural networks (PNNs). In this paradigm of quantum learning, the imaging system functions as a physical learning device that maps input parameters to measurable features, while complex practical tasks are handled by training only the output weights, enabled by the systematic identification of well-estimated features (eigentasks) and their corresponding sample thresholds. Using this framework, we analyze both direct imaging and superresolution strategies for compact sources, defined as sources with sizes bounded below the Rayleigh limit. In particular, we introduce the orthogonalized SPADE method-a nontrivial generalization of existing superresolution techniques-that achieves superior performance when multiple compact sources are closely spaced. This method relaxes the earlier superresolution studies' strong assumption that the entire source must lie within the Rayleigh limit, marking an important step toward developing more general and practically applicable approaches. Using the example of face recognition, which involve complex structured sources, we demonstrate the superior performance of our orthogonalized SPADE method and highlight key advantages of the quantum learning approach-its ability to tackle complex imaging tasks and enhance performance by selectively extracting well-estimated features.
Paper Structure (25 sections, 1 theorem, 114 equations, 32 figures, 2 tables, 3 algorithms)

This paper contains 25 sections, 1 theorem, 114 equations, 32 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Preliminary
  2. Review of quantum-metrology-inspired superresolution
  3. Physical neural networks
  4. Resolvable expressive capacity
  5. Two-point sources
  6. A single compact source (generally distributed source within the Rayleigh limit)
  7. Direct imaging of a single compact source
  8. Image a single compact source using SPADE method
  9. Numerical results for eigenvalues $\beta_k^2$
  10. Deviation of the approximated eigenbasis from the actual eigenbasis
  11. Imaging general source extended outside the Rayleigh limit
  12. Imaging multiple compact sources
  13. Direct imaging
  14. Separate SPADE method
  15. Orthogonalized SPADE method
  16. ...and 10 more sections

Key Result

Proposition 1

Let $\rho(\boldsymbol{\theta})$ be a family of states parameterized by $\boldsymbol{\theta}$ with prior $p(\boldsymbol{\theta})$, a fixed POVM $\{M_i\}_{i=0}^{K-1}$, sample size $S$, and measured features $\eta_i(\boldsymbol{\theta})=\operatorname{Tr}[\rho(\boldsymbol{\theta})M_i]$. In the parameter

Figures (32)

  • Figure 1: Imaging of both Rayleigh resolvable features and sub-Rayleigh features. (a) Multiple compact sources of size $L_i$ are imaged by a lens with a PSF width of $\sigma$. Besides Rayleigh-resolvable features, we would also like to extract information from sub-Rayleigh features, which are associated with the small parameter $\alpha=L/\sigma$. (b) The total REC, $C_T$, which shows a stepwise increase, is plotted as a function of the number of samples, $S$. The threshold of $S$ for each stepwise increase of $C_T$ in the shaded region is determined by the eigenvalue $\beta_k^2$ associated with the corresponding eigentask in learning. Each time $C_T$ increases by 1, there is a corresponding eigenvalue $\beta_k^2$, with the sample number threshold following $S \sim \Theta(\beta_k^2)$. The intensity of each compact source corresponds to resolvable features, which can be imaged with a constant number of samples, scaling as $S \sim \beta^2_{0\leq k\leq Q-1}=\Theta(\alpha^0)$, independent of the source size. In contrast, sub-Rayleigh features that reveal detailed information about each compact source require a number of samples scaling inversely with the source size, following $S \sim \beta^2_{ k\geq Q}= \Theta(\alpha^{-m})$, where $\alpha$ is determined by the compact sources, and $m$ depends on the order of moments.
  • Figure 2: Total REC $C_T$ for direct imaging and the SPADE method as a function of $S$, when imaging one generally distributed compact source with different $\alpha$.
  • Figure 3: Scaling of the $\beta_k^2$ as a function of $\alpha$ for imaging two compact sources with distance $L/2$. We consider three different cases: (a) direct imaging (b) separate SPADE method (c) orthogonalized SPADE method. Width of PSF $\sigma=1$.
  • Figure 4: Total REC $C_T$ for direct imaging, the separate SPADE method and the orthogonalized SPADE method as a function of $S$ for imaging of two compact sources with $\alpha=10^{-1},10^{-2},10^{-3}$, number of compact source $Q=2$, $\sigma=1$. The distance between the centroid of the two compact sources is $L/2$. When $L=2$, the orthogonalized SPADE method demonstrates a clear advantage over both the separate SPADE method and direct imaging in the shaded region. When $L=20$, the performance of SPADE and orthogonalized SPADE is comparable (essentially because they become equivalent when the two compact sources are sufficiently far apart) and both outperform direct imaging. Overall, the orthogonalized SPADE protocol demonstrates excellent performance, achieving a high $C_T$ (compared to the best from direct and SPADE protocols) for various choices of $L = 2$ and $L = 20$, as well as across a wide range of sample sizes $S$.
  • Figure 5: We compare the basis states constructed in Eq. \ref{['eq:bjl']} for the orthogonalized SPADE measurement, $\ket{b_{q}^{(l=1)}} = \int du \, b_{q}^{(1)}(u)\ket{u}$, with the corresponding basis states for the separate SPADE measurement, $\ket{b_{q,1}} = \int du \, b_{q,1}(u)\ket{u}$. Here, $q = 1, 2$ corresponds to the case of two compact sources ($Q = 2$) with centroids located at $\pm L/4$, so that the separation between the two sources is $L/2$. We examine cases where $L = 2, 6, 20$. The shaded regions show two Gaussian PSFs of width $\sigma = 1$ located at the centers of two compact sources, that are used to construct the basis states $\ket{b_q^{(l)}}$ and $\ket{b_{q,m}}$.
  • ...and 27 more figures

Theorems & Definitions (2)

  • Proposition 1: Reparameterization invariance of total REC and eigentasks
  • proof