Quantum-Enhanced Barcode Decoding and Pattern Recognition

Leonardo Banchi, Quntao Zhuang, and Stefano Pirandola

Phys. Rev. Applied 14, 064026 – Published 8 December 2020

Abstract

Quantum hypothesis testing is one of the most fundamental problems in quantum information theory, with crucial implications in areas like quantum sensing, where it has been used to prove quantum advantage in a series of binary photonic protocols, e.g., for target detection or memory cell readout. In this work, we generalize this theoretical model to the multipartite setting of barcode decoding and pattern recognition. We start by defining a digital image as an array or grid of pixels, each pixel corresponding to an ensemble of quantum channels. Specializing each pixel to a black and white alphabet, we naturally define an optical model of a barcode. In this scenario, we show that the use of quantum entangled sources, combined with suitable measurements and data processing, greatly outperforms classical coherent-state strategies for the tasks of barcode data decoding and classification of black and white patterns. Moreover, introducing relevant bounds, we show that the problem of pattern recognition is significantly simpler than barcode decoding, as long as the minimum Hamming distance between images from different classes is large enough. Finally, we theoretically demonstrate the advantage of using quantum sensors for pattern recognition with the nearest-neighbor classifier, a supervised learning algorithm, and numerically verify this prediction for handwritten digit classification.

Received 21 July 2020
Revised 3 November 2020
Accepted 11 November 2020

DOI:https://doi.org/10.1103/PhysRevApplied.14.064026

Physics Subject Headings (PhySH)

Machine learning Optical quantum information processing Quantum channels Quantum entanglement Quantum measurements Quantum optics Quantum protocols Quantum sensing

Atomic, Molecular & OpticalInterdisciplinary PhysicsQuantum Information, Science & Technology

Authors & Affiliations

Leonardo Banchi ^1,2,*, Quntao Zhuang ^3,4, and Stefano Pirandola ⁵

¹Department of Physics and Astronomy, University of Florence, via G. Sansone 1, Sesto Fiorentino (FI) I-50019, Italy
²INFN Sezione di Firenze, via G. Sansone 1, I-50019, Sesto Fiorentino (FI), Italy
³Department of Electrical and Computer Engineering, University of Arizona, Tucson, Arizona 85721, USA
⁴James C. Wyant College of Optical Sciences, University of Arizona, Tucson, Arizona 85721, USA
⁵Department of Computer Science, University of York, York YO10 5GH, United Kingdom

^*banchi.leonardo@gmail.com

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Vol. 14, Iss. 6 — December 2020

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Images

Figure 1
Barcode decoding. Examples of (a) 1D and (b) 2D barcodes. (c) Schematic physical setup for decoding a 2D barcode with $n = 5 \times 5$ pixels. A source shines light on the grid of pixels, each modeled by a quantum channel, which is either $E_{B}$ or $E_{W}$ depending on the pixel gray level, black ( $B$ ) or white ( $W$ ). The reflected or scattered light is collected by a detector, which aims at recognizing all pixel values depending on the detected photons. In a quantum setup, the signal modes shined over the pixels are entangled with idler modes that are directly sent to the detector for a joint measurement. (d) General theoretical setup scheme for optimal barcode discrimination using entangled TMSV states, where the mixing optical circuit, the measurements, and following classical postprocessing must be optimized. (e),(f) Special setup to claim quantum advantage: we compare independent entangled TMSV states on each pixel followed by independent local measurements (e) with classical coherent sources followed by global measurements. Quantum advantage is claimed whenever (e) beats (f).
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Figure 2
Regimes of provable quantum advantage. (a) Threshold value from Eq. (13) as a function of $η_{W}$ and $η_{B}$ . The threshold $ν_{th}$ is negative in the filled gray area and positive for $η_{W} > 1 - η_{B}$ . Contours are from 0.01 in steps of 0.02. (b) Threshold $ν_{th}$ for $η_{W} = 1$ . Whenever $n \leq M N_{S} ν_{th}$ , or, similarly, $M \geq n / N_{s} ν_{th}$ , entangled light beats classical strategies based on coherent states.
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Figure 3
Pattern recognition. (a) Images from the MNIST dataset without pixel recognition error ( $p = 0 %$ ) and with pixel error probabilities $p = 2 %, 4 %, 6 %$ , where each pixel is randomly flipped with probability $p$ . (b) Probability $P_{c c^{'}} (h)$ that one image from class $c$ has Hamming distance $h$ , with $0 \leq h \leq 28 \times 28$ , from another image from class $c^{'}$ . The empirical histogram is evaluated for images from the MNIST dataset that correspond to digits 0 and 1. The Gaussian fit has mean $μ_{01} ≃ 157$ and standard deviation $σ ≃ 27$ . Different digits show a similar behavior with $110 ≲ μ_{c c^{'}} ≲ 167$ , where the minimum is achieved between 1 and 7.
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Figure 4
Pattern recognition with independent on-pixel measurements. (a) Classification error, namely the empirical probability of recognizing the wrong digit using the nearest-neighbor classifier with Hamming distance, as a function of the pixel error probability. (b) Classification error when each pixel is probed using either coherent inputs or entangled TMSV states. The plot is generated by combining the error coming from the single pixel error probability (see Appendix pp2) with the classification error from (a). We focus on the limit $M \to \infty$ , $N_{S} \to 0$ while keeping fixed the total mean number of photons $M N_{S}$ irradiated over each pixel. The colored areas represent the region between the upper and lower bounds, assuming quantum (blue) or classical (red) sources combined with optimal local measurements. These bounds depend on the quantum and classical fidelities from Eqs. (10) and (12). The cyan and orange lines represents the performance with quantum light (cyan) or classical coherent states (orange) using (nonoptimal) photodetection measurements. We set $η_{B} = 0.9$ and $η_{W} = 0.95$ .
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