⚛️ quantum physics

Analysis of Quantum Image Representations for Supervised Classification

This paper compares four quantum image representations, finding that FRQI and QPIE offer superior compression, and demonstrates that quantum kernels based on these representations achieve classification accuracy comparable to classical methods while requiring exponentially fewer storage resources.

Original authors: Marco Parigi, Mehran Khosrojerdi, Filippo Caruso, Leonardo Banchi

Published 2026-01-15

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Marco Parigi, Mehran Khosrojerdi, Filippo Caruso, Leonardo Banchi

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a massive library of digital photos. On a normal computer, storing and organizing these photos takes up a huge amount of space, like trying to fit a library into a shoebox by printing every single page of every book. This paper explores a new way to handle these photos using the strange, magical rules of quantum mechanics.

The authors, a team of physicists from the University of Florence, asked a simple question: If we try to shrink these photos down to their absolute smallest quantum size, which method works best, and does it still let us recognize what's in the picture?

Here is the breakdown of their study using everyday analogies:

1. The Four "Shrinking" Methods

The team tested four different ways to encode (or "shrink") a black-and-white image into a quantum state. Think of these as four different compression algorithms, but for the quantum world:

TNR (Tensor Network Representation): Imagine taking a photo and folding it into a complex origami structure where the creases represent the relationships between pixels. It's a structured way to hold the image, but it requires a specific amount of "paper" (memory) to keep the folds intact.
FRQI (Flexible Representation of Quantum Image): This is like taking a photo and turning the brightness of every pixel into a specific angle on a spinning wheel. All the pixels spin together in a superposition. It's very compact, but reading the picture back out is tricky because you have to guess the angles based on probability.
NEQR (Novel Enhanced Quantum Representation): This method is like writing the brightness of every pixel as a specific code (like a binary barcode) on a separate strip of paper for each pixel. It's very precise—you can read the picture back perfectly—but it requires more strips of paper (memory) than the spinning wheel method.
QPIE (Quantum Probability Image Encoding): This is the most extreme shrunken version. It treats the brightness of the pixels as the "weight" or "probability" of a single quantum state. It uses the absolute minimum amount of space (qubits), but like FRQI, reading the exact original picture back is a game of chance.

2. The "Squeeze" Test (Compression)

The researchers wanted to see how much these methods actually compressed the data. They used a tool called a Gram Matrix, which is essentially a "similarity scorecard."

The Analogy: Imagine you have 100 different photos. If you compress them poorly, they all still look very distinct from one another (low similarity). If you compress them too much, they all start to look like blurry smudges of the same color (high similarity).
The Result: They found that FRQI and QPIE were the "super-squeezers." They compressed the images so tightly that the quantum versions of different photos looked very similar to each other (high overlap). NEQR was the "gentle squeezer," keeping the photos distinct but taking up more space. TNR landed somewhere in the middle.

3. The "Guessing Game" (Classification)

The real test wasn't just about shrinking the photos; it was about whether a computer could still tell them apart. They set up a binary classification game (a "Yes/No" test).

The Task: Show the computer a photo and ask, "Is this a '0' or is it a '1'?" (using the famous MNIST dataset of handwritten numbers).
The Comparison: They compared the quantum methods against a standard classical linear kernel (the traditional, non-quantum way of doing this).

The Big Surprise:
The quantum methods (especially FRQI and QPIE) performed just as well as the classical method in terms of accuracy. They guessed the correct number almost 99% of the time.

However, the trade-off was massive:

Classical Method: To store a 16x16 pixel image, the classical computer needed 2,048 bits of memory.
Quantum Method: The quantum computers only needed 8 to 16 qubits (quantum bits) to store the exact same image.

This is an exponential reduction. It's the difference between storing a library in a warehouse versus storing it in a single matchbox.

4. The Catch (State Preparation)

The paper is careful to point out a major hurdle. While storing the image in the quantum format is incredibly efficient, loading the image into that format in the first place is currently slow and difficult.

The Analogy: Imagine you have a magic box that can shrink a whole house into a marble (the quantum state). The paper shows that once the house is in the marble, you can identify it perfectly. But the process of putting the house into the marble right now takes a long time and a lot of effort (complexity), which currently cancels out some of the speed benefits.

Summary

The paper concludes that:

Quantum Image Representations (QImRs) can shrink images down to a tiny fraction of their classical size.
FRQI and QPIE are the best at this compression, even though they make the images look very similar to each other.
Despite this heavy compression, these quantum methods can classify images (tell a 0 from a 1) just as accurately as traditional computers.
The main benefit is memory efficiency: quantum computers need exponentially less space to hold the data, even if the process of getting the data into the quantum state is still a work in progress.

In short: Quantum computers can hold a photo in a matchbox and still recognize it perfectly, but getting the photo into the matchbox is currently the hard part.

Technical Summary: Analysis of Quantum Image Representations for Supervised Classification

Problem Statement
In the context of big data and artificial intelligence, there is a critical need to improve the efficiency of data storage, processing, and analysis. Quantum Image Processing (QIP) offers a potential solution by leveraging quantum mechanical properties such as superposition and entanglement to store and manipulate visual information more efficiently than classical computers. While classical $N \times N$ images require $N$ bits for storage, quantum representations can theoretically encode the same information using $O(\log N)$ qubits, offering an exponential reduction in memory resources. However, a comprehensive comparison of different Quantum Image Representations (QImRs) regarding their compression properties and their efficacy in supervised learning tasks remains necessary. Specifically, the paper addresses the trade-off between classification accuracy and memory requirements when using various QImRs as kernels in Support Vector Machines (SVM).

Methodology
The authors conduct a comparative analysis of four distinct QImRs for encoding grayscale images:

Tensor Network Representation (TNR): Uses Matrix Product States (MPS) to encode pixel values as real amplitudes in a structured quantum state, utilizing block-structured addressing to reflect spatial relationships.
Flexible Representation of Quantum Images (FRQI): Encodes pixel positions in $2n$ qubits and grayscale intensity in the probability amplitudes of a single color qubit via rotation angles.
Novel Enhanced Quantum Representation (NEQR): Encodes pixel positions in $2n$ qubits and grayscale intensity in $q$ qubits using basis states, allowing for deterministic retrieval.
Quantum Probability Image Encoding (QPIE): Encodes pixel positions in $2n$ qubits and grayscale intensity directly in the probability amplitudes of the state.

The study utilizes the MNIST and Fashion-MNIST datasets, pre-processing images to $16 \times 16$ grayscale resolution. The authors implement binary classification tasks (class 0 vs. classes 1–9) using Support Vector Machines (SVM). The quantum kernels are constructed by calculating the overlap (fidelity) between quantum states encoding different images. These quantum kernels are compared against a classical linear kernel. Simulations are performed using Pennylane for quantum circuit implementation and Scikit-Learn for the SVM classifier.

Key Contributions and Results

Compression Analysis: The authors analyze the compression capabilities of the four representations by examining the Gram matrices of 100 training images. The off-diagonal elements of these matrices represent the overlap between different image states.
- FRQI and QPIE demonstrate the highest compression, with off-diagonal elements largely falling in the range $[0.8, 1]$ , indicating that the encoded quantum states are very close to one another.
- NEQR shows the lowest compression, with off-diagonal elements in the range $[0, 0.4]$ , indicating more distinct states.
- TNR exhibits an intermediate level of compression ( $[0.4, 0.7]$ ).
Classification Performance: The study evaluates the accuracy of SVMs using these quantum kernels. The results indicate that quantum kernels achieve average classification accuracies comparable to the classical linear kernel on both datasets (e.g., $\approx 99\%$ on MNIST and $\approx 94-96\%$ on Fashion-MNIST).
Resource Efficiency: While accuracy is comparable, the storage requirements differ significantly. The classical linear kernel requires 2,048 bits for a $16 \times 16$ $16 \times 16$ image. In contrast, the quantum representations require exponentially fewer resources:
- TNR: 9 qubits
- FRQI: 9 qubits
- NEQR: 16 qubits
- QPIE: 8 qubits
Bond Dimension in TNR: For TNR, the authors investigate the effect of bond dimension on accuracy. They find that accuracy improves with increasing bond dimension, converging at a dimension of 4 for MNIST and 6 for Fashion-MNIST, suggesting moderate bond dimensions are sufficient to capture essential image correlations.

Significance and Claims
The paper claims that quantum kernels based on QImRs provide a viable alternative to classical kernels for supervised image classification, offering comparable accuracy while requiring exponentially fewer resources for image storage. The authors highlight that the non-orthogonality of quantum embedding states, derived from storing pixels via superposition, is central to this enhanced memory storage.

However, the authors maintain a modest stance regarding the immediate practical advantage. They explicitly note that the current best-known protocols for loading exact classical data into an $n$ -qubit state require a number of quantum gates scaling as $O(2^n)$ . This state preparation cost currently predominates the complexity of the quantum algorithms and challenges the realization of a quantum advantage. The paper concludes that while the theoretical storage benefits are clear, efficient data loading remains an open question, and future work should explore RGB images and the impact of noise on quantum hardware implementations.

1. The Four "Shrinking" Methods

2. The "Squeeze" Test (Compression)

3. The "Guessing Game" (Classification)

4. The Catch (State Preparation)

Summary

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