FRQI: Flexible Quantum Image Representation

• FRQI is a quantum image encoding scheme that maps pixel positions and intensities to the amplitudes of a normalized quantum state, offering exponential register compression.
• It employs 2n+1 qubits with multi-controlled rotations, resulting in an exponential increase in circuit depth and gate count, which challenges its practical deployment on NISQ hardware.
• FRQI serves as a foundational tool in quantum image processing and machine learning, inspiring optimized circuit designs and hybrid classical-quantum pipelines to mitigate resource constraints.

The Flexible Representation of Quantum Images (FRQI) is a foundational quantum image encoding scheme that maps both the position and the intensity (color or grayscale) values of classical images into the amplitudes of a normalized quantum state. As a canonical method in quantum image processing, FRQI enables the compact representation, manipulation, and analysis of digital images on quantum hardware, offering exponential compression in register size compared to classical memory, but at the cost of significant circuit depth and experimental challenges.

1. Formalism and State Construction

Let $2^n \times 2^n$ be the target image size ($n \in \mathbb{N}$), resulting in $2^{2n}$ pixels. Each pixel is described by a unique index $i = 0, \ldots, 2^{2n}-1$, with value (gray or RGB) mapped to an angle $\theta_i \in [0, \frac{\pi}{2}]$ (for grayscale), or a suitable function for multi-channel. The FRQI state is defined as

$$|I(\theta)\rangle = \frac{1}{2^n} \sum_{i=0}^{2^{2n}-1} \left( \cos\theta_i\,|0\rangle_{\mathrm{c}} + \sin\theta_i\,|1\rangle_{\mathrm{c}} \right) \otimes |i\rangle_{\mathrm{p}}$$

where:

• $|0\rangle_{\mathrm{c}}, |1\rangle_{\mathrm{c}}$: computational basis states of the single "color" (or "intensity") qubit
• $|i\rangle_{\mathrm{p}}$: computational basis of the $2n$ position qubits encoding row and column index (concatenated as $i = y \cdot 2^n + x$)
• The scaling factor $n \in \mathbb{N}$0 ensures normalization

Typical angle encodings include $n \in \mathbb{N}$1 for 8-bit images ($n \in \mathbb{N}$2 the pixel value), but nonlinear variants exist (e.g., arcsin-based) (Sanchez et al., 2018, Iyengar et al., 2020, Amankwah et al., 2021, Kulkarni et al., 31 Jan 2025, Haque et al., 2022).

2. State Preparation and Circuit Realization

FRQI state preparation consists of:

1. Position register initialization: Apply Hadamard gates $n \in \mathbb{N}$3 to all position qubits, creating $n \in \mathbb{N}$4.
2. Color encoding: For each $n \in \mathbb{N}$5, apply a $n \in \mathbb{N}$6-controlled rotation $n \in \mathbb{N}$7 on the color qubit, conditioned on position register $n \in \mathbb{N}$8: $n \in \mathbb{N}$9.

This process requires $2^{2n}$0 qubits, and for a full $2^{2n}$1 image, a circuit with $2^{2n}$2 multi-controlled rotations, each typically decomposed into $2^{2n}$3 two-qubit gates (e.g., CNOTs) (Iyengar et al., 2020, Kulkarni et al., 31 Jan 2025, Geng et al., 2021). The total circuit depth and gate count thus scale as $2^{2n}$4, imposing a steep resource burden for large $2^{2n}$5.

3. Compression, Resource Considerations, and Efficient Variants

FRQI delivers dramatic compression in register width: storing all pixel data in only $2^{2n}$6 qubits (for comparison, direct Qubit Lattice encoding requires $2^{2n}$7 qubits) (Kulkarni et al., 31 Jan 2025, Parigi et al., 29 Jul 2025). However, the cost is a circuit depth exponential in image dimensions, hampering practical usability.

Circuit optimizations and variants:

• Gray-code/ladder decompositions: Reduce redundant CNOT operations for multi-controlled rotations, lowering total two-qubit gate count from $2^{2n}$8 to $2^{2n}$9 (Amankwah et al., 2021).
• EFRQI/SCMFRQI: Introduce auxiliary qubits and/or reset gates to cut the number of Toffoli gates per pixel; block-based (DCT) encoding reduces the number of required entangling operations by encoding only nonzero coefficients (Haque et al., 2022, Haque et al., 2022).
• Low-rank/Schmidt-truncated FRQI: Using the dominant Schmidt coefficients in a bipartition of the FRQI state achieves up to $i = 0, \ldots, 2^{2n}-1$0 reduction in circuit depth with visually negligible loss for moderate truncation ranks (e.g., on 64×64 images) (Pangeva et al., 9 Jun 2026).
• QPIXL: Clustering angle parameters and decomposing control patterns yields further pragmatic gate count reductions while maintaining high-fidelity reconstruction, lowering CNOT requirements up to 90% in empirical tests for scientific images (Amankwah et al., 2021).

4. Quantum Information and Entanglement Structure

The global FRQI state is pure, but analysis of subsystem entropies reveals nontrivial multipartite entanglement. After tracing out the color qubit, the reduced density matrix spectrum determines the von Neumann entropy $i = 0, \ldots, 2^{2n}-1$1, which quantifies classical-quantum and quantum-quantum (e.g., position-color) correlations.

• Quantum joint entropy is sensitive to pixel angle values (even under monotonic relabeling that would leave classical joint entropy invariant) (Sanchez et al., 2018).
• The total quantum correlation, $i = 0, \ldots, 2^{2n}-1$2, can reach double the classical joint entropy, highlighting multipartite entanglement not visible in classical statistics.
• Classical SVM and quantum-kernel-based SVMs on FRQI representations achieve comparable accuracy (e.g., 97% for 16×16 images) but with FRQI requiring exponentially fewer qubits than the classical bit count; however, the high mutual overlap between distinct FRQI states reflects substantial quantum compression, possibly at the expense of discriminability in some contexts (Parigi et al., 29 Jul 2025).

5. Practical Limitations on Quantum Hardware

Despite its theoretical elegance, FRQI is currently impractical for large images on NISQ devices:

• Decoherence and noise: The requirement for numerous controlled rotations means that decoherence times $i = 0, \ldots, 2^{2n}-1$3 are exceeded for $i = 0, \ldots, 2^{2n}-1$4 (i.e., images bigger than 2×2) on available superconducting platforms (Iyengar et al., 2020, Mastriani, 2020, Geng et al., 2021).
• Measurement constraints: Only $i = 0, \ldots, 2^{2n}-1$5-basis measurements are feasible on current platforms; amplitude information is therefore collapsed to probabilistic binary outcomes, precluding direct retrieval of grayscale values and resulting in severe fidelity loss (Mastriani, 2020).
• Cl2Qu interface: No traceable, scalable classical-to-quantum mapping for general images; practical state-preparation requires explicit control wiring for every nonzero angle (Kulkarni et al., 31 Jan 2025, Geng et al., 2021).
• Noise-mitigation and circuit simplification: Methods such as MARY (a CNOT-count-halved decomposer), calibration matrix inversion, and compressed encoding mitigate, but do not eliminate, circuit infidelity and resource barriers (Geng et al., 2021, Amankwah et al., 2021, Pangeva et al., 9 Jun 2026).

Empirically, on superconducting hardware, only 2×2 images are reliably retrievable via FRQI, even with mitigative error correction, and sampling needs grow exponentially to resolve amplitudes for larger images (Geng et al., 2021, Mastriani, 2020).

6. Applications and Algorithmic Adaptations

FRQI serves as the backbone for a range of quantum image processing and quantum machine learning protocols:

• Edge detection: Integration with Quantum Hadamard Edge Detection (QHED) circuits, using partial measurement to extract two amplitude branches; dynamic thresholding after FRQI-based neighbor difference routines yields noise-robust edge outlines (Shubha et al., 2024).
• Quantum-classical compression pipelines: DCTEFRQI, SCMFRQI, and related schemes employ energy compaction (blockwise DCT) and quantization to minimize quantum gate count, making quantum storage and post-processing tractable (Haque et al., 2022, Haque et al., 2022).
• Quantum kernels and QRNNs: FRQI-encoded images enable quantum support vector machines (QSVM) and QRNNs in classification, exploiting efficient memory embedding, as well as FRQI Pairs, which reduces quantum cell count exponentially by local consumption of coordinate bits (Potempa et al., 12 Dec 2025, Parigi et al., 29 Jul 2025).
• Quantum pixel library frameworks: Modular platforms (e.g. QPIXL++) leverage the FRQI formalism for uniform state representation, image compression, and circuit design across diverse application domains (Amankwah et al., 2021).

FRQI, while not alone sufficient for practical quantum image processing at scale, provides a universal substrate for the development of compressed, hybridized, or hardware-feasible schemes as the field matures.

7. Research Directions and Open Challenges

Key ongoing research areas focus on:

• Resource reduction: Exploiting circuit simplification, low-rank approximations, blockwise state-prep, and clustering to reduce gate count and circuit depth for NISQ-era deployment (Amankwah et al., 2021, Pangeva et al., 9 Jun 2026, Haque et al., 2022).
• Hybrid classical-quantum pipelines: Pairing FRQI-style quantum representations with classical preprocessing (e.g., PCA, DCT), postprocessing, or classical feature extraction to circumvent quantum hardware bottlenecks (Kulkarni et al., 31 Jan 2025, Haque et al., 2022).
• Measurement and retrieval improvements: Development of quantum schemes that permit faithful, low-depth amplitude readout, or encode color/gray information in retrievable basis states rather than amplitudes (Mastriani, 2020, Iyengar et al., 2020).
• Fully quantum learning pipelines: Leveraging FRQI as data loaders in quantum neural networks or kernel-based classification, with continued emphasis on scaling, expressivity, and noise resilience (Shubha et al., 2024, Potempa et al., 12 Dec 2025, Parigi et al., 29 Jul 2025).

While the limitations are significant, FRQI and its variants will remain a central theoretical and methodological pillar for quantum image processing—both as a compact encoding and as a laboratory for new circuit reduction and quantum information-theoretic techniques.

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Table 1. Key Properties of FRQI and Select Variants

Scheme	Qubit Count	Circuit Depth / Gate Count	Major Limitation
FRQI	2n+1	O(n·2^{2n})	Exponential gate cost
EFRQI	2n+2	O(n·2^{2n}) (half CNOT count)	Still exponential
SCMFRQI	q + 2n – 2log₂s + 1	O(K) (K = #nonzero coefficients)	RESET gate, block-approx error
QPIXL	2n+1	O(2^{2n}) after compression	Preprocessing cluster choice

Compression and resource scaling details are drawn from (Amankwah et al., 2021, Haque et al., 2022, Haque et al., 2022, Pangeva et al., 9 Jun 2026).

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References:

• (Sanchez et al., 2018) Application of Flexible Presentation of Quantum Images in Multipartite Correlations
• (Iyengar et al., 2020) Analysis of five techniques for the internal representation of a digital image inside a quantum processor
• (Shubha et al., 2024) Edge Detection Quantumized: A Novel Quantum Algorithm For Image Processing
• (Mastriani, 2020) Quantum Image Processing: the truth, the whole truth, and nothing but the truth about its problems on internal image representation and outcomes recovering
• (Amankwah et al., 2021) Quantum pixel representations and compression for $i = 0, \ldots, 2^{2n}-1$6-dimensional images
• (Haque et al., 2022) A novel state connection strategy for quantum computing to represent and compress digital images
• (Geng et al., 2021) Improved FRQI on superconducting processors and its restrictions in the NISQ era
• (Kulkarni et al., 31 Jan 2025) From Bits to Qubits: Challenges in Classical-Quantum Integration
• (Pangeva et al., 9 Jun 2026) Schmidt Decomposition-Based Methods for Efficient Quantum Image Encoding
• (Parigi et al., 29 Jul 2025) Supervised Quantum Image Processing
• (Potempa et al., 12 Dec 2025) FRQI Pairs method for image classification using Quantum Recurrent Neural Network
• (Haque et al., 2022) Advance quantum image representation and compression using DCTEFRQI approach