Flexible Representation of Quantum Images (FRQI)

• FRQI is a quantum image representation that encodes a 2^n×2^n image using 2n+1 qubits by separating pixel addresses from grayscale values via angle encoding.
• It employs position-conditioned R_y rotations for state preparation, offering exponential storage savings while incurring probabilistic readout and high circuit complexity.
• Subsequent studies have refined its circuit synthesis with low-rank approximations and alternative angle maps to enhance image processing and mitigate hardware limitations.

Flexible Representation of Quantum Images (FRQI) is a quantum image representation in which a $2^n \times 2^n$ image is encoded as a uniform superposition over pixel addresses entangled with exactly one color qubit whose amplitudes are parameterized by pixel-dependent angles. In later overviews it is treated as one of the first methods for encoding classical images into quantum states, and as a foundational member of the family of quantum image representations that exploit address superposition and amplitude-based value storage rather than storing pixel values purely in computational basis states (Lang et al., 2024). Its central attraction is compactness: a grayscale image occupies $2n+1$ qubits, while the principal liabilities are the cost of state preparation, the probabilistic nature of readout, and the consequent difficulty of exact image recovery and routine image processing at scale (Parigi et al., 29 Jul 2025).

1. Foundational role within quantum image representations

FRQI was introduced by P. Q. Le, F. Dong, and K. Hirota in the paper “A flexible representation of quantum images for polynomial preparation, image compression, and processing operations,” and later surveys position it historically as an early and influential representation that motivated a substantial part of subsequent quantum image processing research (Lang et al., 2024). Its defining design choice is to separate pixel location from pixel value asymmetrically: the location register is basis-encoded across $2n$ qubits, whereas all grayscale information is funneled through a single color qubit by angle encoding.

This architecture places FRQI between two broad representational styles. Relative to basis-encoded schemes such as NEQR, FRQI minimizes color-register width and gives a mathematically compact state. Relative to pure amplitude encodings such as QPIE, it retains an explicit color qubit and thereby preserves a distinct locus for color-conditioned transformations. Later application-oriented work describes FRQI’s purpose in quantum image processing as achieving exponential savings in storage for $2^n \times 2^n$ images while supporting quantum-native color and geometric transforms, as well as downstream learning tasks based on state overlaps (Parigi et al., 29 Jul 2025).

2. State definition, normalization, and intensity-to-angle maps

For a $2^n \times 2^n$ image with pixels normalized to angles $\theta_i \in [0,\pi/2]$, FRQI is written as

$$\ket{I} = \frac{1}{2^n}\sum_{i=0}^{2^{2n}-1}\left(\cos\theta_i\ket{0}+\sin\theta_i\ket{1}\right)\otimes\ket{i}.$$

The normalization factor is explicitly emphasized in the literature: $2^{-n}$, not $2^{-2n}$, is the correct prefactor because the address register contains $2^{2n}$ basis states, while each color component already satisfies $2n+1$0 (Lang et al., 2024).

The canonical interpretation is that $2n+1$1 indexes the pixel address and the color qubit stores the grayscale value of the $2n+1$2-th pixel through the amplitudes $2n+1$3 and $2n+1$4. For 8-bit grayscale, one standard linear normalization used in worked examples is

$2n+1$5

which maps $2n+1$6 into $2n+1$7. Under that convention, the values $2n+1$8 map to $2n+1$9, respectively, yielding the standard $2n$0 demonstration state with addresses $2n$1 (Lang et al., 2024).

Later FRQI-based studies preserve the same state form but use variant angle maps according to task requirements. In Sobel- and Harris-style gradient detection, both nonlinear arcsine encoding and linear encoding are discussed, but the experiments use the linear map

$2n$2

because linear encoding preserves relative grayscale differences more uniformly for gradient computation (Sohail et al., 1 May 2026). In a traffic-classification study, a different application-specific convention maps 8-bit grayscale values to $2n$3 via

$2n$4

showing that while $2n$5 is canonical in much of the FRQI literature, alternative angular ranges do appear in downstream models (Innan et al., 2024).

3. Circuit synthesis and asymptotic resource profile

The logical qubit count for a $2n$6 grayscale image is $2n$7: $2n$8 address qubits and exactly one color qubit. A standard preparation circuit begins from $2n$9, applies Hadamard gates to the address register to create the uniform superposition over all positions, and then applies position-conditioned $2^n \times 2^n$0 rotations on the color qubit so that each address $2^n \times 2^n$1 selects the corresponding color amplitude (Lang et al., 2024).

In explicit gate terms, this preparation amounts to a bank of multi-controlled $2^n \times 2^n$2 rotations on the color qubit, one per pixel address, with the controls configured by the address qubits. Because the number of pixels is $2^n \times 2^n$3, direct state loading scales with image size rather than with qubit count. One source states the original FRQI preparation complexity as

$2^n \times 2^n$4

and characterizes retrieval as probabilistic because the color is encoded in the amplitudes of a single qubit (Parigi et al., 29 Jul 2025). A complementary survey formulates the same issue by noting that the number of simple gates is polynomial in the dimension of the underlying Hilbert space but therefore exponential in the number of qubits (Lang et al., 2024).

Subsequent circuit frameworks target this preparation bottleneck rather than the state definition itself. The QPIXL framework gives a unified construction for FRQI using only $2^n \times 2^n$5 and CNOT gates, no ancilla qubits, and a linear number of gates in the number of pixels; it also reports compression reductions of up to $2^n \times 2^n$6 for preparing FRQI states of example scientific images without sacrificing image quality (Amankwah et al., 2021). This does not alter the fact that exact loading remains linear in the pixel count, but it materially changes the constant factors and synthesis practicality.

4. Readout model and implications for image operations

FRQI’s measurement model is the source of both its compactness and its principal operational difficulty. Measuring the address register samples a pixel location $2^n \times 2^n$7, while measuring the color qubit returns a Bernoulli outcome governed by $2^n \times 2^n$8 and $2^n \times 2^n$9. As later reports emphasize, a single run typically reveals only one pixel sample, so reconstructing the entire image requires many circuit executions; for FRQI this is described as theoretically requiring an infinite number of executions for exact recovery (Lang et al., 2024).

The inversion problem becomes more delicate when the angle map is nonlinear. In the Sobel-based study, the authors note that under linear encoding the recovery of grayscale from measured color-qubit statistics involves a nonlinear inverse-sine step, which increases sensitivity to shot noise relative to the arcsine map used for analytical retrieval discussions (Sohail et al., 1 May 2026). This is one reason FRQI readout degrades more rapidly than QPIE under limited shot budgets in that work.

Operationally, FRQI does permit image manipulations at the circuit level. Because color is carried by a single qubit, intensity modifications can be implemented by unitary rotations on that qubit, while geometric transformations can act on the address register. At the same time, a survey that contrasts FRQI with NEQR explicitly characterizes FRQI as non-convenient for expressing image processing operations, meaning that the existence of such operators does not imply convenient or noise-robust implementations in general (Lang et al., 2024).

5. Successors, variants, and compression-oriented refinements

Later representations are best understood as targeted responses to FRQI’s trade-off profile.

Representation	Encoding / qubits	Contrast with FRQI
FRQI	One angle on one color qubit; $2^n \times 2^n$0 qubits	Compact, probabilistic retrieval
NEQR	Basis-encoded color in $2^n \times 2^n$1 or $2^n \times 2^n$2 qubits; $2^n \times 2^n$3 or $2^n \times 2^n$4 qubits	Deterministic readout, larger color register
QPIE	Pixel values in amplitudes of address register; $2^n \times 2^n$5 qubits	Fewer qubits, still probabilistic
TNR	Tensor-network / real-ket form; reported $2^n \times 2^n$6 qubits	Intermediate compression, different contraction structure

Comparative studies consistently describe NEQR as a response to FRQI’s disadvantages, especially probabilistic retrieval and awkward image-operation syntax, whereas QPIE pushes qubit minimization further by eliminating the color qubit entirely (Parigi et al., 29 Jul 2025). Survey literature also notes BRQI as a bitplane-oriented alternative that saves $2^n \times 2^n$7 qubits relative to GNEQR for 8-bit images and improves storage efficiency by a factor of $2^n \times 2^n$8, again reflecting the continued search for representations that relax one or another FRQI bottleneck (Lang et al., 2024).

Refinement work has concentrated on preparation depth. QPIXL provides a practical FRQI-compatible preparation template and compression mechanism (Amankwah et al., 2021). A Schmidt decomposition–based low-rank approximation study goes further by replacing exact FRQI loading with low-rank state preparation. On a $2^n \times 2^n$9 grayscale image, the full-rank FRQI circuit used $\theta_i \in [0,\pi/2]$0 qubits, depth $\theta_i \in [0,\pi/2]$1, and $\theta_i \in [0,\pi/2]$2 CNOTs; with rank $\theta_i \in [0,\pi/2]$3, the depth fell to $\theta_i \in [0,\pi/2]$4 and the CNOT count to $\theta_i \in [0,\pi/2]$5, approximately $\theta_i \in [0,\pi/2]$6 reductions in both metrics, while the reconstructed image retained $\theta_i \in [0,\pi/2]$7 (Pangeva et al., 9 Jun 2026).

Hardware-oriented gate engineering has pursued smaller, local improvements. On superconducting backends, the MARY construction replaces a naive MCRY decomposition, reducing the controlled-rotation cost per pixel from $\theta_i \in [0,\pi/2]$8 CX, $\theta_i \in [0,\pi/2]$9 SX, and $$\ket{I} = \frac{1}{2^n}\sum_{i=0}^{2^{2n}-1}\left(\cos\theta_i\ket{0}+\sin\theta_i\ket{1}\right)\otimes\ket{i}.$$0 single-qubit $$\ket{I} = \frac{1}{2^n}\sum_{i=0}^{2^{2n}-1}\left(\cos\theta_i\ket{0}+\sin\theta_i\ket{1}\right)\otimes\ket{i}.$$1 gates to $$\ket{I} = \frac{1}{2^n}\sum_{i=0}^{2^{2n}-1}\left(\cos\theta_i\ket{0}+\sin\theta_i\ket{1}\right)\otimes\ket{i}.$$2 CX, $$\ket{I} = \frac{1}{2^n}\sum_{i=0}^{2^{2n}-1}\left(\cos\theta_i\ket{0}+\sin\theta_i\ket{1}\right)\otimes\ket{i}.$$3 SX, and $$\ket{I} = \frac{1}{2^n}\sum_{i=0}^{2^{2n}-1}\left(\cos\theta_i\ket{0}+\sin\theta_i\ket{1}\right)\otimes\ket{i}.$$4 $$\ket{I} = \frac{1}{2^n}\sum_{i=0}^{2^{2n}-1}\left(\cos\theta_i\ket{0}+\sin\theta_i\ket{1}\right)\otimes\ket{i}.$$5 gates, while also avoiding CX gates between location qubits (Geng et al., 2021). Another extension translates FRQI from qubits to qutrits through

$$\ket{I} = \frac{1}{2^n}\sum_{i=0}^{2^{2n}-1}\left(\cos\theta_i\ket{0}+\sin\theta_i\ket{1}\right)\otimes\ket{i}.$$6

with the suggestion that qutrit realizations may scale more favorably than the regular FRQI qubit scheme (Khandelwal et al., 2022).

6. Empirical uses, hardware restrictions, and critical assessments

FRQI has been used in several quantum machine learning and quantum image processing pipelines. In supervised kernel classification on Fashion-MNIST downsampled to $$\ket{I} = \frac{1}{2^n}\sum_{i=0}^{2^{2n}-1}\left(\cos\theta_i\ket{0}+\sin\theta_i\ket{1}\right)\otimes\ket{i}.$$7, FRQI showed very high compression in Gram-matrix overlap space, with off-diagonal elements mostly in $$\ket{I} = \frac{1}{2^n}\sum_{i=0}^{2^{2n}-1}\left(\cos\theta_i\ket{0}+\sin\theta_i\ket{1}\right)\otimes\ket{i}.$$8, and achieved $$\ket{I} = \frac{1}{2^n}\sum_{i=0}^{2^{2n}-1}\left(\cos\theta_i\ket{0}+\sin\theta_i\ket{1}\right)\otimes\ket{i}.$$9 average accuracy using $2^{-n}$0 qubits per image, compared with $2^{-n}$1 for a classical linear kernel using $2^{-n}$2 bits per image (Parigi et al., 29 Jul 2025). In vehicle-road cooperation experiments, FRQI-based $2^{-n}$3 classifiers outperformed the corresponding NEQR pipelines, reaching $2^{-n}$4 on Carla and $2^{-n}$5 on Cropped LISA for $2^{-n}$6 inputs, although the best overall model in that study used angle encoding rather than FRQI (Innan et al., 2024). A later QRNN study using the FRQI Pairs method reported $2^{-n}$7 test accuracy on MNIST with $2^{-n}$8 total qubits, consisting of $2^{-n}$9 FRQI qubits for $2^{-2n}$0 images and $2^{-2n}$1 memory qubits (Potempa et al., 12 Dec 2025).

Classical image-processing analogues have also been ported to FRQI. A bilinear interpolation scheme for FRQI images reported better PSNR and SSIM than FRQI nearest-neighbor interpolation in simulation (Yan et al., 2020). Sobel-based edge detection and Harris-style corner detection have been implemented with FRQI and QPIE; in that setting, FRQI produced outputs consistent with classical operators under noiseless simulation, but QPIE was found to be more stable under limited shot budgets (Sohail et al., 1 May 2026). Another work adapted Quantum Hadamard Edge Detection to FRQI by measuring the color qubit, collapsing the address register into a QPIE-like amplitude image, and then reusing the measured qubit as the QHED ancilla; the reported outcome was improved object outlines relative to traditional QHED (Shubha et al., 2024). Beyond processing and learning, FRQI has served as the substrate for entropy-based studies of multipartite correlations, where quantum total correlation $2^{-2n}$2 was reported to reach approximately twice the classical joint entropy and to remain sensitive to color changes that leave classical joint histograms invariant (Sanchez et al., 2018).

The practical status of FRQI on NISQ hardware remains mixed. A superconducting-backend study found that with MARY simplification and measurement-error mitigation, $2^{-2n}$3 FRQI images were reconstructable on real IBM devices with approximately $2^{-2n}$4–$2^{-2n}$5 relative difference on newer heavy-hex hardware, but $2^{-2n}$6 images already failed to preserve recognizable structure on hardware, and simulator studies indicated “usable” reconstruction only up to about $2^{-2n}$7 with $2^{-2n}$8 shots (Geng et al., 2021). Earlier comparative papers were more skeptical, arguing from simulator and hardware tests that FRQI’s non-basis outputs, deep controlled-rotation networks, decoherence exposure, and sampling burden made it technically nonviable on then-current processors (Iyengar et al., 2020, Mastriani, 2020). Taken together, these results suggest that FRQI is most defensible when compact state storage, overlap computation, or simulator-based analysis is more important than exact dense image recovery, and when approximate preparation, compression, or low-rank structure can substantially reduce the raw circuit cost.