NEQR: Enhanced Quantum Image Representation

• NEQR is a quantum image representation method that deterministically encodes pixel intensities and spatial coordinates into dedicated qubits.
• Its state preparation uses Hadamard gates and multi-controlled NOT operations, with Boolean and block-wise optimizations mitigating exponential circuit complexity.
• NEQR enables practical applications in segmentation, quantum neural networks, and cryptographic protocols by ensuring direct pixel access through computational-basis measurements.

The Novel Enhanced Quantum Representation (NEQR) is a quantum image representation framework designed to encode classical grayscale (and, by extensions, color) images in a form directly recoverable from the computational basis of a quantum register. NEQR arose as a response to constraints in probabilistic/amplitude-based encodings such as the Flexible Representation of Quantum Images (FRQI), particularly targeting deterministic pixel retrieval, improved measurement reliability, and decoupling of color amplitude from spatial superposition. In NEQR, pixel intensity values are encoded as binary patterns in dedicated qubits, while spatial coordinates are mapped to other qubit subregisters, yielding a structurally uniform and deterministic quantum state over the image domain.

1. Mathematical Definition and Core Register Structure

For a 2ⁿ × 2ⁿ grayscale image with $q$-bit pixel depth, NEQR constructs a normalized quantum state over $q + 2n$ qubits: $q$ intensity (color) qubits, $n$ row-index qubits, and $n$ column-index qubits. The state is

$$|I_{\textrm{NEQR}}\rangle = \frac{1}{2^n} \sum_{Y=0}^{2^n-1} \sum_{X=0}^{2^n-1} |C_{Y,X}\rangle \otimes |Y\rangle \otimes |X\rangle,$$

where $$|C_{Y,X}\rangle = |C_{Y,X}^{q-1}C_{Y,X}^{q-2}\dots C_{Y,X}^0\rangle$$ is the $q$-bit intensity encoding, and $|Y\rangle$, $|X\rangle$ are the $q + 2n$0-bit binary register encodings for pixel coordinates $q + 2n$1 (Parigi et al., 29 Jul 2025, Haque et al., 2022, Wang et al., 2023, Pangeva et al., 9 Jun 2026, Iyengar et al., 2020, Mastriani, 2020, Ganguly, 2022).

The NEQR state enables deterministic and direct retrieval of any pixel value through measurement in the computational ($q + 2n$2) basis; a measurement reveals both the position and its associated $q + 2n$3-bit intensity value without recourse to quantum state tomography.

2. State Preparation: Circuit Construction and Resource Scaling

NEQR state preparation is decomposed into three principal stages:

1. Initialize all $q + 2n$4 qubits in $q + 2n$5.
2. Apply Hadamard gates to the $q + 2n$6 coordinate qubits, generating a uniform superposition over all possible $q + 2n$7 positions.
3. For each pixel $q + 2n$8 and for each bit $q + 2n$9 in its $q$0-bit intensity, apply a $q$1-controlled-NOT (generalized Toffoli) gate to flip the $q$2-th color qubit if and only if $q$3 (Ganguly, 2022, Haque et al., 2022, Wang et al., 2023, Parigi et al., 29 Jul 2025).

Resource scaling is dominated by step 3: for an image of $q$4 pixels, the total number of such multi-controlled gates is $q$5. Each multi-controlled operation decomposes to $q$6 Toffolis and further to $q$7 CNOTs and single-qubit gates, resulting in an exponential total circuit depth and gate count in $q$8: $q$9 For $n$0 (i.e., 256 × 256 images) and $n$1 (8-bit grayscale), this scales rapidly beyond the capabilities of current NISQ-class quantum processors (Mastriani, 2020, Ganguly, 2022, Parigi et al., 29 Jul 2025).

3. Advantages and Deterministic Readout

NEQR's computational-basis encoding confers several advantages:

• Deterministic Pixel Recovery: Unlike angle-based (FRQI) or amplitude encoding (QPIE), NEQR allows all pixel intensity bits to be read directly and exactly from a single measurement shot, as each bit is mapped to a separate qubit (Parigi et al., 29 Jul 2025, Wang et al., 2023).
• Orthogonality: Each image is a pure basis state superposition; overlapping (i.e., nonorthogonal) patterns are avoided, reducing ambiguities in image reconstruction and supporting lossless quantum signatures (Şahin et al., 2018).
• State Structure: The explicit division between coordinate and color registers creates a modular structure, facilitating implementations of pixelwise operations, thresholding, and blockwise transforms (Haque et al., 2022, Wang et al., 2023).

4. Circuit Optimizations: Gate Compression and Block-wise Encoding

Given NEQR's prohibitive resource scaling, multiple approaches have been developed to reduce the circuit depth and gate count:

• Boolean Optimization via ESOP → PPRM: Each color-bit encoding for the $n$2-th color-qubit constitutes a Boolean function of $n$3 coordinate bits, naturally expressed in Exclusive-Or Sum-of-Products (ESOP) form. By applying Positive Polarity Reed-Muller (PPRM) transformations, the number of high-control-count MCNOT gates can be greatly reduced without introducing ancillary qubits. For exponential MCNOT decompositions the run-time is reduced from $n$4 to $n$5, with observed compression ratios ≳99% on 256 × 256 images. Linear decompositions retain a compression ratio ≳ 50% (Iranmanesh et al., 2024).
• Block-wise SC-NEQR: The SCMNEQR approach partitions images into small blocks, encoding them sequentially. Within each block, an auxiliary qubit (reset after each pixel) is used to minimize Toffoli chains: the connection between state labels and color bits is established with a single reset operation, dramatically decreasing gate depth and overall width. For images partitioned into 16 × 16 blocks, SCMNEQR reduces circuit depth and qubit overhead by ~30–40%, with negligible loss in PSNR (Haque et al., 2022).
• Low-rank State Preparation via Schmidt Truncation: The NEQR state is represented via Schmidt decomposition across a bipartition of coordinate and intensity qubits, then truncated at rank $n$6. This replaces pixelwise controlled-X operations with $n$7 multiplexed state-preparation blocks, reducing preparation depth and CNOT count by up to 70%—with minimal mean square error when $n$8 is chosen appropriately (e.g., $n$9 of $n$0 achieves $n$1) (Pangeva et al., 9 Jun 2026).

Optimization	Resource Saving	Effect on Fidelity
PPRM (QC)	50–99% fewer gates	None (exact)
Block-wise (SC)	~30–40% bit-rate	Negligible loss
Schmidt trunc.	70% gate/depth cut	Small MSE, tunable

5. Applications and Integration in Quantum Image Processing

NEQR underpins various quantum image-processing primitives:

• Segmentation (Thresholding): NEQR’s register separation supports efficient implementation of comparators and segmenters, as the color register can be manipulated in-place based on position-wise logical controls. Scalable, multi-threshold circuits with $n$2 cost (independent of $n$3) enable ternary and higher-order segmentation within a single run (Wang et al., 2023, Wang et al., 2023).
• Quantum Neural Networks (QNNs): NEQR-prepared images serve as data loading mechanisms for QNN architectures. NEQR pre-processing can yield moderate accuracy improvements in image classification tasks, but resource costs remain prohibitive compared to classical pipelines (Ganguly, 2022).
• Cryptographic Protocols: NEQR is used in quantum image signature and authentication schemes, especially those involving quantum Fourier transforms and secret keying. The deterministic encoding and basis separation allow robust information hiding and permutation-based obfuscation for signature protocols (Şahin et al., 2018).
• Compression and Rate-Distortion Optimization: Block-based and functionally optimized NEQR variants can integrate with DCT/quantization for joint quantum-classical lossy or lossless compression (Haque et al., 2022).

6. Experimental Results and Limitations on NISQ Hardware

Empirical studies consistently highlight that NEQR's exponential circuit complexity and the prevalence of multi-controlled-NOT gates severely restrict physical realizability on contemporary NISQ platforms. For small examples (e.g., $n$4, $n$5), NEQR has been validated via simulator and, partially, on superconducting qubit arrays, but scaling beyond $n$614 qubits or circuit depths exceeding $n$7 renders current hardware infeasible due to decoherence and error propagation. Attempts to prepare NEQR states for moderate image sizes ($n$8, $n$9) result in negligible output fidelity, rapidly decaying to random or highly mixed states (Mastriani, 2020, Iyengar et al., 2020, Ganguly, 2022).

A common misconception is that NEQR, due to computational-basis encoding, is inherently robust to noise. In practice, accumulated errors and long gate sequences, as well as unintended entanglement between coordinate and color subspaces, make deterministic retrieval unreliable, especially when compared to linearly scaling Boolean encoding schemes (e.g., QBIP), which exhibit superior hardware robustness (Mastriani, 2020, Iyengar et al., 2020).

7. Comparative Analysis and Future Directions

Relative to established quantum image models, NEQR offers a tradeoff profile summarized as follows:

• Compared to FRQI: NEQR increases the number of required qubits (from $$|I_{\textrm{NEQR}}\rangle = \frac{1}{2^n} \sum_{Y=0}^{2^n-1} \sum_{X=0}^{2^n-1} |C_{Y,X}\rangle \otimes |Y\rangle \otimes |X\rangle,$$0 to $$|I_{\textrm{NEQR}}\rangle = \frac{1}{2^n} \sum_{Y=0}^{2^n-1} \sum_{X=0}^{2^n-1} |C_{Y,X}\rangle \otimes |Y\rangle \otimes |X\rangle,$$1), but supports deterministic basis-state measurement and direct pixel access; FRQI relies on amplitude encoding and requires tomography for image retrieval (Parigi et al., 29 Jul 2025, Iyengar et al., 2020).
• Compared to QBIP: QBIP achieves dramatically reduced circuit complexity and superior fidelity by only encoding Boolean image information, sacrificing full gray/color recovery for efficiency. NEQR is less practical for large-scale image tasks under current hardware but is mathematically more expressive (Mastriani, 2020, Iyengar et al., 2020).
• Optimization Landscape: The design of more efficient NEQR-loading circuits, exploitation of block-encoding, advanced Boolean minimization (e.g., PPRM/ESOP), reset-based ancilla reuse, and low-rank state approximation remain active research areas (Iranmanesh et al., 2024, Pangeva et al., 9 Jun 2026, Haque et al., 2022). Hybrid amplitude–basis schemes and partial loading strategies may further reduce requirements.

Open challenges include scalable, hardware-efficient data loading, robust error mitigation for high-depth circuits, and extension to color and multi-channel images without exponential register growth.

• (Iranmanesh et al., 2024) Gate Optimization of NEQR Quantum Circuits via PPRM Transformation
• (Pangeva et al., 9 Jun 2026) Schmidt Decomposition-Based Methods for Efficient Quantum Image Encoding
• (Haque et al., 2022) Block-wise quantum grayscale image representation and compression scheme using state connection
• (Wang et al., 2023) An improved two-threshold quantum segmentation algorithm for NEQR image
• (Wang et al., 2023) A quantum segmentation algorithm based on local adaptive threshold for NEQR image
• (Parigi et al., 29 Jul 2025) Supervised Quantum Image Processing
• (Ganguly, 2022) Classification of NEQR Processed Classical Images using Quantum Neural Networks (QNN)
• (Şahin et al., 2018) Security of NEQR Quantum Image by Using Quantum Fourier Transform with Blind Trent
• (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
• (Iyengar et al., 2020) Analysis of five techniques for the internal representation of a digital image inside a quantum processor