Quantum Probability Image Encoding (QPIE)

• Quantum Probability Image Encoding (QPIE) is a quantum data-loading strategy that maps classical grayscale images into quantum states via amplitude encoding using logarithmic qubit scaling.
• It uses recursive controlled-rotation circuits to transform normalized pixel intensities into basis state amplitudes, facilitating quantum machine learning, image compression, and feature extraction.
• Advanced techniques like tensor network approximations and JPEG-assisted pipelines mitigate exponential state preparation costs, enhancing feasibility on NISQ devices.

Quantum Probability Image Encoding (QPIE) is a quantum data-loading strategy designed to embed classical grayscale images directly into quantum states via amplitude encoding. It achieves minimal qubit usage—logarithmic in the number of pixels—by mapping normalized pixel intensities to state amplitudes. QPIE has emerged as a key foundation for quantum image processing, supporting downstream applications in quantum machine learning, image compression, and quantum-enhanced feature extraction. Its construction, implementation overhead, and empirical trade-offs have been comprehensively benchmarked against alternatives such as FRQI, NEQR, and tensor network approaches.

1. Mathematical Formulation and State Preparation

Given a non-negative grayscale image $I$ of $N$ pixels ($I_k \ge 0$), QPIE normalizes the intensity vector and encodes it as the amplitudes of a pure quantum state over $n=\log_2 N$ qubits:

$$|\psi_{\text{QPIE}}\rangle = \sum_{k=0}^{N-1} c_k \, |k\rangle,\qquad c_k = \frac{I_k}{\sqrt{\sum_{j=0}^{N-1} I_j^2}}$$

This construction uses $\ell_2$-normalization to satisfy the unit-norm condition $\sum_k |c_k|^2 = 1$. Each computational-basis state $|k\rangle$ corresponds to a unique pixel address, typically via row- or column-major ordering. For square images of $2^n \times 2^n$ pixels, the encoding uses $2n$ qubits, the theoretical minimum for uniquely specifying all amplitudes (Parigi et al., 29 Jul 2025, Lang et al., 2024, Sarkar, 2023).

State preparation proceeds via recursive sequences of single- and multi-controlled $N$0 rotations (binary-tree construction), requiring the computation of partial sums and spherical angle decompositions:

• For $N$1 amplitudes, $N$2 controlled-rotation gates and $N$3 CNOT gates are required (Pangeva et al., 9 Jun 2026, Lang et al., 2024).
• Restricting to real-valued intensities (common in image data) enables the exclusive use of real rotations, omitting arbitrary-phase gates (Sarkar, 2023).

QPIE’s state preparation circuit is equivalent to amplitude encoding for classical data, but specialized for images with a natural mapping between the 2D pixel grid and the computational basis.

2. Resource Scaling and Optimization Strategies

QPIE achieves the minimal qubit count for encoding $N$4-pixel images: $N$5 qubits. However, its generic state-preparation circuit is exponentially costly in gate count and depth:

• Direct state preparation is $N$6 in both gate count and circuit depth for arbitrary images (Parigi et al., 29 Jul 2025, Pangeva et al., 9 Jun 2026).
• Resource optimizations are possible for sparse or structured images, such as skipping zero-intensity regions or exploiting classical precomputation of rotation angles (Lang et al., 2024).

Table 1: Comparison of Qubit and Preparation Complexity (for $N$7 images)

Representation	Qubit Count	Preparation Complexity
TNR	$N$8	(instance-dependent)
FRQI	$N$9	$I_k \ge 0$0 (orig.), $I_k \ge 0$1 (EFRQI)
NEQR	$I_k \ge 0$2 ($I_k \ge 0$3=bit-depth)	$I_k \ge 0$4
QPIE	$I_k \ge 0$5	$I_k \ge 0$6

QPIE achieves the highest compression ratio in terms of bits-per-qubit, with a $I_k \ge 0$7 reduction on 16×16 images compared to their classical representation (Parigi et al., 29 Jul 2025).

3. Advanced State Preparation: Tensor Network and Schmidt Decomposition

To address the exponential circuit depth and CNOT count on near-term hardware, QPIE can be hybridized with tensor network techniques:

• Matrix Product State (MPS) Approximation: For sufficiently smooth images, the amplitude vector can be approximated by an MPS of low bond dimension $I_k \ge 0$8, yielding a circuit of $I_k \ge 0$9 gates and $n=\log_2 N$0 depth (Bohun et al., 2024). For smooth, low-entanglement images, $n=\log_2 N$1 suffices, resulting in dramatically shallower circuits.
• Schmidt Decomposition (Low-Rank Truncation): By truncating the bipartite Schmidt rank to the top-$n=\log_2 N$2 terms (cumulative weight $n=\log_2 N$3), QPIE circuits achieve $n=\log_2 N$4 scaling in entangling gates while incurring only a modest mean-squared error ($n=\log_2 N$5 for $n=\log_2 N$6) (Pangeva et al., 9 Jun 2026). Empirically, an $n=\log_2 N$7 depth reduction was realized on $n=\log_2 N$8 images at little perceptual cost.

Hybrid amplitude-compression methods thus substantively expand the tractable image size on NISQ devices.

4. Hybrid Classical-Quantum Image Preparation (JPEG-Assisted QPIE)

JPEG-assisted QPIE (JQPIE) and its quantization-free variant (QF-JQPIE) further reduce the quantum state-preparation overhead:

• JQPIE loads JPEG-quantized DCT coefficients into the quantum register, implements blockwise decompression with inverse zigzag permutation and quantization, and applies a quantum inverse DCT (Boosari, 5 Feb 2026).
• QF-JQPIE omits quantization, loading DCT coefficients directly and yielding a unitary, ancilla-free quantum pipeline.

This hybrid reduces the main cost from $n=\log_2 N$9 to $$|\psi_{\text{QPIE}}\rangle = \sum_{k=0}^{N-1} c_k \, |k\rangle,\qquad c_k = \frac{I_k}{\sqrt{\sum_{j=0}^{N-1} I_j^2}}$$0 by truncating frequency coefficients, where $$|\psi_{\text{QPIE}}\rangle = \sum_{k=0}^{N-1} c_k \, |k\rangle,\qquad c_k = \frac{I_k}{\sqrt{\sum_{j=0}^{N-1} I_j^2}}$$1 is the number of active coefficients per $$|\psi_{\text{QPIE}}\rangle = \sum_{k=0}^{N-1} c_k \, |k\rangle,\qquad c_k = \frac{I_k}{\sqrt{\sum_{j=0}^{N-1} I_j^2}}$$2 block. Empirical studies observe $$|\psi_{\text{QPIE}}\rangle = \sum_{k=0}^{N-1} c_k \, |k\rangle,\qquad c_k = \frac{I_k}{\sqrt{\sum_{j=0}^{N-1} I_j^2}}$$3–$$|\psi_{\text{QPIE}}\rangle = \sum_{k=0}^{N-1} c_k \, |k\rangle,\qquad c_k = \frac{I_k}{\sqrt{\sum_{j=0}^{N-1} I_j^2}}$$4 reductions in gate count/depth with negligible or positive PSNR/SSIM impact compared to direct QPIE.

5. Practical Applications: Quantum Edge and Feature Extraction

QPIE’s amplitude encoding directly supports quantum implementations of classical image processing primitives:

• Quantum Gradient Computation: QPIE states lend themselves to linear, translation-invariant differencing via a lag-2 permutation pipeline, enabling Sobel-like gradient kernels in superposition (Sohail et al., 1 May 2026).
• Edge and Corner Detection: QPIE with quantum Sobel–Harris pipelines achieves superior edge density/fragmentation and up to 100% corner-detection accuracy at zero false-positive rate (in ideal simulation) versus FRQI. QPIE-based outputs exhibit lower shot noise and more coherent gradient sensitivity, especially under restrictive measurement budgets (Sohail et al., 1 May 2026).
• In all cases, state preparation dominates resource cost, highlighting the necessity of hybrid classical-quantum and compressed-preparation strategies for scalability.

6. Empirical Benchmarks and Quantum Image Kernel Learning

QPIE has been applied to benchmark tasks in quantum machine learning and classification:

• In supervised binary classification with quantum kernels $$|\psi_{\text{QPIE}}\rangle = \sum_{k=0}^{N-1} c_k \, |k\rangle,\qquad c_k = \frac{I_k}{\sqrt{\sum_{j=0}^{N-1} I_j^2}}$$5 (via QPIE), QPIE matches FRQI accuracy (≈97%) on Fashion-MNIST (16×16) at half the qubit cost and only marginally below the classical linear kernel (98%) (Parigi et al., 29 Jul 2025).
• QPIE-encoded images produce moderately compressed Gram matrices (intermediate off-diagonal overlaps $$|\psi_{\text{QPIE}}\rangle = \sum_{k=0}^{N-1} c_k \, |k\rangle,\qquad c_k = \frac{I_k}{\sqrt{\sum_{j=0}^{N-1} I_j^2}}$$6–$$|\psi_{\text{QPIE}}\rangle = \sum_{k=0}^{N-1} c_k \, |k\rangle,\qquad c_k = \frac{I_k}{\sqrt{\sum_{j=0}^{N-1} I_j^2}}$$7), placing it between highly-compressed FRQI and less-compressed NEQR/TNR embeddings for kernel SVMs.

7. Limitations, Trade-offs, and Future Directions

The key advantages of QPIE are minimal qubit usage, expressivity for arbitrary grayscale data, and compatibility with quantum transformations (QFT, wavelets, global filters) (Lang et al., 2024). However:

• Measurement Bottleneck: Exact pixel retrieval is only possible in the infinite-shot (sampling) limit; typically, $$|\psi_{\text{QPIE}}\rangle = \sum_{k=0}^{N-1} c_k \, |k\rangle,\qquad c_k = \frac{I_k}{\sqrt{\sum_{j=0}^{N-1} I_j^2}}$$8 shots are required per pixel, imposing resource demands on quantum readout.
• Exponential State Preparation: Full-rank QPIE preparation is infeasible for large images without compression or structure; hybrid methods are critical for scalability on NISQ processors (Pangeva et al., 9 Jun 2026, Bohun et al., 2024, Boosari, 5 Feb 2026).
• No direct storage use: QPIE’s encoding is transient and suitable only for in-circuit computation; measurement collapses the entire superposition.

Ongoing developments focus on adaptive, structure-exploiting compression (tensor networks, JPEG-DCT domain, Schmidt truncation), hybrid pipelines leveraging classical pre/post-processing, and noise-resilient state-preparation tailored to available hardware error budgets.

• (Parigi et al., 29 Jul 2025) Supervised Quantum Image Processing
• (Pangeva et al., 9 Jun 2026) Schmidt Decomposition-Based Methods for Efficient Quantum Image Encoding
• (Lang et al., 2024) Representation of Classical Data on Quantum Computers
• (Bohun et al., 2024) Entanglement scaling in matrix product state representation of smooth functions and their shallow quantum circuit approximations
• (Sohail et al., 1 May 2026) Quantum Gradient-Based Approach for Edge and Corner Detection Using Sobel Kernels
• (Boosari, 5 Feb 2026) Hybrid Quantum Image Preparation via JPEG Compression
• (Sarkar, 2023) Novel Design of Quantum Circuits for Representation of Grayscale Images
• (Mohsen et al., 2021) Image Compression and Classification Using Qubits and Quantum Deep Learning