Schmidt Decomposition-Based Methods for Efficient Quantum Image Encoding

Abstract: In quantum image processing, a fundamental step is encoding classical image data into quantum states. This can be achieved using methods such as Flexible Representation of Quantum Images (FRQI), Quantum Probability Image Encoding (QPIE), and Novel Enhanced Quantum Representation (NEQR). However, on real quantum hardware, these encodings can quickly lead to circuits with many gates, large circuit depth, and high qubit usage, which is a problem for Noisy Intermediate-Scale Quantum (NISQ) devices. In this work, we investigate whether low-rank state approximation, formulated via Schmidt decomposition, can help reduce this complexity. The method keeps only the most significant parts of a quantum state's entanglement structure, making state preparation more efficient while preserving most of the image information. We compare the three encoding techniques in their original form and with low-rank approximation, evaluating metrics such as circuit depth, CNOT count, MSE, and visual quality of reconstructed images. The results reveal meaningful trade-offs between accuracy and resource efficiency, with the FRQI model achieving a 97 percent reduction in circuit depth while maintaining a near-perfect reconstruction (MSE of about 0.27). This demonstrates the potential of low-rank techniques for advancing practical quantum image processing on near-term hardware.

• The paper demonstrates that a low-rank approximation via Schmidt decomposition effectively reduces circuit complexity in quantum image encoding.
• Empirical evaluations on FRQI, QPIE, and NEQR show reductions up to 97% in circuit depth and CNOT gates with minimal loss in reconstruction fidelity.
• The study highlights the discrete entanglement spectra of natural images, paving the way for hardware-aware optimizations in scalable quantum image processing.

Schmidt Decomposition-Based Low-Rank Approximation for Efficient Quantum Image Encoding

Introduction

Quantum Image Processing (QIP) leverages the inherent parallelism and entanglement in quantum computing to represent and manipulate large-scale visual data exponentially more compactly than classical methods. However, practical QIP is severely constrained by the limited qubit counts, noisy gate operations, and shallow circuit depth of NISQ hardware. State-of-the-art quantum image encoding models—Flexible Representation of Quantum Images (FRQI), Quantum Probability Image Encoding (QPIE), and Novel Enhanced Quantum Representation (NEQR)—each pose distinct trade-offs between accuracy, qubit resources, and preparation complexity. Addressing the common bottleneck of high circuit depth and entanglement, this work introduces a unified, model-independent optimization for quantum image encoding using Low-Rank Approximation (LRA) via the Schmidt decomposition (2606.10874).

Quantum Image Representation and Challenges

Quantum encodings such as FRQI, QPIE, and NEQR reduce the memory requirement for image data from $\mathcal{O}(N)$ classical bits to $\mathcal{O}(\log_2 N)$ qubits by mapping pixels to amplitudes, phases, or basis states in multiqubit systems. FRQI encodes grayscale via phase rotation of a color qubit entangled with position qubits; QPIE encodes direct amplitude values; NEQR represents pixel intensities in binary via basis encoding. Each encoding, while theoretically enabling quantum acceleration, demands highly entangled, deep quantum circuits for state preparation—a process infeasible on current NISQ machines due to rapid decoherence and gate infidelity.

Low-Rank Approximation via Schmidt Decomposition

The core of this work is the hypothesis that natural images, dominated by spatial correlations and redundancies, can be well-compressed in the Schmidt basis. The Schmidt decomposition factorizes a bipartite quantum state into a sum of orthonormal product states, with a handful of dominant Schmidt coefficients containing most of the structured visual information.

By truncating the expansion to $r \ll k$ terms, a low-rank quantum state is produced with controlled approximation error $\epsilon = \sum_{i = r+1}^{k} \lambda_i^2$. This approach ensures that the resources required for state preparation (gate count, circuit depth) scale logarithmically with the Schmidt rank $r$ instead of the exponential image size $k$, reducing practical quantum circuit complexity substantially.

Empirical Evaluation and Numerical Results

The evaluation involves encoding a $64 \times 64$ grayscale image using each representation (FRQI, QPIE, NEQR) and applying LRA at various Schmidt ranks. Key metrics include circuit depth, CNOT gate count, reconstruction fidelity (Mean Squared Error, MSE), compilation time, and visual reconstruction quality.

Figure 1: 64×64 grayscale input image used for all encoding experiments.

For FRQI, the original circuit (full-rank, $r = 64$) exhibits extreme resource requirements—depth over $385{,}000$ and over $221{,}000$ CNOTs. Applying LRA with $\mathcal{O}(\log_2 N)$0 reduces both metrics by approximately $\mathcal{O}(\log_2 N)$1 (to $\mathcal{O}(\log_2 N)$2 and $\mathcal{O}(\log_2 N)$3) with MSE $\mathcal{O}(\log_2 N)$4, corresponding to sub-perceptual errors in pixel intensity.

Figure 2: FRQI reconstructions for increasing Schmidt ranks $\mathcal{O}(\log_2 N)$5.

QPIE is the most qubit/compact-efficient, using only $\mathcal{O}(\log_2 N)$6 qubits for the $\mathcal{O}(\log_2 N)$7 image, and achieves similar compression: $\mathcal{O}(\log_2 N)$8 yields a circuit depth reduction of $\mathcal{O}(\log_2 N)$9 and nearly optimal MSE $r \ll k$0. However, QPIE's amplitude encoding comes with significant transpilation overhead due to the complexity of mapping arbitrary classical data vectors to normalized quantum amplitudes.

Figure 3: QPIE reconstructions for increasing Schmidt ranks $r \ll k$1.

NEQR, while affording deterministic pixel-level reconstructions, is the most resource-intensive, with circuit depths up to $r \ll k$2 and over $r \ll k$3 CNOTs at full rank. At $r \ll k$4, depth and CNOTs are reduced by $r \ll k$5 and $r \ll k$6, respectively, still yielding visually indistinguishable results from the original.

Figure 4: NEQR reconstructions for increasing Schmidt ranks $r \ll k$7.

The sublinear scaling of CNOT count and circuit depth with growing Schmidt rank (log-log profiles) is documented for all representations. The MSE sharply decreases with increasing $r \ll k$8, indicating that most image information is concentrated in a few leading Schmidt components.

Figure 5: CNOT count vs. Schmidt rank (log scale) after applying Low-Rank Approximation (LRA).

Figure 6: Circuit depth vs. Schmidt rank (log scale).

Figure 7: Mean Squared Error (MSE) vs. Schmidt rank (log scale).

A notable emergent property is the discrete progression in attainable fidelities and circuit resources: only for $r \ll k$9 $\epsilon = \sum_{i = r+1}^{k} \lambda_i^2$0 do new information and visible changes appear, reflecting a non-continuous entanglement structure tied to the powers-of-two structure of quantum registers and the redundancies of natural imagery.

Implications and Theoretical Significance

The demonstrated reductions (up to $\epsilon = \sum_{i = r+1}^{k} \lambda_i^2$1 in circuit depth and CNOTs for FRQI) without loss of reconstruction fidelity bold the claim that exact state preparation is both unnecessary and suboptimal for practical QIP on NISQ hardware. This result challenges the prevalent notion that quantum advantage requires full-rank, maximally entangled quantum states. Instead, for most quantum image representations with natural (non-random) data, low-Schmidt-rank truncations preserve essential semantics, reduce noise susceptibility, and facilitate successful execution on current hardware. The results further suggest that discrete entanglement spectra are a general feature of quantum encodings and may inform optimal choices of Schmidt rank for measurable trade-offs in circuit complexity versus fidelity.

In practice, LRA enables more resilient quantum image pipelines, supports integration with hardware-aware quantum circuit optimizations, and lays the foundation for scalable quantum-classical hybrid workflows where only dominant entangled features are utilized. The results are also theoretically relevant for understanding redundancy and compressibility in quantum states representing structured data.

Limitations and Future Developments

The study used noiseless simulators; real NISQ hardware, with its decoherence and gate errors, may further highlight the value of low-rank circuits. The evaluation focused on a single moderate-complexity image; broader tests with diverse and higher-resolution data, as well as alternative data partitioning schemes for the Schmidt split, are prerequisites for generalizing the results. The origin of stepwise fidelity improvements remains incompletely explained, inviting analytical work on the interplay between quantum image encoding structure and entanglement spectra.

Future directions include:

• Hardware-based benchmarking of LRA-optimized image encodings to empirically validate superior robustness against noise and decoherence.
• Development of quantum image processing primitives (filtering, pattern recognition, etc.) that directly operate on low-rank truncated states.
• Automated selection mechanisms for optimal Schmidt truncation based on image/statistical properties or downstream processing requirements.
• Hybrid LRA approaches combining model-dependent decompositions (tailored for specific encoding schemas) with general hardware-aware transpilation techniques.

Conclusion

Low-Rank Approximation through the Schmidt decomposition provides a principled, universally applicable strategy for compressing quantum image states, reducing their entanglement and circuit complexity while preserving visual fidelity. These findings have practical relevance for the realization of scalable quantum image processing on NISQ devices and contribute to a deeper theoretical understanding of redundancy, entanglement structure, and information content in quantum representations of classical data. The discrete structure of improvement with increasing Schmidt rank opens new questions about the architecture of quantum encoding and the practical threshold for quantum advantage in imaging tasks (2606.10874).