Scaling Adaptive Non-Local Observable Quantum Super-Resolution via Matrix Product States

Technical Summary: Scaling Adaptive Non-Local Observable Quantum Super-Resolution via Matrix Product States

Problem Statement
Image super-resolution (SR) aims to reconstruct high-resolution images from low-resolution inputs. While classical deep learning dominates this field, Variational Quantum Circuits (VQCs) have emerged as an alternative framework. A specific architecture, Adaptive Non-Local Observable VQCs (ANO-VQCs), improves upon standard VQCs by treating Hermitian measurement operators as trainable components rather than fixed Pauli observables. This allows for task-dependent combinations of multiqubit correlations.

However, simulating ANO-VQCs for image SR faces a severe scalability bottleneck. Using a one-pixel-to-one-qubit encoding, an input image of size $p \times p$ requires $n = p^2$ qubits. Conventional statevector simulation requires storing $2^n$ complex amplitudes, which becomes intractable for even modest input sizes (e.g., memory bottlenecks occur at $6 \times 6$ inputs or 36 qubits). While exact tensor-network (Exact TN) contraction offers an alternative, it also becomes computationally impractical beyond certain input sizes (e.g., $12 \times 12$ or 144 qubits). There is a need for a scalable simulation backend that can handle larger qubit counts while maintaining accuracy for training variational quantum algorithms.

Methodology
The authors propose a Matrix Product State (MPS) simulation framework tailored for ANO-VQC image SR. The workflow consists of four main stages:

1. Encoding and Circuit Design:

   • Low-resolution images (e.g., $7 \times 7$) are mapped to qubits via one-pixel-to-one-qubit encoding. Pixel values $x_i \in [0, 1]$ are converted to rotation angles $\phi_i = 2\pi x_i - \pi$.
   • The encoding circuit applies Hadamard gates, followed by pixel-dependent $RY(\phi_i)$ rotations and staggered nearest-neighbor CNOT gates.
   • The circuit depth $L$ is defined by the number of repeated encoding-entangling layers. Notably, the encoding gates are input-dependent but not independently trainable; trainable parameters reside in the observables and the classical reconstruction layer.

2. Adaptive Observable Feature Extraction:

   • Instead of fixed Pauli measurements, the framework uses trainable Hermitian observables $H_q$ acting on pairs of qubits (specifically image-local neighbors: horizontal, vertical, and diagonal).
   • To leverage the MPS backend efficiently, each adaptive observable is expanded in the Pauli basis: $H_q = \sum_\alpha c_{q,\alpha} P_\alpha$.
   • The expectation value $\langle H_q \rangle$ is computed as a linear combination of fixed Pauli expectation values $\langle P_\alpha \rangle$. Since the encoding circuit depends only on the input image, these Pauli expectation values are computed once and cached. The trainable parameters ($A_q, B_q, D_q$) only update the coefficients $c_{q,\alpha}$, separating expensive quantum feature evaluation from the optimization of the observable and reconstruction layers.

3. MPS Simulation:

   • The quantum state $|\psi\rangle$ is represented as a chain of local tensors connected by virtual bonds with dimension $\chi$.
   • Storage scales approximately as $O(n\chi^2)$ rather than $O(2^n)$.
   • One-qubit gates update local tensors, while two-qubit gates update adjacent tensors followed by bond dimension truncation.

4. Reconstruction and Training:

   • The extracted quantum feature vector is passed to a classical linear layer to reconstruct the high-resolution image.
   • Training minimizes a weighted loss function combining Mean Squared Error (MSE) and Learned Perceptual Image Patch Similarity (LPIPS).
   • Experiments were conducted on a single NVIDIA RTX 4070 GPU using the Fashion-MNIST dataset ($7 \times 7$ input $\to$ $28 \times 28$ output).

Key Contributions and Results

• Scalability Benchmarks:

  • The study compares statevector simulation, Exact TN contraction, and MPS simulation.
  • Statevector: Hits a memory bottleneck at $6 \times 6$ inputs (36 qubits).
  • Exact TN: Becomes computationally impractical beyond $12 \times 12$ inputs (144 qubits).
  • MPS: Successfully completes forward feature extraction for inputs up to $16 \times 16$ pixels (256 qubits) with a fixed bond dimension $\chi=8$. In the overlapping range, MPS is faster than Exact TN and outperforms statevector simulation for inputs larger than $4 \times 4$.

• Accuracy-Cost Trade-off (Bond Dimension Sweep):

  • For a fixed $7 \times 7$ input (49 qubits), the authors swept the bond dimension $\chi$ from 2 to 2048 across circuit depths $L=1$ to $L=4$.
  • The required $\chi$ to reach the "near-exact" regime (where MPS error approaches machine precision relative to Exact TN) increases with circuit depth:
    • $L=1$: $\chi=2$
    • $L=2$: $\chi=4$
    • $L=3$: $\chi=8$
    • $L=4$: $\chi=16$
  • At $\chi=16$, the Mean Absolute Error (MAE) for $L=4$ drops to $\approx 10^{-15}$, effectively matching Exact TN results without the exponential cost.

• Super-Resolution Performance:

  • Using $\chi=16$ (ensuring near-exact simulation for all tested depths), the authors trained models for $7 \times 7 \to 28 \times 28$ Fashion-MNIST SR.
  • Counter-intuitive Finding: The shallowest circuit ($L=1$) achieved the best performance across all metrics (Loss, LPIPS, PSNR, SSIM).
    • $L=1$: Loss 0.4864, PSNR 20.87 dB, SSIM 0.4248.
    • $L=4$: Loss 0.6984, PSNR 18.80 dB, SSIM 0.2922.
  • Increasing circuit depth did not improve reconstruction quality under this fixed near-exact MPS setting; in fact, deeper circuits degraded performance and increased training time.

Significance and Claims
The paper positions MPS as a scalable and controllable simulation backend for ANO-VQC image SR. Its primary significance lies in extending the practical simulation range of quantum algorithms beyond the limits of statevector and exact tensor-network methods, enabling the study of systems with up to 256 qubits on a single GPU.

The authors modestly claim that their results highlight the importance of jointly selecting circuit depth with observable design and reconstruction layers, noting that deeper circuits do not necessarily yield better SR quality. They conclude that MPS serves as a practical tool for studying large-scale variational quantum circuits, with future work planned to incorporate parallelization and distributed tensor-network computations to further extend the simulation scale.