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Chinese Physics B

Fixed-Topology UCR State Encoding for High-Dimensional Quantum Reinforcement Learning

Junchen Han, Fengtao Xiang, Hao Shi, Ming Zhang

Published Sep 8, 2026• 0 citations
Publisher

Abstract

Abstract In the era of noisy intermediate-scale quantum (NISQ) computing, the stable integration of high-dimensional classical states into quantum neural networks is a critical challenge for quantum reinforcement learning. Existing encoding schemes either rely heavily on classical neural networks, which can overshadow the training contributions of the quantum component, or use standard quantum input methods such as amplitude encoding, which suffer from limited trainability, noise sensitivity, and high implementation costs. To address this challenge, this paper repurposes uniformly controlled rotations (UCR) from static state preparation and static data encoding into a state-encoding interface for quantum neural networks, elevating the process of integrating classical states into quantum neural networks to an independent method-ological layer. Across Gymnasium continuous-control tasks, our UCR encoding achieves the best peak return, final return, and evaluation-curve AUC compared to representative angle encoding and amplitude encoding, with the AUC improving by +5.9 to +53.5 return units over angle encoding. While our UCR encoding is not the method with the fewest gates, it eliminates the classical neural networks on which angle encoding relies during dimensionality reduction, provides a fixed, sample-independent topology, and supports explicit resource accounting. On BipedalWalker-v3, UCR encoding reduces the actor parameter count of angle encoding from 1003 to 229 and increases the final return from 220.4 to 264.9, which is higher than the final return of amplitude encoding. A systematic evaluation of gate-level noise shows that the absolute performance advantage of UCR encoding in complex continuous control tasks persists in the low-noise regime, but this advantage window narrows as the input dimension and quantum circuit depth increase. For high-dimensional classical state inputs, UCR encoding achieves a more stable performance–resource trade-off than typical quantum encoding methods. Motivated by the problem of inputting classical states into quantum neural networks, our approach may be further explored in broader machine learning scenarios and higher-dimensional applications.

DOI
10.1088/1674-1056/aea3dc

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Abstract In the era of noisy intermediate-scale quantum (NISQ) computing, the stable integration of high-dimensional classical states into quantum neural networks is a critical challenge for quantum reinforcement learning. Existing encoding schemes either rely heavily on classical neural networks, which can overshadow the training contributions of the quantum component, or use standard quantum input methods such as amplitude encoding, which suffer from limited trainability, noise sensitivity, and high implementation costs. To address this challenge, this paper repurposes uniformly controlled rotations (UCR) from static state preparation and static data encoding into a state-encoding interface for quantum neural networks, elevating the process of integrating classical states into quantum neural networks to an independent method-ological layer. Across Gymnasium continuous-control tasks, our UCR encoding achieves the best peak return, final return, and evaluation-curve AUC compared to representative angle encoding and amplitude encoding, with the AUC improving by +5.9 to +53.5 return units over angle encoding. While our UCR encoding is not the method with the fewest gates, it eliminates the classical neural networks on which angle encoding relies during dimensionality reduction, provides a fixed, sample-independent topology, and supports explicit resource accounting. On BipedalWalker-v3, UCR encoding reduces the actor parameter count of angle encoding from 1003 to 229 and increases the final return from 220.4 to 264.9, which is higher than the final return of amplitude encoding. A systematic evaluation of gate-level noise shows that the absolute performance advantage of UCR encoding in complex continuous control tasks persists in the low-noise regime, but this advantage window narrows as the input dimension and quantum circuit depth increase. For high-dimensional classical state inputs, UCR encoding achieves a more stable performance–resource trade-off than typical quantum encoding methods. Motivated by the problem of inputting classical states into quantum neural networks, our approach may be further explored in broader machine learning scenarios and higher-dimensional applications.
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Quantum Neural Networks (QNNs) offer a framework for learning functions from classical data, but their expressive power depends critically on how inputs are encoded into quantum states. Angle encoding can require resources that scale with the number of features, whereas amplitude encoding represents ddd features using only O(log⁡2d)\mathcal{O}(\log_2 d)O(log2​d) qubits. This compactness comes at a representational cost: normalization makes conventional amplitude encoding invariant to nonzero real rescaling of the input, up to an unobservable global phase. We show that this invariance induces an exact, architecture-independent restriction on the resulting hypothesis class. No downstream quantum circuit can distinguish inputs differing by nonzero real scaling. We develop a general condition under which deterministic preprocessing can remove this ambiguity and show that nonlinearity alone is insufficient. Based on this analysis, we introduce a fixed nonlinear feature map that provably distinguishes all real inputs under subsequent amplitude encoding, without adding trainable parameters and while retaining logarithmic qubit scaling. We evaluate the resulting QNN on three canonical classification benchmarks and a nonlinear synthetic dataset against conventional amplitude and angle encoding, as well as a classical neural network. Across these tasks, the proposed encoding consistently improves classification accuracy over both quantum encoding baselines by 21$-$44 %. Our results establish normalization-induced scale invariance as a concrete representational bottleneck of amplitude encoding and demonstrate that it can be removed while preserving a compact quantum representation, marking a step toward quantum models capable of capturing arbitrary relationships in classical data.
A central challenge in quantum machine learning is the state preparation bottleneck that describes the prohibitive computational cost of loading high-dimensional classical data into a quantum state. Although amplitude encoding can represent 2n2^n2n-dimensional data using only nnn qubits in principle, preparing arbitrary states remains computationally expensive, typically requiring variational optimization of a parameterized quantum circuit for each individual data instance. In this work, we propose a method that avoids iterative optimization by training a classical neural network to map input data directly to the continuous parameters of a fixed quantum circuit. We demonstrate the generation of quantum image states with high fidelity on data not seen during training. Since all optimization is performed once during training, the resulting model encodes new inputs in a single inference step, providing a scalable pathway for data loading in near-term quantum algorithms. We validate our method on the MNIST and Fashion-MNIST datasets, achieving fidelities up to 0.992 on unseen images and reducing the per-data-instance runtime by more than 5000-fold.
Quantum convolutional neural networks (QCNNs) offer a promising architecture for near-term quantum machine learning by combining hierarchical feature extraction with modest parameter growth. However, any QCNN operating on classical data must rely on an encoding scheme to embed inputs into quantum states, and this choice can dominate both performance and resource requirements. This work presents an implementation-level comparison of three representative encodings -- Angle, Amplitude, and a Hybrid phase/angle scheme -- for QCNNs under depolarizing noise. We develop a fully differentiable PyTorch--Qiskit pipeline with a custom autograd bridge, batched parameter-shift gradients, and shot scheduling, and use it to train QCNNs on downsampled binary variants of MNIST and Fashion-MNIST at 4×44\times 44×4 and 8×88\times 88×8 resolutions. Our experiments reveal regime-dependent trade-offs. On aggressively downsampled 4×44\times 44×4 inputs, Angle encoding attains higher accuracy and remains comparatively robust as noise increases, while the Hybrid encoder trails and exhibits non-monotonic trends. At 8×88\times 88×8, the Hybrid scheme can overtake Angle under moderate noise, suggesting that mixed phase/angle encoders benefit from additional feature bandwidth. Amplitude-encoded QCNNs are sparsely represented in the downsampled grids but achieve strong performance in lightweight and full-resolution configurations, where training dynamics closely resemble classical convergence. Taken together, these results provide practical guidance for choosing QCNN encoders under joint constraints of resolution, noise strength, and simulation budget.
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Quantum computing promises significant computational advantages over classical methods in certain applications but is currently too unstable for executing complex tasks. High-fidelity quantum circuits are critical to overcoming this challenge, yet they are computationally challenging to design. On current noisy intermediate-scale quantum (NISQ) hardware, shallower circuits with fewer two-qubit gates generally run with higher fidelity. Increased fidelity can be sought by sequentially rewriting the circuit, which involves a vast search space. This costly search space exploration could be circumvented by using reinforcement learning (RL) to teach an agent to rewrite the circuit in a sequential decision-making process. This thesis aims to implement a Deep Q-Network (DQN) algorithm to optimize BernsteinVazirani (BVZ) circuits. The DQN algorithm interacts with a Cirq-based environment. The environment provides valid rewrite actions to the DQN algorithm, which gains a scalar reward emphasizing depth and gate reduction. In the thesis we compare two different state encodings as input for the neural network: a compact state summarizing vector representation with three features, and an image-like encoding that maps circuit state topology with tokens through a fixed embedding. Both input encodings use the same action rewrite set and reward design. The network is implemented in PyTorch and trained with Adam optimizer, and runs on GPU via CUDA. Empirically, the agent shows promising initial optimizing performance. On several BVZ circuits with varying qubit count and topology, the agent drives the depth toward the theoretical optimum of three. The simplistic vector encoding already reaches reasonable circuit depth plateaus, with the image-like encoding together with hyperparameter tuning improving reliability, showing avenues for continued improvement. We conclude that DQN is a viable foundation for automated circuit optimization on BVZ circuits. To achieve reliable circuit optimization at scale, future research should investigate even richer encoding methods and reward function shaping. Finally, the results support applying the DQN to be applied to circuit families beyond BVZ.