Abstract Errors in the current generation of quantum processors pose a significant challenge towards practical-scale implementations of quantum machine learning (QML) as they lead to trainability issues arising from noise-induced barren plateaus, as well as performance degradations due to the noise accumulation in deep circuits even when QML models are free from barren plateaus. Quantum error correction (QEC) protocols are being developed to overcome hardware noise, but their extremely high spacetime overheads, mainly due to magic state distillation, make them infeasible for near-term practical implementation. This work proposes the idea of partial QEC for QML models and identifies a sweet spot where distillations are omitted to significantly reduce overhead. By assuming error-corrected two-qubit Controlled- Z s (Clifford operations), we demonstrate that the QML models remain trainable even when single-qubit gates are subjected to ≈ 0.2 % depolarizing noise, corresponding to a gate error rate of ≈ 0.13 % under randomized benchmarking. Further analysis based on various noise models, such as phase-damping and thermal-dissipation channels at low temperature, indicates that the QML models are trainable independent of the mean angle of over-rotation, or can even be improved by thermal damping that purifies a quantum state away from depolarizations. While it may take several years to build quantum processors capable of fully fault-tolerant QML, our work proposes a resource-efficient solution for trainable and high-accuracy QML implementations in noisy environments.
Abstract Errors in the current generation of quantum processors pose a significant challenge towards practical-scale implementations of quantum machine learning (QML) as they lead to trainability issues arising from noise-induced barren plateaus, as well as performance degradations due to the noise accumulation in deep circuits even when QML models are free from barren plateaus. Quantum error correction (QEC) protocols are being developed to overcome hardware noise, but their extremely high spacetime overheads, mainly due to magic state distillation, make them infeasible for near-term practical implementation. This work proposes the idea of partial QEC for QML models and identifies a sweet spot where distillations are omitted to significantly reduce overhead. By assuming error-corrected two-qubit Controlled- Z s (Clifford operations), we demonstrate that the QML models remain trainable even when single-qubit gates are subjected to ≈ 0.2 % depolarizing noise, corresponding to a gate error rate of ≈ 0.13 % under randomized benchmarking. Further analysis based on various noise models, such as phase-damping and thermal-dissipation channels at low temperature, indicates that the QML models are trainable independent of the mean angle of over-rotation, or can even be improved by thermal damping that purifies a quantum state away from depolarizations. While it may take several years to build quantum processors capable of fully fault-tolerant QML, our work proposes a resource-efficient solution for trainable and high-accuracy QML implementations in noisy environments.
At the intersection of quantum computing and machine learning, quantum machine learning (QML) is poised to revolutionize artificial intelligence. However, the vulnerability of the current generation of quantum computers to noise and computational error poses a significant barrier to this vision. Whilst quantum error correction (QEC) offers a promising solution for almost any type of hardware noise, its application requires millions of qubits to encode even a simple logical algorithm, rendering it impractical in the near term. In this chapter, we examine strategies for integrating QEC and quantum error detection (QED) into QML under realistic resource constraints. We first quantify the resource demands of fully error-corrected QML and propose a partial QEC approach that reduces overhead while enabling error correction. We then demonstrate the application of a simple QED method, evaluating its impact on QML performance and highlighting challenges we have yet to overcome before we achieve fully fault-tolerant QML.
Abstract Quantum machine learning (QML) is an emerging field that promises advantages such as faster training, improved reliability and superior feature extraction over classical counterparts. However, its implementation on quantum hardware is challenging due to the noise inherent in these systems, necessitating the use of quantum error correction (QEC) codes. Current QML research remains primarily theoretical, often assuming noise-free environments and offering little insight into the integration of QEC with QML implementations. To address this, we investigate the performance of a simple, parity-classifying Variational Quantum Classifier (VQC) implemented with the [[4,2,2]] error-detecting stabiliser code in a simulated noisy environment. To our knowledge, this is the first implementation of a stabiliser-based error detection code to a QML algorithm. We invoke ancilla qubits to logically encode rotation gates, and classically simulate the logically-encoded VQC under two simple noise models representing gate noise and environmental noise. We demonstrate that the stabiliser code improves the training accuracy at convergence compared to noisy implementations without error detection. However, we find that the effectiveness and reliability of error detection is contingent upon keeping the ancilla qubit error rates below a specific threshold, due to the propagation of ancilla errors to the physical qubits. Our results provide an important insight that extends to QML implemented with full error-correction: when the QEC code requires ancilla qubits for logical rotations but cannot fully correct errors propagated between ancilla and physical qubits, the maximum achievable accuracy of the QML model is constrained. This highlights the need for additional error correction or mitigation strategies to support the practical implementation of QML algorithms with QEC on quantum devices.
Quantum error correction and fault-tolerance have provided the possibility for large scale quantum computations without a detrimental loss of quantum information. A very natural class of gates for fault-tolerant quantum computation is the Clifford gate set and as such their usefulness for universal quantum computation is of great interest. Clifford group gates augmented by magic state preparation give the possibility of simulating universal quantum computation. However, experimentally one cannot expect to perfectly prepare magic states. Nonetheless, it has been shown that by repeatedly applying operations from the Clifford group and measurements in the Pauli basis, the fidelity of noisy prepared magic states can be increased arbitrarily close to a pure magic state~\cite{Bravyi}. We investigate the robustness of magic state distillation to perturbations of the initial states to arbitrary locations in the Bloch sphere due to noise. Additionally, we consider a depolarizing noise model on the quantum gates in the decoding section of the distillation protocol and demonstrate its effect on the convergence rate and threshold value. Finally, we establish that faulty magic state distillation is more efficient than fault-tolerance-assisted magic state distillation at low error rates due to the large overhead in the number of quantum gates and qubits required in a fault-tolerance architecture. The ability to perform magic state distillation with noisy gates leads us to conclude that this could be a realistic scheme for future small-scale quantum computing devices as fault-tolerance need only be used in the final steps of the protocol.
Quantum error correction and fault-tolerance have provided the possibility for large scale quantum computations without a detrimental loss of quantum information. A very natural class of gates for fault-tolerant quantum computation is the Clifford gate set and as such their usefulness for universal quantum computation is of great interest. Clifford group gates augmented by magic state preparation give the possibility of simulating universal quantum computation. However, experimentally one cannot expect to perfectly prepare magic states. Nonetheless, it has been shown that by repeatedly applying operations from the Clifford group and measurements in the Pauli basis, the fidelity of noisy prepared magic states can be increased arbitrarily close to a pure magic state [1]. We investigate the robustness of magic state distillation to perturbations of the initial states to arbitrary locations in the Bloch sphere due to noise. Additionally, we consider a depolarizing noise model on the quantum gates in the decoding section of the distillation protocol and demonstrate its effect on the convergence rate and threshold value. Finally, we establish that faulty magic state distillation is more efficient than fault-tolerance-assisted magic state distillation at low error rates due to the large overhead in the number of quantum gates and qubits required in a fault-tolerance architecture. The ability to perform magic state distillation with noisy gates leads us to conclude that this could be a realistic scheme for future small-scale quantum computing devices as fault-tolerance need only be used in the final steps of the protocol.
Variational quantum algorithms (VQAs) represent the most promising approach for achieving quantum advantage in near-term supervised learning, yet they face two fundamental challenges that severely limit their practical deployment: device noise degrades classification performance, and barren plateaus render gradient-based training ineffective. We address both challenges simultaneously through a novel hybrid architecture that selectively applies stabilizer quantum error correction only to the most noise-sensitive circuit components while preserving the variational structure essential for NISQ-era feasibility. Our approach combines Steane [7, 1, 3] encoding around entangling layers with a learned variational recovery mechanism (VarQEC). We establish rigorous theoretical foundations through two main contributions: First, we prove an explicit error-suppression theorem showing that selective protection reduces logical error scaling from O(p) to O(p 2) in the physical depolarizing rate p, with the exact leading constant p L ≤ 147 9 p 2 + O(p 3) derived through deterministic combinatorial enumeration of harmful Pauli patterns. Second, we demonstrate enhanced trainability by proving that selective encoding mitigates barren plateaus (reducing exponential suppression from physical to logical qubit scaling) while simultaneously improving noise-induced gradient attenuation from O(p) to O(p 2) through rigorous Weingarten calculus analysis. The manuscript provides complete implementation details including gate-level circuits, unambiguous training algorithms, and reproducible numerical experiments, positioning this work as both a theoretical advance and a practical framework for noise-resilient quantum machine learning.
Quantum computation has the promise to solve problems that are not efficiently solvable by classical means. Yet, due to the fragile nature of quantum states, it is unavoidable that quantum systems are subject to decoherence, gate imperfections, and measurement errors. These problems stand as a major roadblock to the construction of large-scale fault-tolerant quantum computers. Conventional quantum error correction (QEC) codes, such as the surface code or Shor code, need to use an extensive amount of redundancy and classical control, which can be both computationally expensive and challenging to scale. In this context, we propose to investigate the use of machine learning (ML) techniques by realizing on-line identification, localization and correction of these quantum errors for fault-tolerant quantum computation by means of fault-tolerant quantum computing processes. Our model introduces a hybrids ML-QEC architecture that combines classical neural networks and decision-tree classifiers with quantum syndrome decoding. These models are trained on error syndromes produced using the approximation of common noise models (for example, depolarizing noise, dephasing and amplitude damping), simulated under different QEC schemes. The machine learning methods are trained with quantum error patterns and have a capability to rapidly infer where errors are likely to occur, as well as recommend correction operations with high confidence. As a result, the system reduces dramatically the delay and computational costs when compared to conventional decoding methods. There is also a self-adaptive learning module in the work, which recommences training on fresh quantum error distributions, meaning the system can adapt to changes in the environment (and also the decay of the hardware) as time progresses. The study further carries out ML predictions inside a real-time loop and cross-checks them with quantum state tomography and entanglement fidelity such as indicators. Experimental characterization on dedicated devices as well as noise models show that our ML-enhanced decoders reach higher fidelity and lower logical error rates compared to baseline decoders, especially in non-uniform or time-dependent noise. This work is a significant stride towards realizing practical fault-tolerant quantum computation by combining quantum error correction with the versatility and scalability of machine learning. With intelligent models to dictate learning, our proposed framework is able to eliminate a lot of trial-and-error manual processes in the conventional heuristic-based approach, and points to a potential path to achieve scalable quantum information processing platforms.