Quantum Error Correction and Detection for Quantum Machine Learning

TL;DR

This work tackles the challenge of protecting quantum machine learning from hardware noise by analyzing the infeasibility of full QEC in the near term and proposing pragmatic strategies. It shows that partial QEC, which protects Clifford operations while omitting distillation for non-Clifford gates, can dramatically reduce overhead and preserve trainability in variational quantum circuits, with a representative result around a net single-qubit error rate of $1.33\times10^{-3}$ and an effective $\epsilon_T$ near $10^{-4}$. Separately, it investigates quantum error detection using the $[[4,2,2]]$ stabiliser code for a two-qubit VQC, demonstrating detection-driven improvements at low noise but revealing limitations due to ancilla-induced error propagation and the existence of threshold ancilla error rates. The findings argue for a hybrid fault-tolerance approach that combines QEC, error mitigation, and careful circuit/algorithm design to enable practical QML on noisy hardware.

Abstract

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.

Figures (8)

• Figure 1: The detailed circuit design of the variational circuit presented in this paper. The data is encoded into the circuit through amplitude encoding to minimize the requirement on the number of qubits and conserve memory for simulation. The parameterized unitary consists of multiple layers of unitaries, with each layer containing a sequence of single-qubit rotations with parameterized rotation axes and angles, denoted as $R_{lm}$ as shorthand for $R_{m {n}_{lm}}(\theta_{lm})$, followed by a sequence of entangling Controlled-$Z$ gates without trainable parameters. An error channel $\mathcal{E}$ is added after every ideal gate $R_{lm}$, constitutes $\Tilde {R}_{lm}$. The inferred probabilities for each potential label are evaluated by measuring the Pauli-$Z$ expected values on each of the qubits. This figure is adapted from Ref. haiyue_partial_QEC
• Figure 2: Demonstration of how logical operators are implemented on patch-based surface codes introduced in Ref. surface_codes. White dots indicate data qubits, black dots indicate syndrome extraction qubits with orange and blue strips representing Pauli-$X$ and $Z$ stabilizers, respectively. For operators induced from the Clifford group, including $X$, $Z$ and $CX$ gates, their logical operators can be encoded from many physical operators directly, without the need for ancilla qubits surface_codesqec_lattice_surgery. For operators outside the Clifford group, such as the $T$ gate, its logical operator $T_L$ cannot be implemented directly, but must be teleported from an ancilla logical qubit in the state $\ket{T_L}=\frac{1}{\sqrt{2}}(\ket{0_L}+e^{i\pi/4}\ket{1_L})$ via magic state injection. It turns out that $\ket{T_L}$ must be prepared from a single, physical qubit $\ket{T}$ state first, and then perform stabilizer measurements qec_lattice_surgery. Or, one could choose to carry out magic state distillation with very high spacetime cost. (b) Without the redundancy of encoding one logical operator from multiple physical operators, the logical error rate for $T$ gates is comparable to the physical $T$ error rate. However, if the state is distilled properly, the logical gate error rate can be suppressed to the same level as other Clifford gates. This figure is adapted from Ref. haiyue_partial_QEC.
• Figure 3: The performance of the QVC in MNIST number classification problems with (a) 75 quantum layers but without a classical layer, and (b) 75 layers with a fully-connected classical layer and noisy two-qubit gate. The classification success rates inferred from the test datasets are plotted against the total number of images trained. The noise-free simulation is highlighted with black, thickened lines to contrast with noisy ones. Since we employ amplitude encoding, the number of pixels per image is exactly $2^n$, which corresponds to the dimension of $\sqrt{2^{n}}\cross \sqrt{2^{n}}$, where $n$ is the number of qubits for the variational circuit. The legend denotes the strengths of the depolarizing channel from 0 to $5.11\cross 10^{-3}$. (c) The corresponding cost function values (left axis) and their average gradients squared (right axis), $\mathbb{E}_k$$\left((\nabla_{\theta_k}\mathcal{L})^2\right)$, of the trainable parameters (classical and quantum), where the arrows indicate the axis correspondence of the plots. A clear trend of flattening loss landscapes and vanishing gradients can be observed as the depolarizing strength increases. When the gradients drop to a scale around $10^{-8}$, i.e. $p_{\text{depol}}=5.11\times10^{-3}$, due to shot noise, the model becomes distinctively not trainable. (d) For the noisy two-qubit gate plot where the classical layer is included, the averaged cost function gradients with respect to all parameters, including both the quantum and classical layers, actually increase after being trained with more images. In contrast, the classical layer gradients by themselves are behaving as expected, as shown in the zoomed figure. This figure is reproduced from Ref. haiyue_crosstalk with permission.
• Figure 4: The gradients squared averaged over all trainable parameters and all iterations in the training versus the depolarizing strength $p$ in logarithmic scale. Error bar takes the standard deviation of the mean average among all iterations. This figure is reproduced from Ref. haiyue_crosstalk with permission.
• Figure 5: The Variational Quantum Classifier (VQC) with an example input state of $\ket{01}$ and rotational parameter $\theta$. This figure is adapted from Ref. adermann_QEC_VQC.