Advantage of Quantum Neural Networks as Quantum Information Decoders

TL;DR

We study readout decoding for quantum information encoded in groundspaces of perturbed topological stabilizer codes, modeled by H = H0 + λV with code distance d. The authors combine Brillouin-Wigner perturbation theory and quantum machine learning to compare standard QEC with QNN-based decoders, proving universal scaling laws and a provable advantage for QNNs: naive decoding yields ε_Q(Q_L) = Θ(λ^4), QEC improves to ε_Q(Q_QEC) = O(λ^{2⌈d/k⌉}) (noiseless) or Θ(λ^{2⌈(d+1−2p)/k⌉}) (noisy), and QNN decoding achieves ε_Q(Q_QNN) = O(λ^{4⌈(d−2p)/k⌉}) with a matching lower bound. These results are corroborated by numerical simulations across codes with distances from 3 to 5 (and up to 11 in some tests), demonstrating depth-enabled improvements and establishing a practical, near-term path to decoding non-stabilizer codes. The work highlights a provable, architecture-agnostic advantage of QNN-based decoders for realistic quantum error correction under disorder and noise, suggesting broad applicability to near-term quantum devices and non-stabilizer codes.

Abstract

A promising strategy to protect quantum information from noise-induced errors is to encode it into the low-energy states of a topological quantum memory device. However, readout errors from such memory under realistic settings is less understood. We study the problem of decoding quantum information encoded in the groundspaces of topological stabilizer Hamiltonians in the presence of generic perturbations, such as quenched disorder. We first prove that the standard stabilizer-based error correction and decoding schemes work adequately well in such perturbed quantum codes by showing that the decoding error diminishes exponentially in the distance of the underlying unperturbed code. We then prove that Quantum Neural Network (QNN) decoders provide an almost quadratic improvement on the readout error. Thus, we demonstrate provable advantage of using QNNs for decoding realistic quantum error-correcting codes, and our result enables the exploration of a wider range of non-stabilizer codes in the near-term laboratory settings.

Figures (5)

• Figure 1: Schematics of decoding realistic quantum codes. From left to right: Stabilizer codes realized as ground states of physical Hamiltonians get perturbed and yield imperfect codewords. A state $|\Psi\rangle$ prepared by the imperfect codewords is presented to the magic box decoder $\hat{O}$ for subsequent decoding. (i) Top row: $\hat{O}=Q_{\text{QEC}}$. $|\Psi\rangle$ is decoded through standard QEC procedure. (ii) Bottom row: $\hat{O}=Q_{\text{QNN}}$. $|\Psi\rangle$ is decoded through a quantum neural network. After decoding, bit strings are sampled from the resulting probability distributions and the outcomes of the two decoding procedures are compared.
• Figure 2: Universality of standard decoding. Performance of standard decoding strategies on perturbed stabilizer codes. On $y$-axis we abbreviate the generalization error in $X$ basis $\varepsilon_X$ by $\varepsilon$. The universal scaling laws $\varepsilon\sim\lambda^\alpha$ are shown in dashed lines. Panel (a) show the error of measuring logical operator $X_L$ as a function of perturbation strength $\lambda$. Dashed line has scaling exponent $\alpha=4$, as predicted by Theorem \ref{['thm1']}. Panel (b) shows the rescaled logarithm of the error for measuring logical operator $X_L$after measuring syndromes and applying error correction. Dashed line has scaling exponent $\alpha = 2(d+1)$, consistent with Theorem \ref{['thm2']}. Panel (c) is the same as panel (b) but for noisy input states. Dashed line has scaling exponent $\alpha = 2(d+1-2p)$, as predicted by Theorem \ref{['thm2']}. Note that all the curves from different codes nearly collapse into a single, universal shape in (a)-(c) after dividing by the scaling exponent of the dashed line. Inset in (b)-(c) shows the raw unrescaled data which have different slopes depending on the code distance.
• Figure 3: QNN performance for 5-qubit code. (a) Our QNN architecture consists of an error-correction circuit $C_Q$ and a decoding circuit $D_Q$. (b) Decoding noiseless states for perturbed 5-qubit code at $\lambda=1$. (c)-(d) Example decoding of imperfect 5-qubit code using $X_L$ (red), $X_{\text{QEC}}$ (green) and $X_{\text{QNN}}$ (blue). (c) Decoding noiseless states. (d) Decoding noisy states corrupted by a randomly chosen single-qubit Pauli operator $P_\alpha \in \{I, X, Y, Z\}$ acting on a randomly chosen qubit. Panels (b)-(c) demonstrate that QNN significantly outperforms the standard approaches across nearly two orders of magnitude in $\lambda$. Panel (d) confirms that the scaling of standard QEC and QNN agree with our analytical predictions in Theorems \ref{['thm2']}-\ref{['thm3']}.
• Figure S1: 5-qubit QNN. (a) The 'error-correction circuit' $C_Q$ has depth $d_C$. The 'decoding circuit' $D_Q$ consitst of a single 2-qubit unitary. (b)-(e) The decoding performance improves as depth $d_C$ increases.
• Figure S2: Different QNN architectures. We consider the same task as in Fig.\ref{['fig:QNN_panel']}(c). $d_C=2,3,4$ corresponds to independent unitaries $U_k$ in Fig. \ref{['appfig:QNN_5qubit']}, $d_C=4$ (Trans. Inv.) correspond to same $U_k$ within the same layer. For reference, we reproduce the decoding performance of the standard decoding $\varepsilon_X(Q_{QEC})$ (green curve) and $\varepsilon_X(Q_L)$ (blue curve) as in Fig.\ref{['fig:QNN_panel']}(c).

Theorems & Definitions (19)

• Theorem 1
• Theorem 2
• Theorem 3
• Proposition 1
• Claim 1
• Lemma 1
• Lemma 2
• Lemma 3
• Theorem 1
• Remark 1