Bayesian Optimization for Quantum Error-Correcting Code Discovery

TL;DR

This paper addresses the costly evaluation bottleneck in discovering practical quantum error-correcting codes by introducing a Bayesian optimization framework that leverages a multi-view chain-complex neural embedding to predict logical error rates. The core idea is to map CSS codes to a rich, topology-preserving embedding that a Gaussian process can use as a surrogate, enabling efficient exploration of a discrete code space via an acquisition function and hill-climbing in the search. Empirical results on BB and HGP code families under code-capacity noise show that BO discovers high-rate and/or low-error codes that surpass or match strong baselines while using far fewer evaluations, moving operating points toward the quantum Hamming bound when appropriate. The framework is general, scalable, and adaptable to diverse code families, decoders, and noise models, paving the way for hardware-aware code design and potentially neural decoders for quantum LDPC codes.

Abstract

Quantum error-correcting codes protect fragile quantum information by encoding it redundantly, but identifying codes that perform well in practice with minimal overhead remains difficult due to the combinatorial search space and the high cost of logical error rate evaluation. We propose a Bayesian optimization framework to discover quantum error-correcting codes that improves data efficiency and scalability with respect to previous machine learning approaches to this task. Our main contribution is a multi-view chain-complex neural embedding that allows us to predict the logical error rate of quantum LDPC codes without performing expensive simulations. Using bivariate bicycle codes and code capacity noise as a testbed, our algorithm discovers a high-rate code [[144,36]] that achieves competitive per-qubit error rate compared to the gross code, as well as a low-error code [[144,16]] that outperforms the gross code in terms of error rate per qubit. These results highlight the ability of our pipeline to automatically discover codes balancing rate and noise suppression, while the generality of the framework enables application across diverse code families, decoders, and noise models.

Figures (8)

• Figure 1: The workflow of the Bayesian optimization algorithm. The algorithm iterates the following three steps until the computational budget is exhausted: (top) fit a probabilistic surrogate model to the obtained data to predict the objective function $F(\mathcal{C})$, e.g. the logical error rate -- $\mathcal{C}$ denotes a quantum code represented as a Tanner graph and $\mu$ and $\sigma$ are the surrogate mean and uncertainty; (bottom right) optimize the acquisition function that uses the surrogate prediction and uncertainty to propose a new candidate code -- here the figure depicts a hill climbing trajectory; (bottom left) evaluate the objective for the new code by performing a simulation of the code in the presence of errors $E$, which produces new data to fit the model to.
• Figure 2: Structure of the chain complex embedding. (a) Chain complex view: $C_i$$(i=0,1,2)$ shown in circles are vector spaces over $\mathbb{F}_2$. Boundary mappings $\partial_i$ are represented by matrices over $\mathbb{F}_2$, and $\partial_i^\top$ denote the corresponding coboundary mappings. (b) Homology view: $H_1(\mathcal{C})$ is shown in a circle, with the mapping $\iota_\wedge: H_1(\mathcal{C}) \to C_1$ represented by a matrix. (c) Dual homology view: the dual counterpart of (b). In all views, circles represent vertex sets in a tripartite graph, and edges across parts correspond to their associated mappings.
• Figure 3: The architecture of the neural embedding $\operatorname{Emb}(x)$. MP Layer denotes a message-passing layer, and MLP Layer denotes a multi-layer perceptron. Given an input $x$, the data is first transformed into a multi-view chain-complex representation, consisting of the chain complex view ($\partial_1,\partial_2$), the homology view ($\iota_\wedge,\partial_2$) and the dual homology view ($\partial_1,\iota_\vee$). Each view is then processed independently by MP Layers to obtain a corresponding vector representation. These vectors are subsequently concatenated (depicted as $+$) and fed into a stack of MLP Layers. The output of the final MLP Layer is taken as the embedding Emb(x).
• Figure 4: A sketch of hill climbing. The vertical axis represents the objective function. The black path indicates the sequence of selected points, with 0 denoting the starting point. Red bars highlight the best value in each iteration, while yellow bars mark the explored neighborhood. The algorithm terminates at step 6, which is returned as the final result.
• Figure 5: The comparison between true values (blue line) and predicted mean $\pm\sigma$. The true LER values of all codes are shown by the blue curve and are sorted in ascending order from left to right.