The Stabilizer Bootstrap of Quantum Machine Learning with up to 10000 qubits

TL;DR

The paper tackles the challenge of identifying regimes of quantum advantage in quantum machine learning (QML) and designing scalable variational ansatze. It introduces the stabilizer bootstrap, a two-phase approach that combines sampling of Clifford-space with Bayesian optimization to pre-optimize quantum circuits, bridging stabilizer formalism with variational quantum algorithms. The work reveals strong stabilizer enhancement, where the nontrivial sampling probability remains constant with increasing qubit count ($p$ constant, e.g., $p=1/4$ in key cases), and weak stabilizer enhancement, where $p$ decays as $p \sim 1/n^\nu$ with $ u$ ranging from $0$ to about $n/\log n$, supported by proofs and large-scale simulations up to $n=10^4$ and data sizes up to $10^3$. It provides a practical framework to gauge quantum advantages, demonstrates scalability on HPC resources, and charts a route toward fault-tolerant quantum applications by linking Clifford-based sampling dynamics to potential quantum gains.

Abstract

Quantum machine learning is considered one of the flagship applications of quantum computers, where variational quantum circuits could be the leading paradigm both in the near-term quantum devices and the early fault-tolerant quantum computers. However, it is not clear how to identify the regime of quantum advantages from these circuits, and there is no explicit theory to guide the practical design of variational ansatze to achieve better performance. We address these challenges with the stabilizer bootstrap, a method that uses stabilizer-based techniques to optimize quantum neural networks before their quantum execution, together with theoretical proofs and high-performance computing with 10000 qubits or random datasets up to 1000 data. We find that, in a general setup of variational ansatze, the possibility of improvements from the stabilizer bootstrap depends on the structure of the observables and the size of the datasets. The results reveal that configurations exhibit two distinct behaviors: some maintain a constant probability of circuit improvement, while others show an exponential decay in improvement probability as qubit numbers increase. These patterns are termed strong stabilizer enhancement and weak stabilizer enhancement, respectively, with most situations falling in between. Our work seamlessly bridges techniques from fault-tolerant quantum computing with applications of variational quantum algorithms. Not only does it offer practical insights for designing variational circuits tailored to large-scale machine learning challenges, but it also maps out a clear trajectory for defining the boundaries of feasible and practical quantum advantages.

Figures (14)

• Figure 1: Overview of our paper. Firstly, we present results for single-layer stabilizer ansätze under basic conditions, which we denote as strong stabilizer enhancement and weak stabilizer enhancement. Finally, we present our experimental results for the stabilizer bootstrap with datasets, analyze the impact of dataset size, and demonstrate outcomes for systems with 10000 qubits or 1000 data sizes maximally.
• Figure 2: Clifford ansätze. For simplification, We refer to the upper (a) entanglement structure as linear entanglement and the below (b) entanglement structure as reverse linear entanglement.
• Figure 3: Result for 10000 qubits with 200 data.
• Figure 4: Domain-wall observables.
• Figure 5: Probability of nontrivial sampling for even and odd number of qubits. The $Z$-axis is the probability of nontrivial sampling, and the $Y$-axis refers to the proportion of $X$ operators in the observable. Therefore, at the points where $x = 0$, drawn by the orange line, the conditions degenerate into the strong stabilizer enhancement as stated in condition (\ref{['condition2']}). Similarly, at the points where $x = 1$, drawn by the purple line, the conditions degenerate into the weak stabilizer enhancement as described in condition (\ref{['condition4']}).

Theorems & Definitions (6)

• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3
• Theorem 4