Training Variational Quantum Algorithms Is NP-Hard, Even Locally

2026-10-04

AI Summary

This study addresses the low training efficiency of variational quantum algorithms (VQAs), whose global optimization is known to be NP-hard. Moving beyond the conventional focus on global complexity, this work provides the first proof that finding local minima and their neighboring solutions in VQA training is strongly NP-hard. Methodologically, Hermitian trigonometric polynomial approximation techniques are employed to construct quantum circuits, enabling a polynomial-time reduction. The results establish a fundamental computational barrier for VQA training, demonstrating that approximating local minima remains computationally intractable even when the objective function can be efficiently evaluated. Ultimately, this research reveals the theoretical limits of underlying optimization in quantum machine learning, highlighting intrinsic hardness beyond mere evaluation costs.

Abstract

Variational quantum algorithms (VQAs) generally rely on classical optimization to train parameterized quantum circuits. This training seeks to minimize an objective function, and its efficiency is central to the practical success of these algorithms. However, globally minimizing such training objectives over the circuit parameters is known to be $\mathsf{NP}$-hard, limiting the prospect of general guarantees for efficient training. In this Letter, we prove that even the weaker task of finding a local minimum of such VQA training objectives is strongly $\mathsf{NP}$-hard, including when the objective admits efficient classical evaluation. Moreover, we show that this hardness persists even for the task of finding a parameter vector within $\ell_p$-distance strictly less than $\pi/2$ of some local minimizer, for every $p\geq 1$. Our central technical result is that approximating a local minimizer of a Hermitian trigonometric polynomial is strongly $\mathsf{NP}$-hard. By explicitly constructing quantum circuits whose training objectives reproduce these hard instances, we obtain a polynomial-time reduction to VQA training. Our results establish a fundamental computational barrier to variational quantum training: even reaching the vicinity of a local minimum remains hard in the worst case.

Problem

• Variational Quantum Algorithms
• NP-hard
• Local Minimum
• Training Objective
• Computational Barrier

Innovation

• Variational Quantum Algorithms
• NP-hardness
• Local Minimum
• Hermitian Trigonometric Polynomial
• Polynomial-time Reduction