An Exponential Separation Between Quantum and Quantum-Inspired Classical Algorithms for Linear Systems
Allan Grønlund, Kasper Green Larsen
TL;DR
This work establishes the first unconditional exponential separation between quantum and quantum-inspired classical (QIC) algorithms for solving linear systems with sparse, well-conditioned matrices. By developing two reductions—one from random walks and another from k-Forrelation—the authors show that any QIC algorithm must make superpolynomial SQ(M) queries to solve Mx=y, while quantum algorithms achieve polylog-time performance under comparable input representations. The construction uses a carefully designed, highly sparse, well-conditioned M and a spectral analysis of a random-walk-based matrix to ensure both a favorable solution mass distribution and a manageable condition number. Together, these results demonstrate that exponential quantum advantages can persist beyond state-preparation constraints when problem structure is suitably engineered, clarifying the landscape of quantum speedups in linear-algebra tasks.
Abstract
Achieving a provable exponential quantum speedup for an important machine learning task has been a central research goal since the seminal HHL quantum algorithm for solving linear systems and the subsequent quantum recommender systems algorithm by Kerenidis and Prakash. These algorithms were initially believed to be strong candidates for exponential speedups, but a lower bound ruling out similar classical improvements remained absent. In breakthrough work by Tang, it was demonstrated that this lack of progress in classical lower bounds was for good reasons. Concretely, she gave a classical counterpart of the quantum recommender systems algorithm, reducing the quantum advantage to a mere polynomial. Her approach is quite general and was named quantum-inspired classical algorithms. Since then, almost all the initially exponential quantum machine learning speedups have been reduced to polynomial via new quantum-inspired classical algorithms. From the current state-of-affairs, it is unclear whether we can hope for exponential quantum speedups for any natural machine learning task. In this work, we present the first such provable exponential separation between quantum and quantum-inspired classical algorithms for the basic problem of solving a linear system when the input matrix is well-conditioned and has sparse rows and columns.