On the quantum computational complexity of classical linear dynamics with geometrically local interactions: Dequantization and universality

2025-05-15

AI Summary

This work investigates the quantum simulation complexity of classical linear dynamical systems with geometrically local interactions—such as PDEs—and asks whether quantum algorithms can achieve exponential speedups for such local classical systems. Method: We combine Hamiltonian simulation techniques, computational complexity analysis, probabilistic Turing machine modeling, and “dequantization” methods. Results: We prove that short-time local evolution is efficiently simulable classically via randomized algorithms—i.e., it lies in BPP—establishing a rigorous “no quantum advantage” regime. In contrast, long-time simulation under polynomial-space constraints is shown to be equivalent to EXPSPACE ∩ BQTIME(exp), implying superpolynomial quantum speedup potential. Our main contribution is the first sharp theoretical delineation between time regimes where local classical dynamics admit no quantum advantage (short-time) and those where quantum supremacy is possible (long-time), thereby providing a fundamental complexity-theoretic criterion for the practical limits of quantum simulation.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Sampling/Simulation-based Search
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems
• Web Mining and Content Analysis: Web data generation and simulation

Abstract

The simulation of large-scale classical systems in exponentially small space on quantum computers has gained attention. The prior work demonstrated that a quantum algorithm offers an exponential speedup over any classical algorithm in simulating classical dynamics with long-range interactions. However, many real-world classical systems, such as those arising from partial differential equations, exhibit only local interactions. The question remains whether quantum algorithms can still provide exponential speedup under this condition. In this work, we thoroughly characterize the computational complexity of quantum algorithms for simulating such geometrically local systems. First, we dequantize the quantum algorithm for simulating short-time (polynomial-time) dynamics of such systems. This implies that the problem of simulating this dynamics does not yield any exponential quantum advantage. Second, we show that quantum algorithms for short-time dynamics have the same computational complexity as polynomial-time probabilistic classical computation. Third, we show that the computational complexity of quantum algorithms for long-time (exponential-time) dynamics is captured by exponential-time and polynomial-space quantum computation. This suggests a super-polynomial time advantage when restricting the computation to polynomial-space, or an exponential space advantage otherwise. This work offers new insights into the complexity of classical dynamics governed by partial differential equations, providing a pathway for achieving quantum advantage in practical problems.

Problem

• Quantum advantage in simulating local-interaction classical systems
• Dequantization of quantum algorithms for short-time dynamics
• Complexity of quantum algorithms for long-time dynamics

Innovation

• Dequantizing quantum algorithms for local dynamics
• Matching quantum and classical computational complexities
• Exploring quantum advantage in exponential-time dynamics