Predicting quantities related to quantum systems accurately with limited classical information is computationally hard for conventional computers in certain cases. Predicting properties, specifically traces of functions applied to Hamiltonians and density matrices, is difficult using standard computing methods when considering cosine and exponential decay as descriptions of system evolution. Predicting certain properties of quantum systems presents computational challenges for standard computers.

The focus lies on ‘Hamiltonian function learning’, whereby algorithms predict characteristics based solely on limited classical information concerning a system’s energy and state; specifically, traces of functions applied to Hamiltonians and density matrices. This difficulty connects to established problems in computer science such as RSA factoring, suggesting potential benefits from utilising quantum computation for these tasks. Predicting properties of quantum systems can be computationally difficult for conventional computers.

Investigating ‘Hamiltonian function learning’ involves teaching a computer to predict characteristics of a physical system, how it vibrates or interacts with light, for example, based on limited information about its components, similar to understanding complex machinery by observing overall behaviour rather than examining internal parts. The team rigorously proved this computational challenge exists when attempting to predict traces, essentially summing contributions from many possibilities to obtain an overall result relating to specific functions applied to the systems.

Establishing such difficulty has implications beyond physics, linking complexity to mathematical challenges like RSA factoring, vital for internet security and online transactions, and raising questions regarding potential advantages offered by quantum computers in solving them.

Linking Quantum Learning Complexity to RSA Factorisation through Reductive Proofs

A technique called reduction established a link between Hamiltonian function learning and factoring RSA moduli, a mathematical challenge underpinning much internet security. Should an efficient classical computer solve the specific type of quantum machine learning task, it would also be capable of efficiently cracking important encryption codes. This ‘reduction’ demonstrates how any solution for the quantum learning problem automatically provides one for the factoring problem, proving its inherent complexity.

Hamiltonian function learning involves predicting properties of a system described by its Hamiltonian representing total energy, alongside its quantum state using classical information. Cosine and exponential functions were specifically investigated to connect them with established mathematical problems; this allowed direct linkage between the quantum machine learning task and the well-understood difficulty of factoring RSA moduli used in internet security protocols. The result is a clear benchmark for complexity and opens avenues for further investigation into related computational challenges.

Demonstrating computational intractability of simulating evolving quantum system dynamics

Utilising up to 40 qubits represents a strong leap from previous implementations struggling even with smaller systems. Rigorous demonstration showed that predicting traces, summations representing system properties relating to specific functions applied to Hamiltonians and quantum states, is computationally hard for standard computers. This builds on earlier work by Morohoshi and colleagues, who proposed such problems but lacked definitive proof. Hard instances were constructed using projectors defined by relatively small quantum circuits, avoiding reliance on more complex local Hamiltonians often seen in this field.

The research now provides robust proof showing the computational difficulty of predicting how quantum systems change; specifically, it concerns the traces of functions applied to their Hamiltonians and quantum states when utilising up to 40 qubits. Notably, the proof holds even if the function governing system behaviour is known beforehand, a significant refinement over previous approaches which required learning alongside other parameters, allowing focused analysis of inherent limitations within classical computation itself.

Quantum supremacy demonstrated through limitations of classical simulation for simple wave behaviours

These findings offer a compelling argument for why certain quantum computations might ultimately prove superior to their classical equivalents, particularly in predicting complex system behaviour described by ‘Hamiltonians’ and ‘quantum states’. Establishing classical hardness for limited cases clarifies precisely where quantum algorithms may outperform traditional methods when predicting how systems behave. This work rigorously connects a specific challenge in quantum computing with established difficulties in classical computation. Specifically, predicting properties of quantum systems is now demonstrably linked to the intractability of factoring large numbers used in RSA encryption schemes. By proving this ‘average-case classical hardness’ for both cosine and exponential functions commonly employed when modelling these evolving systems, clear boundaries have been identified where conventional computers struggle. The findings validate earlier theoretical suggestions that quantum algorithms could offer genuine speedups for certain machine learning tasks involving complex physical simulations.

The researchers demonstrated the computational difficulty of accurately simulating quantum systems using classical computers under defined conditions. This means there are specific calculations concerning how Hamiltonians and quantum states evolve, involving up to 40 qubits, that appear inherently challenging for even powerful traditional machines. Their work establishes a link between predicting properties within quantum mechanics and the established problem of factoring large numbers used in RSA encryption. By rigorously proving this ‘average-case classical hardness’ for cosine and exponential functions, they clarified limitations inherent in classical computation when applied to these types of problems.

👉 More information
🗞 Classical Hardness of Learning Functions of Hamiltonians
✍️ Sota Hashimoto and Akinori Kawachi
🧠 ArXiv: https://arxiv.org/abs/2610.01141

More like this