A quantum physicist's guide to the CPU and GPU versus QPU decision

Quantum many-body physics has been demonstrated for decades how to extract the most quantumness, i.e., entanglement, out of classical computers and high-performance computing. Forty years after Richard Feynman's vision of a quantum simulator, we can now choose out of a variety of quantum simulators and quantum computers based on superconducting qubits, trapped ions, or neutral Rydberg atoms. But when is it worth the effort? Using tensor network methods, we attack the question of when a problem should run on a quantum computer and what the perfect split in a hybrid quantum-classical computation could be.

We give a broad overview of the possible applications in terms of the simulation of quantum systems and explain how tensor network methods encode entanglement; then, we develop a methodology to relate the error of tensor networks to the error of quantum hardware. This framework helps to identify the regimes favorable to classical applications or quantum computation, and more importantly, we can estimate how far the regimes will shift with the changes in the parameters, e.g., the gate fidelity of the quantum hardware. Applied to a quantum Fourier transformation, we gain an insight into the decision-making and relate it to hybrid quantum-classical computation.