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Training variational quantum algorithms Is NP-hard
Publikationstyp
Journal Article
Date Issued
2021-09-17
Sprache
English
Author(s)
Journal
Volume
127
Issue
12
Article Number
120502
Citation
Physical Review Letters 127 (12) : 120502 (2021-09-17)
Publisher DOI
Scopus ID
PubMed ID
34597099
Variational quantum algorithms are proposed to solve relevant computational problems on near term quantum devices. Popular versions are variational quantum eigensolvers and quantum approximate optimization algorithms that solve ground state problems from quantum chemistry and binary optimization problems, respectively. They are based on the idea of using a classical computer to train a parametrized quantum circuit. We show that the corresponding classical optimization problems are NP-hard. Moreover, the hardness is robust in the sense that, for every polynomial time algorithm, there are instances for which the relative error resulting from the classical optimization problem can be arbitrarily large assuming that P≠NP. Even for classically tractable systems composed of only logarithmically many qubits or free fermions, we show the optimization to be NP-hard. This elucidates that the classical optimization is intrinsically hard and does not merely inherit the hardness from the ground state problem. Our analysis shows that the training landscape can have many far from optimal persistent local minima This means gradient and higher order descent algorithms will generally converge to far from optimal solutions.