Reformulation of the No-Free-Lunch Theorem for Entangled Datasets

Journal Article · 17 February 2022 · Physical Review Letters

DOI: https://doi.org/10.1103/physrevlett.128.070501 · OSTI ID:1994114

Sharma, Kunal  ^[1]; Cerezo de la Roca, Marco Vinicio Sebastian ^[2]; Holmes, Zoë Portia ^[2]; Cincio, Lukasz  ^[2]; Sornborger, Andrew Tyler ^[2]; Coles, Patrick Joseph ^[2]

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1. Los Alamos National Laboratory (LANL), Los Alamos, NM (United States); Louisiana State University, Baton Rouge, LA (United States)
2. Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)

The No-Free-Lunch (NFL) theorem is a celebrated result in learning theory that limits one’s ability to learn a function with a training data set. With the recent rise of quantum machine learning, it is natural to ask whether there is a quantum analog of the NFL theorem, which would restrict a quantum computer’s ability to learn a unitary process with quantum training data. However, in the quantum setting, the training data can possess entanglement, a strong correlation with no classical analog. In this work, we show that entangled data sets lead to an apparent violation of the (classical) NFL theorem. This motivates a reformulation that accounts for the degree of entanglement in the training set. As our main result, we prove a quantum NFL theorem whereby the fundamental limit on the learnability of a unitary is reduced by entanglement. We employ Rigetti's quantum computer to test both the classical and quantum NFL theorems. In conclusion, our work establishes that entanglement is a commodity in quantum machine learning.

• Research Organization: Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)
• Sponsoring Organization: USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR); USDOE Laboratory Directed Research and Development (LDRD) Program
• Grant/Contract Number: 89233218CNA000001
• OSTI ID: 1994114
• Report Number(s): LA-UR-20-25010
• Journal Information: Physical Review Letters, Journal Name: Physical Review Letters Journal Issue: 7 Vol. 128; ISSN 0031-9007
• Publisher: American Physical Society (APS)Copyright Statement
• Country of Publication: United States
• Language: English