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

Kunal Sharma, M. Cerezo, Zoë Holmes, Lukasz Cincio, Andrew Sornborger, and Patrick J. Coles

Phys. Rev. Lett. 128, 070501 – Published 18 February 2022

Abstract

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 dataset. 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 Letter, we show that entangled datasets 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. Our Letter establishes that entanglement is a commodity in quantum machine learning.

Received 16 January 2021
Accepted 19 January 2022

DOI:https://doi.org/10.1103/PhysRevLett.128.070501

Physics Subject Headings (PhySH)

Machine learning Quantum algorithms Quantum gates

Quantum Information, Science & Technology

Authors & Affiliations

Kunal Sharma ^1,2,*, M. Cerezo^1,3,*, Zoë Holmes⁴, Lukasz Cincio ¹, Andrew Sornborger⁴, and Patrick J. Coles¹

¹Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
²Hearne Institute for Theoretical Physics and Department of Physics and Astronomy, Louisiana State University, Baton Rouge, Louisiana 70803, USA
³Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
⁴Information Sciences, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA

^*The first two authors contributed equally to this work.

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Vol. 128, Iss. 7 — 18 February 2022

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Images

Figure 1
Depiction of the no-free-lunch setting. (a) In classical supervised learning, one employs training data of size $t$ to train a hypothesis to mimic the action of an unknown function on domain size $d$ . Here we show input data in the form of bitstrings fed into a neural network (NN) to solve a binary classification problem. The NFL theorem indicates that it is the size of the training data rather than the choice of optimization method that limits the average risk. Namely, a small (large) $t$ leads to big (small) generalization errors on average. (b) In quantum supervised learning, the goal is to learn a $d$ -dimensional unitary process with $t$ quantum states serving as training data. For generality, we allow these states to possibly be entangled with a reference system, with the Schmidt rank $r$ quantifying the degree of entanglement. Here we show these states training a quantum neural network (QNN) to classify quantum data (Schrodinger’s cat being dead or alive). Our quantum NFL theorem indicates that $r * t$ is the quantity that limits the average risk, and hence big entanglement (large $r$ ) leads to small generalization errors even when $t$ is small.
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Figure 2
Implementation on quantum hardware. Here we plot the average risk after learning ten single-qubit unitaries on the Rigetti Aspen-4 quantum computer using ten training sets consisting of $t = 1$ , 2 unentangled $r = 1$ (blue squares) and entangled $r = 2$ (red circles) training states. The solid lines indicate the corresponding bounds imposed by our quantum NFL theorem, Eq. (5). Note that, while the optimizations were performed on the quantum computer, the final risk $R_{U} (V_{S_{Q}})$ and optimal cost $C_{U} (V_{S_{Q}})$ (plotted in the inset and defined in the Supplemental Material [48]) were calculated classically to allow an accurate (i.e., noiseless) evaluation of the success of the optimizations. In black we plot the classical deterministic (dotted) and stochastic (dashed) NFL theorems.
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Figure 3
Large-scale test of NFL theorems. We plot the average risk versus $t$ after learning ten six-qubit unitaries on a simulator for 100 training sets. Each training set consisted of $t = 1, \dots, 64$ training pairs of rank $r = 2^{0}, \dots, 2^{6}$ . The markers indicate the optimization results, whereas the solid lines indicate the bounds imposed by our quantum NFL theorem, Eq. (5). The simulation error bars are $O (10^{- 3})$ and therefore smaller than the size of the markers. In black, we plot the classical NFL bounds for deterministic (solid), stochastic (dashed), permutation (dot-dashed), and bistochastic maps.
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