Transition Role of Entangled Data in Quantum Machine Learning

TL;DR

The authors prove that the impact of entangled data on prediction error exhibits a dual effect, depending on the number of permitted measurements, and show that this is not the case when the allowed number of measurements to incoherently learn quantum dynamics is low.

Abstract

Entanglement serves as the resource to empower quantum computing. Recent progress has highlighted its positive impact on learning quantum dynamics, wherein the integration of entanglement into quantum operations or measurements of quantum machine learning (QML) models leads to substantial reductions in training data size, surpassing a specified prediction error threshold. However, an analytical understanding of how the entanglement degree in data affects model performance remains elusive. In this study, we address this knowledge gap by establishing a quantum no-free-lunch (NFL) theorem for learning quantum dynamics using entangled data. Contrary to previous findings, we prove that the impact of entangled data on prediction error exhibits a dual effect, depending on the number of permitted measurements. With a sufficient number of measurements, increasing the entanglement of training data consistently reduces the prediction error or decreases the required size of the training data to achieve the same prediction error. Conversely, when few measurements are allowed, employing highly entangled data could lead to an increased prediction error. The achieved results provide critical guidance for designing advanced QML protocols, especially for those tailored for execution on early-stage quantum computers with limited access to quantum resources.

Figures (10)

• Figure 1: Illustration of quantum NFL setting with the entangled data. The goal of the quantum learner is to learn a unitary $\bm{V}_{\mathcal{X}}$ that can accurately predict the output of the target unitary $\bm{U}_{\mathcal{X}}$ under a fixed observable $\bm{O}$, where the subscript $\mathcal{X}$ refers to the quantum system in which the operator $\bm{O}$ act on. The learning process is as follows. (a) A total number of $N$ entangled bipartite quantum states living in Hilbert space $\mathcal{H}_{\mathcal{X}} \otimes \mathcal{H}_{\mathcal{R}}$ ($\mathcal{R}$ denotes the reference system) are taken as inputs, dubbed entangled data. (b) Quantum learner proceeds incoherent learning. The entangled data separately interacts with the target unitary $\bm{U}_{\mathcal{X}}$ (agnostic) and the candidate hypothesis $\bm{V}_{\mathcal{X}}$ extracted from the same Hypothesis set $H$. (c) The quantum learner is restricted to leverage the finite measured outcomes of the observable $\bm{O}$ on the output states of $\bm{U}_{\mathcal{X}}$ and $\bm{V}_{\mathcal{X}}$ to conduct learning. (d) A classical computer is exploited to infer $\bm{V}^*$ that best estimates $\bm{U}_{\mathcal{X}}$ according to the measurement outcomes. For example, in the case of variational quantum algorithms, the classical computer serves as an optimizer to update the tunable parameters of the ansatz $\bm{V}_{\mathcal{X}}$. (e) The learned unitary $\bm{V}^*$ is used to predict the output of unseen quantum states in Hilbert space $\mathcal{H}_{\mathcal{X}}$ under the evolution of the target unitary $\bm{U}_{\mathcal{X}}$ and the measurement of $\bm{O}$. A large Schmidt rank $r$ can enhance the prediction accuracy when combined with a large number of measurements $m$, but may lead to a decrease in accuracy when $m$ is small.
• Figure 2: Simulation results of quantum NFL theorem when incoherently learning quantum dynamics. (a) The averaged prediction error with a varied number of measurements $m$ and Schmidt rank $r$ when $N=2$ and $N=8$. The z-axis refers to the averaged prediction error defined in Eqn. (\ref{['eq:learning_model']}). (b) The averaged prediction error with the varied sizes of training data. The label '$r = a ~\&~ m = b$' refers that the Schmidt rank is $a$ and the number of measurements is $b$. The label '($\times 2/4^n$)' refers that the plotted prediction error is normalized by a multiplier factor $2/4^n$.
• Figure 3: The $\varepsilon$-packing of the hypothesis space. The left panel and the right panel refer to the $\varepsilon$-packing and the local $\varepsilon$-packing with maximal distance $\gamma\varepsilon$ of $\mathcal{F}$, respectively.
• Figure 4: The paradigm of the reduction from quantum dynamics learning to hypothesis testing.
• Figure 5: Simulation results of quantum NFL theorem with orthogonal training states. The averaged prediction error with a varied number of measurements $m$ and Schmidt rank $r$ when $N=1$, $N=4$, and $N=8$. The z-axis refers to the averaged prediction error defined in Eqn. (\ref{['eq:learning_model']}). The label '($\times 2/d^2$)' refers that the plotted prediction error is normalized by a multiplier factor $2/d^2$.

Theorems & Definitions (37)

• Theorem 1: Quantum NFL theorem in learning quantum dynamics, informal
• Theorem 2: Quantum NFL theorem in learning quantum dynamics for generic measurement, informal
• Definition 1: $\chi^2$-divergence
• Lemma 1: Lemma 2.9 in Ref. lowe2022lower
• Lemma 2
• Lemma 3
• Lemma 4
• Lemma 5
• Lemma 6
• proof : Proof of Lemma \ref{['lem:Tr(WA)Tr(WB)_Orth']}