Separable Power of Classical and Quantum Learning Protocols Through the Lens of No-Free-Lunch Theorem

TL;DR

This work extends the No-Free-Lunch framework to quantum learning by categorizing learning protocols into Classical (CLC-LPs), Restricted Quantum (ReQu-LPs), and Quantum (Qu-LPs) paradigms, each with different access to quantum resources. It derives NFL-style lower bounds showing a quadratic separation in sample complexity across protocols, driven by the ability of quantum protocols to exploit inter-state relative phase information, especially for non-orthogonal training states and non-diagonal observables. The authors provide both theoretical NFL theorems and numerical experiments (using Haar-random unitaries and hardware-efficient PQCs) that confirm phase-alignment and diagonality conditions govern the magnitude of the advantage. These results illuminate how quantum resources and phase information interplay to yield universal advantages in learning quantum dynamics, and offer practical guidance for designing phase-aware quantum learning algorithms with limited quantum access.

Abstract

The No-Free-Lunch (NFL) theorem, which quantifies problem- and data-independent generalization errors regardless of the optimization process, provides a foundational framework for comprehending diverse learning protocols' potential. Despite its significance, the establishment of the NFL theorem for quantum machine learning models remains largely unexplored, thereby overlooking broader insights into the fundamental relationship between quantum and classical learning protocols. To address this gap, we categorize a diverse array of quantum learning algorithms into three learning protocols designed for learning quantum dynamics under a specified observable and establish their NFL theorem. The exploited protocols, namely Classical Learning Protocols (CLC-LPs), Restricted Quantum Learning Protocols (ReQu-LPs), and Quantum Learning Protocols (Qu-LPs), offer varying levels of access to quantum resources. Our derived NFL theorems demonstrate quadratic reductions in sample complexity across CLC-LPs, ReQu-LPs, and Qu-LPs, contingent upon the orthogonality of quantum states and the diagonality of observables. We attribute this performance discrepancy to the unique capacity of quantum-related learning protocols to indirectly utilize information concerning the global phases of non-orthogonal quantum states, a distinctive physical feature inherent in quantum mechanics. Our findings not only deepen our understanding of quantum learning protocols' capabilities but also provide practical insights for the development of advanced quantum learning algorithms.

Figures (6)

• Figure 1: An overview of classical and (restricted) quantum learning protocols. Both the learning protocols shown in the upper panel (a) and the lower panel (b) aim to learn the target function $\mathop{\mathrm{Tr}}\nolimits(U\rho U^{\dagger}O)$ for a given fixed observable $O$. (a) The classical learning protocols employ a tunable unitary $V(\bm{\theta})$ to evolve the input states $\ket{\bm{\psi}_j}$ to the output states $e^{-i\bm{\gamma}_j}\ket{\bm{\psi}_j(\bm{\theta})}:=V(\bm{\theta})\ket{\bm{\psi}_j}$ with $\bm{\gamma}_j$ being the global phase of states $V(\bm{\theta})\ket{\bm{\psi}_j}$, where $\bm{\theta}$ refers to tunable variables which could be discrete circuit structure or continuous parameters. A pre-defined observable $O$ is employed to measure the output states, leading to the information loss of the global phase $\bm{\gamma}_j$. An optimizer then is exploited to update the variables $\bm{\theta}$ according to the disparity of the measurement output and the target output $\mathop{\mathrm{Tr}}\nolimits(OU\ket{\bm{\psi}_j}\bra{\bm{\psi}_j}U^{\dagger})$. (b) ReQu-LPs (or Qu-LPs) employ quantum operations to process the output states of the target unitary $U\ket{\bm{\psi}_j}$ (or $U^{\dagger}\ket{\bm{\psi}_j}$) and the tunable unitary $V(\bm{\theta})\ket{\bm{\psi}_j}$ in the same quantum system, where the output quantum states are stored in quantum memory. This allows for learning the phase information such that the phase difference $\bm{\alpha}_j-\bm{\gamma}_j$ over various training states converges to the same value as the optimization proceeds.
• Figure 2: Paradigm of variational quantum algorithms. (a) Preparing input states by obtaining from quantum experiments or encoding classical data. (b) The input state is evolved by a parameterized quantum circuit $V(\bm{\theta})$ and then is measured to obtain the classical data stored in (c) classical memory. (d) The classical optimizer is employed to update the parameters $\bm{\theta}$ according to the measurement outputs.
• Figure 3: Scheme of various learning protocols. The output states $U\ket{\bm{\psi}_j}$ and $U^{\dagger}\ket{\bm{\psi}_j}$ are stored in quantum memory. The tunable unitary $V(\bm{\theta})$ is implemented on quantum computers in ReQu-LPs and Qu-LPs. However, in the case of CLC-LPs, it can also be implemented on classical computers using tensor network techniques orus2019tensor or deep neural networks gao2017efficient1.
• Figure 4: Numerical results for quantum dynamics learning. (a) The averaged prediction error with varying training dataset size $N$ under various learning protocols and different types of input states when the number of qubits $n=4$. (b) The averaged training error with varying training dataset size $N$ under various LPs when $n=4$. (c) The magnitude of the inter-state relative phase defined in Eqn. (\ref{['eq:pt_co']}) and Eqn. (\ref{['eq:pt_co_dagger']}) for various LPs. The labels 'CLC-LP', 'ReQu-LP', 'Qu-LP' refer to the classical learning protocol, restricted quantum learning protocol, and quantum learning protocol with access to $U^{\dagger}$. $R_U(V)$ and $\mathcal{L(\bm{\theta})}$ refers to the prediction error and training error. The label '$(\times 1/4^n)$’ means that the plotted prediction error is normalized by a multiplier factor $1/4^n$. The label '$\mathcal{S}_{\mathop{\mathrm{Haar}}\nolimits_n}^{(o)}$' and '$\mathcal{S}_{\mathop{\mathrm{Haar}}\nolimits_n}$' refer to the input states being the orthogonal $n$-qubit Haar states and the $n$-qubit Haar states.
• Figure 5: Structure of hardware efficient ansatz (HEA). $\mathop{\mathrm{Rot}}\nolimits(\bm{\theta}_{j1})$ represents the general rotation gate with three rotation angles $\bm{\theta}_x, \bm{\theta}_y, \bm{\theta}_z$.

Theorems & Definitions (18)

• Lemma 1
• Theorem 1: NFL for CLC-LPs
• Theorem 2: NFL theorem for ReQu-LPs
• Theorem 3: NFL for Qu-LPs with access to $U^{\dagger}$
• Proposition 1: Lemma 1 of Ref. cerezo2021cost
• Proposition 2: Lemma 2 of Ref. cerezo2021cost
• Proposition 3: Lemma 3 of Ref. cerezo2021cost
• Proposition 4
• Proposition 5
• Lemma 2