[Paper Review] Storage properties of a quantum perceptron

Aikaterini, Gratsea+2 more|arXiv (Cornell University)|Nov 16, 2021

Neural Networks and Applications4 citations

TL;DR

This paper investigates the storage capacity of a specific quantum perceptron architecture using statistical mechanics and spin glass theory, showing that its maximum storage capacity scales linearly with the number of qubits and is significantly enhanced compared to classical counterparts. The analysis reveals that quantum interference and superposition enable higher pattern retention than classical perceptrons under the same constraints.

ABSTRACT

Driven by growing computational power and algorithmic developments, machine learning methods have become valuable tools for analyzing vast amounts of data. Simultaneously, the fast technological progress of quantum information processing suggests employing quantum hardware for machine learning purposes. Recent works discuss different architectures of quantum perceptrons, but the abilities of such quantum devices remain debated. Here, we investigate the storage capacity of a particular quantum perceptron architecture by using statistical mechanics techniques and connect our analysis to the theory of classical spin glasses. We focus on a specific quantum perceptron model and explore its storage properties in the limit of a large number of inputs. Finally, we comment on using statistical physics techniques for further studies of neural networks.

Motivation & Objective

• To determine the maximum storage capacity of a quantum perceptron model proposed in Tacchino et al. (2019) without relying on a specific learning rule.
• To apply Gardner’s program—originally developed for classical neural networks—to a quantum perceptron architecture.
• To explore the role of quantum effects such as superposition and entanglement in enhancing memory capacity beyond classical limits.
• To establish a theoretical framework linking quantum perceptron design to spin glass models and statistical field theory.
• To compare quantum and classical perceptron storage capacities using Monte Carlo simulations and analytical approximations.

Proposed method

• Adopts Gardner’s program to compute the storage capacity of a quantum perceptron by treating weights as random variables and analyzing the overlap between input patterns and weight configurations.
• Uses a quantum perceptron model with a controlled unitary operation $V_{\vec{w}}$ that maps input states to a computational basis state $|m-1\rangle$ based on the dot product $\vec{i}^\mu \cdot \vec{w}$.
• Employs a multi-controlled NOT gate with a readout qubit to project the output amplitude $|c_{m-1}|^2 = |\vec{i}^\mu \cdot \vec{w}|^2$, enabling measurement-based readout of the perceptron output.
• Applies replica field theory and saddle-point approximations to derive the free energy and effective potential, leading to an expression for the storage capacity $\alpha_c$.
• Performs Monte Carlo simulations for both classical and quantum perceptrons, sampling over random input patterns and weight configurations to estimate average storage capacity $\alpha(N_s) = \langle P \rangle / N_s$.
• Uses numerical simulations with $N_s = N$ for classical and $N_s = m$ for quantum models, employing $10000$ samples and $m = 8, 16, 32$ to assess scalability and error.

Experimental results

Research questions

• RQ1 What is the maximum number of patterns a quantum perceptron can store as a function of its Hilbert space dimension?
• RQ2 How does the storage capacity of a quantum perceptron compare to that of a classical perceptron with Ising weights?
• RQ3 To what extent do quantum interference and superposition enhance the storage capacity beyond classical limits?
• RQ4 What is the critical storage capacity $\alpha_c$ of the quantum perceptron, and how does it depend on the threshold $\kappa$?
• RQ5 Can the statistical mechanics framework used for classical spin glasses be extended to analyze quantum neural network architectures?

Key findings

• The quantum perceptron exhibits a storage capacity that scales linearly with the number of qubits $m$, with the critical capacity $\alpha_c$ reaching a maximum of $1/8$ when $\kappa = 0$.
• The analytical expression for the effective potential predicts a critical capacity $\alpha_c(\kappa=0) = 1/8$, derived from the condition $\alpha_c(2+\kappa)^2/2 = 1$.
• Monte Carlo simulations confirm that the quantum perceptron stores significantly more patterns than its classical counterpart, especially for larger $m$, due to quantum superposition and interference.
• The storage capacity $\alpha_c$ decreases with increasing threshold $\kappa$, as higher thresholds reduce the overlap region between input patterns and weight configurations.
• The model shows that quantum perceptrons can store $P \approx \alpha_c \cdot m$ patterns on average, with $\alpha_c$ determined by the interplay of quantum coherence and pattern overlap.
• Numerical results for $m = 8, 16, 32$ demonstrate that the quantum perceptron's storage capacity grows faster than classical models, suggesting a quantum advantage in memory capacity.