[Paper Review] Quantum Neuron: an elementary building block for machine learning on quantum computers

Yudong Cao, Gian Giacomo Guerreschi+1 more|arXiv (Cornell University)|Nov 30, 2017

Quantum Computing Algorithms and ArchitectureComputer Science27 references113 citations

TL;DR

The paper introduces a quantum neuron built from repeat-until-success circuits to implement nonlinear activation, enabling quantum feedforward networks and Hopfield-like associative memories that can learn from superpositions of inputs.

ABSTRACT

Even the most sophisticated artificial neural networks are built by aggregating substantially identical units called neurons. A neuron receives multiple signals, internally combines them, and applies a non-linear function to the resulting weighted sum. Several attempts to generalize neurons to the quantum regime have been proposed, but all proposals collided with the difficulty of implementing non-linear activation functions, which is essential for classical neurons, due to the linear nature of quantum mechanics. Here we propose a solution to this roadblock in the form of a small quantum circuit that naturally simulates neurons with threshold activation. Our quantum circuit defines a building block, the "quantum neuron", that can reproduce a variety of classical neural network constructions while maintaining the ability to process superpositions of inputs and preserve quantum coherence and entanglement. In the construction of feedforward networks of quantum neurons, we provide numerical evidence that the network not only can learn a function when trained with superposition of inputs and the corresponding output, but that this training suffices to learn the function on all individual inputs separately. When arranged to mimic Hopfield networks, quantum neural networks exhibit properties of associative memory. Patterns are encoded using the simple Hebbian rule for the weights and we demonstrate attractor dynamics from corrupted inputs. Finally, the fact that our quantum model closely captures (traditional) neural network dynamics implies that the vast body of literature and results on neural networks becomes directly relevant in the context of quantum machine learning.

Motivation & Objective

• Motivate and formulate a quantum analogue of the classical neuron that preserves coherence and entanglement.
• Propose a quantum circuit construction that simulates threshold/sigmoid activations using repeat-until-success (RUS) circuits.
• Demonstrate how quantum neurons can form feedforward networks and Hopfield networks.
• Show learning from superpositions of training data and establish connections to classical neural network theory.

Proposed method

• Map a classical neuron input to a quantum rotation on a qubit, enabling a quantum state that encodes the activation.
• Realize nonlinear activations via repeat-until-success (RUS) circuits that implement rotations like Ry(2q(theta)) with q(theta)=arctan(tan^2 theta).
• Use controlled rotations and an ancilla-based RUS scheme to simulate threshold behavior and drive the output toward attractor states.
• Provide a theoretical runtime analysis, including Theorem 1’s expected runtime bound for achieving a desired accuracy.
• Demonstrate that the quantum neuron can simulate classical feedforward networks (Theorem 2) and Hopfield networks (Theorem 3) in quantum form.
• Show learning from a superposition of training data and evaluate training via measurable correlations such as <ZZ> between output and target qubits.

Experimental results

Research questions

• RQ1 Can a quantum circuit realize nonlinear activation compatible with quantum mechanics to emulate classical neurons?
• RQ2 Can networks of quantum neurons perform feedforward computation and approximate classical deep neural networks?
• RQ3 Do quantum Hopfield networks exhibit associative memory and attractor dynamics similar to classical Hopfield networks?
• RQ4 What are the resource costs (time, qubits) for simulating classical networks and Hopfield dynamics with quantum neurons?
• RQ5 Is learning feasible when training data are presented as superpositions rather than classical batches?

Key findings

• A quantum neuron is constructed that realizes threshold-like nonlinear activation using repeat-until-success circuits.
• The paper provides a formal runtime bound for preparing the activated output qubit with specified accuracy (Theorem 1).
• A quantum algorithm is shown to simulate an ell-layer classical deep feedforward neural network with step activation, including qubit/ runtime scaling (Theorem 2).
• A quantum Hopfield network with the quantum neuron can simulate t updates with provable efficiency and preserves attractor dynamics akin to associative memory (Theorem 3).
• Numerical results demonstrate learning XOR and 8-bit parity functions using superposed training data and Nelder–Mead optimization, indicating learning from quantum superpositions is viable.