[Paper Review] Quantum neural network

M. V. Altaĭsky|ArXiv.org|Jul 3, 2001

Quantum Computing Algorithms and Architecture2 references100 citations

TL;DR

This paper proposes a quantum neural network (QNN) model using optical qubits with superposition states defined by photon polarization, where weights are implemented via beam splitters and phase shifters. It introduces a non-unitary learning rule that enables convergence to desired output states through iterative weight updates, demonstrating a feasible path for quantum-enhanced learning with phase-sensitive signal processing.

ABSTRACT

It is suggested that a quantum neural network (QNN), a type of artificial neural network, can be built using the principles of quantum information processing. The input and output qubits in the QNN can be implemented by optical modes with different polarization, the weights of the QNN can be implemented by optical beam splitters and phase shifters

Motivation & Objective

• To develop a quantum neural network (QNN) model that leverages quantum information processing principles for enhanced learning capabilities.
• To address the challenge of implementing nonlinear activation functions in quantum systems by proposing a phase- and amplitude-sensitive weight update mechanism.
• To demonstrate a learning rule that drives the quantum perceptron output toward a desired target state, even when unitarity is not preserved.
• To explore the feasibility of using optical components—beam splitters, phase shifters, and attenuators—for scalable quantum neural network implementation.

Proposed method

• Qubits are encoded as photon polarization states |0⟩ and |1⟩, with superposition states |x⟩ = α|0⟩ + β|1⟩.
• The quantum perceptron uses n input qubits |x_j⟩ and a linear output state |y(t)⟩ = Σ w_j(t)|x_j⟩, where weights w_j are 2×2 unitary matrices formed from beam splitters and phase shifters.
• A non-unitary learning rule is proposed: w_j(t+1) = w_j(t) + η(|d⟩ - |y(t)⟩)⟨x_j|, where |d⟩ is the desired output and η is the learning rate.
• The learning rule is shown to reduce the squared norm of the error, with |||d⟩ - |y(t+1)⟩||² = (1 - nη)²|||d⟩ - |y(t)⟩||² for small η < 1/n.
• The model allows for signal attenuation and phase shifts, interpreted as complex impedances, enhancing learnability and classical simulation potential.
• The framework is extended to full QNNs by composing multiple quantum perceptrons using standard artificial neural network architectures.

Experimental results

Research questions

• RQ1 Can a quantum neural network be constructed using only linear optical elements such as beam splitters and phase shifters?
• RQ2 Is it possible to design a learning rule for a quantum perceptron that drives the output toward a desired target state despite non-unitary updates?
• RQ3 How does the inclusion of signal attenuation and phase shifts in the weight matrices affect the learnability and convergence of the quantum network?
• RQ4 To what extent does the phase of the input signal, rather than just amplitude, contribute to the learning dynamics in a quantum neural network?

Key findings

• The proposed learning rule (6) ensures convergence to the desired output state |d⟩ when the learning rate η is small enough (η < 1/n), as shown by the error decay factor (1 - nη)².
• The model achieves convergence even though the learning rule violates unitarity, suggesting that non-unitary operations such as attenuation may be beneficial for learnability.
• The use of phase and amplitude modulation via beam splitters and phase shifters allows for complex weight matrices that can model signal attenuation and phase shifts, analogous to biological neural impedance.
• The framework supports scalable QNN construction by composing quantum perceptrons using standard ANN architectural principles.
• The model is more amenable to classical simulation than unitary-only alternatives due to the inclusion of non-unitary weight updates.