A quantum neural network framework for scalable quantum circuit approximation of unitary matrices

2024-02-07

arXiv.org

AI Summary

This work addresses the problem of efficiently approximating multi-qubit unitary matrices with quantum circuits. We propose a scalable, Lie-group-theoretic parametrization framework. Methodologically, we replace the conventional Pauli-string basis with a recursively defined block basis as generators; employ the exponential map combined with structured parameterized quantum circuits to achieve compact unitary representations; and introduce a linear-scale recursive construction scheme—where an (n+1)-qubit unitary circuit is built incrementally from an n-qubit circuit by adding only a constant number of CNOT and single-qubit gates. Our approach achieves O(n) depth and width scaling, markedly enhancing scalability, training efficiency, and hardware compatibility. The resulting framework provides a novel paradigm for large-scale quantum algorithm compilation and quantum neural network design.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Learning to Search
• Computer Vision: Learning & Optimization for CV

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Security and Privacy: Large-scale security measurements

Abstract

In this paper, we develop a Lie group theoretic approach for parametric representation of unitary matrices. This leads to develop a quantum neural network framework for quantum circuit approximation of multi-qubit unitary gates. Layers of the neural networks are defined by product of exponential of certain elements of the Standard Recursive Block Basis, which we introduce as an alternative to Pauli string basis for matrix algebra of complex matrices of order $2^n$. The recursive construction of the neural networks implies that the quantum circuit approximation is scalable i.e. quantum circuit for an $(n+1)$-qubit unitary can be constructed from the circuit of $n$-qubit system by adding a few CNOT gates and single-qubit gates.

Problem

• Develop scalable quantum circuit approximation for unitary matrices
• Introduce quantum neural network framework for multi-qubit gates
• Use Lie group theory for parametric unitary matrix representation

Innovation

• Lie group theory for unitary matrices
• Quantum neural network framework
• Scalable recursive quantum circuits