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

TL;DR

This paper introduces a scalable quantum neural network framework based on Lie group theory for approximating multi-qubit unitary matrices, offering a new basis and recursive construction method for quantum circuits.

Contribution

It develops a Lie group theoretic approach and a recursive neural network framework for scalable quantum circuit approximation of unitary matrices, using an alternative basis to Pauli strings.

Findings

The framework enables scalable quantum circuit construction for increasing qubits.

It introduces a new basis for matrix algebra of complex matrices.

The recursive approach simplifies the extension from n-qubit to (n+1)-qubit circuits.

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.