An explicit tensor notation for quantum computing

TL;DR

This paper proposes a tensor-based formalism for quantum computing that enhances the intuitive understanding of quantum states and gates, preserves Hilbert space structure, and reduces computational costs in simulating multi-qubit systems.

Contribution

It introduces a novel tensor notation formalism that improves visualization and efficiency in representing quantum states and operations, linking entanglement with gate representation.

Findings

Formalism preserves Hilbert space structure

Reduces classical simulation costs for multi-qubit states

Connects entanglement generation with gate representation

Abstract

This paper introduces a formalism that aims to describe the intricacies of quantum computation by establishing a connection with the mathematical foundations of tensor theory and multilinear maps. The focus is on providing a comprehensive representation of quantum states for multiple qubits and the quantum gates that manipulate them. The proposed formalism could contribute to a more intuitive representation of qubit states, and to a clear visualisation of the entanglement property. The main advantages of this formalism are that it preserves the fundamental structure of the Hilbert space to which quantum states belong, and also reduces the computational cost associated with classical prediction of the effect of quantum gates on multi-qubit states. A connection between the ability to generate entanglement and the quantum gate representation is also established.