Quantum simulation beyond Hamiltonian paradigm: categorical quantum simulation

TL;DR

This paper introduces categorical quantum simulation, a novel approach that leverages tensor category theory instead of traditional group theory, enabling more efficient simulation of complex quantum systems like $SU(3)$ Yang-Mills theory.

Contribution

The paper proposes a new simulation paradigm based on tensor categories, expanding quantum simulation capabilities beyond the Hamiltonian framework and demonstrating its application to Yang-Mills theory.

Findings

Categorical quantum simulation reduces qubit resources compared to traditional methods.

It enables simulation of systems previously hard to simulate efficiently.

Provides a new encoding method called emergenism encoding.

Abstract

With the development of topological field theory, the mathematical tool of the tensor category was also introduced into physics. Traditional group theory corresponds to a special category,group category. Tensor categories can describe higher-order interactions and symmetric relations, while group theory can only describe first-order interactions. In fact, the quantum circuit itself constitutes a category. However, at present, the field of quantum computing mainly uses group theory as a mathematical tool. If category theory is introduced into the field of quantum simulation, the application scope of quantum computers can be greatly expanded. This paper propose a new dynamic simulation method,categorical quantum simulation. In our paradigm quantum simulation is no longer based on the structure of the group theory, but based on the structure of the tensor category. This could enable many systems that could not be efficiently quantum simulated before.In this article we give an concrete example of the categorical simulation of $SU(3)$ Yang-Mills theory. It shows that categorical quantum simulation provides a new encoding method,emergenism encoding, which saves more qubits resources than reductionism quantum encoding. In addition, many domains can be described in the language of category theory, which allows quantum circuits to directly encode and simulate these domains.