Duality in spin systems via the $SU(4)$ algebra

TL;DR

This paper introduces a diagrammatic approach using $SU(4)$ algebra to map complex spin Hamiltonians to simpler forms, revealing dualities that simplify analysis and facilitate quantum state preparation.

Contribution

It presents a novel $SU(4)$ algebra-based method for identifying dualities in spin systems, enabling easier diagonalization and phase transition analysis.

Findings

Dualities map complex Hamiltonians to simpler ones

Minimum energy gap remains constant with system size

Facilitates efficient adiabatic preparation of quantum states

Abstract

We provide several examples and an intuitive diagrammatic representation demonstrating the use of two-qubit unitary transformations for mapping coupled spin Hamiltonians to simpler ones and vice versa. The corresponding dualities may be exploited to identify phase transition points or to aid the diagonalization of such Hamiltonians. For example, our method shows that a suitable one-parameter family of coupled Hamiltonians whose ground states transform from an initially factorizing state to a final cluster state on a lattice of arbitrary dimension is dual to a family of trivial decoupled Hamiltonians containing local on-site terms only. As a consequence, the minimum enery gap (which determines the adiabatic run-time) does not scale with system size, which facilitates an efficient and simple adiabatic preparation of e.g. the two-dimensional cluster state used for measurement-based quantum computation.