Latent Computational Complexity of Symmetry-Protected Topological Order with Fractional Symmetry

TL;DR

This paper demonstrates that ground states of certain symmetry-protected topological orders with fractional symmetry possess inherent computational complexity, enabling universal quantum computation through simple measurements.

Contribution

It introduces fractional symmetry in SPTO and proves that fixed-point states of 2D $( ext{Z}_2)^m$ SPTO can be used for universal quantum computation.

Findings

Fixed-point states enable universal quantum computation with Pauli measurements.

Fractional symmetry defines a new class of computationally complex topological states.

Potential to establish a 'quantum computational phase' of matter.

Abstract

An emerging insight is that ground states of symmetry-protected topological orders (SPTO's) possess latent computational complexity in terms of their many-body entanglement. By introducing a fractional symmetry of SPTO, which requires the invariance under 3-colorable symmetries of a lattice, we prove that every renormalization fixed-point state of 2D $(\mathbb{Z}_2)^m$ SPTO with fractional symmetry can be utilized for universal quantum computation using only Pauli measurements, as long as it belongs to a nontrivial 2D SPTO phase. Our infinite family of fixed-point states may serve as a base model to demonstrate the idea of a "quantum computational phase" of matter, whose states share universal computational complexity ubiquitously.