Computational Power of Symmetry-Protected Topological Phases

TL;DR

This paper demonstrates that ground states of symmetry-protected topological (SPT) phases in quantum spin chains can serve as universal resources for measurement-based quantum computation, with their computational power uniform across phases.

Contribution

It establishes that SPT phases universally enable a set of quantum gates in MBQC, confirming their potential as computationally useful phases of matter.

Findings

Ground states of SPT phases support a full set of single-qubit gates.

Computational power is uniform within each SPT phase.

Lie groups of gates are determined by algebraic invariants of the phase.

Abstract

We consider ground states of quantum spin chains with symmetry-protected topological (SPT) order as resources for measurement-based quantum computation (MBQC). We show that, for a wide range of SPT phases, the computational power of ground states is uniform throughout each phase. This computational power, defined as the Lie group of executable gates in MBQC, is determined by the same algebraic information that labels the SPT phase itself. We prove that these Lie groups always contain a full set of single-qubit gates, thereby affirming the long-standing conjecture that general SPT phases can serve as computationally useful phases of matter.