Subsystem symmetries, quantum cellular automata, and computational phases of quantum matter

TL;DR

This paper links quantum information theory and condensed matter physics by using quantum cellular automata to classify subsystem symmetry-protected topological phases and demonstrate their universality as resources for measurement-based quantum computation.

Contribution

It introduces a unified framework using quantum cellular automata to characterize subsystem SPT phases and proves these phases are computationally universal for MBQC.

Findings

Subsystem symmetries are characterized by quantum cellular automata.

Most constructed phases are universal resources for MBQC.

The approach enables practical advantages like computational speedup.

Abstract

Quantum phases of matter are resources for notions of quantum computation. In this work, we establish a new link between concepts of quantum information theory and condensed matter physics by presenting a unified understanding of symmetry-protected topological (SPT) order protected by subsystem symmetries and its relation to measurement-based quantum computation (MBQC). The key unifying ingredient is the concept of quantum cellular automata (QCA) which we use to define subsystem symmetries acting on rigid lower-dimensional lines or fractals on a 2D lattice. Notably, both types of symmetries are treated equivalently in our framework. We show that states within a non-trivial SPT phase protected by these symmetries are indicated by the presence of the same QCA in a tensor network representation of the state, thereby characterizing the structure of entanglement that is uniformly present throughout these phases. By also formulating schemes of MBQC based on these QCA, we are able to prove that most of the phases we construct are computationally universal phases of matter, in which every state is a resource for universal MBQC. Interestingly, our approach allows us to construct computational phases which have practical advantages over previous examples, including a computational speedup. The significance of the approach stems from constructing novel computationally universal phases of matter and showcasing the power of tensor networks and quantum information theory in classifying subsystem SPT order.