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.
Highlights
The fields of study of quantum phases of matter and of quantum computation have been evolving alongside each other for over a decade, such that they are deeply intertwined
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)
We show that every phase we construct is computationally universal in the same way as the cluster phase, except for those defined by non-entangling quantum cellular automata (QCA)
Summary
We present a review of quantum cellular automata (QCA) for qubit systems, as described in Refs. [77,78,79], as they will be central to our description of symmetry protected topological order with respect to subsystem symmetries. We use the symbol ξ to represent both the binary and polynomial representations of an element of PN interchangeably. We can represent the CQCA T as a 2 × 2 matrix t of polynomials by arranging. Every CQCA T can be represented as a 2 × 2 matrix t whose entries are Laurent polynomials over Z2, up to phase factors [78]. When Tr(t) = uc + u−c for some positive integer c, the CQCA supports gliders These are operators on which the CQCA acts as translation by ±c sites. Where we have used our assumption that det(t) = 1, and the fact that the polynomials are defined over the field Z2, so addition and subtraction are equivalent This useful equation allows us to reduce any power of t to a linear combination of t and I
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