# Entanglement Generation of Clifford Quantum Cellular Automata 

Johannes G¨ utschow 1 

$^{1}$Institut f¨ ur Theoretische Physik, Universit¨ at Hannover, Appelstraße 2, 30167 Hannover 

(Dated: 17th August 2018) 

Clifford quantum cellular automata (CQCAs) are a special kind of quantum cellular automata (QCAs) that incorporate Clifford group operations for the time evolution. Despite being classically simulable, they can be used as basic building blocks for universal quantum computation. This is due to the connection to translation-invariant stabilizer states and their entanglement properties. We will give a self-contained introduction to CQCAs and investigate the generation of entanglement under CQCA action. Futhermore, we will discuss finite configurations and applications of CQCAs. 

In this work we will give an introduction to Clifford quantum cellular automata (CQCAs) [1, 2] and study their entanglement generation properties. Quantum cel- lular automata (QCAs) [3] are a quantum computational model that only requires global control for the time evo- lution. The only part of a QCA computation that needs local control is the preparation of the input state, which in the case of a QCA capable of universal quantum com- putation [4, 5] encodes the program and the data input. But not all QCAs are universal quantum computers. The CQCAs we deal with here are even classically simulable. Yet they are of great use in several quantum computa- tional schemes. Furthermore, the classical structure al- lows an in-depth analysis of their time evolution, which can give us a good intuition about the properties of gen- eral QCAs. Most of the results presented here are also included in [2]. In this work we focus on entanglement generation and try to present the theory of CQCAs in a self-contained manner. For a more detailed and mathe- matically concise formulation we refer to [1] and [2]. 

We will begin with a short introduction to QCAs and CQCAs as well as the classical description of CQCAs. Then we will introduce translation-invariant pure sta- bilizer states and study their entanglement in a bipar- tite setting. Putting the results of the first two sections together, we will investigate the evolution of stabilizer states under CQCA action and derive a simple expression for the generation of entanglement by CQCAs acting on stabilizer states. Namely, the entanglement generation is linear and depends only on the degree of the trace of the polynomial matrix representing the CQCA. We will then make some short remarks on CQCAs over finite chains and finally give a short introduction to applications of CQCAs in quantum computing [6–8]. 

## II. CLIFFORD QUANTUM CELLULAR AUTOMATA 

## A. Reversible Quantum Cellular Automata 

A reversible quantum cellular automaton (QCA) T is a translation-invariant reversible discrete time operation 

## I. INTRODUCTION 

on a translation-invariant lattice of quantum systems. So, in each discrete step of the time evolution of our lattice of quantum systems, we will do the same thing and we will do it in a way independent of the position in our lattice. Because we deal with infinite lattices, e.g. spin chains, it is convenient to look at the time evolution of observables rather than states to circumvent mathe- matical problems with Hilbert spaces on infinitely many tensor factors. Using the observable evolution, we get the nice property that a QCA is uniquely determined by the image of the observables on a single site [3]. We call this local operation T$_{i}$ if it acts on the observables of the i th system. As we defined our QCA to be translation invariant, all T$_{i}$ are the same and we pick one, namely T$_{0}$, as a representative. We now introduce another prop- erty of QCAs: locality. In each time step we only allow a finite propagation of information on the lattice, i.e. two states that only differ on one site may only differ on a neighborhood of finitely many sites around the original site after the application of the QCA. This means that the image of the observables at site i is a subset of the the observables of a finite number of sites surrounding site i . We call this set the neighborhood N of the automaton. This is depicted in Figure 1. 

In mathematical terms we say that a QCA is a lo- cal, translation-invariant automorphism of the algebra of observables A . An automorphism T has the prop- erty T( AB ) = T( A )T( B ) , ∀ A, B ∈ A . The local op- eration T$_{0}$ is a homomorphism from A$_{i}$ to A ( N + i ), where N + i denotes the neighborhood of the i th cell and A$_{i}$ the observable algebra of the cell. A homomor- phism preserves commutation relations in the sense that [T( A ) , T( B )] = T[ A, B ] holds. But in our local rule T$_{0}$, information about the evolution of neighboring cells does not play a role. However, for T to be an automorphism, the images of two commuting observables still have to commute after the evolution. So, we impose this condi- tion on T$_{0}$ by demanding that all observables in T( A$_{i}$ ) commute with all observables in T( A$_{i}$ + $_{x}$), x ̸ = 0, where T( A$_{i}$ ) denotes the algebra of the images of observables localized on cell i . 

