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# Nearest-neighbor coupling asymmetry in the generation of cluster states 

$^{1}$Department of Physics, Southeast University, Nanjing 211189, P. R. China 

Peng Xue 1, 2 and Barry C. Sanders 2 

$^{2}$Institute for Quantum Information Science, University of Calgary, Alberta T2N 1N4, Canada 

(Dated: October 17, 2018) 

We demonstrate that charge-qubit cluster state generation by capacitive coupling is anisotropic. Specifically, horizontal vs vertical nearest-neighbor inter-qubit coupling differs in a rectangular lat- tice. We show how to ameliorate this anisotropy by applying potential biases to the array of double dots. 

One-way quantum computing is a particularly attrac- tive model for quantum circuits because global entangle- ment is accomplished in a single step, and then all sub- sequent quantum computation is effected simply by se- quential feedback-controlled single-qubit measurements $^{1}$. The globally entangled state that serves as a universal substrate for quantum computation is known as a clus- ter state and was originally proposed for optical lattices 2 and subsequently demonstrated with photons $^{3}$. Efficient quantum circuits for one-way quantum computing was proposed for solid state devices $^{4}$. 

Solid-state charge-qubit cluster states offer the excit- ing prospect of one-way quantum computing with semi- conductors $^{5,6}$. Here we show that proposals for peri- odic generation of charge-qubit cluster states involving double-dot (henceforth ‘ddot’ as this term emphasizes the single-entity nature of the ddot structure) charge qubits are complicated by an overlooked inter-qubit coupling asymmetry in two dimensions. We remedy this compli- cation by showing that the original proposals 5,6 can be recovered simply by applying a potential field bias. 

We proceed first by establishing the second-quantized Hamiltonian description for the array of quantum dots and then showing how the Hamiltonian can be simplified to a first-quantized Hamiltonian over ddot charge qubits. In the slow tunneling-rate regime, we show that the first- quantized Hamiltonian is well approximated by the ubiq- uitous Ising-like Hamiltonian. This Hamiltonian consid- erably simplifies the dynamical description and shows that the necessary bias of the ddot charge qubits to gen- erate cluster states is determined by the number of ddot neighbors. Thus a global bias of a large structure will lead to periodic evolution of excellent approximations to cluster states. Furthermore, by applying different biases only to the ddots on the boundary (and not to the ddots within), the Hamiltonian induces evolution to the ideal cluster state. 

Our aim here is to remedy the deficiency of anisotropies in the evolution of ddot charge-qubits. We show that this problem can be nearly remedied by a global field bias and completely remedied by a global field bias with corrective biases applied to the ddots at the boundary. This method 

PACS numbers: 03.65.Md, 03.67.Lx, 73.63.Kv 

## I. INTRODUCTION 

<img src='content_image/420969.jpg'>

for correcting anisotropy is examined numerically for the case of 40nm GaAs ddots in a two-dimensional lattice formation with a being the distance between two sites of the ddot, d$_{x}$ and d$_{y}$ the distance between both left and right sites of two nearest-neighbor ddots in x direction and y direction in Fig. 1(a), respectively. 

## II. CHARGE-QUBIT CLUSTER STATE 

The charge-qubit cluster state (CQCS) in two dimen- sions is depicted in Fig. 1. The standard depiction of this cluster state is shown in Fig. 1(a) as a periodic rectan- gular lattice of qubits connected to nearest neighbors by solid lines. Each qubit state is in H 2 = span {| 0 〉 , | 1 〉} , with | 0 〉 the logical zero state and | 1 〉 the logical one state. Cluster state generation proceeds first by globally trans- forming every qubit from the state | 0 〉 to the state | + 〉 where |±〉 := ( | 0 〉 ± | 1 〉 ) $_{/}$√$_{2.}$ We refer to {| 0 〉 , | 1 〉} as 

