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used to indicate point-like ddot coordinates rather than the coordinates of a particular quantum dot. Thus, we use ( i, j ) to designate the location of a quantum dot in a two-dimensional array and ( m , n ) to denote the location of a point-like ddot charge qubit. 

Assuming two dots of each ddot pair share one elec- tron, the state od ddot pair is a superposition of two basis states: {| L 〉 , | R 〉} . Here L ( R ) indicates that the electron is in the left (right) dot. In this basis, ˆ H in Eq. (1) for one ddot projected onto ˆ H mn is 

<img src='content_image/376762.jpg'>

for V = V$_{L}$ − V$_{R}$ (the relative energy between the left and right dots of a ddot pair), and T is the flip rate corresponding to the tunneling rate between the two dots of a ddot charge qubit. 

This Hamiltonian can be conveniently rewritten as a linear combination of three types of quantum gates. These gates are the identity 1 1, X = | 0 〉 〈 1 | + | 1 〉 〈 0 | , phase gate Z = | 0 〉 〈 0 | − | 1 〉 〈 1 | with | 0 〉 := | L 〉 and | 1 〉 := | R 〉 . With these simplifications, the Hamiltonian (1) can be projected into the qubit space. This projection becomes clear by studying the single-qubit case comprising one ddot pair. 

In the standard basis the ddot single-qubit Hamilto- nian is ˆ H mn = E 1 + T X + V Z at site ( m , n ). Here E is an energy term for the qubit. The bias V can be con- trolled by applying an electric field potential across the ddot pair. 

In order to connect our mathematical expressions to a typical experimental setting, we consider a GaAs ddot with a single-dot diameter of 40nm $^{12}$. A typical experi- mental parameter range for tunneling is T ≈ 0 − 10 µ eV: here we choose T = 0 . 1 µ eV (160MHz). By tuning the electric field to V = 0, the evolution of the Hamiltonian ˆ H mn effectively implements bit flips via the X operator, and the resultant tunneling or flipping rate is 160MHz. By tuning the electric potential bias to V ≫ T , e.g. V = 10 µ eV, the dynamics is dominated by phase flip- ping at a rate of 16GHz. 

Now let us consider the two-qubit gate such as the CZ gate, which can be implemented via Coulomb interac- tion between two nearest-neighbor charge qubits shown below. The Hamiltonian for two nearest-neighbor charge qubits located at ( m , n ) and at ( $^{$_{m}$′}$,$^{$_{n}$′}$) with (possibly screened) Coulomb interaction is 

<img src='content_image/376763.jpg'>

## C. Two-qubit gates 

The last term describes the Coulomb interaction en- ergy in the qubit basis between the two nearest-neighbor charge qubits located at ( m , n ) and at ( $^{$_{m}$′}$,$^{$_{n}$′}$) to with sums over l and k representing the two charge qubits are 

in the state | l 〉 and | k 〉 . The coefficients ς$_{lk}$ are the inter-site Coulomb interac- tion strengths between the same or opposite sites of the two charge qubits ( m , n ) and ( $^{$_{m}$′}$,$^{$_{n}$′}$). For the case l = k , i.e. the two charge qubits are in the same states and both electrons in the left or right dots, we have ς$_{00}$ = W mnm ′ n ′ LL and ς$_{11}$ = W mnm ′ n ′ RR . For the other case l ̸ = k , i.e. the two charge qubits are in different states and the two electrons in different dots, we have ς$_{01}$ = W mnm ′ n ′ LR and ς$_{10}$ = W mnm ′ n ′ RL . 

In the 2D lattice shown in Fig. 1(a), for two nearest- neighbor charge qubits there are two cases: both qubits located in the same column (along the x direction) and in the same row (along the y direction). We use the su- perscripts x and y to distinguish these two cases. For the two nearest-neighbor charge qubits in the same column in which the structure is symmetric, ς x 00 = ς x 11 = V$_{Q}$/d$_{x}$ , and ς x 01 = ς x 10 = V$_{Q}$/ √ d 2 x + a 2 for V$_{Q}$ = e $^{2}$/ (4 πϵ ) with ϵ the applicable dielectric constant. In contrast, for the two nearest-neighbor charge qubits in the same row in which the structure is asymmetric, ς y 00 = ς y 11 = V$_{Q}$/d$_{y}$ , ς y 01 = V$_{Q}$/ ( d$_{y}$ + a ), and ς y 10 = V$_{Q}$/ ( d$_{y}$ − a ). For the GaAs ddot system considered in the previous 

subsection, we have V$_{Q}$ = 1 . 75 × 10 − $^{29}$N · m $^{2}$. With the experimental parameters a = 400nm, d$_{x}$ = 5 . 5 µ m and d$_{y}$ = 5 . 85 µ m $^{12}$, we numerically estimate the coefficients of the interaction terms to obtain ς x 00 = ς x 11 = 20 . 0 µ eV, ς x 01 = ς x 10 = 19 . 8 µ eV, ς y 00 = ς y 11 = 18 . 7 µ eV, ς y 01 = 17 . 5 µ eV and ς y 10 = 20 . 1 µ eV. 

## IV. GENERATING CLUSTER STATES 

In the one-way quantum computing model, the two- dimensional cluster state is a highly entangled multi- qubit state and processed by performing sequences of adaptive single-qubit measurements, thereby realizing ar- bitrary quantum computations. The two-dimensional cluster state serves as a universal resource for one-way quantum computing, in the sense that any multi-qubit state can be prepared by performing sequences of local operations on a sufficiently large two-dimensional cluster state $^{11}$. 

In previous proposals $^{5,6}$, charge qubits are treated as being symmetrically coupled, which is an appropriate strategy for the one-dimensional case but not at all for the two-dimensional case. Here we show that, by apply- ing local external electric fields, we can generate two- dimensional cluster states without the requirement of symmetry of charge qubits. 

In the two-dimensional case shown in Fig. 1(a), we obtain a more general Hamiltonian case with external electric fields V mn applied on each qubit ( m , n ) located in the $_{m}$th row and the $_{n}$th column of the ddot charge qubit 

