2 A. The extended Hubbard model III. MODELING THE DYNAMICS B. Single-qubit gates ˆ V = 1 2 ∑ i,j,i ′ ,j ′ V$_{ij,i}$ ′ j ′ ̂ Δ n$_{ij,i}$ ′ j ′ (2) the ‘standard basis’ and {| + 〉 , |−〉} as the ‘dual basis’. After all N qubits in the cluster state are prepared in the | + 〉 ⊗ N state, nearest-neighbor qubits then interact via the two-qubit controlled- Z operations, denoted CZ. Here Z is the Pauli ‘phase’ operator. The other Pauli operators are the ‘flip’ operator X , the ‘flip+phase’ op- erator Y = XZ , and the identity operator 1 . This op- eration is represented in the two-qubit standard basis {| 00 〉 , | 01 〉 , | 10 〉 , | 11 〉} as CZ = diag { 1 , 1 , 1 , − 1 } . The unitary operation CZ is independent of whether the line in Fig. 1(a) is horizontal or vertical. Fig. 1(b) shows that the horizontal vs vertical sym- metry is in fact broken by the coupling axis of the charge qubit, which is represented in Fig. 1(b) as an ex- cess charge in the left or right quantum dot. Although the charge-qubit coupling axis could be aligned indepen- dently of the orientation of the overall qubit lattice, we will treat the case that the charge-qubit coupling axis is in the x -direction. This is a physically reasonable case, and extending to the case of arbitrary alignment is in- volved but not difficult. The charge qubit can be created as a semiconductor ddot structure $^{7}$. Other alternatives exist such as the su- perconducting charge qubit 8 or a pair of dangling bonds on a surface $^{9}$. In any case, the logical states typically cor- respond either to the left- and right-well occupancy by the excess charge or, alternatively, to the cases of sym- metric or antisymmetric charge states between the two dots of a ddot charge qubit. For coherently evolving charge qubits, Schr¨ odinger’s equation can be used to describe the dynamics, and the potential in Schr¨ odinger’s equation is depicted in Fig. 1(c). Here we treat the standard basis as corre- sponding to left- and right-occupancy; the dual basis then corresponds to the symmetric and antisymmetric charge- occupancy states. The quantum dots are engineered so that each poten- tial well has only one bound energy state for the excess electron. Due to the Pauli exclusion principle, the num- ber of electrons in each well is either zero or one or else two electrons with opposite spins. The case of two excess electrons in one double-dot structure should be energet- ically forbidden by Coulomb repulsion between the two electrons to preserve the integrity of the charge qubit. The goal is to have one excess electron per closely- spaced quantum dot pair, but the general picture is that each quantum dot can have one or two electrons. The restriction of one excess electron must emerge as an ener- getically favorable configuration rather than be imposed by fiat. The full second-quantized description of the elec- trons in the array of quantum dots is given by the ex- tended Hubbard model (EHM). Eq. (1) is a second-quantized Hamiltonian. To bridge this Hamiltonian over to the multi-qubit description, we restrict the Hilbert space, upon which the first-quantized version of the Hamiltonian acts, to the case of a single excess electron in each double well. Note here that a dot has lattice coordinates expressed here as ( i$_{1}$, j$_{1}$ ) and dot 2 is at ( i$_{2}$, j$_{2}$ ). For an array with close proximity between dots of the ddot pair, the resultant charge-qubit ddot pair can be treated as a point-like object in the quantum computing architecture. This ddot charge qubit has a point-like co- ordinate designated by ( m , n ) where the change of font is In fact Eq. (1) describes not just nearest-neighbor in- teractions but interactions between all dots with all other dots, where the inter-dot couplings T$_{ij,i}$ ′ j ′ and W$_{ij,i}$ ′ j ′ are suitably chosen. For charge qubits corresponding to closely spaced dot pairs, T$_{ij,i}$ ′ j ′ can be neglected for all but the ddot of a given charge qubit. Also W$_{ij,i}$ ′ j ′ is only significant between charge qubits. The interdot (possibly screened) Coulomb repulsion is neglected for the ddots of a charge qubit because W$_{ij,ij}$ is sufficiently large to prevent both dots from being simultaneously excessively charged. with V$_{ij,i}$ ′ j ′ the inter-site ( i, j ) ↔ ( i $^{′}$, j $^{′}$) potential differ- ence. Here E is the effective on-site energy for each site ( i, j ), which can vary due to local field effects. T$_{ij,i}$ ′ j ′ is the coherent tunneling rate between sites ( i, j ) and ( i $^{′}$, j $^{′}$). W$_{ij,i}$ ′ j ′ is the Coulomb repulsion energy be- tween sites ( i, j ) and ( i $^{′}$, j $^{′}$). Finally, for ̂ Δ n$_{ij,i}$ ′ j ′ := ˆ n$_{ij}$ − ˆ n$_{i}$ ′ j ′ the number-difference operator between sites ( i, j ) and ( i $^{′}$, j $^{′}$), the potential bias operator is The EHM applies to an array of quantum dots whose locations in a two-dimensional array are denoted by lat- tice coordinates. For ˆ c$_{ij}$ the annihilation operator at dot site ( i, j ), ˆ c † ij the conjugate creation operator, and ˆ n$_{ij}$ = ˆ c † $_{ij}$ˆ c$_{ij}$ the number operator, the dynamics of the charge-qubit cluster state can conveniently be described by the extended Hubbard Hamiltonian 10 (As spin is con- served, we can, without loss of generality, assume fixed spin and ignore this degree of freedom) ˆ H = ∑ i,j E ˆ n$_{ij}$ + ˆ V (1) + ∑ i,j,i ′ ,j ′ W$_{ij,i}$ ′ j ′ ˆ n$_{ij}$ ˆ n$_{i}$ ′ j ′ − T$_{ij,i}$ ′ j ′ ( ˆ c † $_{ij}$ˆ c$_{i}$ ′ j ′ + h.c. ) .