B. Cylindrical cluster state VI. CONCLUSIONS C. Validity of Ising-like evolution to a cluster state Acknowledgments 6 By changing the structure of the array of charge qubits to that shown in Fig. 2(c), we can generate a large two- dimensional cluster state by simply applying global elec- tric fields instead of local electric fields, resulting in an important simplification to the technical challenges of generating cluster states. From our analysis above we observe that the choice of external electric field strength depends on two factors: the number of connections to nearest neighbors ν$_{x}$ + ν$_{y}$ and the extra energy offset due to asymmetry of the two- dimensional structure. The periodic boundary condition n + N = n dismisses the second factor, and the extra en- ergy offsets of the ddot pairs located on two edge columns cancel each other. The periodic boundary condition also partially dimin- ishes the first factor and makes the connections of all ddot pairs except for those located on the first and last rows to be ν$_{x}$ + ν$_{y}$ = 4. For those ddots located on the first and last rows, ν$_{x}$ + ν$_{y}$ = 3, and the global electric fields on the two rows are set to be (8 k ′ − 3) ς/ (8 k + 1) while all the others are set to be (8 k ′ − 4) ς/ (8 k + 1) . For simplic- ity, choosing k = k ′ = 0, to generate a two-dimensional cluster state, one only needs to apply an external global electric field − 3 ς on the first and last rows, and − 4 ς for the remaining rows, respectively. Our approach is valid if the contribution of the term ˆ H$_{und}$ in Eq. (11) to the evolution is negligible. The fi- delity of the pure two-dimensional cluster state is for N qubits, which gives an upper bound of the maxi- mum number of cluster qubits with a fixed fidelity for ¯ V the absolute value of the average energy off- set of ddot pairs. In other words, approximating the anisotropic evolution by isotropic Ising-like evolution is valid provided that the total number of qubits does not exceed N$_{max}$ , which depends on the acceptable less-than- unity fidelity F . With the evolution time t = π/ 4 ς , the fidelity and the maximum number of cluster qubits can be written as F = 1 − ( πNT/ ¯ V ) 2 and N$_{max}$ ( N $^{2}$) = (1 − F ) ( ¯ V /πT ) $^{2}$. For T = 0 . 1 µ eV, ς = 10 µ eV and ¯ V ≈ 4 ς = 40 µ eV (for large N ), our calculations show that a 2D 162-qubit cluster state with a high fidelity F = 0 . 99 can be produced. For We have considered deterministic unitary evolution of a cluster state in a charge-qubit structure with charge qubits made of double-dot (ddot) structures. Although periodic evolution into charge-qubit cluster states has been considered before, anisotropy presents a critical yet overlooked challenge. At first anisotropy seems to destroy the opportunity to create cluster states in this way. We have shown how to circumvent this problem by applying electric field biases to the ddot structures. In the slow-tunneling regime, the effective single-quantized multi-qubit Hamiltonian can be approximated by the Ising-like Hamiltonian. In this case, electric field biases can overcome the challenge of anisotropy. If the elec- tric fields had to be tailored to each ddot charge qubit, or if the field had to be controlled over time, the strat- egy would be impractical. However, we have shown that a global field over all but the boundary qubits, and five choices of electric field biases on boundary qubits no mat- ter how large the system is, entirely eliminates the prob- lem of anisotropy. Remarkably, by changing the structure of the array of charge qubits, we can generate a large two-dimensional cylindrical cluster state by simply applying an electric field on the first and last rows, and a different one for the remaining rows, respectively. Compared to previous schemes, no assumption of isotropy for charge qubit cou- plings is made in our procedure and in fact is shown not to be valid for two-dimensional cluster states. We aug- ment our theoretical analysis of anisotropy by including numerical analysis for the case of GaAs double dots in a two-dimensional lattice. In particular we show that the energy offsets due to anisotropy are noneligible in this case. For these charge qubits to be useful, noise and deco- herence need to be considered. Also the charge qubits considered here periodically evolve into cluster states and then back to their initial states due to the periodic nature of the unitary evolution. Timing becomes critical in such dynamics, or else the interactions that produce the clus- ter states must be able to be switched off. Measurement- based quantum computing also becomes challenging for such periodically-occurring cluster states. These consid- erations are the seeds for future study. This work has been supported by National Natural Science Foundation of China, Grant No. 10944005, the Southeast University Start-Up Fund, Canada’s Natural Science and Engineering Research Council, the Canadian Innovation Platform “QuantumWorks”, and Alberta’s F = ∣ ini 〈 Ψ | e i ˆ H eff $^{t}$e − i ˆ H Is $^{t}$| Ψ 〉 ini ∣ ∣$^{2}$ ≈ 1 − ( 4 NςtT ¯ V ) 2 , (18) N$_{max}$ ( N $^{2}$) = (1 − F ) ( ¯ V 4 ςtT ) 2 (19) T = 1 µ eV and ς = 10 µ eV, N$_{max}$ = 16 with a fidelity F = 0 . 9 for the 2D cluster state.