Efficiency of the ground-state quantum computer

The energy gap is calculated for a ground-state quantum computer circuit. It is found that, when implementing a quantum algorithm by Hamiltonians containing only pairwise interaction, the inverse of energy gap is proportional to $N^{4k}$, where $N$ is the number of bits involved in the problem, and $N^k$ is the number of control operations performed in a standard quantum paradigm.

Figure 1

Each dot represents a row of two quantum dots. (a) shows two qubits interacting with each other by $h$(CNOT). (b) shows a chain of qubits interacting with each other through $h$(CNOT). The label $I$ stands for identity transformation, CNOT for $h$(CNOT) for arbitrary unitary transformation, and $B$ for boost Hamiltonian $h(B,\lambda )$.

Figure 2

Modified circuit of the dotted box in Fig. 1b, in which four teleportation boxes are inserted before and after $H$(CNOT). The circuit inside each teleportation box is similar to the teleportation circuit in 15. Label $I$ represents identity transformation Hamiltonian $h\left(I\right),H$ for Hadamard transformation Hamiltonian $h\left(H\right)$, and $P\left(0\right)$ for projection Hamiltonian $h(\mid 0⟩,\lambda )$. The dotted lines represent the part of circuit not showing.