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## III. THE CNOT GATE 

As mentioned previously AQC has been shown to be polynomially equivalent to the circuit model of quantum computing.The ‘ground state quantum computing’ (GSQC) formalism, described in [7], offers the most practical method of constructing a H$_{0}$ that encodes an arbitrary M qubit, N step quantum circuit. In the GSQC formalism, each of the M qubits in the circuit is viewed as a single electron that can occupy the states in an array of 2 × ( N +1) quantum dots; where the rows in the array represents either the | 0 〉 or | 1 〉 states of the qubit. The state of the m th qubit during the n th step of the algorithm is given by the probability amplitude of the electron being found on the quantum dots denoted by the indices ( m, n, 0) and ( m, n, 1). This theoretical construction only incurs a polynomial overheard ( O ( N )) in hardware. 

The CNOT gate is one of the simplest entangling quantum gates and when used in con- junction with single qubit rotations it forms a set of universal gates. Therefore, any quantum algorithm can be constructed using a combination of these gates, because of this property we assume that the adiabatic CNOT gate is a representative example of a prototypical adi- abatic quantum algorithm. To construct H$_{0}$ for a CNOT gate using the GSQC formalism, we envisage an array of 8 quantum dots (as shown in the inset of Fig. 3), which corresponds to a system of 4 physical qubits. Following the procedure described in [7] we can then write; 

<img src='content_image/330263.jpg'>

where c † mnj is a fermionic creation operator that creates an electron on the corresponding quantum dot, C † mn = $^{(}$c † mn $_{0}$, c † mn 1 $^{)}$and σ$_{x}$ is a Pauli matrix. The ground state energy of the CNOT Hamiltonian (5) is zero. To specify the initial state of the qubits before the gate op- eration we add a small energy penalty to (5); e.g. of the form H$_{Init}$ = µ $^{(}$c † $_{000}$c$_{000}$ + c † $_{100}$c$_{100}$ ) for the operation | 00 〉 → | 00 〉 . 

We numerically solve the equations (3) and (4). For stability reasons we use the multi- step Adams-Moulton method to solve (3). For the sake of generality, the initial conditions for the Pechukas gas, x$_{n}$ ( λ = 1), v$_{n}$ ( λ = 1) and l$_{nm}$ ( λ = 1), are calculated using a per- turbation theory expansion in terms of Z − 1 as at λ = 1, H (1) = H$_{0}$ + ZH$_{b}$ + δh (1) ≡ 

