# Quantum Circuit Implementation of Multi-Dimensional Non-Linear Lattice Models
## 6 Extension to Two- and Three-Dimensional Lattice Models
### Analysis of Circuit Width for Different Designs
#### 6.2.1. Design \(Ekin2D1\)  
For Design \(Ekin2D1\), the following additional qubits are needed:  
\[|qu3\rangle : \text{`hidden' mantissa qubit for $u$ and $v$ for $N_{M}=4$}\] \[|r7|\dots|r0\rangle : 2N_{M}\text{-qubit register for mantissa squaring output}\] \[|c\rangle : \text{`carry' qubit--initialized as $|0\rangle$}\] \[|anc\rangle : \text{ancilla qubit--initialized as $|0\rangle$}\] \[|eusq2|eusq1|eusq0\rangle : N_{E}\text{-qubit representation of the exponent of $u^{2}$}\] \[|musq2|musq1|musq0\rangle : (N_{M}-1)\text{-qubit representation of the mantissa of $u^{2}$}\] \[|evsq2|evsq1|evsq0\rangle : N_{E}\text{-qubit representation of the exponent of $v^{2}$}\] \[|mvsq2|mvsq1|mvsq0\rangle : (N_{M}-1)\text{-qubit representation of the mantissa of $v^{2}$}\]  
Figure 9: Quantum circuit design \(Ekin2D2\) for computation of \(u^{2}+v^{2}\). The floating-point representation uses four mantissa qubits (\(N_{M}=4\)) and three exponent qubits (\(N_{E}=3\)). Here it is assumed that a number of space-saving measures are used. First, the temporary swapping of \(u\) and \(v\) mantissa qubits so the same set-up for mantissa squaring of \(u\) can be re-used for \(v\). Secondly, the difference in \(u^{2}\) and \(v^{2}\) exponents, temporarily needed in working out the quantum floating-point addition, can be stored in \(|evsq2|evsq1|evsq0\rangle\) and that after un-computation of this difference, the original exponent of \(v^{2}\) is restored. Lastly, it can be assumed that the \(2N_{M}\)-qubit register \(|r7|\ldots|r0\rangle\) can be re-used to perform the addition of the (re-normalized) mantissa qubit-strings of \(u^{2}\) and \(v^{2}\). Therefore, the additional qubit count amounts to the total,  
\[1+2N_{M}+2+2N_{E}+2(N_{M}-1)=1+4N_{M}+2N_{E}\]  
Design _Ekin2D2_  
For Design _Ekin2D2_, the following additional qubits are needed:  
\[|r7|\dots|r0\rangle : 2N_{M}\text{-qubit register for mantissa squaring output}\]0 \[|r7|\dots|r0\rangle : 2N_{M}\text{-qubit register for mantissa squaring output}\]1 \[|r7|\dots|r0\rangle : 2N_{M}\text{-qubit register for mantissa squaring output}\]2 \[|r7|\dots|r0\rangle : 2N_{M}\text{-qubit register for mantissa squaring output}\]3 \[|r7|\dots|r0\rangle : 2N_{M}\text{-qubit register for mantissa squaring output}\]4 \[|r7|\dots|r0\rangle : 2N_{M}\text{-qubit register for mantissa squaring output}\]5 \[|r7|\dots|r0\rangle : 2N_{M}\text{-qubit register for mantissa squaring output}\]6  
Here it is assumed that a number of space-saving measures are used. First, the temporary swapping of \(u\) and \(v\) mantissa qubits so the same set-up for mantissa squaring of \(u\) can be re-used for \(v\). Secondly, the difference in \(u\) and \(v\) exponents, temporarily needed in working out the quantum floating-point addition, can be temporarily stored in \(|ev2|ev1|ev0\rangle\) and that after un-computation of this difference, the original exponent of \(v\) is restored. Lastly, it is assumed that starting from the two \(2N_{M}\)-qubit mantissa-squaring result registers, the exponent difference for \(u\) and \(v\), stored temporarily in \(|ev2|ev1|ev0\rangle\), can be used to set up two \(N_{M}\) input qubit strings representing re-normalized mantissa qubits of \(u^{2}\) and \(v^{2}\) into \(N_{M}\)-qubit full adder, re-using \(|c\rangle\) as the 'carry' qubit. Therefore, for this design, the total number of additional qubits is  
\[|r7|\dots|r0\rangle : 2N_{M}\text{-qubit register for mantissa squaring output}\]7  
since, for practical applications, \(N_{M}\) needs to be significantly greater than \(N_{E}\), it follows that Design _Ekin2D2_ has greater circuit width than Design _Ekin2D1_.