# Quantum Circuit Implementation of Multi-Dimensional Non-Linear Lattice Models
## 5 Quantum-Circuit Implementation for One-Dimensional Burgers Equation  
The quantum-circuit implementation of the non-linear lattice model for the one-dimensional Burgers equation is investigated in this section. In particular, the quantum floating-point-based implementation of its non-linear equilibrium function is employed in the collision step. For \(N_{M}=4\) and \(N_{E}=3\) (using an exponent bias of 5), Figures 6 and 7 show the overall design of this quantum-circuit implementation. In the quantum-circuit implementation, the number of qubits needed is identical to that demonstrated for the modified D1Q3 model in the previous section for the same choice of \(N_{M}\) and \(N_{E}\). For convenience, the qubits associated with storing the equilibrium distribution function \(\tilde{f}^{eq}\) are now named with \(f\) instead of \(g\). Specifically, here the qubits are named:  
\[|ef2|ef1|ef0\rangle : N_{E}\text{-qbit representation of the exponent of }f^{eq}\] \[|sf\rangle : \text{sign bit for considered the component of }f^{eq}\] \[|mf2|mf1|mf0\rangle : (N_{M}-1)\text{-qbit representation of mantissa of }f^{eq}\]  
Figure 5: Quantum-circuit implementation of \(SQ4\) operation (and its un-computation \(ISQ4\)) as used in the modified D1Q3 model and in the non-linear lattice model for 1D Burgers equations. Figure 6: Quasi-\(1\)D Burgers: quantum circuit design for computing \(\tilde{f}^{eq}\) in the lattice model. The floating-point representation uses four mantissa qubits (\(N_{M}=4\)) and three exponent qubits (\(N_{E}=3\)), with an exponent bias set to five. Part 1. Starting from the left-hand side of Figure 6, the steps shown can be summarized as follows:  
* Qubit \(|cut\rangle\) initialized at \(|0\rangle\) is set to \(|1\rangle\) in case \(u^{2}\) gets truncated to 0. For the employed quantum floating-point format (with exponent bias equal to five in contrast to the case \(u^{2}\)), the qubits \(|0\rangle\) and \(|1\rangle\) are set to \(|1\rangle\). Figure 7: Quasi-1D Burgers: quantum circuit design for the evaluation of \(\tilde{f}^{\text{eq}}\) in the lattice model. The floating-point representation uses four mantissa qubits (\(N_{M}=4\)) and three exponent qubits (\(N_{E}=3\)), with an exponent bias set to five. Part 2.  
to eight used for modified D1Q3 model), this occurs for sub-normal \(u\) velocity, i.e., \(|eu2|eu1|eu0\rangle=|000\rangle\);
* For \(|cut\rangle=|0\rangle\), the mantissa qubits of \(u\) get temporarily copied into the workspace qubits in preparation of the squaring operation (termed \(SQ4\) here for \(N_{M}=4\) considered). Once \(2N_{M}\)-qubits defining the square of \(u\) mantissa is set, the temporary copies of \(u\) mantissa qubits are removed;
* For the two discrete-velocity directions defined by \(|dv1|dv0\rangle=|01\rangle\) and \(|dv1|dv0\rangle=|10\rangle\), the final state of the qubits defining the equilibrium function in floating-point format can directly be defined from velocity \(u\). This operation is performed at \(0110u4\) in the circuit;
* For the two discrete-velocity directions defined by \(|dv1|dv0\rangle=|00\rangle\) and \(|dv1|dv0\rangle=|11\rangle\), the equilibrium distribution function combines terms \(\pm u/4\) and \(u^{2}/4\). In the quantum circuit, the operation shown as \(0011a2\) prepares this summation for which a 5-qubit modulo adder (\(MADD5\)) is used for \(N_{M}=4\) by employing the \(2N_{M}\)-qubit mantissa squaring output with \(u\), while introducing shift accounting for the exponents of \(u\) and \(u^{2}\) terms;
* Once the 5-qubit modulo adder has performed the required addition for the two discrete-velocity directions defined by \(|dv1|dv0\rangle=|00\rangle\) and \(|dv1|dv0\rangle=|11\rangle\), operation \(0011b2\) uses this output to define the equilibrium distribution function for these directions in the quantum floating-point format.