## 5 The GHZ\({}_{3}\)MQPEC Protocol in Action  
This section contains an example that demonstrates and explains the operation of the GHZ\({}_{3}\)MQPEC protocol. For practical reasons, this is a small-scale example utilizing 24 qubits. A bigger example, involving more qubits, would result in unintelligible figures that would obstruct the readability and understanding of the quantum circuit. This example, which faithfully follows the abstract quantum circuit of Figure 3, was implemented in Qiskit [70], and the resulting quantum circuit is shown in Figure 6.  
We assume that \(n=3\) millionaires \(\text{Alice}_{0}\), \(\text{Alice}_{1}\), and \(\text{Alice}_{2}\) with private fortunes \(\mathbf{f}_{0}=11\), \(\mathbf{f}_{1}=11\), and \(\mathbf{f}_{2}=01\) want to compare their fortunes in one go with the mediation of [PERSON], whose secret number is \(\mathbf{s}=10\), and [PERSON]. To keep the qubit count low, we opted for input registers that have two qubits, i.e., \(m=2\). So, we employ a parallel array of three circuits \(QC_{0}\), \(QC_{1}\), and \(QC_{2}\), corresponding to \(\text{[PERSON]}_{0}\), \(\text{[PERSON]}_{1}\), and \(\text{[PERSON]}_{2}\), respectively. Below we explain the acronyms used in Figure 6.  
[PERSON]\({}_{0}\), \(\text{[PERSON]}_{1}\), \(\text{[PERSON]}_{2}\), and [PERSON] embed their private numbers in the global state of the entangled system. The embedding is trivially implemented using exclusively CNOT gates. When they measure their input registers, \(\text{[PERSON]}_{0}\), \(\text{Alice}_{1}\), \(\text{Alice}_{2}\), [PERSON], and [PERSON] obtain one of the \(2^{24}=16\),777,216 outcomes, all with equal probability. Due to evident technical constraints, it is not feasible to present all these outcomes, as doing so would lead to a figure that is difficult to interpret. Therefore, we illustrate only 16 of them in Figure 7.  
\(QC_{0}\) consists of the following components:  
* [PERSON]'s input register _alice's_0_IR_ containing two qubits. * [PERSON]'s single-qubit output register _alice's_0_OR_ initialized at \(|-\rangle\). * [PERSON]'s input register _sophia's_0_IR_ containing two qubits. * [PERSON]'s single-qubit output register _[PERSON]'s_0_OR_ initialized at \(|-\rangle\). * [PERSON]'s input register _[PERSON]'s_0_IR_ containing two qubits. * _[PERSON]'s_0_IR_, _[PERSON]'s_0_IR_, and _[PERSON]'s_0_IR_ are initialized in the entangled state \(|GHZ_{3}\rangle^{\otimes 2}\). * The three classical registers (each containing two bits) used by [PERSON], [PERSON], and [PERSON] to store their measurements, are collectively referred to as the _classicalRegister0_ with a total capacity of six bits. \(QC_{1}\) consists of the following components:  
* [PERSON]\({}_{1}\)'s input register _[PERSON]'s_1_IR_ containing two qubits. * [PERSON]\({}_{1}\)'s single-qubit output register _alice's_1_OR_ initialized at \(|-\rangle\). * [PERSON]'s input register _[PERSON]'s_1_IR_ containing two qubits. * [PERSON]'s single-qubit output register _[PERSON]'s_1_OR_ initialized at \(|-\rangle\). * [PERSON]'s input register _[PERSON]'s_1_IR_ containing two qubits. * _[PERSON]'s_1_IR_, _[PERSON]'s_1_IR_, and _[PERSON]'s_1_IR_ are initialized in the entangled state \(|GHZ_{3}\rangle^{\otimes 2}\). * The three classical registers (each containing two bits) used by [PERSON], [PERSON], and [PERSON] to store their measurements, are collectively referred to as the _classicalRegister1_ with a total capacity of six bits. Figure 6: This circuit, created with Qiskit, implements the GHZ\({}_{3}\)MQPEC protocol for 3 millionaires, [PERSON]\({}_{0}\). [PERSON]\({}_{1}\), and [PERSON]\({}_{2}\). Their fortunes were chosen so that they can be represented with \(m=2\) qubits. Figure 7: This figure contains a few potential measurements and their associated probabilities for the circuit shown in Figure 6.  
\(QC_{2}\) consists of the following components:  
* [PERSON]\({}_{2}\)'s input register _alice's_2_IR_ containing two qubits. * [PERSON]\({}_{2}\)'s single-qubit output register _alice's_2_OR_ initialized at \(|-\rangle\).