## 4 The GHZ\({}_{3}\)MQPEC Protocol
### Box 2.
[PERSON]'s transform \(U_{\mathbf{s}}\) is based on the scheme \(U_{\mathbf{s}}\colon\left|y\right\rangle\left|\mathbf{x}\right\rangle\to \left|y\oplus\left(\mathbf{s}\bullet\mathbf{x}\right)\right\rangle\left| \mathbf{x}\right\rangle\), which, for the particular circuit \(\mathbb{QC}_{i}\), becomes  
Figure 2.0  
The intuition behind [PERSON]\({}_{i}\)'s unitary transform \(U_{\mathbf{f}_{i}}\) is quite similar since it implements the scheme \(U_{\mathbf{f}_{i}}\colon\left|y\right\rangle\left|\mathbf{x}\right\rangle \rightarrow\left|y\oplus\left(\mathbf{f}_{i}\bullet\mathbf{x}\right)\right\rangle \left|\mathbf{x}\right\rangle\), which, for our circuit \(\mathbb{QC}_{i}\), assumes the following explicit form:  
Figure 2.1  
The cumulative effect of both unitary transforms sends the quantum circuit into the next state \(\left|\psi_{1}\right\rangle\):  
Figure 2.2 Figure 2.3 Figure 2.4  
At the conclusion of Phase 1, [PERSON] and [PERSON]\({}_{i}\) have untraceably integrated their confidential numbers \(\mathbf{s}\) and \(\mathbf{f}_{i}\), which are known solely to them, into the state \(\left|\psi_{1}\right\rangle\) of the entangled quantum system. The next step is to extract this information, enabling [PERSON] to compare the wealth of the millionaires. To retrieve the embedded private data, [PERSON], [PERSON], and [PERSON]\({}_{i}\) perform the \(m\)-fold Hadamard transform on their input registers, as illustrated in Figure 2.5 As a result, by the end of Phase 2, the system's state has transitioned to \(\left|\psi_{2}\right\rangle\):  
Figure 2.6  
At this point, the use of Equation (4) allows us to expand \(H^{\otimes m}\left|\mathbf{x}\right\rangle_{T}\), \(H^{\otimes m}\left|\mathbf{x}\right\rangle_{S}\), and \(H^{\otimes m}\left|\mathbf{x}\right\rangle_{i}\), as shown below:  
Figure 2.7 Figure 2.8 Figure 2.9  
By substituting the previous relations, we may rewrite \(\left|\psi_{2}\right\rangle\) as  
Figure 2:0 appearing in (14) becomes just 0. * If \(\mathbf{s}\oplus\mathbf{f}_{i}\oplus\mathbf{z}_{2}\oplus\mathbf{z}_{1}\oplus \mathbf{z}_{0}=\mathbf{0}\), or, equivalently, \(\mathbf{z}_{2}\oplus\mathbf{z}_{1}\oplus\mathbf{z}_{0}=\mathbf{s}\oplus \mathbf{f}_{i}\), the sum Figure 2:1 becomes Figure 2:2 (15) Thus, \(\left|\psi_{2}\right\rangle\) can be written in the simpler form given below: Figure 2:3 (16) where Figure 2:4 (17) The relationship expressed by (17) is identified as the **Hadamard Entanglement Property**, which connects the input registers of [PERSON], [PERSON], and [PERSON]\({}_{i}\). This relationship emerges as a consequence of the initial entanglement present among these input registers. By the time Phase 2 is concluded, [PERSON] and [PERSON]\({}_{i}\) have encoded their private values, \(\mathbf{s}\) and \(\mathbf{f}_{i}\), as the modulo 2 sum \(\mathbf{s}\oplus\mathbf{f}_{i}\) within the state of the system, thereby imposing this correlation on the contents of the input registers. The GHZ\({}_{3}\)MQPEC protocol's quantum portion is completed when [PERSON], [PERSON], and [PERSON]\({}_{i}\) perform measurements on their input registers with respect to the computational basis. Through this process, the composite system's state collapses to \(\left|\psi_{f}\right\rangle\):  
Figure 2:5 Figure 2:6  
We note that both output registers \(\mathit{SOR}_{i}\) and \(\mathit{AOR}_{i}\) remain in their initial state \(\left|-\right\rangle\) throughout the execution of the protocol. In that sense, they could be ignored and removed from (18). It is expedient to point out that our reasoning is valid no matter whether the private circuits of the players are in the same locality or are spatially distributed. This is because the previous equations derive their validity from the fact that the input registers are entangled in the first place. In the presence of entanglement, the distance among the players is irrelevant. From now on, the GHZ\({}_{3}\)MQPEC protocol uses only classical authenticated communication channels.