## 4 The GHZ\({}_{3}\)MQPEC Protocol
### Box 2.
\(U_{\mathbf{f}_{i}}\) is the unitary transform used by [PERSON]\({}_{i}\), which is described by Equation (11). * \(H^{\otimes m}\) is the \(m\)-fold Hadamard transform. * [PERSON]\({}_{i}\), \(0\leq i\leq n-1\), [PERSON], and [PERSON] act on their local circuits in a private and secret manner. [PERSON]\({}_{i}\)'s private circuit is structurally similar to [PERSON]'s, but there is a critical difference. [PERSON]\({}_{i}\) encodes her fortune \(\mathbf{f}_{i}\) into the global state of the system using an appropriate unitary transform \(U_{\mathbf{f}_{i}}\). Similarly, [PERSON] embeds her secret number \(\mathbf{s}\) into the global system through her unitary transform \(U_{\mathbf{s}}\). Since, in general, \(\mathbf{f}_{i}\
eq\mathbf{s}\), the unitary transforms \(U_{\mathbf{f}_{i}}\) and \(U_{\mathbf{s}}\) are different. Ideally, assuming sufficient quantum resources, we shall employ a parallel array of \(n\) such circuits \(QC_{0},\ldots,QC_{n-1}\), as visualized in Figure 4. In case this is not feasible, due to a lack of adequate resources, it is possible to employ these circuits in sequential batches, as we show in Figure 5. All quantum circuits \(QC_{i}\), \(0\leq i\leq n-1\), are structurally identical, their only possible difference being in unitary transform \(U_{\mathbf{f}_{i}}\) that embeds [PERSON]\({}_{i}\)'s fortune \(\mathbf{f}_{i}\) into the system. Obviously, if [PERSON]\({}_{i}\) and [PERSON]\({}_{j}\)'s fortunes \(\mathbf{f}_{i}\) and \(\mathbf{f}_{j}\), \(0\leq j\
eq i\leq n-1\), are equal, the corresponding unitary transforms \(U_{\mathbf{f}_{i}}\) and \(U_{\mathbf{f}_{j}}\) are the same; otherwise, they are different. Figure 3: The above circuit \(QC_{i}\), \(0\leq i\leq n-1\), embeds [PERSON]\({}_{i}\)’s fortune \(\mathbf{f}_{i}\) and [PERSON]’s secret number \(\mathbf{s}\) into the global state of the system. The potentially spatially separated private circuits operated by [PERSON]\({}_{i}\), [PERSON], and [PERSON] are linked due to entanglement, indicated by the wavy red line connecting \(TIR_{i},SIR_{i}\), and \(AIR_{i}\), and form one composite system. Utilizing (2), the start state \(\ket{\psi_{0}}\) of the quantum circuit depicted in Figure 3 can be expressed as  
\[\ket{\psi_{0}}=\frac{1}{\sqrt{2^{m}}}\sum_{\mathbf{x}\in\mathbb{B}^{m}}\ket{ \mathbf{x}}_{T}\ket{-}_{S}\ket{\mathbf{x}}_{S}\ket{-}_{i}\ket{\mathbf{x}}_{i}. \tag{9}\]  
As mentioned previously, throughout our analysis, we shall use the subscripts \(T\), \(S\), and \(i\) to clarify that we refer to the kets in the private circuits of Trent, Sophia, and Alice\({}_{i}\), respectively. Hopefully, this practice will enhance readability and avoid any ambiguity. Figure 4: In the ideal scenario, the GHZ\({}_{3}\)MQPEC protocol can be executed entirely in parallel by employing a parallel array of \(n\) circuits \(QC_{0},\ldots,QC_{n-1}\). Figure 5: If the lack of resources precludes the parallel implementation of the protocol, the \(n\) circuits \(QC_{0},\ldots,QC_{n-1}\) can be partitioned in sequential batches. In this example, \(QC_{0},\ldots,QC_{n-1}\) are partitioned into 3 batches. The circuits within each batch are employed simultaneously, but the 3 batches are executed sequentially. The execution of the protocol commences in earnest by having [PERSON] and [PERSON]\({}_{i}\) act on their private circuits via their secret unitary transforms \(U_{\mathbf{s}}\) and \(U_{\mathbf{f}_{i}}\). By doing so, both of them encode their secret numbers into the entangled state of the global circuit. Let us emphasize that [PERSON] and [PERSON]\({}_{i}\)'s actions result in the embedding of their private numbers into the entanglement of the three input registers of the distributed circuit.