Encrypted quantum state tomography with phase estimation for quantum Internet

Abstract

Quantum state tomography (QST) is a fundamental tool requiring privacy in future distributed systems where unknown states are measured for extracting information. Gentle measurement and differential privacy (DP)-based privacy solutions minimize the damage on unknown state and leakage about the quantum information, respectively. In this article, we propose a fundamentally different design for privacy-preserving QST in a multi-party setting. We assume that Alice delegates QST task of a distant source for which she has no access to a third-party player Bob accessing to the source while preserving the source privacy against the operations realized by Bob. Encrypted QST algorithm is proposed which encodes or maps source computational basis states by exploiting phase estimation and feature mapping concept of quantum machine learning (QML). Bob maps basis states to eigenvalues of a specially designed unitary operator in an entangled manner with his ancillary qubits while teleporting the source qubits back to Alice before applying conventional QST. Encoding mechanism is conjectured as having NP-hard decoding complexity based on difficulty of subset-sum problem combined with Hadamard transform. Linear optical design and quantum circuit implementations are presented for future experiments in noisy intermediate-scale quantum (NISQ) devices. Theoretical and numerical supporting evidences are proposed supporting the proposed eigenstructure. EQST promises further applications for multiple source classification tasks and as a novel feature mapping method for future data embedding tasks in QML.

Appendices

Appendix A: Evolution of single photon state in linear optical set-up with multiple WPDs

We assume that $BS_k$ in Fig. 4a applies the following operator:

$$\begin{aligned} U_{BS,k} \, = \, \begin{bmatrix} \cos (\theta _k) &{} \quad \imath \, \sin (\theta _k) \\ \imath \, \sin (\theta _k) &{} \quad \cos (\theta _k)\\ \end{bmatrix} \end{aligned}$$

(A1)

The standard phase shift operator shifts the phase of the photon in the path with the following for the chosen path index $j = 0$ or 1:

$$\begin{aligned} \Phi _{k,j} \, \equiv \, \begin{bmatrix} e^{\imath \, \phi _{k,0}} &{} \quad 0 \\ 0 &{} \quad e^{\imath \, \phi _{k,1}}\\ \end{bmatrix} \end{aligned}$$

(A2)

WPD with the index k applies either $U_{k,0}$ or $U_{k,1}$ depending on the selected path of the photon passing through BS. Although there is no constraint on the unitaries and the dimension of Hilbert space in WPDs, we assume that each WPD state is two dimensional with the corresponding single qubit unitary gates $U_{k,0}$ and $U_{k,1}$. We assume that the initial states in WPDs at time $t_0$ are pure states represented by $|D_n\rangle \, \otimes \, |D_{n-1}\rangle \, \ldots \otimes |D_1\rangle$ in order to better observe the evolution of each path through consecutive WPDs. At time $t_i$, the state is denoted as $|\Psi _i\rangle$. Then, at time $t_0$, the initial state is as follows:

$$\begin{aligned} |\Psi _0\rangle \, = \,|Q\rangle \, \otimes \, |D_n\rangle \, \otimes \, |D_{n-1}\rangle \, \ldots \, \otimes \, |D_1\rangle \end{aligned}$$

(A3)

where $|Q\rangle = \alpha _0 \, |0\rangle \, + \, \alpha _1 \, |1\rangle$ denotes the initial state of the single photon. Assume that $|Q\rangle = |0\rangle$ is chosen to analyze the evolution of the WPD states more simply. In the following, we remove the tensor product $\otimes$ in order to simplify the expressions which can be understood from the context. Furthermore, we reorder positions of WPD states for the following modeling in order to better characterize the evolution since the interaction of WPDs starts with the first WPD. At the end, we re-correct again.

After the first beam splitter $BS_1$, the state is transformed into the following:

$$\begin{aligned} U_{BS,1} \, |0\rangle \,|D_1\rangle \, |D_{2}\rangle \, \ldots \, |D_n\rangle \end{aligned}$$

(A4)

The state becomes $\big (\cos (\theta _1) |0\rangle \, + \, \imath \, \sin (\theta _1) \, |1\rangle \big )\, |D_1 \, D_{2} \, \ldots \, D_n\rangle$. Then, unitary operator is applied on the first WPD state depending on the path chosen by the photon and the state becomes entangled with the photon path as follows:

$$\begin{aligned} \big (\cos (\theta _1) |0\rangle U_{1,0} |D_1\rangle \, + \, \imath \, \sin (\theta _1) |1\rangle U_{1,1} |D_1\rangle \big )\,| D_2 \, \ldots \, D_n \rangle \ \end{aligned}$$

(A5)

Finally, phase shift operator $\Phi _{1,j}$ is applied depending on the path resulting in $|\Psi _1\rangle$ as a superposition of two paths entangling the photon and the first WPD state:

$$\begin{aligned} |\Psi _1\rangle \,= & {} \, \big ( \cos (\theta _1) \, e^{\imath \, \phi _{1,0}} \, |0\rangle U_{1,0} |D_1\rangle \, + \, \imath \, \sin (\theta _1) \, e^{\imath \, \phi _{1,1}} \, |1\rangle U_{1,1} |D_1\rangle \big ) \, \nonumber \\{} & {} \otimes | D_2 \, \ldots \, D_n \rangle \end{aligned}$$

(A6)

Defining $\widetilde{c}_{k,j} \equiv \cos (\theta _k) \, e^{\imath \, \phi _{k,j}}$, $\widetilde{s}_{k,j} \equiv \imath \, \sin (\theta _k) \, e^{\imath \, \phi _{k,j}}$ and $|d_j\rangle _{k} \equiv U_{k,j} |D_k\rangle$ simplifies the expression as $|\Psi _1\rangle = \, \big ( \widetilde{c}_{1,0} \, |0\rangle |d_0\rangle _{1} + \widetilde{s}_{1,1} \, |1\rangle |d_1\rangle _{1} \big ) \, | D_2 \, \ldots \, D_n \rangle$. Similarly, entanglement of the photon with the next WPD detector results in $|\Psi _2\rangle$ consisting of the superposition of four paths:

$$\begin{aligned} |\Psi _2\rangle= & {} \big ( |0\rangle ( \widetilde{c}_{2,0}\, \widetilde{c}_{1,0} \, |d_0\rangle _{1} \, + \, \widetilde{s}_{2,0}\, \widetilde{s}_{1,1} \, |d_1\rangle _{1} ) |d_0\rangle _{2} \, \nonumber \\{} & {} + |1\rangle ( \widetilde{c}_{1,0}\, \widetilde{s}_{2,1} \, |d_0\rangle _{1} \, + \, \widetilde{c}_{2,1}\, \widetilde{s}_{1,1} \, |d_1\rangle _{1} ) |d_1\rangle _{2} \big ) \nonumber \\{} & {} \otimes \, | D_3 \, \ldots \, D_n \rangle \, \hspace{0.2in} \end{aligned}$$

(A7)

It can be simply iterated until the $(n+1)$th step to compute $|\Psi _{n+1}\rangle$ as follows after re-ordering the positions of WPD states:

$$\begin{aligned} \sum _{j_{1} =0}^{1} \,\sum _{j_{2} =0}^{1} \, \ldots \, \sum _{j_{n} =0}^{1} \, \sum _{j_{n+1} =0}^{1} \, A(j_1, \ldots , j_n, j_{n+1}) \, \, |j_{n+1}\rangle _{n+1} \, |d_{j_n}\rangle _n \, \ldots \, |d_{j_1}\rangle _1\hspace{0.15in}{} & {} \end{aligned}$$

(A8)

where we denote the final state of the photon as $|j_{n+1}\rangle _{n+1}$ and $A(j_1, \ldots , j_n, j_{n+1})$ is defined as follows:

$$\begin{aligned} A(j_1, \ldots , j_{n+1})\equiv & {} K_{j_1} \prod _{k=1}^{n} \chi _{k+1, j_{k}, j_{k+1}} \end{aligned}$$

(A9)

where the following amplitudes are defined:

$$\begin{aligned} \chi _{k+1, j_{k}, j_{k+1}}= & {} {\left\{ \begin{array}{ll} \widetilde{c}_{k+1, j_{k+1}}, &{} \text {if } j_{k} \, = \, 0, \, j_{k+1}\,=\,0 \\ \widetilde{s}_{k+1, j_{k+1}}, &{} \text {if } j_{k} \, = \, 1, \, j_{k+1}\,=\,0 \\ \widetilde{s}_{k+1, j_{k+1}}, &{} \text {if } j_{k} \, = \, 0, \, j_{k+1}\,=\,1 \\ \widetilde{c}_{k+1, j_{k+1}}, &{} \text {if } j_{k} \, = \,1, \, j_{k+1}\,=\,1 \\ \end{array}\right. }\end{aligned}$$

(A10)

$$\begin{aligned} K_{j_1}= & {} {\left\{ \begin{array}{ll} \widetilde{c}_{1,0}, &{} \text {if } j_{1} \, = \, 0 \\ \widetilde{s}_{1,1}, &{} \text {if } j_{1} \, = \, 1 \\ \end{array}\right. } \end{aligned}$$

(A11)

Appendix B: Duality of the solution for ancillary qubit state

Assume that the following equalities are satisfied for some unknown $\omega$, $\gamma _{0, 0}$, $\gamma _{0, 1}$ and $\gamma _{1, 0}$ depending on $\phi _k$ for $k \in [1, n+1]$ for varying $\vec {s}$:

$$\begin{aligned} \varrho _{0,0}= & {} \cos (\omega ) \, e^{\imath \, \gamma _{00}} \end{aligned}$$

(B12)

$$\begin{aligned} \varrho _{1,0}= & {} \sin (\omega )\, e^{\imath \, \gamma _{10}} \end{aligned}$$

(B13)

$$\begin{aligned} \varrho _{0,1}= & {} \sin (\omega )\, e^{\imath \, \gamma _{01}} \end{aligned}$$

(B14)

$$\begin{aligned} \varrho _{1,1}= & {} \cos (\omega )\, e^{\imath \, (\gamma _{01} \, + \, \gamma _{10} \,- \, \gamma _{00}\,- \, \pi )} \end{aligned}$$

(B15)

where $\varrho _{j_0, k_0}$ are expressed for $j_0, \, k_0 \, \in [0,1]$ as defined in Sect. 5. Furthermore, assume that equalities (30–33) have a solution with $a_0 = \mu _0$, $b_0 = \mu _1$, $a_1 = \mu _2$ and $b_1 \, = \,0$. Then, by inserting the solutions and using the assumptions in (B12–B15), the satisfied equalities in (30–33) are converted to the following:

$$\begin{aligned} E_1\equiv & {} \mu _2 \,\sin (2 \, \omega ) \big (\mu _0 \, g_1 \, - \, \mu _1 \, g_2 \big ) \, - \, \sin ^2(\omega ) \big ( \mu _0^2 \, + \, \mu _1^2 \, - \, \mu _2^2 \big ) \end{aligned}$$

(B16)

$$\begin{aligned} E_2\equiv & {} \sin ^2(\omega ) \big (\mu _0^2 \,+ \, \mu _1^2 \,- \, \mu _2^2 \big ) \, + \, \mu _2 \, \sin (2 \,\omega ) \big (\mu _1 \, g_2 \, - \, \mu _0 \, g_1 \big ) \end{aligned}$$

(B17)

$$\begin{aligned} E_3\equiv & {} \sin (\omega ) \Big ( \big ( \mu _1^2 \,- \, \mu _0^2 \big ) \, g_3 \, + \, 2 \, \mu _0 \, \mu _1 \, g_4 \, + \, \mu _2^2 \, g_5\Big ) \nonumber \\{} & {} + \, \mu _2 \, \cos (\omega ) \, \big ( \mu _0 f_4 \, - \, \mu _1 \, f_1 \big ) \end{aligned}$$

(B18)

$$\begin{aligned} E_4\equiv & {} \sin (\omega ) \Big ( \big ( \mu _1^2 \, - \, \mu _0^2 \big ) \, g_4 \, - \, 2 \, \mu _0 \, \mu _1 \, g_3 \, + \, \mu _2^2 \, g_6 \Big ) \nonumber \\{} & {} + \mu _2 \, \cos (\omega ) \big ( \mu _0 \, f_1 \, + \, \mu _1 \, f_4 \big ) \end{aligned}$$

(B19)

$$\begin{aligned} E_5\equiv & {} \mu _0^2 + \mu _1^2 + \mu _2^2 - 1 \end{aligned}$$

(B20)

where $g_1 = \cos (\gamma _{00} \,- \,\gamma _{10})$, $g_2 = \sin (\gamma _{00} \, - \, \gamma _{10})$, $g_3 = \cos (\gamma _{00} \, + \, \gamma _{01})$, $g_4 = \sin (\gamma _{00} \, + \, \gamma _{01})$, $g_5 = \cos (\gamma _{00} \,+ \, \gamma _{10})$, $g_6 = \sin (\gamma _{00} \, + \, \gamma _{10})$, $f_1 = \big (\sin (2 \, \gamma _{00}) \, + \, \sin (\gamma _{01} \, + \, \gamma _{10}) \big )$, $f_2 = \, \mu _2^2 \, \cos (\gamma _{00} \, + \, \gamma _{10})$, $f_3 = \, 2 \, \mu _0 \, \mu _1 \, \sin (\gamma _{00} \, + \, \gamma _{01})$, $f_4 = \big (\cos (2 \, \gamma _{00}) \, + \, \cos (\gamma _{01} \, + \, \gamma _{10}) \big )$, $E_k \, = \, 0$ for $k \in [1, 5]$. Similarly, if another solution, i.e., the dual solution, is $a_0 = \mu _2$, $b_0 \, = \,0$, $a_1 = - \, \mu _0$ and $b_1 = \mu _1$, then (B12–B15) are utilized to obtain following which should give $E_k \, = \, 0$ for $k \in [6, 10]$:

$$\begin{aligned} E_6\equiv & {} - \, E_1 \end{aligned}$$

(B21)

$$\begin{aligned} E_7\equiv & {} - \, E_2 \end{aligned}$$

(B22)

$$\begin{aligned} E_8\equiv & {} \sin (\omega ) \Big ( \big ( \mu _1^2 \, - \, \mu _0^2 \big ) \, g_5 \, - \, 2 \, \mu _0 \, \mu _1 \, g_6 \, + \, \mu _2^2 \,g_3 \Big ) \nonumber \\{} & {} + \, \mu _2 \cos (\omega ) \big ( \mu _0 \, f_4 \, + \, \mu _1 \, f_1 \big ) \end{aligned}$$

(B23)

$$\begin{aligned} E_9\equiv & {} \sin (\omega ) \Big ( \big ( \mu _1^2 \, - \, \mu _0^2 \big ) \, g_6 \, + \, 2 \, \mu _0 \, \mu _1 \, g_5 \, + \, \mu _2^2 \, g_4 \Big ) \nonumber \\{} & {} + \, \mu _2 \cos (\omega ) \big ( \mu _0 \, f_1 \, - \, \mu _1 \, f_4 \big ) \end{aligned}$$

(B24)

$$\begin{aligned} E_{10}\equiv & {} E_5 \end{aligned}$$

(B25)

Therefore, for both the solutions to exist, $E_k$ should be zero for $k \in [1, 10]$. Equating $E_{1}$, $E_2$ and $E_3$ to zero gives the following solutions for $\omega$ and $\mu _0$:

$$\begin{aligned} \omega \,= & {} \, \tan ^{-1}\Bigg (-\, \frac{\mu _2 \,\Big ( \mu _0 \big (\cos (2 \, \gamma _{00}) \, + \, \cos (\gamma _{01} \,+ \, \gamma _{10}) \big ) \, - \, \mu _1 \, f_1 \Big )}{\left( \mu _1^2 \,- \, \mu _0^2\right) \, g_3 \, + \, f_2 \,+ f_3} \bigg ) \end{aligned}$$

(B26)

$$\begin{aligned} \mu _0\,= & {} \, - \, \mu _1 \, \cot \left( \frac{\gamma _{01} \,- \, \gamma _{10}}{2}\right) \end{aligned}$$

(B27)

Substituting $\mu _0$ into $E_k$ for $k \in \lbrace 3, 4, 8, 9\rbrace$ results in the following equalities:

$$\begin{aligned} \frac{E_3}{\cos (\gamma _{00} \, + \, \gamma _{10})} \, - \, \frac{E_8}{\cos (\gamma _{00} \, + \, \gamma _{01})}= & {} 0 \end{aligned}$$

(B28)

$$\begin{aligned} \frac{E_4}{\sin (\gamma _{00} \, + \, \gamma _{10})} \, - \, \frac{E_9}{\sin (\gamma _{00} \, + \, \gamma _{01})}= & {} 0 \end{aligned}$$

(B29)

If $\cos (\gamma _{00} \, + \, \gamma _{01})$, $\cos (\gamma _{00} \, + \, \gamma _{10})$, $\sin (\gamma _{00} \, + \, \gamma _{01})$ and $\sin (\gamma _{00} \, + \, \gamma _{10})$ are not equal to zero, i.e., $\gamma _{00} \, + \, \gamma _{01}$ and $\gamma _{00} \, + \, \gamma _{10}$ are not multiples of $\pi \, / \, 2$, then $E_3 = 0$ and $E_4 = 0$ imply that $E_8 = 0$ and $E_{9} = 0$, respectively. It is also easily seen that $E_1 \, + \, E_2 = 0$. Since $E_6 \, = \, - \, E_1$, $E_7 \, = \, - \, E_2$ and $E_{10} = E_5$, the equalities $E_{1} = E_2 = E_5 = E_6 = E_ 7 = E_{10}= 0$ are satisfied together. Therefore, all the equalities are satisfied together for being equal to zero by providing the existence of the duality solution where $\gamma _{00}$, $\gamma _{01}$ and $\gamma _{10}$ depend on $\phi _k$ for $k \in [1, n+1]$ for varying $\vec {s}$ as defined in (B12–B15). They should also satisfy the conditions of $\cos (\gamma _{00} \, + \, \gamma _{01})$, $\cos (\gamma _{00} \, + \, \gamma _{10})$, $\sin (\gamma _{00} \, + \, \gamma _{01})$ and $\sin (\gamma _{00} \, + \, \gamma _{10})$ being not equal to zero.

We have utilized the assumptions about the forms of $\varrho _{k_0, j_0}$ in (B12–B15) while defining $\gamma _{00}$, $\gamma _{01}$ and $\gamma _{10}$. In order to satisfy these equalities, $E_{k}$ values should be equal to zero for $k \in [11, 14]$ defined as follows:

$$\begin{aligned} E_{11}\equiv & {} \vert \varrho _{0,0} \vert ^2 \, + \, \vert \varrho _{1,0} \vert ^2 \, -\, 1 \end{aligned}$$

(B30)

$$\begin{aligned} E_{12}\equiv & {} \vert \varrho _{0,0} \vert ^2 \, + \, \vert \varrho _{0,1} \vert ^2 \, -\, 1 \end{aligned}$$

(B31)

$$\begin{aligned} E_{13}\equiv & {} \vert \varrho _{0,1} \vert ^2 \, + \, \vert \varrho _{1,1} \vert ^2 \, -\, 1 \end{aligned}$$

(B32)

$$\begin{aligned} E_{14}\equiv & {} \varrho _{0,0}^* \, \varrho _{1,0} + \varrho _{0,1}^* \, \varrho _{1,1} \end{aligned}$$

(B33)

$E_{k}$ values for $k \in [11, 14]$ are calculated by using $\varrho _{j_0, k_0} \equiv \vec {u}_{k_0}^T \, \big (\prod _{l=1}^{m-1} \mathbf {\widehat{M}}_{m-l, s_{m-l}} \big ) \, \textbf{K}_{n-m+1} \, \vec {v}_{j_0} \, / \, \sqrt{2^{n+1}}$ with the modeling in Sect. 5. If the most general case is chosen with $m = n+1$, then $\varrho _{j_0, k_0}$ becomes equal to $\vec {u}_{k_0}^T \, \big (\prod _{l=1}^{n} \mathbf {\widehat{M}}_{n+1-l, s_{n+1-l}} \big ) \, \vec {v}_{j_0} \, / \, \sqrt{2^{n+1}}$. One can perform the calculations with symbolic variables $\phi _k$ to check whether $E_{k} \, = \, 0$ for $k \in [11, 14]$. We perform iterations of symbolic calculations until $n \, = \, 6$ for small values of n to observe that $E_{k} \, = \, 0$ for $k \in [11, 14]$. We also perform numerical analysis in Sect. B.1 until $n = 13$ satisfying (B12–B15) with great accuracy. It is an open issue to show that the equalities are satisfied for all n by using trigonometry and by exploiting the rotation matrices in the definition of $\mathbf {\widehat{M}}_{k, s_{k}}$ for a more rigorous proof.

As a result, there exist dual solutions depending on whether $\phi _k$ for $k \in [1, n+1]$ satisfies $E_k = 0$ for $k \in [1, 10]$ while $\gamma _{00} \, + \, \gamma _{01}$ and $\gamma _{00} \, + \, \gamma _{10}$ are not multiples of $\pi \, / \, 2$. It is an open issue to determine the class of $\phi _k$ for $k \in [1, n+1]$ providing the favorable solution and the eigenstructure. Uniformly distributed values of $\phi _k$ for $k \in [1, n+1]$ provide promising solutions in numerical analysis as a supporting numerical evidence as shown next.

1.1 B.1. Numerical Analysis

We obtain the unitary matrix $U_{\vec {\Phi }}$ for varying n and phase shift values by using (18) and then calculate its eigenvalues and eigenvectors. We numerically analyze that it has distinct eigenvalues and also the coefficients $\alpha _0$ and $\alpha _1$ defined for the eigenvectors in (21) and (22) satisfy the duality relation. The main assumptions to be satisfied for the proposed eigenstructure are given in (B12–B15) and the following equalities $E_1$ to $E_{10}$ in (B16–B25). We solve (B12–B15) for $\omega$, $\gamma _{00}$, $\gamma _{01}$ and $\gamma _{10}$ given the uniformly distributed values of $\phi _i$ for $i \in [1, n+1]$ and define the following performance metric for the solution corresponding to a specific $\vec {s} = [s_1 \, \ldots \, s_{m-1}]$ for $m \in [1, n+1]$ indexed with $k \in [1, 2^n]$:

$$\begin{aligned} \epsilon _{1}[k]\equiv & {} \vert \varrho _{0,0} \, - \, \cos (\omega ) \, e^{\imath \, \gamma _{00}} \vert \, + \, \vert \varrho _{1,0} \, - \, \sin (\omega )\, e^{\imath \, \gamma _{10}} \vert \, \nonumber \\{} & {} + \, \vert \varrho _{0,1} \, - \, \sin (\omega )\, e^{\imath \, \gamma _{01}} \vert + \, \vert \varrho _{1,1} \, - \, \cos (\omega )\, e^{\imath \, (\gamma _{01} \, + \, \gamma _{10} \,- \, \gamma _{00}\,- \, \pi )} \vert \qquad \quad \end{aligned}$$

(B34)

We use a classical computer with finite precision, i.e., use MATLAB with 16 digits of precision by default, to solve the nonlinear equalities by using fsolve function in the optimization toolbox. We set tolerance levels for the optimization toolbox equal to $10^{-30}$. It solves nonlinear equations based on trust region methods [32,33,34]. The second error metric we define is based on solutions of (B16–B25) to find $a_{0,k}$, $b_{0,k}$, $a_{1,k}$ and $b_{1,k}$ for the $k^{th}$ eigenvector by using calculated values of $\omega$, $\gamma _{00}$, $\gamma _{01}$ and $\gamma _{10}$. We define the error in satisfying the equalities by also considering that $E_6 = - E_1$, $E_7 = - E_2$ and $E_{10} = E_5$ for $k^{\text{ th }}$ vector $\vec {s}$ for $k \in [1, 2^n]$ as follows:

$$\begin{aligned} \epsilon _{2}[k]\equiv & {} \, \vert E_1 \vert \, + \, \vert E_3 \vert \, + \, \vert E_4 \vert \, + \,\vert E_5 \vert \, + \, \vert E_8 \vert \, + \, \vert E_9 \vert \end{aligned}$$

(B35)

We desire that $\epsilon _1$ and $\epsilon _2$ are small by satisfying the proposed eigenstructure.

Finally, the distinctness of eigenvalues is checked by calculating differences among neighbor pairs of the angles of eigenvalues after sorting them in ascending order. If we define $\equiv e^{\imath \, 2\, \pi \, \lambda _k }$ as the eigenvalue corresponding to kth eigenvector for $k \in [1, 2^{n+1}]$, then eigenvalue distinctness vector $\vec {d}_{E}$ with elements denoted by $d_{E}[k]$ for $k \in [1, 2^{n+1}-1 ]$ is defined as follows:

$$\begin{aligned} d_{E}[k] \equiv \vert \lambda _{k+1} \, - \, \lambda _{k} \vert \end{aligned}$$

(B36)

The minimum value in the vector $\vec {d}_{E}$ determines the minimum distance between angle parameters of neighbor eigenvalues. It determines the number of ancillary qubits in Bob, i.e., $\widetilde{t}$, in order to approximate the eigenvalue with enough precision so that different basis states will be reliably mapped to different eigenvalues.

A numerical analysis example is provided for the following $\vec {\Phi }$:

$$\begin{aligned} \vec {\Phi } = \begin{bmatrix} \phi _1&\ldots&\phi _{n+1} \end{bmatrix} = \pi \begin{bmatrix} 1.755&0.648&1.754&0.615&0.880&1.061&1.290 \end{bmatrix} \end{aligned}$$

(B37)

where $n = 6$ and $\phi _i \, / \, \pi$ has three digits of precision. The values of $\epsilon _1[k]$ and $\epsilon _2[k]$ are shown for $k \in [1, 2^{n}]$ in Fig. 7a and b, respectively. In Fig. 7c, $d_{E}[k]$ for $k \in [1, 2^{n+1} -1 ]$ is shown. It is observed in Fig. 7a and b that error metrics are significantly small satisfying the proposed eigenstructure with accuracy. Eigenvalues are all different with the minimum difference of the angle parameters $\approx \, 5\, \times 10^{-6}$ as shown in Fig. 7c where there are a large number of neighbor pairs with much larger differences reaching $\approx 3\, \times 10^{-2}$.

Fig. 7

The eigenvalues and eigenvectors of $U_{\vec {\Phi }}$ are calculated for $\vec {\Phi } = \begin{bmatrix} \phi _1&\phi _2&\ldots&\phi _{n+1} \end{bmatrix} = \pi \, \begin{bmatrix} 1.755&0.648&1.754&0.615&0.880&1.061&1.290 \end{bmatrix}$ where $n = 6$. The performance metric values satisfying the theoretically modeled architecture are shown for a $\epsilon _1[k]$ and b $\epsilon _2[k]$ for $k \in [1, 2^n]$ by numerically analyzing the accuracy of satisfying (B12–B15) and (B16–B25), respectively. c $d_{E}[k]$ showing the distance between the angular parameters of neighbor pairs of the sorted eigenvalues for $k \in [1, 2^{n+1} -1]$

Next, we calculate performance metrics for many trials of uniformly distributed phase shift values and n values. We denote performance metrics indexed by using also the trial number l and n as $\epsilon _{1,n,l}[k]$, $\epsilon _{2,n,l}[k]$ and $d_{E,n, l}[k]$ for a uniformly distributed $\vec {\Phi }$ in each trial. Then, we define average errors in order to characterize how much the proposed eigenstructure is satisfied for increasing values of n on average of all $\vec {s}$ for $\epsilon _{1}$ or $\epsilon _{2}$. In addition, we find the minimum distance for the angle parameters of the eigenvalues for each trial. We define the following:

$$\begin{aligned} \overline{\epsilon }_{i}[n,l]\equiv & {} \frac{1}{2^{n+1}} \sum _{k = 1}^{2^{n+1}} \epsilon _{i,n,l}[k] \end{aligned}$$

(B38)

$$\begin{aligned} \overline{d}_{E}[n,l]\equiv & {} \min _{k} \lbrace d_{E, n, l}[k] \rbrace \end{aligned}$$

(B39)

where $i \in [1, 2]$. The number of digits of precision in phase shift values is set to 16 by conjecturing that it increases the likelihood to obtain distinct eigenvalues. We are motivated based on the complexity of calculating $\textbf{V}_{\vec {s}}$ in (27) with consecutive multiplication of rotation operators.

Fig. 8

The average performance metrics of $U_{\vec {\Phi }}$ for a $\overline{\epsilon }_{1}[n,l]$ and b $\overline{\epsilon }_{2}[n,l]$ both sorted in ascending order with respect to the value of $\overline{\epsilon }_{2}[n,l]$ for trial indices l. c $\overline{\epsilon }_{1}[n,l]$ sorted in ascending order for l. d $\overline{d}_{E}[n]$ for varying values of n

We sort the values of $\overline{\epsilon }_{2}[n,l]$ in ascending order with respect to l in order to observe the performance of $\overline{\epsilon }_{1}[n,l]$ for specific $\vec {\Phi }$ encountered in different trials where the equalities in (B16–B25) are not satisfied with accuracy. The performance results sorted with respect to the values of $\overline{\epsilon }_{2}[n,l]$ are shown in Fig. 8a and b for $\overline{\epsilon }_{1}[n,l]$ and $\overline{\epsilon }_{2}[n,l]$, respectively, for varying $n \in [3, 13]$ and $l \in [1, 150]$. It is observed in Fig. 8a and b that the proposed eigenstructure is maintained with enough accuracy for many values of uniformly distributed $\vec {\Phi }$ for varying values of n. The value of $\overline{\epsilon }_{1}[n,l]$ for varying n is approximately $\approx 10^{-15}$ as also shown in Fig. 8c where the values of $\overline{\epsilon }_{1}[n,l]$ are sorted in ascending order. while the error occurs due to $\overline{\epsilon }_{2}[n,l]$. The total count of samples providing good results decreases as we increase n, i.e., it becomes more difficult to find a good random $\vec {\Phi }$ providing small $\overline{\epsilon }_{2}[n,l]$. It is observed that the fraction of bad $\vec {\Phi }$ values increases almost linearly with increasing n requiring to get more samples to find a good $\vec {\Phi }$ with increasing n. It is an open issue to determine the conditions on $\vec {\Phi }$ satisfying the proposed eigenstructure for large values of n.

On the other hand, we define the average minimum distance for M trials where the specific trials are selected producing significantly small $\overline{\epsilon }_{1}[n,l]$ and $\overline{\epsilon }_{2}[n,l]$, i.e., smaller than $10^{-15}$. Then, $\overline{d}_{E}[n]$ is defined as follows for the specific set of trials producing good $\vec {\Phi }$ values and the proposed eigenstructure:

$$\begin{aligned} \overline{d}_{E}[n]\equiv & {} \frac{1}{M} \sum _{l = 1}^{M} \overline{d}_{E}[n,l] \end{aligned}$$

(B40)

Numerical analysis of $\overline{d}_{E}[n]$ allows to observe how the minimum distance increases for increasing n values. Angular parameters of the eigenvalues are exponentially closer with increasing n due to the increasing size of the Hilbert space with $2^{n+1}$ and the number of eigenvalues as shown in Fig. 8d. This will increase the number of ancillary qubits of Bob $\widetilde{t}$ as $\mathcal {O}(n)$ with a linear increase in order to increase the precision for detecting different eigenvalues.