l
| Level of entanglement |
F
| Fidelity of entanglement |
N
| Number of nodes in the network |
T
| Number of fidelity types \(F_{j} \), \(j=1,\ldots ,T\) of the entangled states |
\(\mathrm{S}_{O} \)
| Objective space |
\({{\mathcal S}}_{F} \)
| Feasible space |
\(\mathrm{L}_{l} \)
| An l-level entangled connection. For an \(\mathrm{L}_{l} \) link, the hopdistance is \(2^{l-1} \) |
\(d\left( x,y\right) _{\mathrm{L}_{l} } \)
| Hop-distance of an l-level entangled connection between nodes x and y |
\(E_{\mathrm{L}_{l} } \left( x,y\right) \)
| entangled connection \(E_{\mathrm{L}_{l} } \left( x,y\right) \) between nodes x and y |
\(B_{F} \left( E_{\mathrm{L}_{l} } \left( x,y\right) \right) \)
| Entanglement throughput of an \(\mathrm{L}_{l} \)-level entangled connection \(E_{\mathrm{L}_{l} } \left( x,y\right) \) between nodes \(\left( x,y\right) \) |
\(B_{F}^{j} \left( x_{i} \right) \)
| Number of incoming entangled states in an ith node \(x_{i} \), with fidelity-type j, \(i=1,\ldots ,N\) |
\(\mathbf {X}\)
| An \(N\times T\) matrix, \(\mathbf {X}={{\left( B_{F}^{j}\left( {{x}_{i}} \right) \right) }_{N\times T}}\), it describes the number of resource entangled states injected into the nodes from each fidelity-type in the network, \(B_{F}^{j} \left( x_{i} \right) \ge 0\) for all i and j |
\({{\mathcal F}}\left( x_{i} \right) \)
| A primary objective function. It identifies the cumulative entanglement fidelity (a sum of entanglement fidelities in \(x_{i} \)) after an entanglement purification \(\mathrm{P} \left( x_{i} \right) \) and an optimal quantum error correction \(\mathrm{C}\left( x_{i} \right) \) in \(x_{i} \) |
\(\mathrm{P} \left( x_{i} \right) \)
| Entanglement purification in \(x_{i} \) |
\(\mathrm{C}\left( x_{i} \right) \)
| Optimal quantum error correction in \(x_{i} \) |
\(\left\langle B \right\rangle _{F}^{j}\left( {{x}_{i}} \right) \)
| An initialization value for \(B_{F}^{j} \left( x_{i} \right) \) in a particular node \(x_{i} \) |
\(\mathbb {E}\left( {{D}_{i}}\left( \mathbf {X} \right) \right) \)
| A secondary objective function. It refers to the expected amount of cumulative relative entropy of entanglement (a sum of relative entropy of entanglement) in node \(x_{i} \), |
\(w_{j} \left( x_{i} \right) \)
| Quantum memory coefficient for the storage of entangled states from the jth fidelity type in a node \(x_{i} \), evaluated as: \({{w}_{j}}\left( {{x}_{i}} \right) ={{\eta }_{j}}B_{F}^{j}\left( {{x}_{i}} \right) +{{\kappa }_{j}}\left\langle B \right\rangle _{F}^{j}\left( {{x}_{i}} \right) \), where \(\eta _{j} \) and \(\kappa _{j} \) are coefficients to describe the storage characteristic of entangled states with the jth fidelity type |
\(\tau _{j} \left( \mathbf {X}\right) \)
| Differentiation of storage characteristic of entangled states from the jth fidelity type, defined as \(\tau _{j} \left( \mathbf {X}\right) =\sum _{i=1}^{N}\left( w_{j} \left( x_{i} \right) -\Omega \right) ^{2} ,\) where
|
\(f_{C} \left( \mathrm{P} \left( x_{i} \right) \right) \)
| Cost of entanglement purification \(\mathrm{P} \left( x_{i} \right) \) in \(x_{i} \) |
\(f_{C} \left( \mathrm{C}\left( x_{i} \right) \right) \)
| Cost of optimal quantum error correction \(\mathrm{C}\left( x_{i} \right) \) in \(x_{i} \) |
\({{\mathcal C}}\left( \mathbf {X}\right) \)
| Total cost function, defined as \(\begin{array}{rcl} \mathcal {C}\left( \mathbf {X} \right) =&\sum \limits _{i=1}^{N}{{{f}_{C}}\left( P \left( {{x}_{i}} \right) \right) +{{f}_{C}}\left( \text {C}\left( {{x}_{i}} \right) \right) } =\sum \limits _{i=1}^{N}{\sum \limits _{i=1}^{T}{{{f}_{j}}B_{F}^{j}\left( {{x}_{i}} \right) ,}} \end{array}\) where T is the number of fidelity types, N is the number of nodes, \(f_{j} \) is a total cost of purification and error correction associated to the jth fidelity type of entangled states |
\(f_{j} \)
| Total cost of purification and error correction associated to the jth fidelity type of entanglement fidelity |
\(F^{\mathrm{*}} \)
| Critical fidelity coefficient |
\({{\mathcal S}}_\mathrm{low} \), \({{\mathcal S}}_\mathrm{high} \) | Sets with fidelity bounds \({{\mathcal S}}_\mathrm{low} \left( F\right) \) and \({{\mathcal S}}_\mathrm{high} \left( F\right) \) as \({{\mathcal S}}_\mathrm{low} \left( F\right) :\mathop {\max }\limits _{\forall i} F_{i} <F^{{*}} \), and \({{\mathcal S}}_\mathrm{high} \left( F\right) :\mathop {\min }\limits _{\forall i} F_{i} \ge F^{{*}}\) |
\(X_{{{\mathcal S}}_\mathrm{low} } \)
| Set of nodes for which condition \({{\mathcal S}}_\mathrm{low} \left( F\right) :\mathop {\max }\limits _{\forall i} F_{i} <F^{{*}} \) holds |
\(X_{{{\mathcal S}}_\mathrm{high} } \)
| Set of nodes for which condition \({{\mathcal S}}_\mathrm{high} \left( F\right) :\mathop {\min }\limits _{\forall i} F_{i} \ge F^{{*}} \) holds |
\({{\mathcal S}}_{i} \left( \mathbf {X}\right) \)
| Cost of quantum memory usage in node \(x_{i} \), defined as \({{\mathcal S}}_{i} \left( \mathbf {X}\right) =\lambda \sum _{j=1}^{T}\alpha _{i} \frac{1}{\Upsilon _{i} } B_{F}^{j} \left( x_{i} \right) \), where \(\lambda \) is a constant, \(\alpha _{i} \) is a quality coefficient, while \(\Upsilon _{i} \) is a capacity coefficient of the quantum memory |
\({{\mathcal G}}\left( \mathbf {X}\right) \)
| Main objective function, \(\mathcal {G}\left( \mathbf {X} \right) =\max \sum \limits _{i=1}^{N}{{{\mathcal {F}}_{i}}\left( \mathbf {X} \right) \mathbb {E}\left( {{D}_{i}}\left( \mathbf {X} \right) \right) }\) |
\(F_{1} \left( N\right) \)
| Minimization function for cost \({{\mathcal C}}\left( \mathbf {X}\right) \) |
\(F_{2} \left( N\right) \)
| Minimization function for cost \({{\mathcal S}}\left( \mathbf {X}\right) \) |
\(C_{1} \), \(C_{2} \), \(C_{3} \) | Problem constraints |
\({{\mathcal E}}\)
| Epicenter, represents a solution in the feasible space |
\(L_{j} \)
| A random location around epicenter \({{\mathcal E}}\) |
\(D\left( {{\mathcal E}}\right) \)
| Dispersion coefficient of an epicenter \({{\mathcal E}}\) (solution in the feasible space). It determines the number of affected \(L_{j} \), \(j=1,\ldots ,D\left( {{\mathcal E}}\right) \), locations (also represent solutions in the feasible space) around an epicenter \({{\mathcal E}}\) |
\({{\mathcal P}}\)
| Population \({{\mathcal P}}\) (a set of possible solutions) |
m
| Control parameter |
\({{\mathcal E}}_{i} \)
| An ith individual (epicenter) from the \(\left| {{\mathcal P}}\right| \) individuals (epicenters) in the population \({{\mathcal P}}\) |
\(\tilde{f}\left( \cdot \right) \)
| Fitness function |
\(\tilde{f}\left( \langle \mathcal {E} \rangle \right) \)
| A maximum objective value among the \(\left| {{\mathcal P}}\right| \) individuals |
\(\vartheta \)
| A residual quantity |
\(f_{R} \left( \cdot \right) \)
| Rounding function |
q
| Total number of locations, \(q=\sum _{i=1}^{\left| {{\mathcal P}}\right| }D\left( {{\mathcal E}}_{i} \right) \) |
\(\hat{D}\left( {{\mathcal E}}_{i} \right) \)
| Upper bound on \(D\left( {{\mathcal E}}_{i} \right) \) for a given epicenter \({{\mathcal E}}_{i} \) |
\(d\left( {{\mathcal E}}_{i} ,l_{j} \right) \)
| Euclidean distance \(d\left( {{\mathcal E}}_{i} ,l_{j} \right) \) between an ith epicenter \({{\mathcal E}}_{i} \) and the projection point \(l_{j} \) of a jth location point \(L_{j} \),\(j=1,\ldots ,D\left( {{\mathcal E}}\right) \) on the ellipsoid around \({{\mathcal E}}_{i} \) |
\(\dim _{i} \left( \cdot \right) \)
| An ith dimension of \(l_{j} \) |
\(P\left( {{\mathcal E}}_{i} ,L_{j} \right) \)
| Seismic power \(P\left( {{\mathcal E}}_{i} ,L_{j} \right) \) operator for an ith epicenter \({{\mathcal E}}_{i} \). Measures the power in a jth location point \(L_{j} \),\(j=1,\ldots ,D\left( {{\mathcal E}}_{i} \right) \), as\(P\left( {{\mathcal E}}_{i} ,L_{j} \right) =\left( \frac{1}{d\left( {{\mathcal E}}_{i} ,l_{j} \right) } M\left( {{\mathcal E}}_{i} ,L_{j} \right) \right) ^{b_{1} } b_{0} e^{\sigma _{\ln P\left( {{\mathcal E}}_{i} ,L_{j} \right) } } \),where \(b_{0} \) and \(b_{1} \) are regression coefficients, \(\sigma _{\ln P\left( {{\mathcal E}}_{j} \right) } \) is the standard deviation, while \(M\left( {{\mathcal E}}_{i} ,L_{j} \right) \) is the seismic magnitude in a location \(L_{j} \), while \(l_{j} \) is the projection of \(L_{j} \) onto the ellipsoid around \({{\mathcal E}}_{i} \) |
\(M\left( {{\mathcal E}}_{i} ,L_{j} \right) \)
| Magnitude between epicenter \({{\mathcal E}}_{i} \) and location \(L_{j} \) is evaluated as\(M\left( {{\mathcal E}}_{i} ,L_{j} \right) =\left( P\left( {{\mathcal E}}_{i} ,L_{j} \right) \frac{1}{b_{0} e^{\sigma _{\ln P\left( {{\mathcal E}}_{i} ,L_{j} \right) } } } \right) ^{\frac{1}{b_{1} } } d\left( {{\mathcal E}}_{i} ,l_{j} \right) \) |
\(P^{\mathrm{*}} \left( {{\mathcal E}}_{i} \right) \)
| Maximal seismic power for a given epicenter \({{\mathcal E}}_{i} \) |
\(C\left( {{\mathcal E}}_{i} \right) \)
| Cumulative magnitude for an epicenter \({{\mathcal E}}_{i} \) |
\({{\mathcal E}}'\)
| Highest seismic power epicenter with magnitude \(M\left( {{\mathcal E}}',L_{j}^{{{\mathcal E}}'} \right) \) |
\(\tilde{f}\left( {{\mathcal E}}'\right) \)
| Minimum objective values among the \(\left| {{\mathcal P}}\right| \) epicenters |
\({{\mathcal M}}\)
| Control parameter, \({{\mathcal M}}=\sum _{i=1}^{\left| {{\mathcal P}}\right| }M\left( {{\mathcal E}}_{i} ,L_{j}^{{{\mathcal E}}_{i} } \right) \),where \(L_{j}^{{{\mathcal E}}_{i} } \) provides the maximal seismic power for an ith epicenter \({{\mathcal E}}_{i} \) |
\(\Phi \left( {{\mathcal E}}_{i} ,{{\mathcal R}}_{k} ,{{\mathcal R}}_{l} \right) \)
| Poisson range identifier function of \({{\mathcal E}}_{i} \), where \({{\mathcal R}}_{k} \) and \({{\mathcal R}}_{l} \) are random reference points |
\(c_{w} \left( {{\mathcal E}}_{i} ,{{\mathcal R}}_{k} \right) \), \(c_{w} \left( {{\mathcal R}}_{k} ,{{\mathcal R}}_{l} \right) \) | Weighting coefficients between epicenters \({{\mathcal E}}_{i} \) and \({{\mathcal R}}_{k} \), and between \({{\mathcal R}}_{k} \) and \({{\mathcal R}}_{l} \) |
\(\mathfrak {D}\left( {{\mathcal {E}}_{p}} \right) \)
| Poissonian distance function \(\mathfrak {D}\left( {{\mathcal {E}}_{p}} \right) \), where \({{\mathcal E}}_{p} \) is a new solution |
\(r\left( {{\mathcal E}}_{i} \right) \)
| Radius around a current solution \({{\mathcal E}}_{i} \), defined as\(r\left( {{\mathcal E}}_{i} \right) =\chi 10^{Q_{1} \left( 2\tilde{{{\mathcal M}}}\right) -Q_{2} } \),where \(\tilde{{M}}\) is the average magnitude \(\tilde{{M}}=\frac{1}{\left| {{\mathcal P}}\right| } {{\mathcal M}}=\frac{1}{\left| {{\mathcal P}}\right| } \sum _{i=1}^{\left| {{\mathcal P}}\right| }M\left( {{\mathcal E}}_{i} ,L_{j}^{{{\mathcal E}}_{i} } \right) \),while \(Q_{1} \) and \(Q_{2} \) are constants, while \(\chi \) is a normalization term |
\(\dim _{k} \left( {{\mathcal E}}_{i} \right) \)
| Randomly selected kth dimension, \(k=1,\ldots ,\dim \left( {{\mathcal E}}_{i} \right) \) of a current epicenter \({{\mathcal E}}_{i} \), \(i=1,\ldots ,\left| {{\mathcal P}}\right| \) |
\({{\mathcal H}}\left( \dim _{k} \left( {{\mathcal E}}_{i} \right) \right) \)
| Hypocentral, provides a random displacement of \(\dim _{k} \left( {{\mathcal E}}_{i} \right) \) using \(C\left( {{\mathcal E}}_{i} \right) \) |
\(L_{r}^{\dim _{k} \left( {{\mathcal E}}_{i} \right) } \)
| A random location in the kth dimension \(L_{r}^{\dim _{k} \left( {{\mathcal E}}_{i} \right) } \) around \(\dim _{k} \left( {{\mathcal E}}_{i} \right) \) |
\(\mathrm{N} \left( \cdot \right) \)
| Normalization operator \(\mathrm{N} \left( \cdot \right) \) of \(L_{r}^{\dim _{k} \left( {{\mathcal E}}_{i} \right) } \). It keeps the new locations around \(\dim _{k} \left( {{\mathcal E}}_{i} \right) \) in \({{\mathcal S}}_{F} \), where \(B_\mathrm{low}^{k} \) and \(B_{up}^{k} \) are lower and upper bounds on the boundaries of locations in a kth dimension |
S-metric | Hypervolume indicator. A quality measure for the solutions or a contribution of a single solution in a solution set |
\(S\left( {{\mathcal R}}\right) \)
| S-metric for a solution set \({{\mathcal R}}=\left\{ r_{1} ,\ldots ,r_{n} \right\} \) is as\(S\left( {{\mathcal R}}\right) ={{\mathcal L}}\left( \bigcup _{r\in {{\mathcal R}}}\left\{ x_{ref} \angle x\angle \left. x\right| r\right\} \right) ,\)where \({{\mathcal L}}\) is a Lebesgue measure, notation \(b\angle a\) refers to that a dominates b (or b is dominated by a), while \(x_{ref} \) is a reference point dominated by all valid solutions in the solution set |
\(f_{1} \), \(f_{2} \) | Objective functions |
\({{\mathcal C}}_{1} \left( x_{i} \right) \)
| Cost results from the first-type classical communications related to a \(x_{i} \) |
\({{\mathcal C}}_{2} \left( x_{i} \right) \)
| Cost results from the second-type classical communications with respect to \(x_{i} \) |
\({{\mathcal E}}^{\mathrm{*}} \)
| Global optima |
m
| Number of magnitude ranges |
\(n_{i} \)
| Number of locations belonging to an ith magnitude range |
\({{\mathcal B}}\left( n_{i} \right) \)
| Power law distribution function for a log-scaled \(n_{i} \),\({{\mathcal B}}\left( n_{i} \right) :\log _{10} \left( n_{i} \right) =ab\ tilde{M}_{i} \),where \(\tilde{M}_{i} \) is a log scaled \(M_{i} \), while a and b are constants |
\(\tilde{n}_{i} \)
| Poisson estimate of \(n_{i} \), as \(\tilde{n}_{i} =\sigma _{i}^{2} =\lambda _{i} \),where \(\sigma _{i}^{2} \) is the observational variance, while \(\lambda _{i} \) is the mean of a Poisson distribution |
\(\sigma _{q}^{2} \)
| Estimated uncertainty, \(\sigma _{q}^{2} =\lambda \left( q\right) =\sum _{i=1}^{m}f\left( \tilde{M}_{i} \right) \), where \(f\left( \cdot \right) \) is a fitting function |
\(\lambda \left( q\right) \)
| Mean total number, \(\lambda \left( q\right) =\sum _{i=1}^{m}\lambda _{i} \approx q\), where \(\lambda _{i} \) is an ith component mean |
\({{\mathcal B}}\left( \lambda _{i} \right) \)
| Power law distribution function for \(\lambda _{i} =\tilde{n}_{i} \) |
\(k_{it} \)
| Number of iterations |