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Table 2. Summary of abbreviations and acronyms used throughout this paper. concerning the quantum annealing process, which is at the heart of the D-Wave quantum platforms, and what kind of problems can be solved on it. Preskill [151, 152] debated the concept of “quantum supremacy” and future potential of quantum computing. concerning the quantum annealing process, which is at the heart of the D-Wave quantum platforms, and what kind of

Table 2 Summary of abbreviations and acronyms used throughout this paper. concerning the quantum annealing process, which is at the heart of the D-Wave quantum platforms, and what kind of problems can be solved on it. Preskill [151, 152] debated the concept of “quantum supremacy” and future potential of quantum computing. concerning the quantum annealing process, which is at the heart of the D-Wave quantum platforms, and what kind of

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Table 1. Summary of notation used throughout this paper. many scientific debates, respectively. Subsequently, in section 6, we move to the core topic of this survey, i.e. QNNs approaches. Finally, the conclusions are drawn in section 7.
Table 1. Summary of notation used throughout this paper. many scientific debates, respectively. Subsequently, in section 6, we move to the core topic of this survey, i.e. QNNs approaches. Finally, the conclusions are drawn in section 7.
Fig. 1. Bloch sphere representation of a qubit: a quantum state |y) = a |0) + |1), where a = |a|e’?@ and B = [Ble?P, is represented as the point of the Bloch sphere with spherical coordinates given by the angles 0 and g, where @ = arccos (|a|* — |B|*) € [0, z] and Y = B- Ya € [0, 272). Vice versa, a point a unit sphere mh spherical coordinates x = sin@ cos g, y = sin @ sin g, and z = cos 0 corresponds to the quantum state |y/) = cos 5 8 10) + ef sin a @ 1), three-dimensional representation that can be easily visualized. Specifically, we could use the following transformation
Fig. 1. Bloch sphere representation of a qubit: a quantum state |y) = a |0) + |1), where a = |a|e’?@ and B = [Ble?P, is represented as the point of the Bloch sphere with spherical coordinates given by the angles 0 and g, where @ = arccos (|a|* — |B|*) € [0, z] and Y = B- Ya € [0, 272). Vice versa, a point a unit sphere mh spherical coordinates x = sin@ cos g, y = sin @ sin g, and z = cos 0 corresponds to the quantum state |y/) = cos 5 8 10) + ef sin a @ 1), three-dimensional representation that can be easily visualized. Specifically, we could use the following transformation
Fig. 2. Q-sphere for a 4-qubit system. The represented state is the uniform superposition of all possible states of the 4-qubit system.
Fig. 2. Q-sphere for a 4-qubit system. The represented state is the uniform superposition of all possible states of the 4-qubit system.
Table 3. Basic one-qubit gates: Notations typically used in literature, the corresponding matrices with respect to the basis {|0), |1)}, the action on a quantum state, and the corresponding action on the spherical coordinates in the Bloch sphere representation. where |) and |y) are the control and target qubit, respectively.
Table 3. Basic one-qubit gates: Notations typically used in literature, the corresponding matrices with respect to the basis {|0), |1)}, the action on a quantum state, and the corresponding action on the spherical coordinates in the Bloch sphere representation. where |) and |y) are the control and target qubit, respectively.
Fig. 3. Quantum circuit implementing Deutsch’s algorithm [57, 140]. Boolean function f : {0,1} — {0,1} is balanced or constant. As we can see from Figure 3, a quantum circuit is depicted as a set of qubits, represented by lines, to which the gates
Fig. 3. Quantum circuit implementing Deutsch’s algorithm [57, 140]. Boolean function f : {0,1} — {0,1} is balanced or constant. As we can see from Figure 3, a quantum circuit is depicted as a set of qubits, represented by lines, to which the gates
Table 4. Frequently used multi-qubit gates. number of operations applied to a given qubit from the beginning to the end of the computations. In Figure 3, we report A Leap among Entanglement and Neural Networks: A Quantum Survey
Table 4. Frequently used multi-qubit gates. number of operations applied to a given qubit from the beginning to the end of the computations. In Figure 3, we report A Leap among Entanglement and Neural Networks: A Quantum Survey
Fig. 4. Schematic adiabatic evolution for the Deutsch’s algorithm [57]. so that the final configuration of the qubit system is represented by eigenvectors of Gz, i.e., each qubit can be founc either in the |0) or |1) state. Such a configuration is nothing more than a classical sequence of bits. Thus, by starting from an initial quantum superposition, the adiabatic process evolves the state into a classical configuration which car then be read to return the solution to the problem at hand. One of the first “proof of concenpt” to show the effectiveness of such a computation model was first realized in [69] for solving instances of the satisfiability problem. Apart from such a first example, many optimization problems can be recast as an energy minimization problem, such as the Grove! search algorithm [80] for which the adiabatic approach offers a quadratic speedup [70, 71] over the classical formulation We report in Figure 4 a schematic representation of an adiabatic path representing the Deutsch’s algorithm [57]. The reader might compare such an image with Figure 3 to notice the difference between the two designs. 5 THE QUEST FOR QUANTUM SUPREMACY
Fig. 4. Schematic adiabatic evolution for the Deutsch’s algorithm [57]. so that the final configuration of the qubit system is represented by eigenvectors of Gz, i.e., each qubit can be founc either in the |0) or |1) state. Such a configuration is nothing more than a classical sequence of bits. Thus, by starting from an initial quantum superposition, the adiabatic process evolves the state into a classical configuration which car then be read to return the solution to the problem at hand. One of the first “proof of concenpt” to show the effectiveness of such a computation model was first realized in [69] for solving instances of the satisfiability problem. Apart from such a first example, many optimization problems can be recast as an energy minimization problem, such as the Grove! search algorithm [80] for which the adiabatic approach offers a quadratic speedup [70, 71] over the classical formulation We report in Figure 4 a schematic representation of an adiabatic path representing the Deutsch’s algorithm [57]. The reader might compare such an image with Figure 3 to notice the difference between the two designs. 5 THE QUEST FOR QUANTUM SUPREMACY
A Leap among Entanglement and Neural Networks: A Quantum Survey problem Hamiltonian Hc, one can use QAOA to find its ground state by applying the evolution operator U:
A Leap among Entanglement and Neural Networks: A Quantum Survey problem Hamiltonian Hc, one can use QAOA to find its ground state by applying the evolution operator U:
Code: X the code is not publicly available, / the code is available either on GitHub (*), upon request to the authors of the original work (0), or in the supplementary material of the original work (e). Table 5. Summary comparison among various quantum perceptron proposals. The symbol “nr” stands for “not reported” meanin; ‘hat the authors did not explicit reported that specific information. In the column “Simulation” we reported whether the methox vas implemented on an actual QPU or tested using numerical simulations only (reporting in brackets the library used when th nformation was available). vector x = [x0,X1,...; Xn-1]" it evaluates the non-linear transformation g(x) = f (wl x + b) that, once thresholded defines the activation status of the perceptron. In such a context, f represents the non-linear operation whose output lies in [1, u] where typically / = —1 or 0 and u = 1. Such an operation can be mapped to the quantum realm by defining the perceptron as a qubit whose state is defined as: Ry(az + $)|0) = cos(a% + 4)|0) + sin(a4 + 4)|1), where a € [-1,1] and Ry represents a rotation around the y-axis as described by Gy. Note that as far as the extreme values of a are concerned, they recover a classical behavior being the system in the state |0) or |1). Instead, in all other cases it behave: quantum-mechanically. Input data are encoded into a state vector acting as the control register over the ancilla qubit: and the non-linearity is realized by exploiting the RUS [143] design. Specifically, the authors implemented a sigmoid-like tangent-based, non-linear function: arctan(tan?(x)). Stemming from the same approach, Hu et al. [97] implemented < different RUS circuit to represent a sigmoid-based activation function, f(x) = arcsin(/sigmoid(x)), thus avoiding the drawbacks of using a periodic functions, such as the tangent. Moreover, being based on the sigmoid, it accepted any real number as input rather than in [0, 27] only. vector x = [x0,X1,...,;Xn-1]° it evaluates the non-linear transformation g(x) = fw x + b) that, once thresholded,
Code: X the code is not publicly available, / the code is available either on GitHub (*), upon request to the authors of the original work (0), or in the supplementary material of the original work (e). Table 5. Summary comparison among various quantum perceptron proposals. The symbol “nr” stands for “not reported” meanin; ‘hat the authors did not explicit reported that specific information. In the column “Simulation” we reported whether the methox vas implemented on an actual QPU or tested using numerical simulations only (reporting in brackets the library used when th nformation was available). vector x = [x0,X1,...; Xn-1]" it evaluates the non-linear transformation g(x) = f (wl x + b) that, once thresholded defines the activation status of the perceptron. In such a context, f represents the non-linear operation whose output lies in [1, u] where typically / = —1 or 0 and u = 1. Such an operation can be mapped to the quantum realm by defining the perceptron as a qubit whose state is defined as: Ry(az + $)|0) = cos(a% + 4)|0) + sin(a4 + 4)|1), where a € [-1,1] and Ry represents a rotation around the y-axis as described by Gy. Note that as far as the extreme values of a are concerned, they recover a classical behavior being the system in the state |0) or |1). Instead, in all other cases it behave: quantum-mechanically. Input data are encoded into a state vector acting as the control register over the ancilla qubit: and the non-linearity is realized by exploiting the RUS [143] design. Specifically, the authors implemented a sigmoid-like tangent-based, non-linear function: arctan(tan?(x)). Stemming from the same approach, Hu et al. [97] implemented < different RUS circuit to represent a sigmoid-based activation function, f(x) = arcsin(/sigmoid(x)), thus avoiding the drawbacks of using a periodic functions, such as the tangent. Moreover, being based on the sigmoid, it accepted any real number as input rather than in [0, 27] only. vector x = [x0,X1,...,;Xn-1]° it evaluates the non-linear transformation g(x) = fw x + b) that, once thresholded,
Loss Function: ¢ j runs on the output qubits only, § sum runs over an ensemble S of input states, * y; is the ground truth label and the sum runs over the dataset of size |D|, o average fidel evaluated over a batch size of size |B|, + k runs over the output classes, @ Psuce(|Yo)), Perr(|Wo)), and Pinc(|Wo)) are the probabilities of correct/erroneous/inconclusive measurement outcom Implementation - Sim.: Numerical Simulation. + used the QuTiP library, +} used the QxBranch library, § the used PennyLane library, + used a QASM library. Implementation - QPU.: « used IBM Q 4, e used IBM Q 5 “Vigo”, o used Rigetti Q 4. Application Field: Quantum Information (QI), Chemistry (Chem.), Computer Science (CS), Physics (Phys.), Information Theory (Inf. Theory) Input Type: Quantum (Q). Classical (C) + We onlv considered the Ouantum Granh Recurrent Neural Networks Table 6. Summary comparison among various quantum neural networks proposals. The symbol “nr” stands for “not reported” meaning that the authors did not explicit reported that specific information or that the specific attribute cannot be applied to the referred work.
Loss Function: ¢ j runs on the output qubits only, § sum runs over an ensemble S of input states, * y; is the ground truth label and the sum runs over the dataset of size |D|, o average fidel evaluated over a batch size of size |B|, + k runs over the output classes, @ Psuce(|Yo)), Perr(|Wo)), and Pinc(|Wo)) are the probabilities of correct/erroneous/inconclusive measurement outcom Implementation - Sim.: Numerical Simulation. + used the QuTiP library, +} used the QxBranch library, § the used PennyLane library, + used a QASM library. Implementation - QPU.: « used IBM Q 4, e used IBM Q 5 “Vigo”, o used Rigetti Q 4. Application Field: Quantum Information (QI), Chemistry (Chem.), Computer Science (CS), Physics (Phys.), Information Theory (Inf. Theory) Input Type: Quantum (Q). Classical (C) + We onlv considered the Ouantum Granh Recurrent Neural Networks Table 6. Summary comparison among various quantum neural networks proposals. The symbol “nr” stands for “not reported” meaning that the authors did not explicit reported that specific information or that the specific attribute cannot be applied to the referred work.
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