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A Leap among Entanglement and Neural Networks: A Quantum Survey

Profile image of Fabrizio FalchiFabrizio Falchi

2021

Abstract

In recent years, Quantum Computing witnessed massive improvements both in terms of resources availability and algorithms development. The ability to harness quantum phenomena to solve computational problems is a long-standing dream that has drawn the scientific community’s interest since the late ’80s. In such a context, we pose our contribution. First, we introduce basic concepts related to quantum computations, and then we explain the core functionalities of technologies that implement the Gate Model and Adiabatic Quantum Computing paradigms. Finally, we gather, compare and analyze the current state-of-the-art concerning Quantum Perceptrons and Quantum Neural Networks implementations.

Figures (11)
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.
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
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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Federico Spedalieri

Entropy, 2018

Training deep learning networks is a difficult task due to computational complexity, and this is traditionally handled by simplifying network topology to enable parallel computation on graphical processing units (GPUs). However, the emergence of quantum devices allows reconsideration of complex topologies. We illustrate a particular network topology that can be trained to classify MNIST data (an image dataset of handwritten digits) and neutrino detection data using a restricted form of adiabatic quantum computation known as quantum annealing performed by a D-Wave processor. We provide a brief description of the hardware and how it solves Ising models, how we translate our data into the corresponding Ising models, and how we use available expanded topology options to explore potential performance improvements. Although we focus on the application of quantum annealing in this article, the work discussed here is just one of three approaches we explored as part of a larger project that considers alternative means for training deep learning networks. The other approaches involve using a high performance computing (HPC) environment to automatically find network topologies with good performance and using neuromorphic computing to find a low-power solution for training deep learning networks. Our results show that our quantum approach can find good network parameters in a reasonable time despite increased network topology complexity; that HPC can find good parameters for traditional, simplified network topologies; and that neuromorphic computers can use low power memristive hardware to represent complex topologies and parameters derived from other architecture choices.

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"Overview of Quantum Computing in Quantum Neural Network and Artificial Intelligence"
Dr. K A R U N A M U R T H Y . A

“Quing: International Journal of Innovative Research in Science and Engineering, 2023

In recent years, quantum computing has emerged as a potentially gamechanging technology, with applications across various disciplines, including AI and machine learning. In recent years, the combination of quantum computing and neural networks has led to the development of quantum neural networks (QNNs). This paper explores the potential of QNNs and their applications in solving complex problems that are challenging for classical neural networks. This paper explores the fundamental principles of quantum computing, the architecture of QNNs, and their advantages over classical neural networks. Furthermore, this will highlight key research areas and challenges in the development and utilization of QNNs. Through an in-depth analysis, it demonstrates the QNNs hold significant promise for addressing complex computational problems and advancing the field of artificial intelligence.

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Neural networks with quantum gated nodes
Fariel Shafee

Engineering Applications of Artificial Intelligence, 2007

We study a quantum neural network with superposed qubits replacing classical neurons with deterministic states, and also with quantum gate operators in place of the classical action potentials observed in biological contexts. With our choice of logic gates interconnecting the neural lattice, we find that the state of the system behaves in ways reflecting both the strength of coupling between neurons as well as the initial conditions, and depending on whether there is a threshold for emission from excited to ground state, the system shows either chaotic oscillations or coherent ones with periodicity that depends on the strength of coupling in a unique way. The spatial pattern of the initial input affects the subsequent dynamic behavior of the system in an interesting unambiguous way, which indicates that it can serve as a dynamic memory system analogous to biological ones, but with an unlimited lifetime.

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A Quantum Neural Network Computes Entanglement
Elizabeth Behrman

arXiv (Cornell University), 2002

An outstanding problem in quantum computing is the calculation of entanglement, for which no closed-form algorithm exists. Here we solve that problem, and demonstrate the utility of a quantum neural computer, by showing, in simulation, that such a device can be trained to calculate the entanglement of an input state, something neither an algorithmic quantum computer nor a classical neural net can do. Quantum neural network Two maximally entangled states are the Bell state, |00>+|11>, and the EPR state, |10>+|01>. Neither can be represented as a product state (a|1>+b|0>) A (c|1>+d|0>) B. They can be represented by specifying 4 c-numbers, but if we write the state as a product state not all those numbers are independent, since the product is equal to ac|11>+ad|10>+bc|01>+bd|00>. So, for example, with the Bell state, one cannot specify complex numbers a,b,c,d such that ac and bd are not zero but ad and bc are. This is also equivalent to saying [5] that the density matrix for

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