Quantum information technologies and intelligent learning systems are both emerging technologies that will have a transformative impact on our society. The researchers looked into the question of the extent to which these fields can actually learn and benefit from each other. We have recently seen significant advances in both directions of influence: on the one hand, quantum computing is finding a vital application in providing accelerations for machine learning problems, on the other hand machine learning can become instrumental in quantum technologies. advanced. Works exploring the use of artificial intelligence for the design of quantum experiments itself and for performing bits of authentic research on their own have reported their first successes.

Quantum information technologies and intelligent learning systems are both emerging technologies that will have a transformative impact on our society. Small quantum computers have already been built as proofs of concept on which the main obstacles to large-scale quantum computing are thoroughly investigated. Among its potential uses: quality control will allow breaking classical cryptographic codes, simulating large quantum systems, and faster search and optimization. For example, variants of Grover’s algorithm can be exploited to obtain quadratic speed in search problems, and some recent developments in Quantum Machine Learning have led to exponential improvements in some learning activities. These ideas have the potential to exert a transformative effect on AI research. In addition, the technical aspects of quality control, which place some physical limits on the observation of the internal functioning of a quantum machine and hinder the verification of quantum calculations, can represent an additional challenge for AI alignment problems. Quantum Computing (QC) is able to easily analyze frequency data (FFT) and this allows to solve the factorization problem in exponentially faster times if compared with those of the most powerful, to date, classic supercomputer.

### Quantum Computing

#### DiVincenzo’s criteria

A quantum computer has very different technical characteristics from a classical computer. For the realization of a quantum computer the DiVincenzo criteria must be taken into consideration. These allow us to set general guidelines to follow, in particular by defining the four fundamental characteristics that such a machine must possess:

- It must be able to represent QuBits;
- Able to carry out a group of unitary transformations on these;
- Prepare an initial state with sufficient confidence;
- Being able to measure the final result;

The speed at which the various computations are performed on the QuBit, between when the state is initialized and then measured, must be such that the information is not lost due to decoherence (see next paragraph). This makes it possible only a finite number of operations that can be done before losing the information.

A further important detail for an efficient use of a quantum computer is scalability. A processor of this kind should not present problems or unsustainable costs as the number of QuBits that are used increases. It is the counterpart of what happens in classical calculators, where it is possible to increase the power and speed of the calculations that can be performed by increasingly miniaturizing the processors, without experiencing any loss of performance. This is as long as physical size scales are not reached for which miniaturization is definitively limited.

#### Decoherence

A phenomenon to be taken into consideration, in the procedures for creating a QuBit, is the decoherence that leads to the loss of information. Similar in some respects, it has an effect comparable to the noise of classical electronics, it is not related to an external disturbance but it depends a lot on the methodology used with which we develop the QuBit. It is a probabilistic event, which causes a spontaneous and inevitable loss of information. Similarly, as an electron in an excited state, it spontaneously tends to lose energy, to return to the ground state. This would not happen in a completely isolated system and is due to uncontrolled interactions with the environment whose wave function is not completely isolated from that of the system. It should be emphasized that decoherence is still a quantum phenomenon, which can be better represented by the Bloch Sphere (Figure 1) which is also an excellent representation for a QuBit. This phenomenon leads to the loss of the superposition state between two states, becoming a statistical mixture, which can be described by the density operator.

Figure 1 – Bloch sphere, geometric representation of the Qubit. Sphere of unit radius. The radius, obtained by joining any point on the surface with the center of the sphere, represents the state that the qubit can assume. When the ray points towards the north of the sphere, by definition the state of the qubit coincides with “0”. When the ray points towards the south of the sphere, the state of the qubit coincides with “1”. In all other cases, with the ray that sweeps over the surface of the sphere, the relative state of the qubit can be defined by referring to two specific quantities: the angles formed by the ray relative to the vertical axis of the sphere and its projection onto the plane that it crosses it horizontally, in the center.

Figure 2 – Decoherence.

#### Qubit

Assuming to consider an atom with only one electron in the last occupied orbit, the electron can be moved (excited) to an outermost orbit illuminating it with a light of a given frequency and duration. In this case the electron performs a quantum leap into a higher energy state. If the state is stable it can be used, together with the lower energy state, to represent the numbers 0 and 1, respectively. If the “excited” atom is hit by a further pulse of light, similar to the previous one, the electron returns to the lowest energy state releasing a photon. In the particular case in which the duration of the first light pulse lasts half the time necessary to switch the state of the electron, the latter will be simultaneously in both orbits. The electron will then be in a “superposition” of the two states. According to this logic, a unit of information can be stored, the qubit. So the laws of quantum mechanics imply that in addition to states 1 and 0, a third state is also possible, an intermediate or indeterminate state, which cannot be considered 1 or 0. It means that a qubit (the quantum equivalent of the bit) can be not only in two states corresponding to the logical values 0 and 1, but it can be both in state 0 and in state at the same time 1. It follows that two qubits can be simultaneously in the four states 00, 01, 10 and 11. This makes the qubit much more versatile in computational problems than the actual bit. A qubit can be defined by mathematical notation meaning by this that if measured it can be 0 with probability | a | 2 and 1 with probability | b | 2, being a and b complex numbers. The symbol | ⟩ Represents an oriented vector. The – state is different from the + state, as can be seen from Figure 3. The state of a qubit is represented by a vector that reaches any point on a circle.

Figure 3 – a and b real numbers.

We assume that the two states are equally likely. In the event that the atom is illuminated for half the time, compared to the time required for switching, a superposition of states is obtained. So, if the bit is | 0>, it becomes | 0> + | 1>. Upon receiving another light pulse equal to the previous one, the bit changes to the state | 1>. A full pulse is equivalent to the NOT operator, while a half-duration pulse is equivalent to the square root of NOT. The relationship is valid: x In addition to the NOT operator, other types of operators exist in quantum mechanics, which can be applied to the quantities | 0> and | 1> and their combinations to transform them. One of these is the Hadamard operator (H). Analyzing the quantum circuit proposed by Deutsch as a universal circuit for the construction of a quantum computer, indicating with: c – control input; t – target input; it can be deduced that if we measure c and we obtain 0 also t will be at 0; vice versa, if measuring c, 1 t is obtained, it will also be 1. This is because the value 0 of c does not switch t, while the value 1 causes it to switch. The entanglement principle comes into play: a quantum phenomenon, not reducible to classical mechanics, for which under certain conditions two or more physical systems represent subsystems of a larger system whose quantum state cannot be described individually, but only as a superposition of several States.

Figure 4 – Quantum circuit implementing Deutsch’s algorithm. U indicates transformation. If two qubits are both in the superposition of 0 and 1 they are defined entangled if the measurement result of one of them is always related to the measurement result of the other qubit. Entanglement, together with superposition, is the keystone of the entire operation of the quantum computer. Without entanglement, in fact, how could the results obtained be correlated with the input values. With the three fundamental mechanisms of superposition, entanglement and interference it is possible to build an entire quantum circuit logic, at least at a conceptual level, with which the extraordinary computational capacity of a quantum computer can be highlighted.

#### Applications of quantum computing in AI

Research resulting from the interaction between quantum theory and AI can be roughly classified into two categories:

- using some ideas from quantum theory to solve some problems in AI;
- the application of some ideas developed in AI to quantum theory.

In some researches, similarities have been observed between the mathematical structure used by the AI community in the semantic analysis of natural language and those used in quantum mechanics. Observing these similarities is useful since it can provide suggestions on how some ideas of quantum mechanics can be borrowed in semantic analysis or even more generally in AI. Furthermore, if some semantic aspects of natural languages can be expressed correctly in the framework of quantum theory, eg. ambiguity by superposition, therefore the fact that quantum algorithms are suitable for the simulation of quantum systems suggests that quantum computing could significantly accelerate the processing of natural language.

Nelson, McEvoy and Pointer noted that word associations in natural languages can show “ghostly actions at a distance”. Bruza et al. proposed a model of word associations in terms of tensor products so that “distant spectral activation” can be described in a similar way to quantum entanglement.

Among the various meeting points between Quantum Computing and AI, we have:

#### Quantum Bayesian networks

Problems in which the world presents true randomness and problems in which the world is not accidental, but not perfectly known, can be modeled through belief networks (or Bayes networks) and decision networks (or influence diagrams) . A belief network is a graph without direct cycles in which:

- Nodes are made up of a set of random variables.
- The nodes are connected by a set of arcs that represent each other’s causal influences.
- Each node is characterized by a table of conditional probabilities which quantifies the effects that the parents have on it.

The belief network inference mechanism calculates the posterior probability distribution for a set of query variables, given the exact values for some test variables. Once the network that models the system has been built, the goal of its analysis is to collect evidence and modify its behavior based on this evidence.

To model this behavior, a statistical theory of evidence is needed, which is built on Bayes’ theorem:

Where:

- the probability that the hypothesis is true given evidence E;
- the a priori probability that the hypothesis is true in the absence of specific evidence.

This equation is the basis of all modern Artificial Intelligence systems for probabilistic inference, since it allows to express the conditional independence between variables without resorting to a joint probability distribution table, greatly simplifying the calculation of query results.

In fact, having defined the topology of a network of beliefs, it is sufficient to specify the table of conditional probabilities for each node, and the Bayesian update process (inference) incorporates evidence one piece at a time by modifying the previous belief in the unknown variables. Decision networks extend belief networks by incorporating actions and utilities: the preferences of an agent among the states of the world are summarized by a utility function, which associates with each state a single number that expresses its desirability. The utilities combine with the probabilities of the actions to provide the expected utility of each action.

By maximizing a utility function that correctly reflects the performance measures by which its behavior is judged, an agent achieves the best possible performance.

Tucci introduced a quantum generalization of Bayesian networks in which its nodes are assigned complex amplitudes rather than probabilities and used it to calculate probabilities for some physical experiments. Pearl introduced the notion of causal Bayesian networks which augments Bayesian networks with a set of local operations that specify how probability distributions behave with respect to external interventions. To provide a graph model of causality in the quantum world, Laskey defined a notion of quantum causal networks in which local operations are represented by super-operators which are a popular mathematical formalism of the dynamics of open quantum systems.

#### Recognition and discrimination of quantum states and quantum operations

Pattern recognition is the automated recognition of patterns and regularities in data has applications in statistical data analysis, signal processing, image analysis, information retrieval, bioinformatics, data compression, computer graphics and machine learning. It is an important area of AI and object discrimination can be seen as a special case of model recognition. However, only the recognition and discrimination of classical objects were considered by artificial intelligence researchers. Over the past 20 years, a great deal of work on discrimination and recognition of quantum states and quantum operations has been conducted by physicists without knowing much about the existing work on AI. The unambiguous discrimination of quantum states can be formulated: A system is prepared in a number of known finite series of pure quantum states and we hope to determine which quantum state the system actually is in with the requirement that once a result is reported, it must be real. This problem was first considered by Ivanovic for the case of n = 2. The general case was examined by Chefles. It has been shown that the optimal success probability of discrimination is mathematically equivalent to the well-known semidefinite programming problem. An estimate of the probability of success has been provided. The problem of the discrimination of quantum states has been generalized to the case of mixed states. Recently, the discrimination of quantum operations has received considerable attention. The problem of discrimination of unitary transformations was solved by Acín and D’Ariano, Presti and Paris. In particular, a complete characterization of the perfect distinguishability of quantum operations was achieved by discovering a necessary and sufficient feasible condition under which an unknown quantum operation secretly chosen from a finite set of quantum operations can be perfectly identified and by designing an optimal protocol for such discrimination with a minimum number of requests. A particularly interesting problem is the discrimination of quantum operations acting on a multipartite quantum system from local operations and classical communication. Surprisingly, it has been shown that entanglement is not necessary for this type of unitary discrimination of operators. The pattern recognition problem for quantum states was considered by Sasaki and Carlini: given a set of model quantum states, decide which of them is closest to an input state. An essential difference between quantum and classical model recognition is that in the quantum case multiple copies of the model and input states may be needed since quantum measurements are employed in the recognition strategy and usually change the states of the measured systems. A Bayesian learning method has been proposed to carry out the task of recognizing quantum patterns.

#### Learning of quantum states and quantum operations

Let’s consider a simple example of supervised conceptual learning. In the classical case, the training data set is given in the form of D = {(, c ()): i = 1,…, n}, where are instances and c () = 1 or 0 ∀ i. In the quantum case, the instances are replaced by quantum states, |. If the descriptions of quantum instances | are given in a classical way, the quantum learning problem immediately degenerates into a classical learning problem. More interesting is the case where no classical descriptions of these quantum states are available. To learn a concept from the quantum training set it is necessary to extract classical information from them and then certain quantum measurements must be performed on these quantum states. As these quantum measurements will destroy the original quantum states, multiple copies of these quantum states may be needed. This is contrary to the classic case. State quantum tomography can be seen as a kind of quantum learning. The scenario is as follows: There is a physical process that can repeatedly produce a quantum state. We prepare all necessary state copies by applying this process. The goal is to learn a description of the state from the results of the measurements performed on these copies. A similar problem for quantum operations is known as quantum process tomography of which a theory has been developed by Chuang and Nielsen and Poyatos, Cirac and Zoller.

#### Quantum neural networks

Quantum neural networks are computational neural network models based on the principles of quantum mechanics. The first ideas on quantum neural computing were independently published in 1995 by Subhash Kak and Ron Chrisley, engaging with the theory of quantum mind, which postulates that quantum effects play a role in cognitive function. However, typical research on quantum neural networks involves combining classical artificial neural network models with the advantages of quantum information in order to develop more efficient algorithms. An important motivation for these investigations is the difficulty of training classical neural networks, especially in big data applications. Most quantum neural networks are developed as feed-forward networks. Similar to their classic counterparts, this structure receives input from one qubit layer and transfers it to another qubit layer. This qubit level evaluates this information and takes the output to the next level. Eventually, the path leads to the final qubit layer. Layers don’t have to be the same width, which means they don’t have to have the same number of qubits as the layer before or after it. This facility is trained on which path to take similar to classical artificial neural networks. This is discussed in a lower section. Quantum neural networks refer to three different categories: quantum computer with classical data, classical computer with quantum data, and quantum computer with quantum data.

#### Quantum genetic algorithms

The quantum genetic algorithm (QGA) is the product of the combination of quantum computation and genetic algorithms, and is a new evolutionary probability algorithm. In 1996, the quantum genetic algorithm was first proposed by Narayanan and Moore, and was successfully used to solve the TSP problem. QGA is essentially a kind of genetic algorithm and can be applied in the field where the conventional genetic algorithm can be applied. The efficiency of QGA is significantly better than the conventional genetic algorithm. QGA has a small population value, fast convergence rate, large global optimization capability, and good robustness. The quantum state vector is introduced into the genetic algorithm to express the genetic code and quantum logic gates are used to carry out chromosomal evolution. This way you get better results.

#### Conclusions

The three research classes for artificial intelligence researchers at the intersection of quantum computing, quantum theory and artificial intelligence are: • Progettare algoritmi quantistici per risolvere i problemi nell’IA in modo più efficiente; • Develop more effective methods to formalize problems in AI by borrowing ideas from quantum theory; • To develop new artificial intelligence techniques to address problems in the quantum world. The first class of research is still in the early stage of development and not much progress has been made. As for the second class, research has become very active in recent years, especially through the International Symposium on Quantum Interaction. But it seems that some of these works are rather superficial and that a more in-depth theoretical analysis of the formal methods developed is needed. In particular, more experimental research is needed to test its effectiveness. On the other hand, third-class research is making steady progress. The potential of quantum computing in artificial intelligence will soon be evident, but we still don’t know how to translate that potential into reality.

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