Quantum Neural Machine Learning: Theory and Experiments DOI: http://dx.doi.org/10.5772/intechopen.84149

actualized, and the probability for this actualization coincides with the diagonal density value ρm,m which corresponds to the echo intensity. This is a basic result from quantum mechanics that extends to any observable, including observables with

Therefore, embedded within quantum mechanics' formalism, we find an account of Born's probability rule. Furthermore, given a Hamiltonian operator for the quantum system H^ <sup>S</sup>, and a time lapse of Δt, quantum mechanics defines the

> �i <sup>η</sup>H^ <sup>S</sup>Δ<sup>t</sup>

�i <sup>η</sup>H^ <sup>S</sup>Δ<sup>t</sup>

!

In the case of the illustrative general example, given in Eq. (18), we get:

the k-th initial eigenvalue to the m-th final eigenvalue and <sup>l</sup> exp <sup>i</sup>=ηH^ <sup>S</sup>Δ<sup>t</sup> � � � � �

backward in time propagating amplitude from the n-th final eigenvalue to the l-th

Cramer was, however, the first to fully address the consequences of this dynamics and propose the concept of echo, within the context of quantum mechanics, addressing it related to Born's rule, deriving Born's rule from within the quan-

While Cramer [7] addresses the echo in terms of the encounter of a forwardpropagating retarded wave (which we addressed above under the probe dynamics, proceeding forward from the beginning to the endpoint of the unitary evolution) and the backward-propagating advanced wave (which we addressed above under the response dynamics, proceeding from the endpoint to the beginning of the unitary evolution), by working with the density operator, instead of the wave function, we get a clearer picture of the corresponding dynamics, which accounts, in the case of any quantum physical system, for both the off-diagonal terms

(as failed echoes) and the diagonal terms of the density operator (as the echoes where the probe was met by a matching response) with the echo intensity giving Born's probability rule. This result is generalizable and independent of the interpretation of quantum mechanics that one follows; that is, all interpretations of quantum

It is important to clarify what an interpretation of quantum mechanics is and why there are different interpretations of the same theoretical body and equations.

<sup>1</sup> One may notice the change in the time lapse signal so that the conjugate transposition corresponds to

It turns out that the main interpretations do not disagree on the formalism,

ρ^ð Þ t<sup>0</sup> e i

j ik h jl e i <sup>η</sup>H^ <sup>S</sup>Δ<sup>t</sup> j i n

�<sup>k</sup> � � is a forward in time propagating amplitude from

, and this basic dynamics is a general result that stems from

<sup>η</sup>H^ <sup>S</sup>Δ<sup>t</sup> (19)

j i m h j n (20)

�<sup>n</sup> � � is a

It is important to stress that this echo dynamics is not specific to QUANNs, but is present in any quantum system; any density operator characterizing a quantum system exhibits, in the formalism, this main dynamics, so the echo dynamics is a characteristic of the physics of quantum systems and accounts for Born's probability rule in quantum mechanics—that is, the probability of an alternative eigenvalue to be observed is equal to the corresponding diagonal component of a density

both discrete as well as continuous spectra.

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unitary propagation of a density operator at time t<sup>0</sup> as:

∑ k,l

ρ^ð Þ¼ t<sup>0</sup> þ Δt ∑

where <sup>m</sup> exp �i=ηH^ <sup>S</sup>Δ<sup>t</sup> � � � � �

Schrödinger's unitary evolution.

mechanics agree with the above results.

initial eigenvalue<sup>1</sup>

tum formalism.

time reversal.

100

m, <sup>n</sup>

ρ^ð Þ¼ t<sup>0</sup> þ Δt e

ρk,lð Þ t<sup>0</sup> h j m e

operator.

methods, and how the mathematics is built and applied for prediction of experimental results. The interpretations do not stem from any ambiguity or lack of robustness in the formalism and in the application of the formalism, they stem from the fact that not everything is accounted for by the formalism, and that is where the interpretations come in.

To better frame this issue, one must consider the nature of the theory that one is dealing with, what it explains, and what is outside its theoretical scope.

Quantum mechanics is, in fact, a probabilistic theory of the quantized dynamics of fundamental physical fields, fields that work at the level of the building blocks of physical nature. The physical theory and methods that form the basic structure of quantum mechanics developed progressively from empirical observations and statistical findings on fundamentally random outcomes of physical experiments dealing with the quantum level.

This means that physicists found the basic rules for (dynamical) probability assignments that robustly capture the main probabilistic dynamics of quantum fields.

To understand the nature of the theory, it is important to stress that it was born out of laboratory experiments, that it was built out of the statistical patterns found in an observed stochastic dynamics, and that it was aimed at predicting the statistical distributions of that stochastic dynamics. The current formulation of quantum mechanics essentially encompasses a set of rules for obtaining the probabilities associated with the dynamics of quantum systems.

The theory does not state anything beyond that. A point that allowed many physicists to pragmatically take the theory as it is, not dwelling on the why quantum systems work that way, that is, to take the theory as a rule book that works, is robustly tested empirically, applying it to problems following what is usually called a shut up and calculate stance.

When one starts to ask on the why quantum systems work that way, the interpretations enter into play, but they go beyond the physical setting of the theory in the sense that they are related to ontological questions; that is, each interpretation regards the ontological issue of physical reality and why the quantum dynamics follows the echoes with probabilities coincident with the echo intensities.

In the pragmatic stance, one just takes the formalism as a recipe, calculates the echo intensities without dwelling further on it. Any result in quantum mechanics applying the formalism is valid and empirically testable and the formalism has time and again, during twentieth and twenty-first centuries, been shown to be robust in its predictions.

One way out of the ontological questioning would be to assume that we are dealing with human representations, that we cannot speak of a reality independent of human representations and experiments, that is, that the question of what reality really is outside those representations and experiments cannot be answered and, therefore, one just postulates that the field follows the echoes. This was the approach of the Copenhagen school, including Bohr and Born, leading to Born's rule that the probabilities are coincident with the echoes, a rule that is introduced, usually, in quantum mechanics' classes as a postulate, a very detailed description of this can be found in [7].

Contrasting with the Copenhagen school are the ontological schools, so called because they assume a reality independent of human representations and experiments.

Quantum mechanics itself does not state anything about this, so there is room for proposals; Cramer [7], for instance, considers these interpretations as actually new physical theories that go beyond the strict formalism and introduce new conjecture that cannot be tested under the formalism itself. The ontological interpretations that include the Bohmian and Everettian lines are all consistent with the

formalism, that is, they agree with the formalism and mathematical methods of quantum mechanics and, therefore, cannot be tested using just the formalism.

intensity, so that the following of a given line of force is similar to a bifurcation dynamics where the field will follow, stochastically, one of the branches with a probability that coincides with the force intensity associated with each branch [6]. There is a consequence that comes from assuming the Bohmian framework, namely, from the Bohm's conjecture that a subquantum level randomness averages out at the quantum level, but may lead to small deviations from the theoretical probabilities [9, 10]; if such a conjecture holds, then deviations in quantum physical experiments with actual quantum computers may always take place, such that, even if we were to reduce the interaction with the environment to zero (or close to zero), we could still have deviations due to subquantum level fluctuations, so that the field would tend to follow the lines of force with probabilities that would hold on average

While Bohm's proposal is potentially testable, at the present stage of scientific

subquantum proposal regarding quantum physical systems, and, in particular, to test, empirically, the possibility that deviations from the main lines of force that agree with a theory's prediction are not due to environmental noise and, rather, to

All main interpretations, as reviewed above, agree with quantum mechanics' general predictions, even Bohm, who considers that the predictions will hold empirically on average, therefore, the interpretations do not have, at present, a direct consequence on the results of technological implementation of quantum computers, as long as one is not dealing with fundamental ontological issues regarding the computational nature of quantum fields, but rather with the technological application of quantum algorithms, one is free to choose any interpretation

We consider, nonetheless, that future research directions on Bohm's conjectural line may prove fruitful both at a theoretical and technological level, concerning the issue of quantum errors. This point, however, goes beyond the current chapter's scope. The results that follow, as of any work using the formalism of quantum mechanics, hold for any interpretation of the theory. However, having made that point, we will return to Bohm's conjecture regarding some of the results obtained in

3. Implementing quantum artificial neural networks on IBM's quantum

The development of quantum computing devices has opened up the possibility of transitioning from the purely theoretical approach to QUANNs to an experimental implementation of these networks. A particular example is IBM's quantum processors, available via cloud, under IBM Q Experience, using superconducting

The term transmon stands for transmission-line shunted plasma oscillation. A transmon qubit [11, 12] is an attempt at a technological implementation of a qubit for quantum computation, using superconductivity and Josephson junctions, gaining in charge noise insensitivity [11, 12]. The control, coupling, and measurement are implemented by means of microwave resonators and circuit quantum

IBM has different transmon-based quantum computers in different locations around the world and provides access to these computers via cloud; this availability allows researchers to implement quantum experiments on actual quantum computers via cloud using IBM Q Experience, opening also the way for programmers to

and technological development, we have not yet found a way to test the

but with some deviations that might occur in each case.

Quantum Neural Machine Learning: Theory and Experiments

DOI: http://dx.doi.org/10.5772/intechopen.84149

since it is consistent with the main formalism and results.

the next section, regarding the issue of quantum computing errors.

subquantum level fluctuations.

computers

electrodynamics.

103

transmon quantum processing units.

In the case of Cramer, his proposed transactional interpretation (TI) of quantum mechanics [7] considers a probabilistic selection in terms of a quantum handshake (Cramer's transaction), where there is a sequential hierarchical selection for a quantum handshake linking the beginning and endpoint of the quantum dynamics, where each alternative is evaluated probabilistically for the formation of a quantum handshake or not; if no handshake is selected for a given alternative, the quantum dynamics proceeds to the next alternative. In each case, the probability for a quantum handshake is equal to the echo intensity, thus deriving Born's rule from within the formalism, instead of assuming it as a postulate.

Everett [8] assumed that all alternatives for a quantum system are actualized simultaneously in different cosmic branches. This led to the many worlds interpretation (MWI). MWI's proposal is, thus, that reality is multidimensional and the formalism is considered to be describing such a multidimensional reality that is a single Cosmos with many worlds (many branching lines). This conjecture cannot be tested empirically; it is consistent with the formalism and agrees with the predictions of quantum mechanics. Namely, the statistical measure associated with repeated experiments made on quantum systems tends to coincide with the echo intensities since the echo intensities coincide with the existence intensity of each world, recovering a statistical measure upon repeated experiments, as argued by Everett in [8] regarding Born's rule.

Bohm initially worked on the pilot wave model for quantum mechanics but just as a first approximation. Indeed, in [9], the author addressed the pilot wave model as a first approximation but then criticized it, in particular, in regard to the assumption of a particle being separate from the field; even more, in [9], Bohm defended that, at a lower level, the particle does not move as a permanently existing entity, but is formed in a random way by suitable concentrations of the field's energy. Furthermore, he considered that any quantum field was characterized by a nonlocal dynamics, and that the equations of quantum mechanics were just an approximation, an average that emerged at the quantum level, proposing the concept of quantum force and hypothesizing the existence of a subquantum level, so that both the quantum and subquantum levels play a fundamental role in the field's dynamics.

Gonçalves in [6] addressed the relation between the echo and Bohm's proposal recovering the Bohmian concept of quantum force [9, 10].

In this interpretation, the echo is associated with a dynamics of a quantum field for the evaluation of each alternative; the probing and response dynamics, thus, play a fundamental role, allowing a quantum field, any quantum field, to compute each alternative in parallel, leading to an echo associated with each alternative.

As argued in [6], the intensity (modulated) echoes would, thus, have a functional role as signalizers of an order to be risen (in the QUANN case, this order corresponds to a specific quantum neural firing pattern); the field's quantum and subquantum levels would, then, work in tandem, mobilizing the forces needed to make rise one specific alternative, and the resulting field lines of force, therefore, coincide, in their intensities, with the echo intensities.

This quantum computational dynamics, present in quantum mechanics' formalism, works as a basic form of quantum "learning" dynamics, where the quantum field "learns" about each alternative in the probe (forward propagating) and response (back propagating) dynamics and, then, the field's lines of force are formed along the echoes resulting from the encounters of matching probe and response vectors, with a force intensity that matches the corresponding echo's intensity; the field then follows one of these lines of force with a probability that coincides with the echo
