**5. Does quantum physics allow discrete quantum computing?**

Discrete quantum computing could in principle make error control easier. But in order to take advantage of the fact that quantum states are discrete, Quantum Physics must allow the construction of self-correcting systems. A system with these characteristics associates a basin of attraction with each discrete state so that whenever the *n*qubit falls into said basin of attraction, the system automatically corrects it, transforming it into the associated discrete state. However, this process does not fulfill the Schrödinger equation because it is not unitary. And it cannot be the result of a quantum measurement either because the probability that the result was not the associated discrete state would not be zero. Then, how can Quantum Physics implement discrete quantum computing?

We believe that Quantum Physics can take one step further in the description of physical systems. Quantum Physics still fails to explain fundamental physical concepts, to the point that physicists as relevant as Feynman said "I think I can safely say that nobody understands quantum mechanics" and Quantum Mechanics has a reputation for being especially mysterious.

An example of a surprising result is the the no-cloning theorem [45–47], which states that it is impossible to create an independent and identical copy of an arbitrary unknown quantum state. This result of Quantum Physics contrasts with the self-reproducing systems of nature and is also derived from the Schrödinger equation, that predicts a unitary evolution of physical systems.

Quantum Physics has so far failed to explain the concepts for which it has acquired the fame of mysterious. We must assume that these mysteries are intrinsic to the nature of physical systems or that there is a road for Quantum Physics to explain them and open new paths for its development. Next we are going to analyze some of the less understandable concepts of Quantum Physics.

The first concept that is difficult to understand is the wave-particle duality. These concepts are inherently incompatible and nevertheless both are necessary to explain many results of Quantum Physics. If we assume that physical systems have a coherent physical description, we must conclude that elementary particles are neither waves nor particles. Therefore they must be something else.

On the other hand, the postulates of Quantum Physics introduce two processes to describe the evolution of physical systems: the Schrödinger equation and quantum measurements. The first predicts a unitary evolution of physical systems while the second seems to violate the prediction of the first. Many researchers assume that the result of the measurement of a quantum system is a random process whose probabilities depend on the measured system and not on the device that performs the measurement, and that the result is random, that is, there are no hidden variables that determine the result deterministically. In this interpretation the measurement process violates the Schrödinger equation. Other interpretations regard quantum states as statistical information about quantum systems, thus asserting that abrupt and discontinuous changes of quantum states are not problematic, simply reflecting updates of the available information. These reinforce the mysterious character of Quantum Physics and change its objective of describing physical systems for that of only obtaining information.

Finally, we want to comment on the interpretations made of the wave function obtained by solving the Schrodinger equation. It is common to hear that the wave function, for example of an electron, does not indicate that the particle is at all points where the wave function is not zero and that it is not an indicator of our ignorance of the position of the particle. On the one hand we give all the credit to the Schrödinger equation and on the other we take it away from the wave function.

As we see the controversy continues to haunt Quantum Physics. From our point of view, Quantum Physics has found a prediction system for the results of the measurements of physical systems, but it does not describe them. This prevents Quantum Physics from advancing in the deductive knowledge of physical systems, leaving only the advance based on experimentation. Does Quantum Physics really describe everything we can know about physical systems? We do not believe it.

What can be done to get out of this loop? We believe that we should focus on the initial problem: the wave-particle duality. As we have indicated before, this dilemma indicates that elementary particles are neither waves nor particles. Therefore the first objective is to determine its nature. To do this, we must look for questions that can be answered through the design of experiments and that shed light on the nature of elementary particles. In our opinion the first important question is the following: In how many points of space can an elementary particle be simultaneously?

Physics, in addition to the problems of Quantum Physics already mentioned, also has serious problems to combine two of its most notable theories: General Relativity and Quantum Physics. Undoubtedly, any theory that goes in the direction of discretizing space must also consider the discretization of time. In our study we only intend to contribute ideas so that Quantum Physics can overcome the controversies that it is not able to explain. We do not start from the hypothesis that Quantum Physics must be a discretized theory, but we believe that it must be a theory that allows self-correction and that this property must allow the implementation of a discrete quantum computation.

In Quantum Physics, different types of discretization have been proposed, in addition to the one presented in this article. Thus, in [48] a discretization of the quantum state space is proposed in order to explain Born's rule for probabilities. The proposal, despite being very similar to the one we have presented in this article, has very different objectives. In [48] it is used to try to explain two of the most important interpretations of Quantum Physics: Many Worlds and Copenhagen interpretation. In our case the objective is to define a discrete quantum computing model allowing effective control of quantum errors. And this objective leads us to pose an important question, aimed at explaining the wave-particle duality: In how many points of space can an elementary particle be simultaneously?

#### **5.1 Hypothesis on the nature of elementary particles**

Elementary particles cannot be in only one position in space because they cannot explain their behavior as waves. Then, in how many positions can they be simultaneously? The answer can be a finite number greater than one, a countable infinite number, or even an uncountable infinite number. Due to the principle of simplicity, we are inclined to take as a working hypothesis that the answer is a finite number greater than one.

And what does it mean for a particle to be simultaneously at various points in space? In our hypothesis the particle orbit between all its possible positions but being in only one at each time. Therefore simultaneity must be taken in a non-strict sense. That a particle orbits in different points means that it disappears from one point and appears in another and so on. The particle does not travel from one point to another through ordinary space and, in this sense, it may violate the special relativistic principle of speed limitation. Colloquially speaking the particle travels through a "wormhole", without deforming space through large concentrations of mass.

#### *Discretization, the Road to Quantum Computing? DOI: http://dx.doi.org/10.5772/intechopen.98827*

And, why do we choose this elementary particle model as a hypothesis? Because as we have said, the particle must be able to be in more than one point simultaneously and there are already experimental results of quantum nonlocality [49–53]. As far as we know, quantum nonlocality does not allow for faster-than-light communication and it is generally assumed that is compatible with special relativity and its universal speed limit of objects. We believe that quantum nonlocality in some sense violates the aforementioned principle of special relativity. We do not believe that the physical characteristics of the systems should be subordinated to the ability to transmit information.

From our point of view, the multi-position structure of the particles generates nonlocality in the usual space and breaks its Euclidean behavior. In this way physical systems can interact non-locally in space through their multi-position structure.

Another question that arises naturally from our working hypothesis is how scattered can the points that define an elementary particle be in space? Non-point particles can naturally explain their intrinsic angular momentum and this, in turn, give us information about the structure of the particles. For example, a particle that could be in three points in space would have an angular momentum proportional to the area of the triangle determined by its positions. This would indicate that the dispersion of the particles would occur on typical scales of Quantum Physics.

The multi-position particle hypothesis would again bring up some problems that originated Quantum Theories, such as, for example, the stability of atoms. This problem would be solved by the spatial scattering of the electrons around the nucleus. In this case the far electromagnetic field generated by the electrons would decrease faster than the inverse of the square of the distance and this would prevent the electrons from losing their energy in the form of electromagnetic radiation.

Our hypothesis would force us to readapt Quantum Theory. Therefore, we should plan experiments that allow us to contrast it. Is this possible?

#### **5.2 How to test the hypothesis experimentally?**

We would like to propose a couple of experiments that could theoretically provide information on our hypothesis about the structure of elementary particles. The first is a variation of the flagship experiment in which the wave-particle duality

**Figure 4.** *k-slit experiment.*

of elementary particles is tested: the double-slit experiment. The second uses a known quantum effect: the quantum tunneling.

**Experiment 1.** *k***-slits.** We launch, one by one, elementary particles towards a barrier orthogonal to the direction of the movement of the particles (see **Figure 4**). In the barrier there are *k* parallel slits at a distance *d* one from the following: *s*1, *s*2, … , *sk*. Behind we place a screen parallel to the barrier and at a distance *D* from it. On this screen we place the detectors to obtain the interference pattern of the particles.

The objective of this experiment is to determine if the particles, according to our hypothesis, can be simultaneously in exactly *k* � 1 positions. If this hypothesis is true, a particle cannot pass through the *k* slits. It can pass through *k* � 1 slits at most. Therefore, the interference pattern will depend on whether the hypothesis is true.

We start the experiment by choosing *k* ¼ 3. If the hypothesis that the particles are in exactly *k* � 1 positions simultaneously is not corroborated, we increase the value of *k* by 1 and carry out the experiment again. And when is our hypothesis confirmed? When the interference pattern obtained is *P*ð Þ true instead of *P*ð Þ false :

$$\mathbf{1}.P(\text{true}) = \frac{P(\text{s1, }\dots, \text{s}\_{k-1}) + P(\text{s1, }\dots, \text{s}\_{k-2}, \text{sy}) + \dots + P(\text{s2, }\dots, \text{s}\_k)}{k}.$$

$$\text{2.P}(\text{false}) = P(s\_1, s\_2, \dots, s\_k).$$

It would be necessary to estimate if the measurements can be precise enough to distinguish the two patterns and, in the first, if the probability of the *k* possible cases is the same or not.

**Experiment 2. Quantum tunneling.** We launch, one by one, elementary particles towards a potential barrier orthogonal to the direction of the movement of the particles (see **Figure 5(a)**). The energy of the particles is insufficient to jump the potential barrier and its width is small enough to allow the particles to have an appreciable probability of passing the barrier by tunneling. The particles are prepared in two different states. In the first state the intrinsic angular momentum of the particles is parallel to the direction of motion and, in the second state, it is orthogonal.

The objective of this experiment is to determine if the state of the particles influences the probability of quantum tunneling. If this influence is confirmed, it would mean that the orientation of the intrinsic angular momentum of the particles determines in some way the internal structure of the particle against the potential barrier. This could be explained quite understandably with the hypothesis that the particles are in exactly 3 positions at the same time. In this case the particle is always in a plane and the intrinsic angular momentum can orient that plane. If the three

**Figure 5.** *Quantum tunneling experiment.*

*Discretization, the Road to Quantum Computing? DOI: http://dx.doi.org/10.5772/intechopen.98827*

positions that define the particle reach the barrier simultaneously, the particle will not be able to pass (see **Figure 5(b)**). But if one of the positions arrives earlier, this position could cross the barrier while the particle orbits in the other positions (see **Figure 5(c)**). Thus, when the particle orbits in this position it will already be on the other side of the barrier.

We believe that it is not difficult to design more experiments that can shed light on our hypothesis of elementary particles. At this moment we are studying the dynamics of these multi-position particles.
