Transition amplitude spaces and quantum logics with vector-valued states

Sylvia Pulmannová

Zeno Toffano, François Dubois

We propose a linear algebraic method, named Eigenlogic, for two-, many-valued and fuzzy logic using observables in Hilbert space. All logical connectives are represented by observables where the truth values correspond to eigenvalues and the atomic input propositional cases, i.e. the “models” of a propositional system, to the respective eigenvectors. In this way propositional logic can be formalized by using combinations of tensored elementary quantum observables. The outcome of a “measurement” of a logical observable will give the truth value of the associated logical proposition, and becomes “interpretable” when applied to vectors of its eigenspace, leading to an original insight into the quantum measurement postulate.Recently the concept of “quantum predicate” has been proposed in leading to similar concepts.We develop logical observables for binary logic and extend them to many-valued logic.For binary logic and truth values {0, 1} logical observables are commuting projector operators.For truth values {+1,−1} the logical observables are isometries formally equivalent to the ones of a composite quantum spin 12 system, these observables are reversible quantum logic gates.The analogy of many-valued logic with quantum angular momentum is then established using a general algebraic method, based on classical interpolation framework suggested by the finite-elementmethod. Logical observables for three-valued logic are formulated using the orbital angular momentum observable Lz with ` = 1. The representative 3-valued 2-argument logical observables for theternary threshold logical connectives Min and Max are then explicitly obtained.Also in this approach fuzzy logic arises naturally when considering vectors outside the eigensystem.The fuzzy membership function is obtained by the quantum mean value (Born rule) of the logical projector observable and turns out to be a probability measure. Fuzziness arises because ofthe quantum superposition of atomic propositional cases, the truth of a proposition being in this case a probabilistic value ranging from completely false to completely true.This method could be employed for developing algorithms in high-dimensional vector spaces for example in modern semantic theories, such as distributional semantics or in connectionist modelsof cognition. For practical implementation, due to the exponential growth of the vector space dimension, adapted logical reduction methods must then be used. LSA (Latent Semantic Analysis)algorithms are often used in quantum-like approaches, this was done using the HAL (Hyperspace Analogue to Language) algorithm.Our approach is also of interest for quantum computation because several of the observables in Eigenlogic are well-known quantum gates (for example CONTROL-Z) and other ones can be derived by unitary transformations. Ternary-logic quantum gates using qu-trits lead to less complex circuits, our formulation of multi-valued logical observables could help the development of new multi-valued quantum gate architectures.

Mirko Navara, Pavel Pták

Nasir Kenarangui, Arthur Powalka, László B. Kish

Abstract Instantaneous noise-based logic (INBL) leverages the products and superpositions of uncorrelated noise signals to construct a 2^N-dimensional Hilbert space (‘hyperspace’). This approach has been investigated for complex computational tasks traditionally associated with quantum computing, such as the Deutsch–Jozsa algorithm and phonebook search problems. In this study, we delve into the statistical characteristics of this computational framework. To facilitate simulations, we developed an INBL engine. N noise bits represented by 2N uncorrelated reference noises (random telegraph waves (RTWs)) form a 2^N-dimensional space when used in superpositions. N-long binary strings are the hyperspace vectors formed by the products of the relevant reference noises that represent the corresponding bit values. The superpositions of the 2^N possible products collectively define the 2^N-dimensional universe. To prevent the occurrence of zeros in the time function of the universe signal, we implemented an asymmetric amplitude scheme for the 2N reference noise bits, utilizing different absolute amplitude values for the N uncorrelated waves of low (L) and high (H) bit values. The simulation results indicate that the distribution of discrete amplitudes within the universe waveform (time function) deviates from Gaussianity. This finding reinforces that we cannot assert independence among hyperspace vectors, even though they exhibit zero cross-correlation. Another observation is rare random spikes in the universe waveform; surprisingly, the distribution of their magnitudes (absolute values) conforms to a Gaussian curve. Both the magnitude and the mean frequency of these spikes depend on the number of noise bits N. As N increases, the number of discrete amplitude levels also rises, leading to a broader distribution. Conversely, with a growing N, both the relative magnitudes and frequency of spikes in the superposition diminish. The observed critical statistical properties can affect the implementation of INBL in practical applications, particularly by suggesting potential implications for the energy and supply voltage requirements in processors, although a detailed hardware-level energy model is beyond the scope of this work and is left for future study.

Jarosław Pykacz

Ir J.A.J. van Leunen

The origin of physical dynamics and the reason of existence of special relativity are explored. This endeavour is started by analysing the logic of nature. Next, only mathematics is used in order to explore the dynamics of this model of physical reality. The model that is described here annihilates the old reality and creates a new reality at each dynamic step. Hilbert space cannot treat dynamics. It contains nothing that supports dynamics. In the contrary, dynamics manages the Hilbert spaces. Like traditional quantum logic, Hilbert space cannot treat physical fields. By embedding the separable Hilbert space in a rigged Hilbert space, it can house fields by representing them as blurred sets of Hilbert vectors. The field is the convolution of the blur with a set of Dirac delta functions that represent Hilbert vectors. When the blur is differentiable, then the field is differentiable as well. The field values are attached to the Hilbert vectors. In this way traditional quantum logic can be expanded, such that it also treats fields. This extended quantum logic still cannot handle dynamics. The logic only describes a static status quo. Dynamics let nature step from one status quo to the next. It does that by letting nature transform from configuration space to Fourier space. There the fields control the difference between the past and the future status quo. The Fourier transform converts the rather complicated differentiation into a simple multiplication and since the multiplication factors are close to unity, this comes down to still simpler addition. After the confrontation in Fourier space, nature returns back to configuration space. Feynman's path integral approach exploits this fact. The up and down Fourier transforms reshuffle the Hilbert vectors. All Hilbert vectors are affected. The Hilbert vectors represent virtual or actual quanta and present themselves as shot noise.