Qubits, Mellin Gates, and Wavefunctions

The solutions to the Schrödinger equation, known as quantum waves or wave functions, represent the fundamental description of a quantum system's state. While conventionally treated as single-valued functions to ensure a consistent probabilistic interpretation, exploring novel multi-valued function representations opens a mathematically rich avenue. Such non-standard formalisms might be necessary to capture complex topological properties or subtle non-local correlations in ways that conventional Hilbert space models struggle with, potentially providing new insights into the nature of quantum reality itself and the completeness of the current quantum theory.

This exploration of a wavefunction's mathematical form naturally leads to new interpretations of the postulates of quantum mechanics. The postulates govern how quantum states are described, how observables correspond to operators, and how measurements are made. By re-examining requirements like single-valuedness or linearity, we can conceive of new structures for quantum information. Specifically, the mathematics defining a quantum system's evolution, the Hamiltonian (which corresponds to a quantum gate), could be derived from operations like the Mellin transform. Since the Mellin transform is scale-invariant and closely tied to multiplicative number theory, a quantum gate based on it might facilitate computations leveraging number theoretic properties, leading to novel qubits and entanglement schemes beyond the standard 0 and 1 Pauli basis.

In the computational realm, Physics-Informed Neural Networks (PINNs) are being applied to solve and analyze complex quantum systems. PINNs embed the differential equations of physics (like the Schrödinger equation) directly into the neural network's loss function, allowing it to learn solutions without vast labeled data. Applying PINN analysis to the Fourier decomposition of continuous variable quantum waves has revealed fascinating results regarding their coherence and shape. Fourier analysis of waves involves comparing phase differences between components, and rigorous scrutiny often suggests that such waves, while often modeled as, are not truly sinusoidal in a purely mathematical sense due to environmental interactions, non-linearities, or the discrete nature of quantum energy levels, suggesting a more complex, structured waveform.

A parallel line of research with profound implications for cryptography lies in the study of number theoretic functions. The properties of the L-functions associated with the Riemann zeta function are deeply connected to the distribution of prime numbers. The statistical distribution of the non-trivial zeros of these functions has been conjectured to exhibit the same statistics as the eigenvalues of random Hermitian matrices, a phenomenon known as the Hilbert–Pólya conjecture. This connection motivates the research into the mathematical properties of L-functions as potential sources of high-quality pseudo-randomness.

The generation of complex, yet predictable, pseudo-random sequences is critical to modern security. The current field of research seeks to leverage the deep arithmetic complexity of L-functions and related structures to develop cryptographically secure pseudo-random number generators. This has direct, significant implications for the security of public-key cryptography, particularly for breaking the RSA decryption algorithm. This cutting-edge research, linking abstract number theory to computational security and the fundamental properties of quantum systems, is one of the interesting fields that consultants for General Physics (www.generalphysicscorp.com) are actively investigating.