Quantum Rep., Volume 5, Issue 4 (December 2023) – 4 articles

We propose a correspondence between the partition functions of ideal gases consisting of both bosons and fermions and the algebraic bases of supersymmetric polynomials on the Banach space of absolutely summable two-sided sequences ${\mathcal{\ell }}_1\left({\mathbb{Z}}_0\right).$ Such an approach allows us to interpret some of the combinatorial identities for supersymmetric polynomials from a physical point of view. We consider a relation of equivalence for ${\mathcal{\ell }}_1\left({\mathbb{Z}}_0\right)$, induced by the supersymmetric polynomials, and the semi-ring algebraic structures on the quotient set with respect to this relation. The quotient set is a natural model for the set of energy levels of a quantum system. We introduce two different topological semi-ring structures into this set and discuss their possible physical interpretations. Full article

We establish an analogy between the Fokker–Planck equation describing evolutionary landscape dynamics and the Schrödinger equation which characterizes quantum mechanical particles, showing that a population with multiple genetic traits evolves analogously to a wavefunction under a multi-dimensional energy potential in imaginary time. Furthermore, we discover within this analogy that the stationary population distribution on the landscape corresponds exactly to the ground-state wavefunction. This mathematical equivalence grants entry to a wide range of analytical tools developed by the quantum mechanics community, such as the Rayleigh–Ritz variational method and the Rayleigh–Schrödinger perturbation theory, allowing us not only the conduct of reasonable quantitative assessments but also exploration of fundamental biological inquiries. We demonstrate the effectiveness of these tools by estimating the population success on landscapes where precise answers are elusive, and unveiling the ecological consequences of stress-induced mutagenesis—a prevalent evolutionary mechanism in pathogenic and neoplastic systems. We show that, even in an unchanging environment, a sharp mutational burst resulting from stress can always be advantageous, while a gradual increase only enhances population size when the number of relevant evolving traits is limited. Our interdisciplinary approach offers novel insights, opening up new avenues for deeper understanding and predictive capability regarding the complex dynamics of evolving populations. Full article

We investigate the use of amplitude amplification on the gate-based model of quantum computing as a means for solving combinatorial optimization problems. This study focuses primarily on quadratic unconstrained binary optimization (QUBO) problems, which are well-suited for qubit superposition states. Specifically, we demonstrate circuit designs which encode QUBOs as ‘cost oracle’ operations $U_{\mathrm{C}}$, which distribute phases across the basis states proportional to a cost function. We then show that when $U_{\mathrm{C}}$ is combined with the standard Grover diffusion operator $U_{\mathrm{s}}$, one can achieve high probabilities of measurement for states corresponding to optimal and near optimal solutions while still only requiring O($\frac{\pi }{4}\sqrt{2^N/M}$) iterations. In order to achieve these probabilities, a single scalar parameter $p_{\mathrm{s}}$ is required, which we show can be found through a variational quantum–classical hybrid approach and can be used for heuristic solutions. Full article

Arguments have been made that the violation of the CHSH and similar inequalities shows that reality at the quantum level must be non-local. The derivation of Bell inequality is re-examined, and it is shown that violations of these inequalities merely demonstrate the existence of contextuality—they say nothing about the causal influences underlying such contextuality. It is argued that contextual systems do not possess enduring (propositional) properties, merely contingent properties. An example of a classical situation is presented: a two-player co-operative game, the random variables of which are consistently connected in the sense of Dzhafarov, which is contextual, and violates the CHSH inequality. In fact, it also violates the Tsirel’son bound. The key is that this system is generated, and its properties are disposed of, not determined. Full article