Related papers: On computation with 'probabilities' modulo k
Since the seminal work of Keating and Snaith, the characteristic polynomial of a random Haar-distributed unitary matrix has seen several of its functional studied or turned into a conjecture; for instance: $ \bullet $ its value in $1$…
Probabilistic computers replace logic gates with networks of interacting random variables, creating bidirectional systems that can back-derive inputs from outputs. Such architectures enable efficient generation of random samples,…
Quantum theory provides a comprehensive framework for quantifying uncertainty, often applied in quantum finance to explore the stochastic nature of asset returns. This perspective likens returns to microscopic particle motion, governed by…
Power-law distributions are essential in computational and statistical investigations of extreme events and complex systems. The usual technique to generate power-law distributed data is to first infer the scale exponent $\alpha$ using the…
We expand a set of notions recently introduced providing the general setting for a universal representation of the quantum structure on which quantum information stands. The dynamical evolution process associated with generic quantum…
Ensembles of random fuzzy non-commutative geometries may be described in terms of finite (\(N^2\)-dimensional) Dirac operators and a probability measure. Dirac operators of type \((p,q)\) are defined in terms of commutators and…
The problem of identifying a planted assignment given a random $k$-SAT formula consistent with the assignment exhibits a large algorithmic gap: while the planted solution becomes unique and can be identified given a formula with $O(n\log…
We use recent bounds on bilinear sums with modular square roots to study the distribution of solutions to congruences $x^2 \equiv p \pmod q$ with primes $p\le P$ and integers $q \le Q$. This can be considered as a combined scenario of Duke,…
Classical approach of solvability problem has shed much light on what we can solve and what we cannot solve mathematically. Starting with quadratic equation, we know that we can solve it by the quadratic formula which uses square root.…
The computational efficiency of quantum mechanics can be defined in terms of the qubit circuit model, which is characterized by a few simple properties: each computational gate is a reversible transformation in a connected matrix group;…
We consider the XXZ spin-1/2 Heisenberg chain with antiperiodic boundary conditions. The inhomogeneous version of this model can be solved by Separation of Variables (SoV), and the eigenstates can be constructed in terms of Q-functions,…
More than 15 years ago, a new approach to quantum mechanics was suggested, in which Hermiticity of the Hamiltonian was to be replaced by invariance under a discrete symmetry, the product of parity and time-reversal symmetry, $\mathcal{PT}$.…
Numerous algorithms call for computation over the integers modulo a randomly-chosen large prime. In some cases, the quasi-cubic complexity of selecting a random prime can dominate the total running time. We propose a new variant of the…
Separable Bayesian Networks, or the Influence Model, are dynamic Bayesian Networks in which the conditional probability distribution can be separated into a function of only the marginal distribution of a node's neighbors, instead of the…
In this report we discuss some results of non--commutative (quantum) probability theory relating the various notions of statistical independence and the associated quantum central limit theorems to different aspects of mathematics and…
We consider a finite dimensional $\kk G$-module $V$ of a $p$-group $G$ over a field $\kk$ of characteristic $p$. We describe a generating set for the corresponding Hilbert Ideal. In case $G$ is cyclic this yields that the algebra $\kk[V]_G$…
We study the seeded domino problem, the recurring domino problem and the $k$-SAT problem on finitely generated groups. These problems are generalization of their original versions on $\mathbb{Z}^2$ that were shown to be undecidable using…
Random unitaries are a central object of study in quantum information, with applications to quantum computation, quantum many-body physics, and quantum cryptography. Recent work has constructed unitary designs and pseudorandom unitaries…
In simple -- but selected -- quantum systems, the probability distribution determined by the ground state wave function is infinitely divisible. Like all simple quantum systems, the Euclidean temporal extension leads to a system that…
For a Haar random set $\mathcal{S}\subset U(d)$ of quantum gates we consider the uniform measure $\nu_\mathcal{S}$ whose support is given by $\mathcal{S}$. The measure $\nu_\mathcal{S}$ can be regarded as a…