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Many finite groups, including all finite non-abelian simple groups, can be symmetrically generated by involutions. In this paper we give an algorithm to symmetrically represent elements of finite groups and to transform symmetrically…

Random numbers are widely used for information security, cryptography, stochastic modeling, and quantum simulations. Key technical challenges for physical random number generation are speed and scalability. We demonstrate a method for…

This paper considers global optimization with a black-box unknown objective function that can be non-convex and non-differentiable. Such a difficult optimization problem arises in many real-world applications, such as parameter tuning in…

An algorithm for irreducible decomposition of representations of finite groups over fields of characteristic zero is described. The algorithm uses the fact that the decomposition induces a partition of the invariant inner product into a…

The well-known Gumbel-Max Trick for sampling elements from a categorical distribution (or more generally a non-negative vector) and its variants have been widely used in areas such as machine learning and information retrieval. To sample a…

In 1986 S. Sattolo introduced a simple algorithm for uniform random generation of cyclic permutations on a fixed number of symbols. This algorithm is very similar to the standard method for generating a random permutation, but is less well…

We present an algorithm for efficiently approximating of qubit unitaries over gate sets derived from totally definite quaternion algebras. It achieves $\varepsilon$-approximations using circuits of length $O(\log(1/\varepsilon))$, which is…

We propose an approach for fast random number generation based on homemade optical physical unclonable functions (PUFs). The optical PUF is illuminated with input laser wavefront of continuous modulation to obtain different speckle…

Black-box quantum state preparation is a fundamental primitive in quantum algorithms. Starting from Grover, a series of techniques have been devised to reduce the complexity. In this work, we propose to perform black-box state preparation…

We present deterministic algorithms for the Hidden Subgroup Problem. The first algorithm, for abelian groups, achieves the same asymptotic worst-case query complexity as the optimal randomized algorithm, namely O($\sqrt{ n}\,$), where $n$…

A quantum random number generator (QRNG) can generate true randomness by exploiting the fundamental indeterminism of quantum mechanics. Most approaches to QRNG employ single-photon detection technologies and are limited in speed. Here, we…

We show how to generate random derangements efficiently by two different techniques: random restricted transpositions and sequential importance sampling. The algorithm employing restricted transpositions can also be used to generate random…

A remarkable aspect of quantum theory is that certain measurement outcomes are entirely unpredictable to all possible observers. Such quantum events can be harnessed to generate numbers whose randomness is asserted based upon the underlying…

We provide algorithms for performing computations in generalized numerical semigroups, that is, submonoids of $\mathbb{N}^{d}$ with finite complement in $\mathbb{N}^{d}$. These semigroups are affine semigroups, which in particular implies…

We introduce a Sinkhorn-type algorithm for producing quantum permutation matrices encoding symmetries of graphs. Our algorithm generates square matrices whose entries are orthogonal projections onto one-dimensional subspaces satisfying a…

Cryptographic random number generation is critical for any quantum safe encryption. Based on the natural uncertainty of some quantum processes, variety of quantum random number generators or QRNGs have been created with physical quantum…

Motivated by the randomized generation of slowly synchronizing automata, we study automata made of permutation letters and a merging letter of rank $ n\!-\!1 $. We present a constructive randomized procedure to generate synchronizing…

We study the scaling of the convergence of several statistical properties of a recently introduced random unitary circuit ensemble towards their limits given by the circular unitary ensemble (CUE). Our study includes the full distribution…

We propose a new approach to nondeterministic random number generation. In theory, the randomness originated from the uncorrelated nature of consecutive laser pulses with Poissonian photon number distribution and that of the consecutive…

Quantum physics can be exploited to generate true random numbers, which play important roles in many applications, especially in cryptography. Genuine randomness from the measurement of a quantum system reveals the inherent nature of…