Computing A Glimpse of Randomness

A Chaitin Omega number is the halting probability of a universal Chaitin (self-delimiting Turing) machine. Every Omega number is both computably enumerable (the limit of a computable, increasing, converging sequence of rationals) and random (its binary expansion is an algorithmic random sequence). In particular, every Omega number is strongly non-computable. The aim of this paper is to describe a procedure, which combines Java programming and mathematical proofs, for computing the exact values of the first 64 bits of a Chaitin Omega: 0000001000000100000110001000011010001111110010111011101000010000. Full description of programs and proofs will be given elsewhere.Comment: 16 pages; Experimental Mathematics (accepted

A QUBO formulation for the Tree Containment problem

Phylogenetic (evolutionary) trees and networks are leaf-labeled graphs that are widely used to represent the evolutionary relationships between entities such as species, languages, cancer cells, and viruses. To reconstruct and analyze phylogenetic networks, the problem of deciding whether or not a given rooted phylogenetic network embeds a given rooted phylogenetic tree is of recurring interest. This problem, formally know as Tree Containment, is NP-complete in general and polynomial-time solvable for certain classes of phylogenetic networks. In this paper, we connect ideas from quantum computing and phylogenetics to present an efficient Quadratic Unconstrained Binary Optimization formulation for Tree Containment in the general setting. For an instance (N,T) of Tree Containment, where N is a phylogenetic network with n_N vertices and T is a phylogenetic tree with n_T vertices, the number of logical qubits that are required for our formulation is O(n_N n_T).Comment: final version accepted for publication in Theoretical Computer Scienc

A Hybrid Quantum-Classical Paradigm to Mitigate Embedding Costs in Quantum Annealing

Despite rapid recent progress towards the development of quantum computers capable of providing computational advantages over classical computers, it seems likely that such computers will, initially at least, be required to run in a hybrid quantum-classical regime. This realisation has led to interest in hybrid quantum-classical algorithms allowing, for example, quantum computers to solve large problems despite having very limited numbers of qubits. Here we propose a hybrid paradigm for quantum annealers with the goal of mitigating a different limitation of such devices: the need to embed problem instances within the (often highly restricted) connectivity graph of the annealer. This embedding process can be costly to perform and may destroy any computational speedup. In order to solve many practical problems, it is moreover necessary to perform many, often related, such embeddings. We will show how, for such problems, a raw speedup that is negated by the embedding time can nonetheless be exploited to give a real speedup. As a proof-of-concept example we present an in-depth case study of a simple problem based on the maximum weight independent set problem. Although we do not observe a quantum speedup experimentally, the advantage of the hybrid approach is robustly verified, showing how a potential quantum speedup may be exploited and encouraging further efforts to apply the approach to problems of more practical interest.Comment: 30 pages, 6 figure

Magnetic trapping of metastable 3P2^3P_2 atomic strontium

We report the magnetic trapping of metastable $^3P_2$ atomic strontium. Atoms are cooled in a magneto-optical trap (MOT) operating on the dipole allowed $^1S_0-^1P_1$ transition at 461 nm. Decay via $^1P_1\to {^1D_2}\to {^3P_2}$ continuously loads a magnetic trap formed by the quadrupole magnetic field of the MOT. Over $10^8$ atoms at a density of $8 \times 10^9$ cm$^{-3}$ and temperature of 1 mK are trapped. The atom temperature is significantly lower than what would be expected from the kinetic and potential energy of atoms as they are transferred from the MOT. This suggests that thermalization and evaporative cooling are occurring in the magnetic trap.Comment: This paper has been accepted by PR

Collisions of cold magnesium atoms in a weak laser field

We use quantum scattering methods to calculate the light-induced collisional loss of laser-cooled and trapped magnesium atoms for detunings up to 30 atomic linewidths to the red of the 1S_0-1P_1 cooling transition. Magnesium has no hyperfine structure to complicate the theoretical studies. We evaluate both the radiative and nonradiative mechanisms of trap loss. The radiative escape mechanism via allowed 1Sigma_u excitation is dominant for more than about one atomic linewidth detuning. Molecular vibrational structure due to photoassociative transitions to bound states begins to appear beyond about ten linewidths detuning.Comment: 4 pages with 3 embedded figure

Inter-reader agreement of the PI-QUAL score for prostate MRI quality in the NeuroSAFE PROOF trial

Objectives: The Prostate Imaging Quality (PI-QUAL) score assesses the quality of multiparametric MRI (mpMRI). A score of 1 means all sequences are below the minimum standard of diagnostic quality, 3 implies that the scan is of sufficient diagnostic quality, and 5 means that all three sequences are of optimal diagnostic quality. We investigated the inter-reader reproducibility of the PI-QUAL score in patients enrolled in the NeuroSAFE PROOF trial. Methods: We analysed the scans of 103 patients on different MR systems and vendors from 12 different hospitals. Two dedicated radiologists highly experienced in prostate mpMRI independently assessed the PI-QUAL score for each scan. Interobserver agreement was assessed using Cohen’s kappa with standard quadratic weighting (κw) and percent agreement. Results: The agreement for each single PI-QUAL score was strong (κw = 0.85 and percent agreement = 84%). A similar agreement (κw = 0.82 and percent agreement = 84%) was observed when the scans were clustered into three groups (PI-QUAL 1–2 vs PI-QUAL 3 vs PI-QUAL 4–5). The agreement in terms of diagnostic quality for each single sequence was highest for T2-weighted imaging (92/103 scans; 89%), followed by dynamic contrast-enhanced sequences (91/103; 88%) and diffusion-weighted imaging (80/103; 78%). Conclusion: We observed strong reproducibility in the assessment of PI-QUAL between two radiologists with high expertise in prostate mpMRI. At present, PI-QUAL offers clinicians the only available tool for evaluating and reporting the quality of prostate mpMRI in a systematic manner but further refinements of this scoring system are warranted