6 research outputs found
On the Complexity of Generalized Discrete Logarithm Problem
Generalized Discrete Logarithm Problem (GDLP) is an extension of the Discrete
Logarithm Problem where the goal is to find such for a given . Generalized discrete logarithm is
similar but instead of a single base element, uses a number of base elements
which does not necessarily commute with each other. In this paper, we prove
that GDLP is NP-hard for symmetric groups. Furthermore, we prove that GDLP
remains NP-hard even when the base elements are permutations of at most 3
elements. Lastly, we discuss the implications and possible implications of our
proofs in classical and quantum complexity theory
Quantum Adversarial Learning in Emulation of Monte-Carlo Methods for Max-cut Approximation: QAOA is not optimal
One of the leading candidates for near-term quantum advantage is the class of
Variational Quantum Algorithms, but these algorithms suffer from classical
difficulty in optimizing the variational parameters as the number of parameters
increases. Therefore, it is important to understand the expressibility and
power of various ans\"atze to produce target states and distributions. To this
end, we apply notions of emulation to Variational Quantum Annealing and the
Quantum Approximate Optimization Algorithm (QAOA) to show that QAOA is
outperformed by variational annealing schedules with equivalent numbers of
parameters. Our Variational Quantum Annealing schedule is based on a novel
polynomial parameterization that can be optimized in a similar gradient-free
way as QAOA, using the same physical ingredients. In order to compare the
performance of ans\"atze types, we have developed statistical notions of
Monte-Carlo methods. Monte-Carlo methods are computer programs that generate
random variables that approximate a target number that is computationally hard
to calculate exactly. While the most well-known Monte-Carlo method is
Monte-Carlo integration (e.g. Diffusion Monte-Carlo or path-integral quantum
Monte-Carlo), QAOA is itself a Monte-Carlo method that finds good solutions to
NP-complete problems such as Max-cut. We apply these statistical Monte-Carlo
notions to further elucidate the theoretical framework around these quantum
algorithms
Circuit Transformations for Quantum Architectures
Quantum computer architectures impose restrictions on qubit interactions. We propose efficient circuit transformations that modify a given quantum circuit to fit an architecture, allowing for any initial and final mapping of circuit qubits to architecture qubits. To achieve this, we first consider the qubit movement subproblem and use the ROUTING VIA MATCHINGS framework to prove tighter bounds on parallel routing. In practice, we only need to perform partial permutations, so we generalize ROUTING VIA MATCHINGS to that setting. We give new routing procedures for common architecture graphs and for the generalized hierarchical product of graphs, which produces subgraphs of the Cartesian product. Secondly, for serial routing, we consider the TOKEN SWAPPING framework and extend a 4-approximation algorithm for general graphs to support partial permutations. We apply these routing procedures to give several circuit transformations, using various heuristic qubit placement subroutines. We implement these transformations in software and compare their performance for large quantum circuits on grid and modular architectures, identifying strategies that work well in practice
A Rare Anomaly of Testicular Descend: Transverse Testicular Ectopia and Review of the Literature
Crossed ectopia or transverse testicular ectopia (TTE) is an extremely rare anomaly of testicular descent in which both gonads migrate along the same inguinal canal to the hemiscrotum. Fewer than 100 cases have been reported in the literature. Most cases of TTE involve a concomitant inguinal hernia on the contralateral side. A radiological evaluation and laparoscopy are essential for appropriate diagnosis and treatment. Although a correct diagnosis is not preoperatively made in most cases, we present a 15-year-old boy in whom TTE was diagnosed by magnetic resonance imaging. A laparoscopic evaluation and treatment (orchiectomy) was uneventfully performed