The Identity Problem for Matrix Semigroups in SL2(Z) is NP-complete

In this paper, we show that the problem of determining if the identity matrix belongs to a finitely generated semigroup of $2\times 2$ matrices from the modular group $\text{PSL}_2(\mathbb Z)$ and thus the Special Linear group $\text{SL}_2(\mathbb Z)$ is solvable in $\mathbf{NP}$. From this fact, we can immediately derive that the fundamental problem of whether a given finite set of matrices from $\text{SL}_2(\mathbb Z)$ or $\text{PSL}_2(\mathbb Z)$ generates a group or free semigroup is also decidable in $\mathbf{NP}$. The previous algorithm for these problems, shown in 2005 by Choffrut and Karhum\"aki, was in \EXPSPACE mainly due to the translation of matrices into exponentially long words over a binary alphabet $\{s,r\}$ and further constructions with a large nondeterministic finite state automaton that is built on these words. Our algorithm is based on various new techniques that allow us to operate with compressed word representations of matrices without explicit expansions. When combined with the known $\mathbf{NP}$-hard lower bound, this proves that the membership problem for the identity problem, the group problem and the freeness problem in $\text{SL}_2(\mathbb Z)$ are $\mathbf{NP}$-complete

Simulated Quantum Computation of Global Minima

Finding the optimal solution to a complex optimization problem is of great importance in practically all fields of science, technology, technical design and econometrics. We demonstrate that a modified Grover's quantum algorithm can be applied to real problems of finding a global minimum using modest numbers of quantum bits. Calculations of the global minimum of simple test functions and Lennard-Jones clusters have been carried out on a quantum computer simulator using a modified Grover's algorithm. The number of function evaluations $N$ reduced from O(N) in classical simulation to $O(\sqrt{N})$ in quantum simulation. We also show how the Grover's quantum algorithm can be combined with the classical Pivot method for global optimization to treat larger systems.Comment: 6 figures. Molecular Physics, in pres

Operator-sum representation of time-dependent density operators and its applications

We show that any arbitrary time-dependent density operator of an open system can always be described in terms of an operator-sum representation regardless of its initial condition and the path of its evolution in the state space, and we provide a general expression of Kraus operators for arbitrary time-dependent density operator of an $N$-dimensional system. Moreover, applications of our result are illustrated through several examples.Comment: 4 pages, no figure, brief repor

Angiographic embolization in the treatment of arterial pelvic hemorrhage: evaluation of prognostic mortality-related factors

PURPOSE: The control of arterial bleeding associated with pelvic ring and acetabular fractures (PRAF) remains a challenge for emergency trauma care. The aim of the present study was to uncover early prognostic mortality-related factors in PRAF-related arterial bleedings treated with transcatheter angiographic embolization (TAE). METHODS: Forty-nine PRAF patients (46 pelvic ring and three acetabular fractures) with arterial pelvic bleeding controlled with TAE (within 24 h) were evaluated. RESULTS: All large arterial disruptions (n = 7) were seen in type C pelvic ring injuries. The 30-day mortality in large vessel (iliac artery) bleeding was higher (57 %) than in medium- or small-size artery bleeding (24 %). Overall 30-day mortality was 29 %. No statistically significant difference in the first laboratory values between the survivors and nonsurvivors was found. However, after excluding patients dying of head injuries (n = 5), a reasonable cut-off value was identified for the base excess (BE; lower than −10 mmol/l) obtained on admission. CONCLUSIONS: PRAF patients with exsanguinating bleeding from the large pelvic artery have the worst prognosis. Very low BE values (<−10.0 mmol/l) on admission for exsanguinating patients have a negative predictive value for survival, thus anticipating a poor outcome in bleeding controlled with TAE only and an increased risk of death. In critical cases, an aggressive bleeding control protocol prompts extraperitoneal pelvic packing prior to TAE. PRAF-related rupture of the external iliac artery is rare and indicates surgical techniques in controlling and restoring blood supply to the lower leg

Quantum, Stochastic, and Pseudo Stochastic Languages with Few States

Stochastic languages are the languages recognized by probabilistic finite automata (PFAs) with cutpoint over the field of real numbers. More general computational models over the same field such as generalized finite automata (GFAs) and quantum finite automata (QFAs) define the same class. In 1963, Rabin proved the set of stochastic languages to be uncountable presenting a single 2-state PFA over the binary alphabet recognizing uncountably many languages depending on the cutpoint. In this paper, we show the same result for unary stochastic languages. Namely, we exhibit a 2-state unary GFA, a 2-state unary QFA, and a family of 3-state unary PFAs recognizing uncountably many languages; all these numbers of states are optimal. After this, we completely characterize the class of languages recognized by 1-state GFAs, which is the only nontrivial class of languages recognized by 1-state automata. Finally, we consider the variations of PFAs, QFAs, and GFAs based on the notion of inclusive/exclusive cutpoint, and present some results on their expressive power.Comment: A new version with new results. Previous version: Arseny M. Shur, Abuzer Yakaryilmaz: Quantum, Stochastic, and Pseudo Stochastic Languages with Few States. UCNC 2014: 327-33

Finite automata with advice tapes

We define a model of advised computation by finite automata where the advice is provided on a separate tape. We consider several variants of the model where the advice is deterministic or randomized, the input tape head is allowed real-time, one-way, or two-way access, and the automaton is classical or quantum. We prove several separation results among these variants, demonstrate an infinite hierarchy of language classes recognized by automata with increasing advice lengths, and establish the relationships between this and the previously studied ways of providing advice to finite automata.Comment: Corrected typo

Implementing Shor's algorithm on Josephson Charge Qubits

We investigate the physical implementation of Shor's factorization algorithm on a Josephson charge qubit register. While we pursue a universal method to factor a composite integer of any size, the scheme is demonstrated for the number 21. We consider both the physical and algorithmic requirements for an optimal implementation when only a small number of qubits is available. These aspects of quantum computation are usually the topics of separate research communities; we present a unifying discussion of both of these fundamental features bridging Shor's algorithm to its physical realization using Josephson junction qubits. In order to meet the stringent requirements set by a short decoherence time, we accelerate the algorithm by decomposing the quantum circuit into tailored two- and three-qubit gates and we find their physical realizations through numerical optimization.Comment: 12 pages, submitted to Phys. Rev.

Decidability of Univariate Real Algebra with Predicates for Rational and Integer Powers

We prove decidability of univariate real algebra extended with predicates for rational and integer powers, i.e., $(x^n \in \mathbb{Q})$ and $(x^n \in \mathbb{Z})$. Our decision procedure combines computation over real algebraic cells with the rational root theorem and witness construction via algebraic number density arguments.Comment: To appear in CADE-25: 25th International Conference on Automated Deduction, 2015. Proceedings to be published by Springer-Verla