An optimized quantum circuit for converting from sign–magnitude to two’s complement

Nowadays, one of the critical issues to implement quantum algorithms is the required number of elementary gates, qubits and delay. Current quantum computers and simulators are mainly prototypes, and there is a lack of computational resources. Therefore, it is necessary to optimize the quantum operations to reduce the necessary number of gates and qubits. This work presents a novel reversible circuit which is able to convert signed binary numbers to two’s complement of N digits in a quantum environment. The depth of the circuit is O(log N). It is based on the fastest out-of-place carry look-ahead addition quantum circuit currently available. This addition circuit has been adapted to make the conversion using the minimum number of gates and qubits, being faster than other adder circuits. A robust metric has been used to measure the quantum cost, delay, ancilla inputs and garbage outputs of the proposed converter. Moreover, it has been compared with others described in the literature

Efficient Reversible Quantum Design of Sign-Magnitude to two's complement converters

Despite the great interest that the scientific community has in quantum computing, the scarcity and high cost of resources prevent to advance in this field. Specifically, qubits are very expensive to build, causing the few available quantum computers are tremendously limited in their number of qubits and delaying their progress. This work presents new reversible circuits that optimize the necessary resources for the conversion of a sign binary number into two's complement of N digits. The benefits of our work are two: on the one hand, the proposed two's complement converters are fault tolerant circuits and also are more efficient in terms of resources (essentially, quantum cost, number of qubits, and T-count) than the described in the literature. On the other hand, valuable information about available converters and, what is more, quantum adders, is summarized in tables for interested researchers. The converters have been measured using robust metrics and have been compared with the state-of-the-art circuits. The code to build them in a real quantum computer is given

Quantum annealing solution for the unrelated parallel machine scheduling with priorities and delay of task switching on machines

Quantum computing has emerged in recent years as an alternative to classical computing, which could improve the latter in solving some types of problems. One of the quantum programming models, Adiabatic Quantum Computing, has been successfully used to solve problems such as graph partitioning, traffic routing, and task scheduling. In this paper, the focus is on the scheduling of the problem of unrelated parallel machines, where the processing time of tasks on any of the available processing elements is known. Moreover, the proposed model is extended in two relevant aspects for this kind of problem: the existence of some degree of priority of tasks, and the introduction of a delay or penalty every time a processing unit or machine changes the type of task that executes. In all cases, the problem is expressed as Quadratic Unconstrained Binary Optimization, which can be subsequently solved using quantum annealers. The quantum nonlinear programming framework discussed in this work consists of three steps: quadratic approximation of cost function, a binary representation of parameter space, and solving the resulting Quadratic Unconstrained Binary Optimization on the quantum annealer platform D-Wave. One of the novelties in tackling this problem is the compaction of the model bearing in mind the repetitions of each task, to allow solving larger scheduling problems with the quantum resources available in the experimentation platform. An estimation of the number of qubits required in relation to the scheduling parameters is analyzed. The models have been implemented on the D-Wave platform and validated with respect to other traditional methods. Furthermore, the proposed extensions to consider priorities and to switch the delay of tasks have been analyzed using a case study

Finite size effects in active microrheology in colloids

Active microrheology has emerged in recent years as a new technique to probe microscopically the mechanical properties of materials, particularly, viscoelastic ones. In this technique, a colloidal tracer is pulled through the material, and its dynamics is monitored. The interpretation of results usually relies on the Stokes–Einstein approximation, which is valid for a continuous medium in equilibrium. In this work, we have studied with simulations a suspension of quasi-hard colloidal spheres, where a large tracer is pulled by a constant force. The Navier–Stokes equation for a continuous bath predicts important finite size effects, decaying as the inverse box size, which require simulations of different systems to extract the microviscosity of a bulk system. A strategy to optimize the scheduling of the simulation tasks on a multi GPU–CPU cluster based on the adaptation of a genetic algorithm is presented here, and used to study the effect of different conditions on the friction experienced by the tracer (adding the tracer volume to the total system volume, fixing the center of mass of the system, varying the fluid friction coefficient and tracer size). It is observed that the theoretical prediction is not followed, but deviations are observed for large systems in all cases. These are attributed to the finite size of the bath particles, and the intrinsic dynamics of colloidal systems, as shown by the analysis of the velocity profile in the bath

Implementation of three efficient 4‑digit fault‑tolerant quantum carry lookahead adders

Adders are one of the most interesting circuits in quantum computing due to their use in major algorithms that benefit from the special characteristics of this type of computation. Among these algorithms, Shor’s algorithm stands out, which allows decomposing numbers in a time exponentially lower than the time needed to do it with classical computation. In this work, we propose three fault-tolerant carry lookahead adders that improve the cost in terms of quantum gates and qubits with respect to the rest of quantum circuits available in the literature. Their optimal implementation in a real quantum computer is also presented. Finally, the work ends with a rigorous comparison where the advantages and disadvantages of the proposed circuits against the rest of the circuits of the state of the art are exposed. Moreover, the information obtained from such a comparison is summarized in tables that allow a quick consultation to interested researchers

A quantum circuit to generate random numbers within a specific interval

Random numbers are of vital importance in fields such as cyptography and scientific simulations. However, it is well known how difficult it is for classical computers to generate random numbers. This is not the case for quantum computers, which are able to genuinely generate random numbers thanks to the property of superposition and their counter-intuitive concept of measurement. However, despite the simplicity of designing a circuit that generates a random number between 0 and 2^N-1 (being N the number of available qubits), designing a quantum circuit to generate a number within a specific interval is far from trivial. This paper proposes a customizable circuit design to generate random numbers. The circuit is non- hardware dependent, it allows fault-tolerance, and it can be used by current quantum devices. Therefore, it is a valuable tool for all those quantum applications and algorithms that need to work with random numbers. Moreover, a comparator circuit has also been designed as part of this work. This comparator is the best currently available in the literature in terms of qubits, T-count, and T-depth. It is therefore a useful tool for any other circuit or algorithm where this operation is needed

A review on reversible quantum adders

Reversible adders are essential circuits in quantum computing systems. They are a fundamental part of the algorithms implemented for such systems, where Shor's celebrated factoring algorithm is one of the most prominent examples in which reversible arithmetic is needed. There is a wide variety of works in the existing literature which tackle the design of an adder for quantum systems, and today there is still a great interest in the creation of new designs and the perfection of the existing ones. Similar to how it happens in classical digital systems, there are different methodologies to approach the addition using reversible circuits. Some methodologies focus on minimizing the necessary resources, others on optimizing computing time, etc. In this work we analyze the reversible adders in the state-of-the-art for quantum computing, classifying them according to their type, and finally, comparing each other using referenced and validated metrics that allow highlighting the strengths and weaknesses of each adder

Fault‑tolerant quantum algorithm for dual‑threshold image segmentation

The intrinsic high parallelism and entanglement characteristics of quantum computing have made quantum image processing techniques a focus of great interest. One of the most widely used techniques in image processing is segmentation, which in one of their most basic forms can be carried out using thresholding algorithms. In this paper, a fault-tolerant quantum dual-threshold algorithm has been proposed. This algorithm has been built using only Clifford+T gates for compatibility with error detection and correction codes. Because fault-tolerant implementation of T gates has a much higher cost than other quantum gates, our focus has been on reducing the number of these gates. This has allowed adding noise tolerance, computational cost reduction, and fault tolerance to the state-of-the-art dual-threshold segmentation circuits. Since the dual-threshold image segmentation involves the comparison operation, as part of this work we have implemented two full comparator circuits. These circuits optimize the metrics T-count and T-depth with respect to the best circuit comparators currently available in the literature

Dynamics and friction of a large colloidal particle in a bath of hard spheres: Langevin dynamics simulations and hydrodynamic description

The analysis of the dynamics of tracer particles in a complex bath can provide valuable information about the microscopic behavior of the bath. In this work, we study the dynamics of a forced tracer in a colloidal bath by means of Langevin dynamics simulations and a theory model within continuum mechanics. In the simulations, the bath is comprised of quasihard spheres with a volume fraction of 50 % immersed in a featureless quiescent solvent, and the tracer is pulled with a constant small force (within the linear regime). The theoretical analysis is based on the Navier-Stokes equation, where a term proportional to the velocity arises from coarse-graining the friction of the colloidal particles with the solvent. As a result, the final equation is similar to the Brinkman model, although the interpretation is different. A length scale appears in the model, k − 1 0 , where the transverse momentum transport crosses over to friction with the solvent. The effective friction coefficient experienced by the tracer grows with the tracer size faster than the prediction from Stokes's law. Additionally, the velocity profiles in the bath decay faster than in a Newtonian fluid. The comparison between simulations and theory points to a boundary condition of effective partial slip at the tracer surface. We also study the fluctuations in the tracer position, showing that it reaches diffusion at long times, with a subdiffusive regime at intermediate times. The diffusion coefficient, obtained from the long-time slope of the mean-squared displacement, fulfills the Stokes-Einstein relation with the friction coefficient calculated from the steady tracer velocity, confirming the validity of the linear response formalism

On solving the unrelated parallel machine scheduling problem: active microrheology as a case study

Modern computational platforms are characterized by the heterogeneity of their processing elements. Additionally, there are many algorithms which can be structured as a set of procedures or tasks with different computational cost. Balancing the computational load among the available processing elements is one of the main keys for the optimal exploitation of such heterogeneous platforms. When the processing time of any procedure executed on any of the available processing elements is known, this workload-balancing problem can be modeled as the well-known scheduling on unrelated parallel machines problem. Solving this type of problems is a big challenge due to the high heterogeneity on both, the tasks and the machines. In this paper, the balancing problem has been formally defined as a global optimization problem which minimizes the makespan (parallel runtime) and a heuristic based on a Genetic Algorithm, called Genetic Scheduler (GenS), has been developed to solve it. In order to analyze the behavior of GenS for several heterogeneous clusters, an example taken from the field of statistical mechanics has been considered as a case study: an active microrheology model. Given this type of problem and a heterogeneous cluster, we seek to minimize the total runtime to extend and analyze in depth the case of study. In such context, a task consists of the simulation of a tracer particle pulled into a cubic box with smaller bath particles. The computational load depends on the total number of the bath particles. Moreover, GenS has been compared to other dynamic and static scheduling approaches. The experimental results of such a comparison show that GenS outperforms the rest of the tested alternatives achieving a better distribution of the computational workload on a heterogeneous cluster. So, the scheduling strategy developed in this paper is of potential interest for any application which requires the execution of many tasks of different duration (a priori known) on a heterogeneous cluster