scholarly journals Factoring using 2n+2 qubits with Toffoli based modular multiplication

2017 ◽  
Vol 17 (7&8) ◽  
pp. 673-684
Author(s):  
Thomas Haner ◽  
Martin Roetteler ◽  
Krysta M. Svore
Keyword(s):  
Hardware Implementation ◽  
Quantum Algorithm ◽  
Modular Multiplication ◽  
Time Costs ◽  
Space And Time ◽  
Gate Count ◽  
Circuit Depth ◽  
The Cost ◽  
Shor’S Quantum Algorithm

We describe an implementation of Shor’s quantum algorithm to factor n-bit integers using only 2n+2 qubits. In contrast to previous space-optimized implementations, ours features a purely Toffoli based modular multiplication circuit. The circuit depth and the overall gate count are in O(n 3 ) and O(n 3 log n), respectively. We thus achieve the same space and time costs as Takahashi et al. [1], while using a purely classical modular multiplication circuit. As a consequence, our approach evades most of the cost overheads originating from rotation synthesis and enables testing and localization of some faults in both, the logical level circuit and an actual quantum hardware implementation. Our new (in-place) constant-adder, which is used to construct the modular multiplication circuit, uses only dirty ancilla qubits and features a circuit size and depth in O(n log n) and O(n), respectively.

Faster phase estimation

2014 ◽  
Vol 14 (3&4) ◽  
pp. 306-328
Author(s):  
Krysta M. Svore ◽  
Matthew B. Hastings ◽  
Michael Freedman
Keyword(s):  
Quantum Algorithm ◽  
Quantum Circuit ◽  
Quantum Phase ◽  
Phase Estimation ◽  
Post Processing ◽  
Minimum Number ◽  
Circuit Depth ◽  
Lower Depth ◽  
The Cost ◽  
Random Measurements

We develop several algorithms for performing quantum phase estimation based on basic measurements and classical post-processing. We present a pedagogical review of quantum phase estimation and simulate the algorithm to numerically determine its scaling in circuit depth and width. We show that the use of purely random measurements requires a number of measurements that is optimal up to constant factors, albeit at the cost of exponential classical post-processing; the method can also be used to improve classical signal processing. We then develop a quantum algorithm for phase estimation that yields an asymptotic improvement in runtime, coming within a factor of $\log^*$ of the minimum number of measurements required while still requiring only minimal classical post-processing. The corresponding quantum circuit requires asymptotically lower depth and width (number of qubits) than quantum phase estimation.

scholarly journals A Systematic Hardware Sharing Method for Unified Architecture Design of H.264 Transforms

2015 ◽  
Vol 2015 ◽  
pp. 1-14
Author(s):  
Po-Hung Chen ◽  
Hung-Ming Chen ◽  
Ing-Chao Lin
Keyword(s):  
Hardware Implementation ◽  
Low Cost ◽  
Clock Cycle ◽  
Video Codec ◽  
Hardware Cost ◽  
Compression Efficiency ◽  
Gate Count ◽  
Novel Method ◽  
The Cost ◽  
Hadamard Transforms

Multitransform techniques have been widely used in modern video coding and have better compression efficiency than the single transform technique that is used conventionally. However, every transform needs a corresponding hardware implementation, which results in a high hardware cost for multiple transforms. A novel method that includes a five-step operation sharing synthesis and architecture-unification techniques is proposed to systematically share the hardware and reduce the cost of multitransform coding. In order to demonstrate the effectiveness of the method, a unified architecture is designed using the method for all of the six transforms involved in the H.264 video codec: 2D 4 × 4 forward and inverse integer transforms, 2D 4 × 4 and 2 × 2 Hadamard transforms, and 1D 8 × 8 forward and inverse integer transforms. Firstly, the six H.264 transform architectures are designed at a low cost using the proposed five-step operation sharing synthesis technique. Secondly, the proposed architecture-unification technique further unifies these six transform architectures into a low cost hardware-unified architecture. The unified architecture requires only 28 adders, 16 subtractors, 40 shifters, and a proposed mux-based routing network, and the gate count is only 16308. The unified architecture processes 8 pixels/clock-cycle, up to 275 MHz, which is equal to 707 Full-HD 1080 p frames/second.

scholarly journals Constant-optimized quantum circuits for modular multiplication and exponentiation

2012 ◽  
Vol 12 (5&6) ◽  
pp. 361-394
Author(s):  
Igor L. Markov ◽  
Mehdi Saeedi
Keyword(s):  
Laboratory Experiments ◽  
Constant Factor ◽  
Quantum Circuits ◽  
Modular Exponentiation ◽  
Modular Multiplication ◽  
Worst Case ◽  
Gate Count ◽  
Shor's Algorithm ◽  
Circuit Depth ◽  
Additive Term

Reversible circuits for modular multiplication $Cx\%M$ with $x<M$ arise as components of modular exponentiation in Shor's quantum number-factoring algorithm. However, existing generic constructions focus on asymptotic gate count and circuit depth rather than actual values, producing fairly large circuits not optimized for specific $C$ and $M$ values. In this work, we develop such optimizations in a bottom-up fashion, starting with most convenient $C$ values. When zero-initialized ancilla registers are available, we reduce the search for compact circuits to a shortest-path problem. Some of our modular-multiplication circuits are asymptotically smaller than previous constructions, but worst-case bounds and average sizes remain $\Theta(n^2)$. In the context of modular exponentiation, we offer several constant-factor improvements, as well as an improvement by a constant additive term that is significant for few-qubit circuits arising in ongoing laboratory experiments with Shor's algorithm.

scholarly journals A Short Path Quantum Algorithm for Exact Optimization

Quantum ◽  
2018 ◽  
Vol 2 ◽  
pp. 78 ◽  
Author(s):  
M. B. Hastings
Keyword(s):  
Hybrid Algorithm ◽  
Optimal Solution ◽  
Quantum Algorithm ◽  
Low Energy ◽  
Weighted Sum ◽  
Natural Hybrid ◽  
Sum Of Products ◽  
Time Required ◽  
The Cost ◽  
Exact Optimization

We give a quantum algorithm to exactly solve certain problems in combinatorial optimization, including weighted MAX-2-SAT as well as problems where the objective function is a weighted sum of products of Ising variables, all terms of the same degree D; this problem is called weighted MAX-ED-LIN2. We require that the optimal solution be unique for odd D and doubly degenerate for even D; however, we expect that the algorithm still works without this condition and we show how to reduce to the case without this assumption at the cost of an additional overhead. While the time required is still exponential, the algorithm provably outperforms Grover's algorithm assuming a mild condition on the number of low energy states of the target Hamiltonian. The detailed analysis of the runtime dependence on a tradeoff between the number of such states and algorithm speed: fewer such states allows a greater speedup. This leads to a natural hybrid algorithm that finds either an exact or approximate solution.

scholarly journals Revisiting Sum of Residues Modular Multiplication

2010 ◽  
Vol 2010 ◽  
pp. 1-9 ◽  
Author(s):  
Yinan Kong ◽  
Braden Phillips
Keyword(s):  
Public Key Cryptography ◽  
Number System ◽  
Residue Number System ◽  
The Other ◽  
Public Key ◽  
Modular Multiplication ◽  
Residue Number ◽  
The Cost

In the 1980s, when the introduction of public key cryptography spurred interest in modular multiplication, many implementations performed modular multiplication using a sum of residues. As the field matured, sum of residues modular multiplication lost favor to the extent that all recent surveys have either overlooked it or incorporated it within a larger class of reduction algorithms. In this paper, we present a new taxonomy of modular multiplication algorithms. We include sum of residues as one of four classes and argue why it should be considered different to the other, now more common, algorithms. We then apply techniques developed for other algorithms to reinvigorate sum of residues modular multiplication. We compare FPGA implementations of modular multiplication up to 24 bits wide. The sum of residues multipliers demonstrate reduced latency at nearly 50% compared to Montgomery architectures at the cost of nearly doubled circuit area. The new multipliers are useful for systems based on the Residue Number System (RNS).

APPLICATION OF THE JUMPING FROGS METHOD FOR THE OPTIMIZATION OF THREE-LEVEL PLANS OF A MULTIPLE FACTOR EXPERIMENT

2019 ◽  
pp. 35-42
Author(s):  
N. Koshevoy ◽  
E. Kostenko ◽  
V. Muratov
Keyword(s):  
Mathematical Model ◽  
Mathematical Models ◽  
High Speed ◽  
Minimum Cost ◽  
Optimization Method ◽  
Experimental Designs ◽  
Minimal Cost ◽  
Taboo Search ◽  
Time Costs ◽  
The Cost

he planning of the experiment allows us to solve the problem of obtaining a mathematical model with minimal cost and time costs. The cost of implementing an experiment is significantly affected by the order of alternating levels of change in factors. Thus, it is required to find a procedure for the implementation of experiments that provides the minimum cost (time) for conducting a multivariate experiment. This task becomes especially relevant when studying long and expensive processes. The purpose of this article is the further development of the methodology of optimal planning of the experiment in terms of cost (time), which includes a set of methods for optimizing the plans of the experiment and hardware and software for their implementation. Object of study: optimization processes for the cost of three-level plans for multivariate experiments. Subject of research: optimization method for cost and time costs of experimental designs based on the use of the jumping frog method. Experimental research methods are widely used to optimize production processes. One of the main goals of the experiment is to obtain the maximum amount of information about the influence of the studied factors on the production process. Next, a mathematical model of the object under study is built. Moreover, it is necessary to obtain these models at the minimum cost and time costs. The design of the experiment allows you to get mathematical models with minimal cost and time costs. For this, a method and software were developed for optimizing three-level plans using the jumping frog method. Three-level plans are used in the construction of mathematical models of the studied objects and systems. An analysis is made of the known methods for the synthesis of three-level plans that are optimal in cost and time costs. The operability of the algorithm was tested when studying the roughness of the silicon surface during deep plasma-chemical etching of MEMS elements. Its effectiveness is shown in comparison with the following methods: swarm of particles, taboo search, branches and borders. Using the developed method and software for optimizing three-level plans using the jumping frog method, one can achieve high winnings compared to the initial experimental plan, optimal or close to optimal results compared to particle swarm, taboo search, branches and borders methods, and also high speed of solving the optimization problem in comparison with previously developed optimization methods for three-level experimental designs.

scholarly journals PERAN FACEBOOK DALAM KAMPANYE POLITIK PEMILUKADA DI KABUPATEN OGAN KEMIRING ULU

2018 ◽  
Vol 2 (1) ◽  
pp. 42-50
Author(s):  
Hendra Alfani
Keyword(s):  
Social Media ◽  
General Election ◽  
Access To Information ◽  
Political Campaign ◽  
Media Technology ◽  
Space And Time ◽  
Limited Space ◽  
New Space ◽  
Television Media ◽  
The Cost

Facebook, one of social media, becomes a new space for everyone to interact and socialize without being limited by space and time, like in conventional media (mainstream). In its development, social media is also used to support political campaign in general election of regional head (Pemilukada). In addition to reaching the audience without being limited space and time, the cost is also cheap and efficient. Facebook has been chosen by candidates for regional heads as a medium of communication and socialization, not just relying on television media, advertisements in newspapers, banners, billboards, and other media. Candidates also take advantage of social media technology through facebook with the aim to provide access to information for public through cyberspace. The use of facebook is also done by the pair of candidates in the regional head election of Ogan Komering Ulu (OKU) South Sumatra Province in 2015. This study found the use of facebook is quite significant by both pairs of candidates in the election OKU in order socialization and political campaign.

ORGANIZATION OF WORK OF THE ADMISSION COMMISSION OF THE INSTITUTION OF HIGHER EDUCATION IN THE SYSTEM OF ELECTRONIC ADMINISTRATION AND WAYS TO INCREASE ITS PRODUCTIVITY

2017 ◽  
Author(s):  
Tetyana Kaminska ◽  
◽  
Roman Ovcharyk ◽  
Igor Okhrimenko ◽  
◽  
...  
Keyword(s):  
Higher Education ◽  
Computer Programs ◽  
Organization Of Work ◽  
Time Costs ◽  
Cost Of Time ◽  
The Cost ◽  
Electronic Administration

On the basis of comparison of applied computer programs of work of the admissions committee of the HEI, the main possibilities of them are grouped. On the basis of arm "KKIBP Admission Commission" the main indicators of productivity are determined, the cost of time costs is calculated, appropriate conclusions are formed.

Space-efficient quantum multiplication polynomials for binary finite fields with sub-quadratoc Toffoli gate count

2020 ◽  
Vol 20 (9&10) ◽  
pp. 721-735
Author(s):  
Iggy van Hoof
Keyword(s):  
Finite Fields ◽  
Irreducible Polynomial ◽  
Quadratic Number ◽  
Cnot Gate ◽  
Toffoli Gate ◽  
Essential Step ◽  
Gate Count ◽  
Toffoli Gates ◽  
The Cost

Multiplication is an essential step in a lot of calculations. In this paper we look at multiplication of 2 binary polynomials of degree at most n-1, modulo an irreducible polynomial of degree n with 2n input and n output qubits, without ancillary qubits, assuming no errors. With straightforward schoolbook methods this would result in a quadratic number of Toffoli gates and a linear number of CNOT gates. This paper introduces a new algorithm that uses the same space, but by utilizing space-efficient variants of Karatsuba multiplication methods it requires only O(n^{\log_2(3)}) Toffoli gates at the cost of a higher CNOT gate count: theoretically up to O(n^2) but in examples the CNOT gate count looks a lot better.

Sign in / Sign up

Export Citation Format

Share Document