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.

scholarly journals An Edge-Sense Bidirectional Pyramid Network for Stereo Matching of VHR Remote Sensing Images

2020 ◽  
Vol 12 (24) ◽  
pp. 4025
Author(s):  
Rongshu Tao ◽  
Yuming Xiang ◽  
Hongjian You
Keyword(s):  
Remote Sensing ◽  
Stereo Matching ◽  
Tall Buildings ◽  
Disparity Estimation ◽  
Complex Structures ◽  
Learning Networks ◽  
Remote Sensing Images ◽  
Essential Step ◽  
Disparity Range ◽  
The Cost

As an essential step in 3D reconstruction, stereo matching still faces unignorable problems due to the high resolution and complex structures of remote sensing images. Especially in occluded areas of tall buildings and textureless areas of waters and woods, precise disparity estimation has become a difficult but important task. In this paper, we develop a novel edge-sense bidirectional pyramid stereo matching network to solve the aforementioned problems. The cost volume is constructed from negative to positive disparities since the disparity range in remote sensing images varies greatly and traditional deep learning networks only work well for positive disparities. Then, the occlusion-aware maps based on the forward-backward consistency assumption are applied to reduce the influence of the occluded area. Moreover, we design an edge-sense smoothness loss to improve the performance of textureless areas while maintaining the main structure. The proposed network is compared with two baselines. The experimental results show that our proposed method outperforms two methods, DenseMapNet and PSMNet, in terms of averaged endpoint error (EPE) and the fraction of erroneous pixels (D1), and the improvements in occluded and textureless areas are significant.

First description of the algorithms

2011 ◽  
Author(s):  
Jean-Marc Couveignes ◽  
Bas Edixhoven
Keyword(s):  
Finite Fields ◽  
Modular Forms ◽  
Minimal Polynomial ◽  
Galois Representations ◽  
Essential Step ◽  
Sufficient Precision ◽  
Informal Description

This chapter provides the first, informal description of the algorithms. It explains how the computation of the Galois representations V attached to modular forms over finite fields should proceed. The essential step is to approximate the minimal polynomial P of (3.1) with sufficient precision so that P itself can be obtained.

scholarly journals Constructing irreducible polynomials with prescribed level curves over finite fields

2001 ◽  
Vol 27 (4) ◽  
pp. 197-200
Author(s):  
Mihai Caragiu
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Irreducible Polynomial ◽  
Irreducible Polynomials ◽  
Level Curves ◽  
Curves Over Finite Fields ◽  
Irreducibility Criterion ◽  
Absolutely Irreducible Polynomial

We use Eisenstein's irreducibility criterion to prove that there exists an absolutely irreducible polynomialP(X,Y)∈GF(q)[X,Y]with coefficients in the finite fieldGF(q)withqelements, with prescribed level curvesXc:={(x,y)∈GF(q)2|P(x,y)=c}.

scholarly journals How to factor 2048 bit RSA integers in 8 hours using 20 million noisy qubits

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 433
Author(s):  
Craig Gidney ◽  
Martin Ekerå
Keyword(s):  
Finite Fields ◽  
Large Scale ◽  
Nearest Neighbor ◽  
Quantum Computer ◽  
Superconducting Qubit ◽  
Logical Qubits ◽  
Surface Code ◽  
Neighbor Connectivity ◽  
Approximate Cost ◽  
The Cost

We significantly reduce the cost of factoring integers and computing discrete logarithms in finite fields on a quantum computer by combining techniques from Shor 1994, Griffiths-Niu 1996, Zalka 2006, Fowler 2012, Ekerå-Håstad 2017, Ekerå 2017, Ekerå 2018, Gidney-Fowler 2019, Gidney 2019. We estimate the approximate cost of our construction using plausible physical assumptions for large-scale superconducting qubit platforms: a planar grid of qubits with nearest-neighbor connectivity, a characteristic physical gate error rate of 10−3, a surface code cycle time of 1 microsecond, and a reaction time of 10 microseconds. We account for factors that are normally ignored such as noise, the need to make repeated attempts, and the spacetime layout of the computation. When factoring 2048 bit RSA integers, our construction's spacetime volume is a hundredfold less than comparable estimates from earlier works (Van Meter et al. 2009, Jones et al. 2010, Fowler et al. 2012, Gheorghiu et al. 2019). In the abstract circuit model (which ignores overheads from distillation, routing, and error correction) our construction uses 3n+0.002nlg⁡n logical qubits, 0.3n3+0.0005n3lg⁡n Toffolis, and 500n2+n2lg⁡n measurement depth to factor n-bit RSA integers. We quantify the cryptographic implications of our work, both for RSA and for schemes based on the DLP in finite fields.

scholarly journals Efficient magic state factories with a catalyzed|CCZ⟩to2|T⟩transformation

Quantum ◽  
2019 ◽  
Vol 3 ◽  
pp. 135 ◽  
Author(s):  
Craig Gidney ◽  
Austin G. Fowler
Keyword(s):  
Fault Tolerant ◽  
Code Distance ◽  
Construction Techniques ◽  
Arbitrary Phase ◽  
Shor's Algorithm ◽  
The Mean ◽  
Toffoli Gates ◽  
Surface Code ◽  
The Cost ◽  
Phase Angles

We present magic state factory constructions for producing|CCZ⟩states and|T⟩states. For the|CCZ⟩factory we apply the surface code lattice surgery construction techniques described in \cite{fowler2018} to the fault-tolerant Toffoli \cite{jones2013, eastin2013distilling}. The resulting factory has a footprint of12d×6d(wheredis the code distance) and produces one|CCZ⟩every5.5dsurface code cycles. Our|T⟩state factory uses the|CCZ⟩factory's output and a catalyst|T⟩state to exactly transform one|CCZ⟩state into two|T⟩states. It has a footprint25%smaller than the factory in \cite{fowler2018} but outputs|T⟩states twice as quickly. We show how to generalize the catalyzed transformation to arbitrary phase angles, and note that the caseθ=22.5∘produces a particularly efficient circuit for producing|T⟩states. Compared to using the12d×8d×6.5d|T⟩factory of \cite{fowler2018}, our|CCZ⟩factory can quintuple the speed of algorithms that are dominated by the cost of applying Toffoli gates, including Shor's algorithm \cite{shor1994} and the chemistry algorithm of Babbush et al. \cite{babbush2018}. Assuming a physical gate error rate of10−3, our CCZ factory can produce∼1010states on average before an error occurs. This is sufficient for classically intractable instantiations of the chemistry algorithm, but for more demanding algorithms such as Shor's algorithm the mean number of states until failure can be increased to∼1012by increasing the factory footprint∼20%.

scholarly journals A shared-cube approach to ESOP-based synthesis of reversible logic

2011 ◽  
Vol 24 (3) ◽  
pp. 385-402 ◽  
Author(s):  
Noor Nayeem ◽  
Jacqueline Rice
Keyword(s):  
Power Loss ◽  
Heat Dissipation ◽  
Logic Synthesis ◽  
Reversible Logic ◽  
Experimental Results ◽  
Quantum Cost ◽  
Toffoli Gate ◽  
Gate Count ◽  
Computing Industry ◽  
Synthesis Approach

Reversible logic is being suggested as a possibility for overcoming potential power loss and heat dissipation problems that the computing industry may soon be at a loss to overcome. However, for reversible logic to be a solution we must have techniques for synthesizing function descriptions to reversible circuits. This paper presents an improved ESOP-based reversible logic synthesis approach which leverages situations where cubes are shared by multiple outputs and ensures that the implementation of each cube requires just one Toffoli gate. It has the potential to minimize both gate count and quantum cost, and in fact our experimental results show that this technique can reduce the quantum cost up to 75% compared to results from the existing work.

scholarly journals Quantum Circuit Design of Toom 3-Way Multiplication

2021 ◽  
Vol 11 (9) ◽  
pp. 3752
Author(s):  
Harashta Tatimma Larasati ◽  
Asep Muhamad Awaludin ◽  
Janghyun Ji ◽  
Howon Kim
Keyword(s):  
Circuit Design ◽  
Quantum Circuit ◽  
Lower Number ◽  
Unique Property ◽  
Large Numbers ◽  
Toffoli Gates ◽  
Asymptotic Complexity ◽  
Lower Depth ◽  
The Cost ◽  
Division Operation

In classical computation, Toom–Cook is one of the multiplication methods for large numbers which offers faster execution time compared to other algorithms such as schoolbook and Karatsuba multiplication. For the use in quantum computation, prior work considered the Toom-2.5 variant rather than the classically faster and more prominent Toom-3, primarily to avoid the nontrivial division operations inherent in the latter circuit. In this paper, we investigate the quantum circuit for Toom-3 multiplication, which is expected to give an asymptotically lower depth than the Toom-2.5 circuit. In particular, we designed the corresponding quantum circuit and adopted the sequence proposed by Bodrato to yield a lower number of operations, especially in terms of nontrivial division, which is reduced to only one exact division by 3 circuit per iteration. Moreover, to further minimize the cost of the remaining division, we utilize the unique property of the particular division circuit, replacing it with a constant multiplication by reciprocal circuit and the corresponding swap operations. Our numerical analysis shows that the resulting circuit indeed gives a lower asymptotic complexity in terms of Toffoli depth and qubit count compared to Toom-2.5 but with a large number of Toffoli gates that mainly come from realizing the division operation.

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 The Mathematical Safe Problem and Its Solution (Part 1)

2020 ◽  
pp. 15-38
Author(s):  
S. Kryvyi ◽  
H. Hoherchak
Keyword(s):  
Mathematical Model ◽  
Finite Fields ◽  
Computer Games ◽  
Time Complexity ◽  
Linear Equations ◽  
Irreducible Polynomial ◽  
Finite Rings ◽  
Systems Of Linear Equations ◽  
Residue Fields ◽  
Mathematical Safe

Introduction. The problem of the mathematical safe arises in the theory of computer games and cryptographic applications. The article considers the formulation of the mathematical safe problem and the approach to its solution using systems of linear equations in finite rings and fields. The purpose of the article is to formulate a mathematical model of the mathematical safe problem and its reduction to systems of linear equations in different domains; to consider solving the corresponding systems in finite rings and fields; to consider the principles of constructing extensions of residue fields and solving systems in the relevant areas. Results. The formulation of the mathematical safe problem is given and the way of its reduction to systems of linear equations is considered. Methods and algorithms for solving this type of systems are considered, where exist methods and algorithms for constructing the basis of a set of solutions of linear equations and derivative methods and algorithms for constructing the basis of a set of solutions of systems of linear equations for residue fields, ghost rings, finite rings and finite fields. Examples are given to illustrate their work. The principles of construction of extensions of residue fields by the module of an irreducible polynomial, and examples of operations tables for them are considered. The peculiarities of solving systems of linear equations in such fields are considered separately. All the above algorithms are accompanied by proofs and estimates of their time complexity. Conclusions. The considered methods and algorithms for solving linear equations and systems of linear equations in finite rings and fields allow to solve the problem of a mathematical safe in many variations of its formulation. The second part of the paper will consider the application of these methods and algorithms to solve the problem of mathematical safe in its various variations. Keywords: mathematical safe, finite rings, finite fields, method, algorithm, solution.

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