scholarly journals Quantum Modular Adder over GF(2n − 1) without Saving the Final Carry

2021 ◽  
Vol 11 (7) ◽  
pp. 2949
Author(s):  
Aeyoung Kim ◽  
Seong-Min Cho ◽  
Chang-Bae Seo ◽  
Sokjoon Lee ◽  
Seung-Hyun Seo
Keyword(s):  
Quantum Computing ◽  
Finite Field ◽  
Design Principle ◽  
Quantum Circuit ◽  
Galois Field ◽  
Basic Operation ◽  
Minimum Information ◽  
Information Unit ◽  
Number Systems

Addition is the most basic operation of computing based on a bit system. There are various addition algorithms considering multiple number systems and hardware, and studies for a more efficient addition are still ongoing. Quantum computing based on qubits as the information unit asks for the design of a new addition because it is, physically, wholly different from the existing frequency-based computing in which the minimum information unit is a bit. In this paper, we propose an efficient quantum circuit of modular addition, which reduces the number of gates and the depth. The proposed modular addition is for the Galois Field GF(2n−1), which is important as a finite field basis in various domains, such as cryptography. Its design principle was from the ripple carry addition (RCA) algorithm, which is the most widely used in existing computers. However, unlike conventional RCA, the storage of the final carry is not needed due to modifying existing diminished-1 modulo 2n−1 adders. Our proposed adder can produce modulo sum within the range 0,2n−2 by fewer qubits and less depth. For comparison, we analyzed the proposed quantum addition circuit over GF(2n−1) and the previous quantum modular addition circuit for the performance of the number of qubits, the number of gates, and the depth, and simulated it with IBM’s simulator ProjectQ.

Quantum computing and polynomial equations over Z_2

2005 ◽  
Vol 5 (2) ◽  
pp. 102-112
Author(s):  
C.M. Dawson ◽  
H.L. Haselgrove ◽  
A.P. Hines ◽  
D. Mortimer ◽  
M.A. Nielsen ◽  
...  
Keyword(s):  
Quantum Computing ◽  
Finite Field ◽  
Quantum Computation ◽  
Quantum Computer ◽  
Complexity Classes ◽  
Polynomial Equations ◽  
Computational Power ◽  
Number Of Solutions

What is the computational power of a quantum computer? We show that determining the output of a quantum computation is equivalent to counting the number of solutions to an easily computed set of polynomials defined over the finite field Z_2. This connection allows simple proofs to be given for two known relationships between quantum and classical complexity classes, namely BQP/P/\#P and BQP/PP.

Quantum CFD Simulations for Heat Transfer Applications

2020 ◽  
Author(s):  
Chao Lu ◽  
Zhao Hu ◽  
Bei Xie ◽  
Ning Zhang
Keyword(s):  
Heat Transfer ◽  
Quantum Computing ◽  
Linear System ◽  
Linear Equation ◽  
Linear Systems ◽  
Quantum Algorithm ◽  
Quantum Circuit ◽  
Quantum Phase ◽  
Cfd Simulations ◽  
Analytical Result

Abstract In this paper, computational heat transfer (CHT) equations were solved using the state-of-art quantum computing (QC) technology. The CHT equations can be discretized into a linear equation set, which can be possibly solved by a QC system. The linear system can be characterized by Ax = b. The A matrix in this linear system is a Hermitian matrix. The linear system is then solved by using the HHL algorithm, which is a quantum algorithm to solve a linear system. The quantum circuit requires an Ancilla qubit, clock qubits, qubits for b and a classical bit to record the result. The process of the HHL algorithm can be described as follows. Firstly, the qubit for b is initialized into the phase as desire. Secondly, the quantum phase estimation (QPE) is used to determine the eigenvalues of A and the eigenvalues are stored in clock qubits. Thirdly, a Rotation gate is used to rotate the inversion of eigenvalues and information is passed to the Ancilla bit to do Pauli Y-rotation operation. Fourthly, revert the whole processes to untangle qubits and measure all of the qubits to output the final results for x. From the existing literature, a few 2 × 2 matrices were successfully solved with QC technology, proving the possibility of QC on linear systems [1]. In this paper, a quantum circuit is designed to solve a CHT problem. A simple 2 by 2 linear equation is modeled for the CHT problem and is solved by using the quantum computing. The result is compared with the analytical result. This result could initiate future studies on determining the quantum phase parameters for more complicated QC linear systems for CHT applications.

A novel quantum computing model based on entanglement degree

2020 ◽  
Vol 34 (35) ◽  
pp. 2050401
Author(s):  
Mohammed Zidan
Keyword(s):  
Quantum Computing ◽  
Quantum Circuit ◽  
Circuit Model ◽  
Computing Model ◽  
Model Based ◽  
Proposed Model ◽  
Degree Of Entanglement

This paper shows a novel quantum computing model that solves quantum computing problems based on the degree of entanglement. We show two main theorems: the first theorem shows the quantum circuit that can be used to quantify the concurrence value between two adjacent qubits. The second theorem shows the quantum circuit of a proposed operator, called [Formula: see text] operator, which can be used to differentiate between the non-orthogonal states in the form [Formula: see text], with arbitrary accuracy, using the concurrence value. Then, the mathematical machinery for implementing the proposed model and its techniques using the circuit model is investigated extensively.

QE rings in characteristic pn

1983 ◽  
Vol 48 (1) ◽  
pp. 140-162 ◽  
Author(s):  
Chantal Berline ◽  
Gregory Cherlin
Keyword(s):  
Finite Field ◽  
Jacobson Radical ◽  
Prime Power ◽  
Galois Field ◽  
Power Characteristic ◽  
Witt Ring ◽  
Boolean Power ◽  
Three Components

AbstractWe show that all QE rings of prime power characteristic are constructed in a straightforward way out of three components: a filtered Boolean power of a finite field, a nilpotent Jacobson radical, and the ring Zp. or the Witt ring W2(F4) (which is the characteristic four analogue of the Galois field with four elements).

Machine invention of quantum computing circuits by means of genetic programming

2008 ◽  
Vol 22 (3) ◽  
pp. 275-283 ◽  
Author(s):  
Lee Spector ◽  
Jon Klein
Keyword(s):  
Quantum Computing ◽  
Genetic Programming ◽  
Quantum Computer ◽  
Quantum Circuit ◽  
Quantum Circuits ◽  
Practical Significance ◽  
Programming System ◽  
Computer Simulator ◽  
Classical Quantum ◽  
Better Than

AbstractWe demonstrate the use of genetic programming in the automatic invention of quantum computing circuits that solve problems of potential theoretical and practical significance. We outline a developmental genetic programming scheme for such applications; in this scheme the evolved programs, when executed, build quantum circuits and the resulting quantum circuits are then tested for “fitness” using a quantum computer simulator. Using the PushGP genetic programming system and the QGAME quantum computer simulator we demonstrate the invention of a new, better than classical quantum circuit for the two-oracle AND/OR problem.

scholarly journals Efficient implementation of quantum circuits with limited qubit interactions

2017 ◽  
Vol 17 (13&14) ◽  
pp. 1096-1104
Author(s):  
Stephen Brierley
Keyword(s):  
Quantum Computing ◽  
Efficient Method ◽  
Quantum Circuit ◽  
Efficient Implementation ◽  
Circuit Model ◽  
Quantum Circuits ◽  
Interaction Graph ◽  
Time Overhead

The quantum circuit model allows gates between any pair of qubits yet physical instantiations allow only limited interactions. We address this problem by providing an interaction graph together with an efficient method for compiling quantum circuits so that gates are applied only locally. The graph requires each qubit to interact with 4 other qubits and yet the time-overhead for implementing any n-qubit quantum circuit is 4 log n. Building a network of quantum computing nodes according to this graph enables the network to emulate a single monolithic device with minimal overhead.

Analyzing Heuristic-based Randomized Search Strategies for the Quantum Circuit Compilation Problem

2020 ◽  
Vol 174 (3-4) ◽  
pp. 259-281
Author(s):  
Angelo Oddi ◽  
Riccardo Rasconi
Keyword(s):  
Quantum Computing ◽  
Nearest Neighbor ◽  
Quantum Algorithm ◽  
Quantum Circuit ◽  
Quantum Circuits ◽  
Quantum Gates ◽  
Ranking Functions ◽  
Solution Quality ◽  
Circuit Realization ◽  
Definition Of

In this work we investigate the performance of greedy randomised search (GRS) techniques to the problem of compiling quantum circuits to emerging quantum hardware. Quantum computing (QC) represents the next big step towards power consumption minimisation and CPU speed boost in the future of computing machines. Quantum computing uses quantum gates that manipulate multi-valued bits (qubits). A quantum circuit is composed of a number of qubits and a series of quantum gates that operate on those qubits, and whose execution realises a specific quantum algorithm. Current quantum computing technologies limit the qubit interaction distance allowing the execution of gates between adjacent qubits only. This has opened the way to the exploration of possible techniques aimed at guaranteeing nearest-neighbor (NN) compliance in any quantum circuit through the addition of a number of so-called swap gates between adjacent qubits. In addition, technological limitations (decoherence effect) impose that the overall duration (makespan) of the quantum circuit realization be minimized. One core contribution of the paper is the definition of two lexicographic ranking functions for quantum gate selection, using two keys: one key acts as a global closure metric to minimise the solution makespan; the second one is a local metric, which favours the mutual approach of the closest qstates pairs. We present a GRS procedure that synthesises NN-compliant quantum circuits realizations, starting from a set of benchmark instances of different size belonging to the Quantum Approximate Optimization Algorithm (QAOA) class tailored for the MaxCut problem. We propose a comparison between the presented meta-heuristics and the approaches used in the recent literature against the same benchmarks, both from the CPU efficiency and from the solution quality standpoint. In particular, we compare our approach against a reference benchmark initially proposed and subsequently expanded in [1] by considering: (i) variable qubit state initialisation and (ii) crosstalk constraints that further restrict parallel gate execution.

Improved quantum ternary arithmetic

2016 ◽  
Vol 16 (9&10) ◽  
pp. 862-884
Author(s):  
Alex Bocharov ◽  
Shawn X. Cui ◽  
Martin Roetteler ◽  
Krysta M. Svore
Keyword(s):  
Quantum Computing ◽  
Finite Field ◽  
Quantum Systems ◽  
Arithmetic Circuits ◽  
Levels Of Abstraction ◽  
Circuit Depth

Qutrit (or ternary) structures arise naturally in many quantum systems, notably in certain non-abelian anyon systems. We present efficient circuits for ternary reversible and quantum arithmetics. Our main result is the derivation of circuits for two families of ternary quantum adders. The main distinction from the binary adders is a richer ternary carry which leads potentially to higher resource counts in universal ternary bases. Our ternary ripple adder circuit has a circuit depth of O(n) and uses only 1 ancilla, making it more efficient in both, circuit depth and width, when compared with previous constructions. Our ternary carry lookahead circuit has a circuit depth of only O(log n), while using O(n) ancillas. Our approach works on two levels of abstraction: at the first level, descriptions of arithmetic circuits are given in terms of gates sequences that use various types of non-Clifford reflections. At the second level, we break down these reflections further by deriving them either from the two-qutrit Clifford gates and the non-Clifford gate C(X) : |i, ji 7→ |i, j + δi,2 mod 3i or from the two-qutrit Clifford gates and the non-Clifford gate P9 = diag(e −2π i/9 , 1, e 2π i/9 ). The two choices of elementary gate sets correspond to two possible mappings onto two different prospective quantum computing architectures which we call the metaplectic and the supermetaplectic basis, respectively. Finally, we develop a method to factor diagonal unitaries using multi-variate polynomials over the ternary finite field which allows to characterize classes of gates that can be implemented exactly over the supermetaplectic basis.

scholarly journals Construction and Determination of Irreducible Polynomials in Galois elds, GF(2m)

2019 ◽  
pp. 1-6
Author(s):  
Abraham Aidoo ◽  
Kwasi Baah Gyam ◽  
Fengfan Yang
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Monic Polynomial ◽  
Galois Field ◽  
Irreducible Polynomials ◽  
Primitive Polynomials

This work is about Construction of Irreducible Polynomials in Finite fields. We defined some terms in the Galois field that led us to the construction of the polynomials in the GF(2m). We discussed the following in the text; irreducible polynomials, monic polynomial, primitive polynomials, eld, Galois eld or nite elds, and the order of a finite field. We found all the polynomials in $$F_2[x]$$ that is, $$P(x) =\sum_{i=1}^m a_ix^i : a_i \in F_2$$ with $$a_m \neq 0$$ for some degree $m$ whichled us to determine the number of irreducible polynomials generally at any degree in $$F_2[x]$$.

scholarly journals Construction of Irreducible Polynomials in Galois elds, GF(2m) Using Normal Bases

2019 ◽  
pp. 1-15
Author(s):  
Abraham Aidoo ◽  
Kwasi Baah Gyam
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
General Rule ◽  
Galois Field ◽  
Irreducible Polynomials ◽  
Galois Fields ◽  
Normal Bases ◽  
Primitive Polynomials

This thesis is about Construction of Polynomials in Galois fields Using Normal Bases in finite fields. In this piece of work, we discussed the following in the text; irreducible polynomials, primitive polynomials, field, Galois field or finite fields, and the order of a finite field. We found the actual construction of polynomials in GF(2m) with degree less than or equal to m − 1 and also illustrated how this construction can be done using normal bases. Finally, we found the general rule for construction of GF(pm) using normal bases and even the rule for producing reducible polynomials.

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