scholarly journals Demonstration of Shor’s factoring algorithm for N $$=$$ 21 on IBM quantum processors

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Unathi Skosana ◽  
Mark Tame
Keyword(s):  
Necessary Condition ◽  
Phase Estimation ◽  
Proof Of Concept ◽  
Residual Phase ◽  
Shor's Algorithm ◽  
Functional Correctness ◽  
Toffoli Gates ◽  
Previous Demonstration ◽  
Quantum Order ◽  
Factoring Algorithm

AbstractWe report a proof-of-concept demonstration of a quantum order-finding algorithm for factoring the integer 21. Our demonstration involves the use of a compiled version of the quantum phase estimation routine, and builds upon a previous demonstration. We go beyond this work by using a configuration of approximate Toffoli gates with residual phase shifts, which preserves the functional correctness and allows us to achieve a complete factoring of $$N=21$$ N = 21 . We implemented the algorithm on IBM quantum processors using only five qubits and successfully verified the presence of entanglement between the control and work register qubits, which is a necessary condition for the algorithm’s speedup in general. The techniques we employ may be useful in carrying out Shor’s algorithm for larger integers, or other algorithms in systems with a limited number of noisy qubits.

scholarly journals Trust assessment of power system states

2020 ◽  
Vol 3 (S1) ◽  
Author(s):  
Michael Brand ◽  
Davood Babazadeh ◽  
Carsten Krüger ◽  
Björn Siemers ◽  
Sebastian Lehnhoff
Keyword(s):  
Power Systems ◽  
Power System ◽  
Data Acquisition ◽  
State Estimation ◽  
Trust Model ◽  
Proof Of Concept ◽  
Process Data ◽  
Acquisition Process ◽  
Functional Correctness ◽  
Data Acquisition Process

Abstract Modern power systems are cyber-physical systems with increasing relevance and influence of information and communication technology. This influence comprises all processes, functional, and non-functional aspects like functional correctness, safety, security, and reliability. An example of a process is the data acquisition process. Questions focused in this paper are, first, how one can trust in process data in a data acquisition process of a highly-complex cyber-physical power system. Second, how can the trust in process data be integrated into a state estimation to achieve estimated results in a way that it can reflect trustworthiness of that input?We present the concept of an anomaly-sensitive state estimation that tackles these questions. The concept is based on a multi-faceted trust model for power system network assessment. Furthermore, we provide a proof of concept by enriching measurements in the context of the IEEE 39-bus system with reasonable trust values. The proof of concept shows the benefits but also the limitations of the approach.

Quantum circuits for simple periodic functions

2014 ◽  
Vol 14 (9&10) ◽  
pp. 763-776
Author(s):  
Omar Gamel ◽  
Daniel F.V. James
Keyword(s):  
Quantum Computing ◽  
Periodic Function ◽  
Quantum Circuits ◽  
Special Importance ◽  
Periodic Functions ◽  
Single Period ◽  
Shor's Algorithm ◽  
Toffoli Gates ◽  
One To One

Periodic functions are of special importance in quantum computing, particularly in applications of Shor's algorithm. We explore methods of creating circuits for periodic functions to better understand their properties. We introduce a method for constructing the circuit for a simple monoperiodic function, that is one-to-one within a single period, of a given period $p$. We conjecture that to create a simple periodic function of period $p$, where $p$ is an $n$-bit number, one needs at most $n$ Toffoli gates.

scholarly journals Optimising Matrix Product State Simulations of Shor's Algorithm

Quantum ◽  
2019 ◽  
Vol 3 ◽  
pp. 116 ◽  
Author(s):  
Aidan Dang ◽  
Charles D. Hill ◽  
Lloyd C. L. Hollenberg
Keyword(s):  
Dimensional Structure ◽  
Matrix Product ◽  
Product State ◽  
One Dimensional ◽  
Matrix Product State ◽  
Shor's Algorithm ◽  
The One ◽  
High Level ◽  
Space Requirements ◽  
Factoring Algorithm

We detail techniques to optimise high-level classical simulations of Shor's quantum factoring algorithm. Chief among these is to examine the entangling properties of the circuit and to effectively map it across the one-dimensional structure of a matrix product state. Compared to previous approaches whose space requirements depend on r, the solution to the underlying order-finding problem of Shor's algorithm, our approach depends on its factors. We performed a matrix product state simulation of a 60-qubit instance of Shor's algorithm that would otherwise be infeasible to complete without an optimised entanglement mapping.

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%.

Residual‐phase estimation by homomorphic transformation

1989 ◽  
Author(s):  
Abdulmohsin Aldolaijan ◽  
Ken Larner
Keyword(s):  
Phase Estimation ◽  
Residual Phase

scholarly journals Implementation of Shor’s Quantum Factoring Algorithm using ProjectQ Framework

2019 ◽  
Vol 8 (6S3) ◽  
pp. 52-56
Keyword(s):  
Quantum Computing ◽  
Computer Science ◽  
Polynomial Time ◽  
Quantum Computer ◽  
Quantum Algorithm ◽  
Prime Factorization ◽  
Shor's Algorithm ◽  
Computer Simulators ◽  
Testing And Evaluation ◽  
Factoring Algorithm

Prime number factorization is a problem in computer science where the solution to that problem takes super-polynomial time classically. Shor’s quantum factoring algorithm is able to solve the problem in polynomial time by harnessing the power of quantum computing. The implementation of the quantum algorithm itself is not detailed by Shor in his paper. In this paper, an approach and experiment to implement Shor’s quantum factoring algorithm are proposed. The implementation is done using Python and a quantum computer simulator from ProjectQ. The testing and evaluation are completed in two computers with different hardware specifications. User time of the implementation is measured in comparison with other quantum computer simulators: ProjectQ and Quantum Computing Playground. This comparison was done to show the performance of Shor’s algorithm when simulated using different hardware. There is a 33% improvement in the execution time (user time) between the two computers with the accuracy of prime factorization in this implementation is inversely proportional to the number of qubits used. Further improvements upon the program that has been developed for this paper is its accuracy in terms of finding the factors of a number and the number of qubits used, as previously mentioned.

scholarly journals A 2D nearest-neighbor quantum architecture for factoring in polylogarithmic depth

2013 ◽  
Vol 13 (11&12) ◽  
pp. 937-962
Author(s):  
Paul Pham ◽  
Krysta M. Svore
Keyword(s):  
Nearest Neighbor ◽  
Upper Bounds ◽  
Phase Estimation ◽  
Constant Depth ◽  
Shor's Algorithm ◽  
Circuit Depth ◽  
Quantum Architecture ◽  
The Cost ◽  
2D Model

We present a 2D nearest-neighbor quantum architecture for Shor's algorithm to factor an $n$-bit number in $O(\log^3n)$ depth. Our implementation uses parallel phase estimation, constant-depth fanout and teleportation, and constant-depth carry-save modular addition. We derive upper bounds on the circuit resources of our architecture under a new 2D model which allows a classical controller and parallel, communicating modules. We provide a comparison to all previous nearest-neighbor factoring implementations. Our circuit results in an exponential improvement in nearest-neighbor circuit depth at the cost of a polynomial increase in circuit size and width.

scholarly journals Entanglement and its role in Shor's algorithm

2006 ◽  
Vol 6 (7) ◽  
pp. 630-640
Author(s):  
V.M. Kendon ◽  
W.J. Munro
Keyword(s):  
Numerical Simulation ◽  
Fourier Transform ◽  
Quantum Information ◽  
Quantum Information Processing ◽  
Quantum Computer ◽  
Quantum Algorithms ◽  
Quantum Fourier Transform ◽  
Shor's Algorithm ◽  
Critical Resource ◽  
Factoring Algorithm

Entanglement has been termed a critical resource for quantum information processing and is thought to be the reason that certain quantum algorithms, such as Shor's factoring algorithm, can achieve exponentially better performance than their classical counterparts. The nature of this resource is still not fully understood: here we use numerical simulation to investigate how entanglement between register qubits varies as Shor's algorithm is run on a quantum computer. The shifting patterns in the entanglement are found to relate to the choice of basis for the quantum Fourier transform.

scholarly journals Quantum Factoring Algorithm: Resource Estimation and Survey of Experiments

2020 ◽  
pp. 39-55
Author(s):  
Noboru Kunihiro
Keyword(s):  
Large Scale ◽  
Quantum Circuit ◽  
The Other ◽  
Quantum Circuits ◽  
Small Scale ◽  
Quantum Computers ◽  
Resource Estimation ◽  
Shor's Algorithm ◽  
Order Of An Element ◽  
Factoring Algorithm

Abstract It is known that Shor’s algorithm can break many cryptosystems such as RSA encryption, provided that large-scale quantum computers are realized. Thus far, several experiments for the factorization of the small composites such as 15 and 21 have been conducted using small-scale quantum computers. In this study, we investigate the details of quantum circuits used in several factoring experiments. We then indicate that some of the circuits have been constructed under the condition that the order of an element modulo a target composite is known in advance. Because the order must be unknown in the experiments, they are inappropriate for designing the quantum circuit of Shor’s factoring algorithm. We also indicate that the circuits used in the other experiments are constructed by relying considerably on the target composite number to be factorized.

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