# Experimental realization of Shor's quantum factoring algorithm using nuclear magnetic resonance

@article{Vandersypen2001ExperimentalRO, title={Experimental realization of Shor's quantum factoring algorithm using nuclear magnetic resonance}, author={Lieven M. K. Vandersypen and Matthias Steffen and Gregory Breyta and Costantino S. Yannoni and Mark H. Sherwood and Isaac L. Chuang}, journal={Nature}, year={2001}, volume={414}, pages={883-887} }

The number of steps any classical computer requires in order to find the prime factors of an l-digit integer N increases exponentially with l, at least using algorithms known at present. Factoring large integers is therefore conjectured to be intractable classically, an observation underlying the security of widely used cryptographic codes. Quantum computers, however, could factor integers in only polynomial time, using Shor's quantum factoring algorithm. Although important for the study of…

## 1,472 Citations

### Factoring larger integers with fewer qubits via quantum annealing with optimized parameters

- Computer ScienceScience China Physics, Mechanics & Astronomy
- 2019

This study optimize the problem Hamiltonian to reduce the number of qubits involved in the final Hamiltonian while maintaining the QUBO coefficients in a reasonable range, enabling the improved algorithm to factorize larger integers with fewer qubits.

### Exact search algorithm to factorize large biprimes and a triprime on IBM quantum computer

- Computer Science, Physics
- 2018

It has been concluded that the solution to this problem depends on the level of simplification chosen, not the size of the number factored, and in principle, the results can be extended to factorize any multi-prime integer with minimum quantum resources.

### Realization of a scalable Shor algorithm

- Computer ScienceScience
- 2016

The realization of a scalable Shor algorithm, as proposed by Kitaev, is presented, which has been realized scalably within an ion-trap quantum computer and returns the correct factors with a confidence level exceeding 99%.

### Shor’s factoring algorithm and modern cryptography. An illustration of the capabilities inherent in quantum computers

- Computer Science
- 2005

This paper endeavors to explain, in a fashion comprehensible to the nonexpert, the RSA encryption protocol; the various quantum computer manipulations constituting the Shor algorithm; how theShor algorithm performs the factoring; and the precise sense in which a quantum computer employing Shor’s algorithm can be said to accomplish the factored of very large numbers with less computational effort than a classical computer.

### Quantum Fourier Transform and Its Application in Shor’s Algorithm

- Physics
- 2020

Quantum Fourier transform (QFT) plays an eminent role in quantum computation. It creates a superposition of different quantum states, allowing simultaneous calculation, which would take many steps…

### Using Shor's algorithm on near term Quantum computers: a reduced version

- Computer ScienceQuantum Mach. Intell.
- 2022

This work introduces a reduced version of Shor’s algorithm that proposes a step forward in increasing the range of numbers that can be factorized on noisy Quantum devices and finds noteworthy results in most cases.

### Experimental realisation of Shor's quantum factoring algorithm using qubit recycling

- Physics, Computer Science2013 Conference on Lasers & Electro-Optics Europe & International Quantum Electronics Conference CLEO EUROPE/IQEC
- 2013

This work demonstrates a scalable version of Shor's quantum factoring algorithm in which then qubit control register is replaced by a single qubit that is recycled n times: the total number of qubits is one third of that required in the standard protocol.

### Demonstration of Shor's factoring algorithm for N [Formula: see text] 21 on IBM quantum processors.

- Computer ScienceScientific reports
- 2021

This work implemented the quantum order-finding algorithm for factoring the integer 21 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.

### Quantum Factoring Algorithm: Resource Estimation and Survey of Experiments

- Computer ScienceInternational Symposium on Mathematics, Quantum Theory, and Cryptography
- 2020

This study investigates the details of quantum circuits used in several factoring experiments and indicates 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.

### Streamlining Shor's algorithm for potential hardware savings

- Physics
- 2013

We constructed a virtual quantum computer by running a complete, scaling, quantum-gate\char21{}by\char21{}quantum-gate implementation of Shor's algorithm on a 128-core classical cluster computer. In…

## References

SHOWING 1-10 OF 34 REFERENCES

### Quantum Computers, Factoring, and Decoherence

- PhysicsScience
- 1995

Here it is shown how the decoherence process degrades the interference pattern that emerges from the quantum factoring algorithm, a problem of practical significance for cryptographic applications.

### Algorithms for quantum computation: discrete logarithms and factoring

- Computer ScienceProceedings 35th Annual Symposium on Foundations of Computer Science
- 1994

Las Vegas algorithms for finding discrete logarithms and factoring integers on a quantum computer that take a number of steps which is polynomial in the input size, e.g., the number of digits of the integer to be factored are given.

### An algorithmic benchmark for quantum information processing

- PhysicsNature
- 2000

An experimental realization of an algorithmic benchmark using an NMR technique that involves coherent manipulation of seven qubits is reported, which can be used as a reliable and efficient method for creating a standard pseudopure state, the first step for implementing traditional quantum algorithms in liquid state NMR systems.

### Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer

- Computer ScienceSIAM Rev.
- 1999

Efficient randomized algorithms are given for factoring integers and finding discrete logarithms, two problems that are generally thought to be hard on classical computers and that have been used as the basis of several proposed cryptosystems.

### Implementation of a three-quantum-bit search algorithm

- Physics, Computer Science
- 2000

The experimental implementation of Grover’s quantum search algorithm on a quantum computer with three quantum bits is reported, made possible by the introduction of two techniques which significantly reduce the complexity of the experiment and by the surprising degree of cancellation of systematic errors.

### Ensemble quantum computing by NMR spectroscopy

- Computer ScienceProc. Natl. Acad. Sci. USA
- 1997

A new computational model is presented, which differs from a QC only in that the result of a measurement is the expectation value of the observable, rather than a random eigenvalue thereof, which can solve nondeterministic polynomial-time complete problems inPolynomial time.

### Molecular scale heat engines and scalable quantum computation

- PhysicsSTOC '99
- 1999

N procedure that extracts the asymptotically optimal fraction of purified bits is given, and a quasi-linear time implementation of the procedure is given in a model motivated by NMR quantum computing.

### An approximate Fourier transform useful in quantum factoring", IBM Research Report RC19642 ,; R. Cle

- Computer Science
- 1994

We define an approximate version of the Fourier transform on $2^L$ elements, which is computationally attractive in a certain setting, and which may find application to the problem of factoring…

### Quantum Computation and Shor's Factoring Algorithm

- Physics, Computer Science
- 1999

A brief introduction to the model of quantum computation and to its main success so far: Peter Shor's eecient quantum algorithm for factoring integers.

### A silicon-based nuclear spin quantum computer

- PhysicsNature
- 1998

Quantum computers promise to exceed the computational efficiency of ordinary classical machines because quantum algorithms allow the execution of certain tasks in fewer steps. But practical…