# 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…

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## References

SHOWING 1-10 OF 42 REFERENCES

Efficient networks for quantum factoring.

- Physics, MedicinePhysical review. A, Atomic, molecular, and optical physics
- 1996

The number of memory quantum bits (qubits) and the number of operations required to perform factorization, using the algorithm suggested by Shor are estimated.

Quantum Computers, Factoring, and Decoherence

- Computer Science, 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

- Mathematics, 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

- Computer Science, MedicineNature
- 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 Science, MathematicsSIAM Rev.
- 1999

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

Experimental realization of an order-finding algorithm with an NMR quantum computer.

- Computer Science, PhysicsPhysical review letters
- 2000

A nuclear magnetic resonance quantum computer which combines the quantum Fourier transform with exponentiated permutations is realization, demonstrating a quantum algorithm for order finding which has the same structure as Shor's algorithm and its speed-up over classical algorithms scales exponentially.

Implementation of a three-quantum-bit search algorithm

- Physics
- 2000

We report the experimental implementation of Grover’s quantum search algorithm on a quantum computer with three quantum bits. The computer consists of molecules of 13C-labeled CHFBr2, in which the…

Ensemble quantum computing by NMR spectroscopy

- Chemistry, MedicineProc. 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

- Computer ScienceSTOC '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

- Mathematics, Physics
- 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…