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

Abstract

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 quantum computers, experimental demonstration of this algorithm has proved elusive. Here we report an implementation of the simplest instance of Shor's algorithm: factorization of $N=15$ (whose prime factors are 3 and 5). We use seven spin-1/2 nuclei in a molecule as quantum bits, which can be manipulated with room temperature liquid state nuclear magnetic resonance techniques. This method of using nuclei to store quantum information is in principle scalable to many quantum bit systems, but such scalability is not implied by the present work. The significance of our work lies in the demonstration of experimental and theoretical techniques for precise control and modelling of complex quantum computers. In particular, we present a simple, parameter-free but predictive model of decoherence effects in our system.

accepted version

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• An approximate Fourier transform useful in quantum factoring

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• Quantum Algorithms in Cybernetics
• Entanglement For Ideal Quantum Ensembles
• Ensemble Algorithm for the Selection Problem by NMR Ensemble Quantum Computers
• Topological Properties and Characterizations
• Geometric Measure of Entanglement and Schmidt Decomposition of Multipartite Systems
• The Rise of Quantum Computing -- Take a BITE for Built Environment and Urban Microclimate Research
• Fault-Tolerant Encoding of Logical Qudits in Spin Systems
• A Time Optimization Framework for the Implementation of Robust and Low-latency Quantum Circuits
• Number-theoretic expressions obtained through analogy between prime factorization and optical interferometry
• Quantum reservoir computing for predicting and characterizing chaotic maps
• Quantum Information Processing with Continuous Variables and Atomic Ensembles
• Photon Frequency Bears More At Less
• Characterizing And Exploiting Hybrid Entanglement
• Quantum Fourier Transform in Oscillating Modes
• Quantum Dots: Coulomb Blockade, Mesoscopic Fluctuations, and Qubit Decoherence
• Quantum Error Correction of Time-Correlated Errors