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Optical Fiber Communication Conference

Experimental realization of a Shor-type quantum algorithm

Isaac L. Chuang

Published Jan 1, 2001• 1 citation
Publisher

Abstract

I will describe our recent implementation of a five-qubit NMR quantum computer capable of executing a quantum algorithm for order-finding. This is the first demonstration of a quantum algorithm with the structure of Shor's algorithm for efficient prime factorization, a key step in the experimental realization of quantum computers because its structure appears to be common to all quantum algorithms that achieve an exponential speed-up over their classical counterparts. The experiment was made possible by the synthesis of a 5-spin molecule with remarkable NMR properties, and by the introduction of new techniques for state initialization and quantum control.

DOI
10.1364/icqi.2001.fqipa3

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I will describe our recent implementation of a five-qubit NMR quantum computer capable of executing a quantum algorithm for order-finding. This is the first demonstration of a quantum algorithm with the structure of Shor's algorithm for efficient prime factorization, a key step in the experimental realization of quantum computers because its structure appears to be common to all quantum algorithms that achieve an exponential speed-up over their classical counterparts. The experiment was made possible by the synthesis of a 5-spin molecule with remarkable NMR properties, and by the introduction of new techniques for state initialization and quantum control.
We report the realization of a nuclear magnetic resonance quantum computer which combines the quantum Fourier transform with exponentiated permutations, demonstrating a quantum algorithm for order finding. This algorithm has the same structure as Shor's algorithm and its speed-up over classical algorithms scales exponentially. The implementation uses a particularly well-suited five quantum bit molecule and was made possible by a new state initialization procedure and several quantum control techniques.
We have taken significant steps towards the realization of a practical quantum computer: using nuclear spins and magnetic resonance techniques at room temperature, we provided proof of principle of quantum computing in a series of experiments which culminated in the implementation of the simplest instance of Shor's quantum algorithm for prime factorization (15=3x5), using a seven-spin molecule. This algorithm achieves an exponential advantage over the best known classical factoring algorithms and its implementation represents a milestone in the experimental exploration of quantum computation. These remarkable successes have been made possible by the synthesis of suitable molecules and the development of many novel techniques for initialization, coherent control and readout of the state of multiple coupled nuclear spins. Furthermore, we devised and implemented a model to simulate both unitary and decoherence processes in these systems, in order to study and quantify the impact of various technological as well as fundamental sources of errors. In summary, this work has given us a much needed practical appreciation of what it takes to build a quantum computer. While liquid NMR quantum computing has well-understood scaling limitations, many of the techniques that originated from this research may find use in other, perhaps more scalable quantum computer implementations.
Quantum computation offers the extraordinary promise of solving mathematical and physical problems which are simply beyond the reach of classical computers. However, the experimental realization of quantum computers is extremely challenging, because of the need to initialize, manipulate and measure the state of a set of coupled quantum systems while maintaining fragile quantum coherence. In this thesis work, we have taken significant steps towards the realization of a practical quantum computer: using nuclear spins and magnetic resonance techniques at room temperature, we provided proof of principle of quantum computing in a series of experiments which culminated in the implementation of the simplest instance of Shor’s quantum algorithm for prime factorization (15 = 3 × 5), using a seven-spin molecule. This algorithm achieves an exponential advantage over the best known classical factoring algorithms and its implementation represents a milestone in the experimental exploration of quantum computation. These remarkable successes have been made possible by the synthesis of suitable molecules and the development of many novel techniques for initialization, coherent control and readout of the state of multiple coupled nuclear spins. Furthermore, we devised and implemented a model to simulate both unitary and decoherence processes in these systems, in order to study and quantify the impact of various technological as well as fundamental sources of errors. In summary, this work has given us a much needed practical appreciation of what it takes to build a quantum computer. Furthermore, while liquid NMR quantum computing has well-understood scaling limitations, many of the techniques that originated from this research may find use in other, perhaps more scalable quantum computer implementations.
A quantum Fourier transform (QFT) has been implemented on a three qubit nuclear magnetic resonance (NMR) quantum computer to extract the periodicity of an input state. Implementation of a QFT provides a first step towards the realization of Shor's factoring and other quantum algorithms. The experimental implementation of the QFT on a periodic state is presented along with a quantitative measure of its efficiency measured through state tomography. Experimentally realizing the QFT is a clear demonstration of the ability of NMR to control quantum systems.
Quantum algorithms are computational routines that exploit quantum mechanics to solve problems exponentially faster than the best classical algorithms. Shor's quantum factoring algorithm is a key example and the prime motivator in the international effort to realise a quantum computer. However, due to the large number of resources required, to date, there have been only four small scale demonstrations. Here we address this resource demand and demonstrate a scalable version of Shor's 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. Encoding the work register in higher-dimensional states, we implement a two-photon compiled algorithm to factor N = 21. Significantly, the algorithmic output exhibits structure that is distinguishable from noise, in contrast to previous demonstrations.
Shor's algorithm can find prime factors of a large number more efficiently than any known classical algorithm. Understanding the properties that gives the speedup is essential for a general and scalable construction. Here we present a realization of Shor's algorithm, that does not need any of the simplifications presently needed in current experiments and also gives smaller systematic errors than any former experimental implementation. Our realization is based on classical pass-transistor logic, runs at room temperature, and uses the same amount of resources as a scalable quantum computer. In this paper, the focus is not on the result of the factorization, but to compare our realization with current state-of-the-art experiment, factoring 15. Our result gives further insight to the resources needed for quantum computation, aiming for a true understanding of the subject.