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Physical Review Letters

Quantum Factorization of 143 on a Dipolar-Coupling Nuclear Magnetic Resonance System

Nanyang Xu, Jing Zhu, Dawei Lu, Xianyi Zhou, Xinhua Peng, Jiangfeng Du

Published Mar 30, 2012• 180 citations
Publisher

Abstract

Quantum algorithms could be much faster than classical ones in solving the factoring problem. Adiabatic quantum computation for this is an alternative approach other than Shor's algorithm. Here we report an improved adiabatic factoring algorithm and its experimental realization to factor the number 143 on a liquid-crystal NMR quantum processor with dipole-dipole couplings. We believe this to be the largest number factored in quantum-computation realizations, which shows the practical importance of adiabatic quantum algorithms.

DOI
10.1103/physrevlett.108.130501

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Quantum algorithms could be much faster than classical ones in solving the factoring problem. Adiabatic quantum computation for this is an alternative approach other than Shor's algorithm. Here we report an improved adiabatic factoring algorithm and its experimental realization to factor the number 143 on a liquid-crystal NMR quantum processor with dipole-dipole couplings. We believe this to be the largest number factored in quantum-computation realizations, which shows the practical importance of adiabatic quantum algorithms.
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.
Adiabatic quantum computation is of fundamental importance in the field of quantum computation as it offers an alternative approach to the gate-based model for the manipulation of quantum systems. Recently, an interesting work [arXiv:1805.10549] indicated that we can solve a linear equation system via an algorithm inspired by adiabatic quantum computing. Here we demonstrate the algorithm in a four-qubit nuclear magnetic resonance system by determining the solution of an eight-dimensional linear equation Ax=bA\mathbf{x}=\mathbf{b}Ax=b. The result is by far the maximum-dimensional linear equation solution with a limited number of qubits in experiments, which include some ingenious simplifications. Our experiment provides the possibility of solving so many practical problems related to linear equations systems and has the potential applications in designing the future quantum algorithms.
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.
An important quantum search algorithm based on the quantum random walk performs an oracle search on a database of NNN items with O(phN)O(\sqrt{\mathit{ph}N})O(phN​) calls, yielding a speedup similar to the Grover quantum search algorithm. The algorithm was implemented on a quantum information processor of three-qubit liquid-crystal nuclear magnetic resonance (NMR) in the case of finding 1 out of 4, and the diagonal elements' tomography of all the final density matrices was completed with comprehensible one-dimensional NMR spectra. The experimental results agree well with the theoretical predictions.
Quantum computers can potentially solve real and relevant mathematical and physical problems that are intractable using classical machines. However, the experimental realization of quantum computers represents a significant challenge, because several opposing experimental requirements must be met. A set of coupled quantum bits must be manipulated and measured while coherently retaining entangled quantum states. Yet the manipulation and measurement processes almost inevitably lead to the decay of these fragile states. This thesis work takes significant steps towards building a practical quantum computer using nuclear spins and liquid state nuclear magnetic resonance (NMR) techniques. I present experimental results for proof of principle of quantum computing in a series of small implementations of quantum algorithms, culminating in the implementation of an adiabatic quantum optimization algorithm. The performance of adiabatic algorithms compared with classical optimization methods is unknown, but small quantum computers could provide crucial insight into answering this question. Furthermore, adiabatic algorithms also shed new light on the usefulness of quantum resources for computational tasks and hence they represent an important class of algorithms. The second part of this thesis presents several methods developed for improving the control over NMR quantum computing experiments. Even though liquid state NMR quantum computers have well known and accepted scaling limitations, the developed tools are of general use. I will show how several tools are directly transferable to two other implementations of quantum computers; one using optical methods and the other using loops of superconducting material. This work provides insight into what is needed to build a practical quantum computer, and delivers several tools and techniques that will be useful in future large-scale implementations of quantum computers.