arXiv:quant-ph/0608148

Chain of nuclear spins system quantum computer taking into account second neighbor Ising spins interaction and numerical simulation of Shor factorization of N=4

Gustavo V. López, Lorena Lara

Published Aug 18, 2006• 0 citations

Abstract

For a one-dimensional chain of four nuclear spins (1/2) and taking into account first and second neighbor interactions among the spin system, we make the numerical simulation of Shor prime factorization algorithm of the integer number N=4 to study the influence of the second neighbor interaction on the performance of this algorithm. It is shown that the optimum Rabi's frequency to control the non-resonant effects is dominated by the second neighbor interaction coupling parameter ($J'$), and that a good Shor quantum factorization is achieved for a ratio of second to first coupling constant of $J'/J\ge 0.04$.

DOI: 10.48550/arxiv.quant-ph/0608148

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arXiv:quant-ph/0608148

Chain of nuclear spins system quantum computer taking into account second neighbor Ising spins interaction and numerical simulation of Shor factorization of N=4

Gustavo V. López, Lorena Lara

Published Aug 18, 2006• 0 citations

arXiv HTML PDF

Abstract

J'

), and that a good Shor quantum factorization is achieved for a ratio of second to first coupling constant of

J'/J\ge 0.04

DOI: 10.48550/arxiv.quant-ph/0608148

Related papers

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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.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.

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arXiv:0809.4416Physical Review ADec 12, 200814 citations

J'

), and that a good Shor quantum factorization is achieved for a ratio of second to first coupling constant of

J'/J\ge 0.04

J'/J\ge 0.04

. We also show that the optimum Rabi's frequency to control the non-resonant effects is dominated by the second neighbor interaction coupling parameter (

J'

π/2

and

π

pulses), and we study the efficiency of getting this entangled state as a function of the ratio of the second neighbor interaction coupling constant to the first neighbor interaction coupling constant (

J'/J

). We found that for

J'/J\ge 0.04

, the entangled state is well defined.

{n}_{q}=18

computational qubits allowing to factor numbers up to

N=205\phantom{\rule{0.2em}{0ex}}193

. We confirm that the algorithm remains operational up to a critical coupling strength

{ϵ}_{c}

which drops only polynomially with

{\mathrm{log}}_{2}\phantom{\rule{0.2em}{0ex}}N

. The obtained numerical dependence of

{ϵ}_{c}

{\mathrm{log}}_{2}\phantom{\rule{0.2em}{0ex}}N

is in a good agreement with the analytical estimates that allows to obtain the scaling for functionality of Shor's algorithm on realistic quantum computers with a large number of qubits.