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arXiv:quant-ph/0410198 · Physical Review A

Simulation of the Burgers equation by NMR quantum-information processing

Zhiying Chen, Jeffrey Yepez, David G. Cory

Published Oct 23, 2006• 22 citations
arXivHTMLPDF

Abstract

We report on the implementation of Burgers equation as a type-II quantum computation on a NMR quantum-information processor. Since the flow field evolving under the Burgers equation develops sharp features over time, this is a better test of liquid-state NMR implementations of type-II quantum computers than the previous examples using the diffusion equation. In particular, we show that Fourier approximations used in the encoding step are not the dominant error. Small systematic errors in the collision operator accumulate and swamp all other errors. We propose, and demonstrate, that the accumulation of this error can be avoided to a large extent by replacing the single collision operator with a set of operators with random errors and similar fidelities. Experiments have been implemented on 16 two-qubit sites for eight successive time steps for the Burgers equation.

DOI
10.1103/physreva.74.042321

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We report on the implementation of Burgers equation as a type-II quantum computation on a NMR quantum-information processor. Since the flow field evolving under the Burgers equation develops sharp features over time, this is a better test of liquid-state NMR implementations of type-II quantum computers than the previous examples using the diffusion equation. In particular, we show that Fourier approximations used in the encoding step are not the dominant error. Small systematic errors in the collision operator accumulate and swamp all other errors. We propose, and demonstrate, that the accumulation of this error can be avoided to a large extent by replacing the single collision operator with a set of operators with random errors and similar fidelities. Experiments have been implemented on 16 two-qubit sites for eight successive time steps for the Burgers equation.
A quantum algorithm is presented for modeling the time evolution of a continuous field governed by the nonlinear Burgers equation in one spatial dimension. It is a microscopic-scale algorithm for a type-II quantum computer, a large lattice of small quantum computers interconnected in nearest neighbor fashion by classical communication channels. A formula for quantum state preparation is presented. The unitary evolution is governed by a conservative quantum gate applied to each node of the lattice independently. Following each quantum gate operation, ensemble measurements over independent microscopic realizations are made resulting in a finite-difference Boltzmann equation at the mesoscopic scale. The measured values are then used to re-prepare the quantum state and one time step is completed. The procedure of state preparation, quantum gate application, and ensemble measurement is continued ad infinitum. The Burgers equation is derived as an effective field theory governing the behavior of the quantum computer at its macroscopic scale where both the lattice cell site and the time step interval become infinitesimal. A numerical simulation of shock formation is carried out and agrees with the exact analytical solution.
Nuclear magnetic resonance (NMR) provides an experimental setting to explore physical implementations of quantum information processing (QIP). Here we introduce the basic background for understanding applications of NMR to QIP and explain their current successes, limitations and potential. NMR spectroscopy is well known for its wealth of diverse coherent manipulations of spin dynamics. Ideas and instrumentation from liquid state NMR spectroscopy have been used to experiment with QIP. This approach has carried the field to a complexity of about 10 qubits, a small number for quantum computation but large enough for observing and better understanding the complexity of the quantum world. While liquid state NMR is the only present-day technology about to reach this number of qubits, further increases in complexity will require new methods. We sketch one direction leading towards a scalable quantum computer using spin 1/2 particles. The next step of which is a solid state NMR-based QIP capable of reaching 10-30 qubits.
Nuclear magnetic resonance (NMR) provides an experimental setting to explore physical implementations of quantum information processing (QIP). Here we introduce the basic background for understanding applications of NMR to QIP and explain their current successes, limitations and potential. NMR spectroscopy is well known for its wealth of diverse coherent manipulations of spin dynamics. Ideas and instrumentation from liquid state NMR spectroscopy have been used to experiment with QIP. This approach has carried the field to a complexity of about 10 qubits, a small number for quantum computation but large enough for observing and better understanding the complexity of the quantum world. While liquid state NMR is the only present-day technology about to reach this number of qubits, further increases in complexity will require new methods. We sketch one direction leading towards a scalable quantum computer using spin 1/2 particles. The next step of which is a solid state NMR-based QIP capable of reaching 10—30 qubits.
Quantum information processing has been the subject of countless discoveries since the early 1990's. It is believed to be the way of the future for computation: using quantum systems permits one to perform computation exponentially faster than on a regular classical computer. Unfortunately, quantum systems that not isolated do not behave well. They tend to lose their quantum nature due to the presence of the environment. If key information is known about the noise present in the system, methods such as quantum error correction have been developed in order to reduce the errors introduced by the environment during a given quantum computation. In order to harness the quantum world and implement the theoretical ideas of quantum information processing and quantum error correction, it is imperative to understand and quantify the noise present in the quantum processor and benchmark the quality of the control over the qubits. Usual techniques to estimate the noise or the control are based on quantum process tomography (QPT), which, unfortunately, demands an exponential amount of resources. \n \nThis thesis presents work towards the characterization of noisy processes in an efficient manner. The protocols are developed from a purely abstract setting with no system-dependent variables. To circumvent the exponential nature of quantum process tomography, three different efficient protocols are proposed and experimentally verified. The first protocol uses the idea of quantum error correction to extract relevant parameters about a given noise model, namely the correlation between the dephasing of two qubits. Following that is a protocol using randomization and symmetrization to extract the probability that a given number of qubits are simultaneously corrupted in a quantum memory, regardless of the specifics of the error and which qubits are affected. Finally, a last protocol, still using randomization ideas, is developed to estimate the average fidelity per computational gates for single and multi qubit systems. \n \nEven though liquid state NMR is argued to be unsuitable for scalable quantum information processing, it remains the best test-bed system to experimentally implement, verify and develop protocols aimed at increasing the control over general quantum information processors. For this reason, all the protocols described in this thesis have been implemented in liquid state NMR, which then led to further development of control and analysis techniques.
Abstract Computational fluid dynamics (CFD) simulations are a vital part of the design process in the aerospace industry. Although reliable CFD results can be obtained with turbulence models, direct numerical simulation of complex bodies in three spatial dimensions (3D) is impracticable due to the massive amount of computational elements. For instance, a 3D direct numerical simulation of a turbulent boundary-layer over the wing of a commercial jetliner that resolves all relevant length scales using a serial CFD solver on a modern digital computer would take approximately 750 million years or roughly 20% of the earth’s age. Over the past 25 years, quantum computers have become the object of great interest worldwide as powerful quantum algorithms have been constructed for several important, computationally challenging problems that provide enormous speed-up over the best-known classical algorithms. In this paper, we adapt a recently introduced quantum algorithm for partial differential equations to Burgers’ equation and develop a quantum CFD solver that determines its solutions. We used our quantum CFD solver to verify the quantum Burgers’ equation algorithm to find the flow solution when a shockwave is and is not present. The quantum simulation results were compared to: (i) an exact analytical solution for a flow without a shockwave; and (ii) the results of a classical CFD solver for flows with and without a shockwave. Excellent agreement was found in both cases, and the error of the quantum CFD solver was comparable to that of the classical CFD solver.
Quantum computers exploit quantum mechanical phenomena as resources to process information.They are therefore far more vulnerable to errors than their classical counterparts.The main obstacle to the practical realization of a working large-scale quantum computer is decoherence, i.e., the process by which a quantum system decays to a classical state through interaction with the environment.We herein discuss two techniques to implement robust quantum information processing, and report on their implementation in a liquid state NMR Quantum Computing experiment.We describe a method to implement a two-qubit gate between spins 2 and 3 of a three-spin linear chain molecule by applying a soft pulse to spin 1, with appropriate choices of the rf-field amplitude ω 1 and duration τ [1].The Bloch-Siegert effect for such a soft pulse is negligible in general, which makes NMR pulse programming and quantum gates design much simpler than with conventional hard π-pulses; it also makes this method suitable for use with both homo-and heteronuclear molecules.We then discuss a simple Operator Quantum Error Correction (OQEC) scheme with three qubits [2].This scheme provides the simplest possible Noiseless Subsystem, in terms of the number of CNOT gates, that is robust against (restricted) fully correlated noise.We demonstrated our OQEC scheme experimentally using a three-qubit NMR Quantum Computer, in which the ancillae are in the maximally mixed state.In this experiment, we made practical use of the soft pulse technique [1] to implement the two-qubit gates in the pulse sequences realizing encoding and decoding operations.