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Concepts in Magnetic Resonance

Observations of quantum dynamics by solution-state NMR spectroscopy

Marco A. Pravia, Evan M. Fortunato, Yaakov S. Weinstein, Mark D. Price, Grum Teklemariam, Richard J. Nelson, Yehuda Sharf, Shyamal S. Somaroo, Ching-Hsiang Tseng, Timothy F. Havel, David G. Cory

Published Jan 1, 1999• 26 citations
Publisher

Abstract

NMR is emerging as a valuable testbed for the investigation of foundational questions in quantum mechanics. The present article outlines the preparation of a class of mixed states, called pseudo-pure states, that emulate pure quantum states in highly mixed solution-state NMR samples. It also describes the NMR observation of spinor behavior in spin ½ nuclei, the simulation of wave function collapse using a magnetic field gradient, the creation of entangled (i.e., Bell) pseudo-pure states, and a brief discussion of quantum computing logic gates including the Quantum Fourier Transform. These experiments show that solution-state NMR can be used to demonstrate quantum dynamics at a level suitable for laboratory exercises. ©1999 John Wiley & Sons, Inc. Concepts Magn Reson 11: 225–238, 1999

DOI
10.1002/(sici)1099-0534(1999)11:4<225::aid-cmr3>3.0.co;2-e

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NMR is emerging as a valuable testbed for the investigation of foundational questions in quantum mechanics. The present article outlines the preparation of a class of mixed states, called pseudo-pure states, that emulate pure quantum states in highly mixed solution-state NMR samples. It also describes the NMR observation of spinor behavior in spin ½ nuclei, the simulation of wave function collapse using a magnetic field gradient, the creation of entangled (i.e., Bell) pseudo-pure states, and a brief discussion of quantum computing logic gates including the Quantum Fourier Transform. These experiments show that solution-state NMR can be used to demonstrate quantum dynamics at a level suitable for laboratory exercises. ©1999 John Wiley & Sons, Inc. Concepts Magn Reson 11: 225–238, 1999
NMR is emerging as a valuable testbed for the investigation of foundational questions in quantum mechanics. The present paper outlines the preparation of a class of mixed states, called pseudo-pure states, that emulate pure quantum states in the highly mixed environment typically used to describe solution-state NMR samples. It also describes the NMR observation of spinor behavior in spin 1/2 nuclei, the simulation of wave function collapse using a magnetic field gradient, the creation of entangled (or Bell) pseudo-pure states, and a brief discussion of quantum computing logic gates, including the Quantum Fourier Transform. These experiments show that liquid-state NMR can be used to demonstrate quantum dynamics at a level suitable for laboratory exercises.
NMR is emerging as a valuable testbed for the investigation of foundational questions in quantum mechanics. The present paper outlines the preparation of a class of mixed states, called pseudo-pure states, that emulate pure quantum states in the highly mixed environment typically used to describe solution-state NMR samples. It also describes the NMR observation of spinor behavior in spin 1/2 nuclei, the simulation of wave function collapse using a magnetic field gradient, the creation of entangled (or Bell) pseudo-pure states, and a brief discussion of quantum computing logic gates, including the Quantum Fourier Transform. These experiments show that liquid-state NMR can be used to demonstrate quantum dynamics at a level suitable for laboratory exercises.
Nuclear magnetic resonance (NMR) is a direct macroscopic manifestation of the quantum mechanics of the intrinsic angular momentum of atomic nuclei. It is best known for its extraordinary range of applications, which include molecular structure determination, medical imaging, and measurements of flow and diffusion rates. Most recently, liquid-state NMR spectroscopy has been found to provide a powerful experimental tool for the development and evaluation of the coherent control techniques needed for quantum information processing. This burgeoning new interdisciplinary field has the potential to achieve cryptographic, communications, and computational feats far beyond what is possible with known classical physics. Indeed, NMR has made the demonstration of many of these feats sufficiently simple to be carried out by high school summer interns working in our laboratory (see the last two authors). In this paper the basic principles of quantum information processing by NMR spectroscopy are described, along with several illustrative experiments suitable for incorporation into the undergraduate physics curriculum. These experiments are spin–spin interferometry, an implementation of the quantum Fourier transform, and the quantum simulation of a harmonic oscillator.
A new method of preparing the pseudo-pure state of a spin system for quantum computation in liquid nuclear magnetic resonance (NMR) was put forward and demonstrated experimentally. Applying appropriately connected line-selective pulses simultaneously and a field gradient pulse techniques we acquired straightforwardly all pseudo-pure states for two qubits in a single experiment much efficiently. The signal intensity with the pseudo-pure state prepared in this way is the same as that of temporal averaging. Our method is suitable for the system with arbitrary numbers of qubits. As an example of application, a highly structured search algorithm—-Hogg’s algorithm was also performed on the pseudo-pure state | 00 〉 prepared by our method.
This article reviews recent work done by us an some initial steps towards the implementation of quantum computation using liquid state NMR. We describe how special kinds of states required for such computation (called pseudo-pure states) can be created from a thermal ensemble of spins. We demonstrate the implementation of several quantum logic gates through one- and two-dimensional NMR methods, using transition- and spin- selective pulses. Finally, we discuss the implementation of the Deutsch-Jozsa algorithm using NMR.
A quantum computer (QC) can operate in parallel on all its possible inputs at once, but the amount of information that can be extracted from the result is limited by the phenomenon of wave function collapse. We present a new computational model, which differs from a QC only in that the result of a measurement is the expectation value of the observable, rather than a random eigenvalue thereof. Such an expectation value QC can solve nondeterministic polynomial-time complete problems in polynomial time. This observation is significant precisely because the computational model can be realized, to a certain extent, by NMR spectroscopy on macroscopic ensembles of quantum spins, namely molecules in a test tube. This is made possible by identifying a manifold of statistical spin states, called pseudo-pure states, the mathematical description of which is isomorphic to that of an isolated spin system. The result is a novel NMR computer that can be programmed much like a QC, but in other respects more closely resembles a DNA computer. Most notably, when applied to intractable combinatorial problems, an NMR computer can use an amount of sample, rather than time, which grows exponentially with the size of the problem. Although NMR computers will be limited by current technology to exhaustive searches over only 15 to 20 bits, searches over as much as 50 bits are in principle possible, and more advanced algorithms could greatly extend the range of applicability of such machines.