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arXiv:quant-ph/9905041 · Physical Review Letters

Realization of Logically Labeled Effective Pure States for Bulk Quantum Computation

Lieven M. K. Vandersypen, Costantino S. Yannoni, Mark Sherwood, Isaac L. Chuang

Published Oct 11, 1999• 51 citations
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Abstract

We report the first use of ``logical labeling'' to perform a quantum computation with a room-temperature bulk system. This method entails the selection of a subsystem that behaves as if it were at zero temperature---except for a decrease in signal strength---conditioned upon the state of the remaining system. No averaging over differently prepared molecules is required. In order to test this concept, we execute a quantum search algorithm in a subspace of two nuclear spins, labeled by a third spin, using solution nuclear magnetic resonance and employing a novel choice of reference frame to uncouple nuclei.

DOI
10.1103/physrevlett.83.3085

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•Jan 1, 2000•38 citations
We report the first use of ``logical labeling'' to perform a quantum computation with a room-temperature bulk system. This method entails the selection of a subsystem that behaves as if it were at zero temperature---except for a decrease in signal strength---conditioned upon the state of the remaining system. No averaging over differently prepared molecules is required. In order to test this concept, we execute a quantum search algorithm in a subspace of two nuclear spins, labeled by a third spin, using solution nuclear magnetic resonance and employing a novel choice of reference frame to uncouple nuclei.
Quantum state preparation plays an equally important role with quantum operations and measurements in quantum information processing. The previous methods of preparing initial state for bulk quantum computation all have inevitable disadvantages, such as, requiring multiple experiments, causing loss of signals, or requiring molecules with restrictive structure. In this work, three kinds of quantum circuits are introduced to prepare the pseudo-pure states of (n−1n-1n−1) qubits in the Hilbert space of nnn coupled spins which merely need the assist of one ancilla spin and two experiments independent of nnn. Being without gradient fields effectively avoids the reduction of the signals. Our methods have no special requirements on the structure of the used molecules. To test these methods more comprehensively, we experimentally demonstrate the preparation of the labeled pseudo-pure states using heteronuclear 2-qubit and homonuclear 4-qubit nuclear magnetic resonance quantum information processor.
In bulk quantum computation one can manipulate a large number of indistinguishable quantum computers by parallel unitary operations and measure expectation values of certain observables with limited sensitivity. The initial state of each computer in the ensemble is known but not pure. Methods for obtaining effective pure input states by a series of manipulations have been described by Gershenfeld and Chuang (logical labeling) [Science 275, 350 (1997)] and Cory et al. (spatial averaging) [Proc. Natl. Acad. Sci. USA 94, 1634 (1997)] for the case of quantum computation with nuclear magnetic resonance. We give a different technique called temporal averaging. This method is based on classical randomization, requires no ancilla quantum bits, and can be implemented in nuclear magnetic resonance without using gradient fields. We introduce several temporal averaging algorithms suitable for both high-temperature and low-temperature bulk quantum computing and analyze the signal-to-noise behavior of each. Most of these algorithms require only a constant multiple of the number of experiments needed by the other methods for creating effective pure states.
We show that quantum computation is possible with mixed states instead of pure states as inputs. This is performed by embedding within the mixed state a subspace that transforms like a pure state and that can be identified by labelling it based on logical (spin), temporal, or spatial degrees of freedom. This permits quantum computation to be realized with bulk ensembles far from the ground state. Experimental results are presented for quantum gates and circuits implemented with liquid nuclear magnetic resonance techniques and verified by quantum state tomography.
Based on the logical labelling method, we prepare an effective pure state in a subsystem of a three spin system via liquid nuclear magnetic resonance technique. Then with this subspace effective pure state we implement the Deutsch–Jozsa algorithm. The tomography for the subspace effective pure state and the corresponding spectrum of the output for the Deutsch–Jozsa algorithm agree with theoretical predictions, which shows that we have successfully implemented the Deutsch–Jozsa algorithm in a subsystem of a nuclear spin system and demonstrated a subspace quantum computation.
Preparation of effective pure states ∣00⋯0⟩\mid 00 \cdots 0 \rangle∣00⋯0⟩ on an NMR quantum computer with any number of spins (Analytical Study of Quantum Information and Related Fields)
A method is proposed to prepare an effective pure state corresponding to the pure state ∣0⟩|0\rangle∣0⟩ ∣0⟩|0\rangle∣0⟩ ⋯∣0⟩\cdots|0\rangle⋯∣0⟩ for any number of spins on a nuclear magnetic resonance quantum computer, where ∣0⟩|0\rangle∣0⟩ stands for the spin-up state of a nuclear spin.The method is based on the spatial labeling proposed by Cory et al.
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.