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arXiv:quant-ph/9706053 · Physical Review A

Effective pure states for bulk quantum computation

Emanuel H. Knill, Isaac L. Chuang, Raymond Laflamme

Published May 1, 1998• 245 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.57.3348

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This Week in Science

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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.
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.
Certain difficult computational problems, such as finding prime factors, could be performed much faster if quantum computing, in which interactions of pure quantum states correspond to logical operations, could be realized in a practical manner. Most schemes implemented or proposed require isolating or controlling single atoms to maintain the coherence of quantum states. Gershenfeld and Chuang (p. 350; see the news story by Taubes, p. 307) propose a different approach in which the manipulation of nuclear spins in a chemical compound with nuclear magnetic resonance (NMR) methods performs quantum computation. Such an approach might seem improbable in that the nuclear spins of a bulk sample form a statistical ensemble that would rapidly undergo decoherence. However, the authors show that the deviation from the equilibrium spin states can be manipulated so that it acts like a pure, low-dimensional quantum state.
A method is proposed to prepare an effective pure state corresponding to the pure state $|0〉|0〉\ensuremath{\cdot}\ensuremath{\cdot}\ensuremath{\cdot}|0〉$ for any number of spins on a nuclear magnetic resonance quantum computer, where ∣0〉|0〉∣0〉 stands for the spin-up state of a nuclear spin. The method is based on the spatial labeling proposed by Cory et al. [Proc. Natl. Acad. Sci. USA 94, 1634 (1997)].
Abstract A fast method to prepare effective pure states for NMR‐quantum computer in homo‐nuclear spin system is proposed. The temporal averaging method has been introduced to prepare an effective pure state in NMR‐quantum computer, but the method requires many experiments, whose number increases exponentially with the number of quantum bits (qubits), and it seems that a huge number of logical gates are required. The proposed method avoids this problem by eliminating a part of elements in the density matrices. This method requires only one ancillary qubit and two different experiments; the steps have been confirmed to be proportional to the number of qubits up to four‐qubit. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004
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.