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arXiv:1706.08053

Preparation of Logically Labeled Pure States with Only Two Turns for Bulk Quantum Computation

Tao Xin, Liang Hao, Shi‐Yao Hou, Guanru Feng, Gui‐Lu Long

Published Jun 25, 2017• 0 citations
arXivHTMLPDF

Abstract

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 ($n-1$) qubits in the Hilbert space of $n$ coupled spins which merely need the assist of one ancilla spin and two experiments independent of $n$. 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.

DOI
10.48550/arxiv.1706.08053

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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.
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.
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.
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.
Quantum-information processing by liquid-state NMR spectroscopy uses pseudopure states to mimic the evolution and observations on true pure states. A new method of preparing pseudopure states is described, which involves the selection of spatially labeled states of an ancilla spin with which the spin system of interest is correlated. This permits a general procedure to be given for the preparation of pseudopure states on any number of spins, subject to the limitations imposed by the loss of signal from the selected subensemble. The preparation of a single pseudopure state is demonstrated by carbon and proton NMR of 13C{}^{13}\mathrm{C}13C-labeled alanine. With a judicious choice of magnetic-field gradients, the method further allows encoding of up to 2N{2}^{N}2N pseudopure states in independent spatial modes in an N+1N+1N+1 spin system. Fast encoding and decoding schemes are demonstrated for the preparation of four such spatially labeled pseudopure states.
Basing on the temporal averaging, schedules to prepare 3-qubit and 4-qubit pesudo-pure states in nuclear magnetic resonance (NMR) quantum computation are presented in this paper. Utilizing Lambda(n) (not) gate, our schedules reduced the experiment number that is needed to prepare pesudo-pure state in a homonuclear spin system: only three experiments are needed to prepare 3-qubit peudo-pure state; and only five experiments are needed to prepare 4-qubit peudo-pure state. Then we implemented the schedule for 3-qubit psudo-pure state in a chain-coupled spin system to confirm the schedule, and gave the simulation result of the schedule for the 4-qubit peudo-pure state.
A quantum circuit is introduced to describe the preparation of a labeled pseudo-pure state by multiplet-component excitation scheme which has been experimentally implemented on a 4-qubit nuclear magnetic resonance quantum processor. Meanwhile, we theoretically analyze and numerically investigate the low-power selective single-pulse implementation of a controlled-rotation gate, which manifests its validity in our experiment. Based on the labeled pseudo-pure state prepared, a 3-qubit Bernstein-Vazirani algorithm has been experimentally demonstrated by spectral implementation. The "answers" of the computations are identified from the split peak positions in the spectra of the observer spin, which are equivalent to projective measurements required by the algorithms.