Nuclear magnetic resonance spectroscopy: An experimentally accessible paradigm for quantum computing

We present experimental results which demonstrate that nuclear magnetic resonance spectroscopy is capable of efficiently emulating many of the capabilities of quantum computers, including unitary evolution and coherent superpositions, but without attendant wave-function collapse. Specifically, we have: (1) Implemented the quantum XOR gate in two different ways, one using Pound-Overhauser double resonance, and the other using a spin-coherence double resonance pulse sequence; (2) Demonstrated that the square root of the Pound-Overhauser XOR corresponds to a conditional rotation, thus obtaining a universal set of gates; (3) Devised a spin-coherence implementation of the Toffoli gate, and confirmed that it transforms the equilibrium state of a four-spin system as expected; (4) Used standard gradient-pulse techniques in NMR to equalize all but one of the populations in a two-spin system, so obtaining the pseudo-pure state that corresponds to |00>; (5) Validated that one can identify which basic pseudo-pure state is present by transforming it into one-spin superpositions, whose associated spectra jointly characterize the state; (6) Applied the spin-coherence XOR gate to a one-spin superposition to create an entangled state, and confirmed its existence by detecting the associated double-quantum coherence via gradient-echo methods.Comment: LaTeX + epsfig + amsmath packages, 27 pages, 12 figures, to appear in Physica D; revision updates list of authors and reference

Experimental Realization of Br\"{u}schweiler's exponentially fast search algorithm in a homo-nuclear system

Compared with classical search algorithms, Grover quantum algorithm [ Phys. Rev. Lett., 79, 325(1997)] achieves quadratic speedup and Bruschweiler hybrid quantum algorithm [Phys. Rev. Lett., 85, 4815(2000)] achieves an exponential speedup. In this paper, we report the experimental realization of the Bruschweiler$ algorithm in a 3-qubit NMR ensemble system. The pulse sequences are used for the algorithms and the measurement method used here is improved on that used by Bruschweiler, namely, instead of quantitatively measuring the spin projection of the ancilla bit, we utilize the shape of the ancilla bit spectrum. By simply judging the downwardness or upwardness of the corresponding peaks in an ancilla bit spectrum, the bit value of the marked state can be read out, especially, the geometric nature of this read-out can make the results more robust against errors.Comment: 10 pages and 3 figure

Multiqubit Spin

It is proposed that the state space of a quantum object with a complicated discrete spectrum can be used as a basis for multiqubit recording and processing of information in a quantum computer. As an example, nuclear spin 3/2 is considered. The possibilities of writing and reading two quantum bits of information, preparation of the initial state, implementation of the "rotation" and "controlled negation" operations, which are sufficient for constructing any algorithms, are demonstrated.Comment: 7 pages, PostScript, no figures; translation of Pis'ma Zh. Eksp. Teor. Fiz. 70, No. 1, pp. 59-63, 10 July 1999; (Submitted 29 April 1999; resubmitted 2 June 1999

Effective Pure States for Bulk Quantum Computation

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) and Cory et al. (spatial averaging) 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 qubits 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.Comment: 24 pages in LaTex, 14 figures, the paper is also avalaible at http://qso.lanl.gov/qc

Two-qubit Quantum Logic Gate in Molecular Magnets

We proposed a scheme to realize a controlled-NOT quantum logic gate in a dimer of exchange coupled single-molecule magnets, $[\textrm{Mn}_4]_2$. We chosen the ground state and the three low-lying excited states of a dimer in a finite longitudinal magnetic field as the quantum computing bases and introduced a pulsed transverse magnetic field with a special frequency. The pulsed transverse magnetic field induces the transitions between the quantum computing bases so as to realize a controlled-NOT quantum logic gate. The transition rates between the quantum computing bases and between the quantum computing bases and other excited states are evaluated and analyzed.Comment: 7 pages, 2 figure

Fetching marked items from an unsorted database in NMR ensemble computing

Searching a marked item or several marked items from an unsorted database is a very difficult mathematical problem. Using classical computer, it requires $O(N=2^n)$ steps to find the target. Using a quantum computer, Grover's algorithm uses $O(\sqrt{N=2^n})$ steps. In NMR ensemble computing, Brushweiler's algorithm uses $\log N$ steps. In this Letter, we propose an algorithm that fetches marked items in an unsorted database directly. It requires only a single query. It can find a single marked item or multiple number of items.Comment: 4 pages and 1 figur

Simulations of Quantum Logic Operations in Quantum Computer with Large Number of Qubits

We report the first simulations of the dynamics of quantum logic operations with a large number of qubits (up to 1000). A nuclear spin chain in which selective excitations of spins is provided by the gradient of the external magnetic field is considered. The spins interact with their nearest neighbors. We simulate the quantum control-not (CN) gate implementation for remote qubits which provides the long-distance entanglement. Our approach can be applied to any implementation of quantum logic gates involving a large number of qubits.Comment: 13 pages, 15 figure

Controllability and universal three-qubit quantum computation with trapped electron states

We show how to control and perform universal three-qubit quantum computation with trapped electron quantum states. The three qubits are the electron spin, and the first two quantum states of the cyclotron and axial harmonic oscillators. We explicitly show how the universal gates can be performed. As an example of a non-trivial quantum algorithm, we outline the implementation of the Deutsch-Jozsa algorithm in this system.Comment: 4 pages, 1 figure. Typos corrected. The original publication is available at http://www.springerlink.co

Sub-Riemannian Geometry and Time Optimal Control of Three Spin Systems: Quantum Gates and Coherence Transfer

Many coherence transfer experiments in Nuclear Magnetic Resonance Spectroscopy, involving network of coupled spins, use temporary spin-decoupling to produce desired effective Hamiltonians. In this paper, we show that significant time can be saved in producing an effective Hamiltonian, if spin-decoupling is avoided. We provide time optimal pulse sequences for producing an important class of effective Hamiltonians in three spin networks. These effective Hamiltonians are useful for coherence transfer experiments and implementation of quantum logic gates in NMR quantum computing. It is demonstrated that computing these time optimal pulse sequences can be reduced to geometric problems that involve computing sub-Riemannian geodesics on Homogeneous spaces

Expressing the operations of quantum computing in multiparticle geometric algebra

We show how the basic operations of quantum computing can be expressed and manipulated in a clear and concise fashion using a multiparticle version of geometric (aka Clifford) algebra. This algebra encompasses the product operator formalism of NMR spectroscopy, and hence its notation leads directly to implementations of these operations via NMR pulse sequences.Comment: RevTeX, 10 pages, no figures; Physics Letters A, in pres