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

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

APL And The Numerical Solution Of High-Order Linear Differential Equations

An Nth‐order linear ordinary differential equation is rewritten as a first‐order equation in an N×N matrix. Taking advantage of the matrix manipulation strength of the APL language this equation is then solved directly, yielding a great simplification over the standard procedure of solving N coupled first‐order scalar equations. This eases programming and results in a more intuitive algorithm. Example applications of a program using the technique are given from quantum mechanics and control theory

3. Launching the New Enterprise

As the academic year of 1945-46 approached, the intensity of activity in preparation for actually opening the school in the fall term became overwhelming. Incredible though it may seem, Ives and Day were able in a period of a few weeks to assemble the nucleus of a faculty, several of whom formed a continuing source of counsel and advice both during the school’s formative years and thereafter. Includes: The First Dean and the School’s Dedication; A Participant’s View of the Early Years; Ives Moves On; Several Views of Martin P. Catherwood; The Founders

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

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

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

Experimental Demonstration of Greenberger-Horne-Zeilinger Correlations Using Nuclear Magnetic Resonance

The Greenberger-Horne-Zeilinger (GHZ) effect provides an example of quantum correlations that cannot be explained by classical local hidden variables. This paper reports on the experimental realization of GHZ correlations using nuclear magnetic resonance (NMR). The NMR experiment differs from the originally proposed GHZ experiment in several ways: it is performed on mixed states rather than pure states; and instead of being widely separated, the spins on which it is performed are all located in the same molecule. As a result, the NMR version of the GHZ experiment cannot entirely rule out classical local hidden variables. It nonetheless provides an unambiguous demonstration of the "paradoxical" GHZ correlations, and shows that any classical hidden variables must communicate by non-standard and previously undetected forces. The NMR demonstration of GHZ correlations shows the power of NMR quantum information processing techniques for demonstrating fundamental effects in quantum mechanics.Comment: Latex2.09, 8 pages, 1 eps figur