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arXiv:quant-ph/0507173 · Physical Review A

Natural three-qubit interactions in one-way quantum computing

Mark Tame, Mauro Paternostro, M. S. Kim, Vlatko Vedral

Published Feb 3, 2006• 29 citations
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Abstract

We address the effects of natural three-qubit interactions on the computational power of one-way quantum computation. A benefit of using more sophisticated entanglement structures is the ability to construct compact and economic simulations of quantum algorithms with limited resources. We show that the features of our study are embodied by suitably prepared optical lattices, where effective three-spin interactions have been theoretically demonstrated. We use this to provide a compact construction for the Toffoli gate. Information flow and two-qubit interactions are also outlined, together with a brief analysis of relevant sources of imperfection.

DOI
10.1103/physreva.73.022309

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We address the effects of natural three-qubit interactions on the computational power of one-way quantum computation. A benefit of using more sophisticated entanglement structures is the ability to construct compact and economic simulations of quantum algorithms with limited resources. We show that the features of our study are embodied by suitably prepared optical lattices, where effective three-spin interactions have been theoretically demonstrated. We use this to provide a compact construction for the Toffoli gate. Information flow and two-qubit interactions are also outlined, together with a brief analysis of relevant sources of imperfection.
Each year, the gap between theoretical proposals and experimental endeavours to create quantum computers gets smaller, driven by the promise of fundamentally faster algorithms and quantum simulations. This occurs by the combination of experimental ingenuity and ever simpler theoretical schemes. This thesis comes from the latter perspective, aiming to find new, simpler ways in which components of a quantum computer could be built. We first search for ways to create quantum gates, the primitive building blocks of a quantum computer. We find a novel, low-control way of performing a two-qubit gate on qubits encoded in a decoherence-free subspace, making use of many-body interactions that may already be present. This includes an analysis of the effect of control errors and magnetic field fluctuations on the gate. We then present novel ways to create three-qubit Toffoli and Fredkin gates in a single step using linear arrays of qubits, including an assessment of how well these gates could perform, for quantum or classical computation, using state-of-the-art ion trap and silicon donor technology. We then focus on a very different model from the normal circuit model, combining ideas from measurement-based quantum computation (MBQC) and holonomic quantum computation, showing that all MBQC patterns with a property called \emph{gflow} can be converted into a holonomic computation. The manifestation of the properties of MBQC in this adiabatically driven model is then explored. Finally, we investigate ways in which quantum information can be communicated between distant parties, using minimally engineered spin chains. The viability of using 1D Wigner crystals as a quantum channel is analysed, as well as schemes using ideal uniform spin chains with next-neighbour interactions, and edge-locking effects.
Each year, the gap between theoretical proposals and experimental endeavours to create quantum computers gets smaller, driven by the promise of fundamentally faster algorithms and quantum simulations. This occurs by the combination of experimental ingenuity and ever simpler theoretical schemes. This thesis comes from the latter perspective, aiming to find new, simpler ways in which components of a quantum computer could be built. We first search for ways to create quantum gates, the primitive building blocks of a quantum computer. We find a novel, low-control way of performing a two-qubit gate on qubits encoded in a decoherence-free subspace, making use of many-body interactions that may already be present. This includes an analysis of the effect of control errors and magnetic field fluctuations on the gate. We then present novel ways to create three-qubit Toffoli and Fredkin gates in a single step using linear arrays of qubits, including an assessment of how well these gates could perform, for quantum or classical computation, using state-of-the-art ion trap and silicon donor technology. We then focus on a very different model from the normal circuit model, combining ideas from measurement-based quantum computation (MBQC) and holonomic quantum computation. We generalise an earlier model to show that all MBQC patterns with a property called gflow can be converted into a holonomic computation. The manifestation of the properties of MBQC in this adiabatically driven model is then explored. Finally, we investigate ways in which quantum information can be communicated between distant parties, using minimally engineered spin chains. The viability of using 1D Wigner crystals as a quantum channel is analysed, as well as schemes using ideal uniform spin chains with nextneighbour interactions, and edge-locking effects.
The regular structures obtained by optical lattice technology and their behavior are analyzed from the quantum information perspective. Initially, we demonstrate that a triangular optical lattice of two atomic species, bosonic or fermionic, can be employed to generate a variety of novel spin-1/2 models that include effective three-spin interactions. Such interactions can be employed to simulate specific one-or two-dimensional physical systems that are of particular interest for their condensed matter and entanglement properties. In particular, connections between the scaling behavior of entanglement and the entanglement properties of closely spaced spins are drawn. Moreover, three-spin interactions are well suited to support quantum computing without the need to manipulate individual qubits. By employing Raman transitions or the interaction of the atomic electric dipole moment with magnetic field gradients, one can generate Hamiltonians that can be used for the physical implementation of geometric or topological objects. This work serves as a review article that also includes many new results.
We make a detailed analysis of error mechanisms, gate fidelity, and scalability of proposals for quantum computation with neutral atoms in addressable (large lattice constant) optical lattices. We have identified possible limits to the size of quantum computations, arising in three-dimensional (3D) optical lattices from current limitations on the ability to perform single-qubit gates in parallel and in 2D lattices from constraints on laser power. Our results suggest that 3D arrays as large as $100\ifmmode\times\else\texttimes\fi{}100\ifmmode\times\else\texttimes\fi{}100$ sites (i.e., $\ensuremath{\sim}{10}^{6}$ qubits) may be achievable, provided two-qubit gates can be performed with sufficiently high precision and degree of parallelizability. The parallelizability of long-range interaction-based two-qubit gates is qualitatively compared to that of collisional gates. Different methods of performing single-qubit gates are compared, and a lower bound of $1\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}5}$ is determined on the error rate for the error mechanisms affecting 133Cs^{133}\text{C}\text{s}133Cs in a blue-detuned lattice with Raman-transition-based single-qubit gates, given reasonable limits on experimental parameters.
We assess the effects of a realistic intrinsic model for imperfections in cluster states by introducing noisy cluster states and characterizing their role in the one-way computational model. A suitable strategy to counter-affect these non-idealities is represented by the use of small clusters, stripped of any redundancy, which leads to the search for compact schemes for one-way quantum computation. In light of this, we quantitatively address the behavior of a simple four-qubit cluster which simulates a controlled-NOT under the influences of our model for decoherence. Our scheme can be particularly useful in an all-optical setup and the strategy we address can be directly applied in those experimental situations where small cluster states can be constucted.