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Physical Review A

Bell states in a resonant quantum waveguide network

Gürsoy B. Akgüç, L. E. Reichl, Anil Shaji, Michael G. Snyder

Published Apr 5, 2004• 23 citations
Publisher

Abstract

We show that a network of ballistic electron waveguides can generate entangled Bell-like states from separable states when it is resonant. The network we study is grouped into individual qudits $(d=4)$ made up of pairs of waveguides. Rotation gates in the network produce coherent superpositions of qudit states. A Coulomb gate entangles the qudits. We construct a unitary matrix which characterizes the network dynamics and allows a more systematic study of that dynamics.

DOI
10.1103/physreva.69.042303

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We show that a network of ballistic electron waveguides can generate entangled Bell-like states from separable states when it is resonant. The network we study is grouped into individual qudits (d=4)(d=4)(d=4) made up of pairs of waveguides. Rotation gates in the network produce coherent superpositions of qudit states. A Coulomb gate entangles the qudits. We construct a unitary matrix which characterizes the network dynamics and allows a more systematic study of that dynamics.
We present a small network for the testing of the entanglement of two ballistic electron waveguide qubits. The network produces different output conditional on the presence or absence of entanglement. The structure of the network allows for the determination of successful entanglement operations through the measurement of the output of a single qubit. We also present a simple model of a dynamic Coulomb-like interaction and use it to describe some characteristics of a proposed scheme for the entanglement of qubits in ballistic electron waveguides.
We present a study of ballistic electron waveguides as used to construct simple quantum computational networks.We begin with an introduction to ballistic electron waveguides, some basic theory of quantum computers, and the possible role of electron waveguides in quantum computation.Following that, our study of various waveguide networks and the physics of the specific components of those networks.We continue with a discussion of shot noise in network elements and its possible role in helping to build good computational gates.Finally we conclude with a summary of research and possible future projects.
We investigate the selective generation of Bell-pair entanglement in three qubit networks using a standard quantum collision model. By varying the network geometry, the node contacted by the ancilla, and the interaction Hamiltonians, we identify configurations in which high pairwise concurrence and Bell state fidelity are generated between different target qubits, including nonadjacent qubits and qubits not directly coupled to the ancilla. For comparison, we also consider the continuous time unitary evolution of the complete ancilla network system, sampled at the same discrete time intervals. This reference calculation clarifies how discarding each outgoing ancilla and removing its correlations from subsequent dynamics modifies the attainable entanglement and preparation time. Our results provide a proof of principle demonstration of geometry dependent Bell pair generation and routing in small qubit networks.
We analyze a quantum network, constructed with ballistic electron waveguides, which can add one-bit binary numbers. We consider two different adder networks, one constructed with Hadamard gates and the other constructed with NOT\sqrt{\mathrm{NOT}}NOT​ gates. Reflection of electron probability at the gates causes a loss in fidelity of the overall calculation. A fidelity requirement for the implementation of individual rotation gates corresponding to a probability p>1∕2p>1∕2p>1∕2 of completing the addition successfully is found.
We show how a nonlocal N-qubit controlled unitary gate can be implemented locally and effectively by using non-symmetric quantum channels. We construct respectively two quantum networks for realizing conclusively this nonlocal quantum gate. The first one hires (N–2) symmetric qubit-qubit Bell states and a non-symmetric qubit-qudit Bell state as quantum channels. The basic idea of this scheme is to use (N–2) additional levels of this qudit to “hide” certain computational states of (N–1) nonlocal control states from the conditional dynamics, which results in an effective nonlocal N-qubit controlled unitary gate design. In this scheme, however, either the number of the additional levels or that of 1-qudit gates needs to increase with N. The other one can improve significantly the local implementation of this nonlocal gate if we harness (N–1) non-symmetric qubit-qutrit Bell states as quantum channels. This scheme uses respectively (N–1) qutrits’s additional levels to expose one and only one initial computational state of (N–1) nonlocal control states to the conditional dynamics. In comparison with the first one, the procedure is greatly simplified, and the total gate time is reduced. The fact that the quantum network that does the proposed implementation is built entirely of local single-body and two-body gates, and has only (3N–4) two-body gates is notable.
We present some results from simulation of a network of nodes connected by c-NOT gates with nearest neighbors. Though initially we begin with pure states of varying boundary conditions, the updating with time quickly involves a complicated entanglement involving all or most nodes. As a normal c-NOT gate, though unitary for a single pair of nodes, seems to be not so when used in a network in a naive way, we use a manifestly unitary form of the transition matrix with c?-NOT gates, which invert the phase as well as flipping the qubit. This leads to complete entanglement of the net, but with variable coefficients for the different components of the superposition. It is interesting to note that by a simple logical back projection the original input state can be recovered in most cases. We also prove that it is not possible for a sequence of unitary operators working on a net to make it move from an aperiodic regime to a periodic one, unlike some classical cases where phase-locking happens in course of evolution. However, we show that it is possible to introduce by hand periodic orbits to sets of initial states, which may be useful in forming dynamic pattern recognition systems.