Quantum computers are expected to far surpass the capabilities of today's most powerful supercomputers, particularly in areas such as the theoretical simulation of quantum systems, cryptography, and information processing. The cluster state is a special, highly entangled quantum state that forms the universal resource on which measurement-based quantum computation can be performed. This paper provides a brief review of the theoretical foundations of cluster state quantum computation and how it evolved from the traditional model of digital computers. It then proposes a scheme for the generation of such entanglement in a solid-state medium through the suppression of resonant tunneling of a ballistic electron by a single-electron charge qubit. To investigate the viability of the scheme for the creation of cluster states, numerical calculations are performed in which the entanglement interaction is modeled in detail.

The simulation of quantum systems is an integral part of modern science. Such simulations are difficult, however, due to the massive computational resources they require. Quantum computers [

We begin this paper by giving a brief review of the theoretical foundations of cluster state quantum computation and how it evolved from the traditional model of digital computers. In the next section, we outline a scheme to generate controlled entanglement in a solid-state system using resonant tunneling and give details of the simulation and numerical techniques used to test it. In Section

It was Turing, in his revolutionary 1936 paper “On Computable Numbers, with an Application to the Entscheidungsproblem’’ [

Unlike the universal Turing machine, a modern computer is a digital device based on the laws of electrodynamics. In place of paper tape, computers use binary digits, called

It is the universality of digital computers that has made them such an invaluable tool in modern life and enabled complex simulations of physical systems to be performed. This has led scientists to ask an important question: “what kind of physical systems can be efficiently simulated on a digital computer?’’ Turing envisaged his universal machine performing computations using an infinite paper tape over some finite, but arbitrarily long, amount of time. As infinite computational resources are impossible, the efficiency with which a computer uses its resources and the time a computation takes to complete impose a physical limit on what can and can not be “reasonably’’ simulated. This constraint, called

It was the question of efficiency, and in particular the intractability of simulating quantum systems, that lead Feynman in 1982 [

One of the defining features of quantum computation is

To construct a quantum computer, input must first be encoded onto a logical unit that can be manipulated in the required manner during computation. The quantum binary digit, or

The Bloch Sphere—a geometric representation of the Hilbert space of a qubit. Any pure qubit state, modulo a global phase, can be represented by a point on the surface of the sphere given by

As the state of a single qubit, modulo a global phase, must always reside on the Bloch Sphere, the action of any single-qubit unitary operation

To perform quantum computation a number of qubits

Due to its similarity with traditional digital circuits, the most widely used representation of quantum computation is the

An important example of a quantum circuit is the single-qubit “teleporter’’ which is represented by the following circuit diagram:

In quantum circuit diagrams, time flows from left to right with the horizontal lines, called

It is a remarkable property of the quantum circuit model that any unitary evolution operator needed for quantum computation can be perfectly simulated using only two-qubit gates

The universality, both exact and approximate, of one- and two-qubit gates makes the quantum circuit model a powerful tool for the design and analysis of quantum algorithms. Practically, however, efficient physical implementations of the model have proved difficult to construct. Problems arise due to the need for both fault tolerant multiqubit gates and robust scalable qubits within a single system. For systems with strongly interacting qubits, multiqubit gates are more easily constructed; however, the qubits tend to experience high levels of decoherence making them less robust. Conversely, for systems with weakly interacting qubits, the qubits are more robust but multiqubit gates are much harder to construct. This has lead physicists to consider a new model of quantum computation that can be more easily implemented—the cluster state or one-way computation model.

The cluster state model for quantum computation was proposed by Raussendorf and Briegel in 2001 [

The physical resource of cluster state computation is the titular “cluster state,’’ which is a family of states of entangled qubits that can be defined by some graph

Recall that a single-qubit rotation

where the CNOT gate has been replaced by the CZ gate using the relation

where

where the measurement basis is given in each vertex and the measurement round denoted by the subscript. By adapting the sign of the rotations based on the results of the previous measurements and compensating for any prefactor using a classical computation, an operation equivalent, modulo a global phase, to any single-qubit rotation

The cluster state model of quantum computation offers a unique way of designing quantum algorithms. More than this, it provides a robust scheme for implementing a practical computing device. Such a device does not suffer from many of the disadvantages associated with a quantum circuit implementation as it uses only single qubit measurements to perform computation. Furthermore, it has been shown that a cluster state quantum computer would be more resilient to decoherence effects [

Multiqubit entanglement forms the universal resource on which all cluster state quantum computation is performed. To date, many physical systems have been proposed for the generation of such states including photons [

In this paper, we present a detailed analysis on the entanglement generated between a single-electron charge qubit and a ballistic electron based on resonant tunneling. The modeled system consists of a charge qubit formed by a single-electron confined to a GaAs DQD. A GaAs quantum wire containing symmetric rectangular resonant tunneling barriers, formed by InAs slices, is positioned near to the centre of one of the dots and a ballistic electron pumped through the wire by a single-electron source [

A schematic diagram of the system for generating entanglement between a ballistic electron and double quantum dot (DQD) single-electron charge qubit. The ballistic electron is pumped through the resonant tunneling barriers in the quantum wire by a single-electron source. Resonant tunneling occurs when the qubit electron is localized to the dot furthest from the wire, the

Both the qubit and quantum wire being modeled are confined to the conduction band of a GaAs semiconductor heterostructure. Lattice and screening effects are accounted for by using a linear approximation for the effective electron mass and permittivity of

The qubit modeled in this work is based on the single-electron lateral coupled DQDs experimentally implemented by Petta et al. [

To describe the initial state of the isolated qubit electron a two-dimensional wavefunction

The values of

The qubit potential and electron probability density in the two localized computational basis states

The ballistic electron being modeled is confined to a GaAs quantum wire containing double symmetric resonant tunneling barriers formed by two InAs slices embedded in the quantum wire heterostructure. Currently, GaAs quantum wires have been produced down to diameters of

As initially the ballistic electron experiences no interactions with the qubit electron, its isolated wavefunction

To simulate the total system, the electrons in the qubit and quantum wire are described by the three-dimensional joint wavefunction

The dynamics of the system are governed by the time-dependent Schrödinger equation:

The numerical techniques used in this simulation are based on those of Hines et al. [

Due to the complexity of the system being investigated, analytic solutions for the action of the evolution operator do not exist and so the Chebyshev-Fourier method [

The quantum mechanical phenomenon of coherent resonant tunneling describes the complete transmission of a ballistic electron through a set of barriers that is classically disallowed. It occurs when the incident energy of the ballistic electron closely matches one of the virtual energy eigenstates of the potential well formed by the barriers. At these specific energies the transmission probability is sharply peaked and the barriers are effectively “transparent’’ to the ballistic electron’s wavefunction. The particular energies at which these peaks occur is determined solely by the geometry of the barrier potential and for the one-dimensional case of an isolated ballistic electron and symmetric rectangular double barriers analytic solutions can be derived [

The calculated transmission probabilities for specific incident electron energies on a symmetric rectangular double barrier of height 1.4 meV and separation 90 nm with barrier widths 40 nm (dashed), 50 nm (solid), and 60 nm (dotted). Values were calculated analytically for an isolated ballistic electron.

The calculated transmission probabilities for specific incident electron energies on a symmetric rectangular double barrier of height 1.4 meV and width 50 nm with barrier separations 80 nm (dashed), 90 nm (solid), and 100 nm (dotted). Values were calculated analytically for an isolated ballistic electron.

It has been shown [

While there are many measures for quantifying the degree of entanglement, the most widely accepted, and that used here, is the concurrence or “entanglement of formation’’ [

To optimize the entanglement generated by a single resonant tunneling event, a simplex minimization method was used to alter the separation and width of the resonant tunneling barriers. For each set of barrier parameters, a series of simulations were run using the maximally mixed pure qubit state

Using the optimized system parameters, the probability of the ballistic electron becoming trapped was found to be negligible, the final state of the ballistic electron can be seen in Figure

The probability density for the ballistic electron after entanglement (

To investigate how resonant tunneling leads to this high level of entanglement, the transmission probabilities for the initial qubit states

The concurrence of the system for the qubit in the

The total potential experienced by the ballistic electron when the qubit is in the initial states

To analyze the magnitude of this contribution to the entanglement, the mutual information about the initial state of the qubit

A comparison of the concurrence of the system (solid line) and the classical mutual information (dashed line) between the initial state of the qubit and the final position of the electron in the quantum wire over a range of initial ballistic electron energies and with optimized barrier parameters.

The generation of a cluster state requires not only maximal entanglement but also entanglement between all nearest neighboring qubits. If the resonant tunneling system is to be used to perform such entanglement in a

To test the stability of the system, a series of simulations were run using small perturbations in each of the optimized parameters and the linear response of the concurrence calculated. It was found that a 1 nm change in the width and separation of the barriers gives an average change in the concurrence of

In this paper we have provided a brief review of the theoretical foundations of cluster state quantum computation and how it evolved from the traditional model of digital computers. We also reported on numerical simulations of a solid-state system for generating controlled entanglement between a single-electron charge qubit and a ballistic electron using the suppression of coherent resonant tunneling in a quantum wire. It was found that a high level of entanglement could be achieved, with a concurrence of 0.998, and that the state of the qubit was unaffected by the process. The primary mechanism of entanglement was the difference in the transmission probabilities corresponding to the two localized basis states of the qubit. These differences occurred due to the Coulombic interaction between the electrons changing the ballistic electron’s energy which shifted the resonance peaks. If the ballistic electron can be used to entangle further qubits, then this scheme provides a viable method for the generation of cluster states.

The concurrence was found to be highly sensitive to the system parameters especially the barrier separation. This poses a significant limitation on any practical implementation of this scheme as it requires the precise fabrication of the resonant tunneling barriers and means that the system will be heavily affected by imperfections in their interface. The stability may be improved by using a different geometry for the barriers or a greater number but the effect this would have on the concurrence is unclear and more work needs to be done in this area. Furthermore, while the simulation was closely modeled on experimentally realizable components and treated the ballistic and qubit electrons and their interactions in detail, it did not include decoherence effects that would be present in a real system. This means that our results provide an idealized upper limit for the possible entanglement that such a practical implementation could produce.

The authors would like to thank N. Menicucci and C. Hines for valuable discussions. This work was supported by iVEC—The hub of advanced computing in Western Australia, The University of Western Australia, and the Australian Research Council.