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arXiv:1003.4590 · Quantum Information and Computation

Dirac four-potential tunings-based quantum transistor utilizing the Lorentz force

Agung Trisetyarso

Published Nov 1, 2012• 9 citations
arXivHTMLPDF

Abstract

We propose a mathematical model of \textit{quantum} transistor in which bandgap engineering corresponds to the tuning of Dirac potential in the complex four-vector form. The transistor consists of $n$-relativistic spin qubits moving in \textit{classical} external electromagnetic fields. It is shown that the tuning of the direction of the external electromagnetic fields generates perturbation on the potential temporally and spatially, determining the type of quantum logic gates. The theory underlying of this scheme is on the proposal of the intertwining operator for Darboux transfomations on one-dimensional Dirac equation amalgamating the \textit{vector-quantum gates duality} of Pauli matrices. Simultaneous transformation of qubit and energy can be accomplished by setting the $\{\textit{control, cyclic}\}$-operators attached on the coupling between one-qubit quantum gate: the chose of \textit{cyclic}-operator swaps the qubit and energy simultaneously, while \textit{control}-operator ensures the energy conservation.

DOI
10.26421/qic12.11-12-7

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We propose a mathematical model of \textit{quantum} transistor in which bandgap engineering corresponds to the tuning of Dirac potential in the complex four-vector form. The transistor consists of nnn-relativistic spin qubits moving in \textit{classical} external electromagnetic fields. It is shown that the tuning of the direction of the external electromagnetic fields generates perturbation on the potential temporally and spatially, determining the type of quantum logic gates. The theory underlying of this scheme is on the proposal of the intertwining operator for Darboux transfomations on one-dimensional Dirac equation amalgamating the \textit{vector-quantum gates duality} of Pauli matrices. Simultaneous transformation of qubit and energy can be accomplished by setting the {control, cyclic}\{\textit{control, cyclic}\}{control, cyclic}-operators attached on the coupling between one-qubit quantum gate: the chose of \textit{cyclic}-operator swaps the qubit and energy simultaneously, while \textit{control}-operator ensures the energy conservation.
This paper systematically reviews the theoretical progression from blackbody radiation to the foundations of quantum computing. Beginning with Planck’s energy quantization and the representation of electromagnetic wave electric field components, the wave-particle duality of photons is analogized to introduce the de Broglie relation between momentum and wave vector for matter waves, leading to the establishment of the matter wave function and the Schrödinger equation as its governing partial differential equation. The consistency between the rigorous solution for a one-dimensional infinitely deep rectangular potential well and the conclusions for a one-dimensional hydrogen atom is demonstrated. Subsequently, within the Dirac notation formalism, the wave function is expressed, and the correspondence between spatial differentiation and momentum is derived, formalizing the momentum operator (often referred to as first quantization). To lay the physical groundwork for manipulating two-qubit states in quantum computing, the potential energy function of an electron’s orbital magnetic moment in an external magnetic field is introduced, starting from the torque and work on a current-carrying rectangular coil. This is extended to the magnetic moment and potential energy of a spinning electron, where the orientation of the magnetic field determines the form of the potential and accordingly selects the three components of the Pauli operators. The properties of quantum gates corresponding to the Pauli operators are then examined for single-qubit states. A two-qubit state, constructed via the coupling of two arbitrary single-qubit states represented in a two-level system or electron spin, is shown to be uniquely expressible as a superposition of four eigen two-qubit states. The paper further investigates joint manipulations of two-qubit states, including evolution under X and Z gates conditioned on high- and low-level states, the representation of control-flow circuits (with the low-level as control), and the formulation of measurement and CNOT gates. Throughout this work, essential mathematical tools—including integration, matrix operations, complex Euler’s formula, and solutions to simple second-order ordinary differential equations—are incorporated to provide a coherent theoretical framework for state representation and gate operations in quantum computing.
We present the recent works \cite{trisetyarso2011} on the application of Darboux transformation on one-dimensional Dirac equation related to the field of Quantum Information and Computation (QIC). The representation of physical system in one-dimensional equation and its transformation due to the Bagrov, Baldiotti, Gitman, and Shamshutdinova (BBGS)-Darboux transformation showing the possibility admitting the concept of relativity and the trade-off of concurrent condition of quantum and classical physics play into the area of QIC. The applications in cavity quantum electrodynamics and on the proposal of quantum transistor are presented.
Controlling the time evolution of the population of two states in cavity quantum electrodynamics is necessary by tuning the modified Rabi frequency in which the extra classical effect of electromagnetic field is taken into account. The theoretical explanation underlying the perturbation of potential on spatial regime of bloch sphere is by the use of Bagrov–Baldiotti–Gitman–Shamshutdinova–Darboux transformations [Bagrov et al., “Darboux transformation for two-level system,” Ann. Phys. 14, 390 (2005)] on the electromagnetic field potential in one-dimensional stationary Dirac model in which the Pauli matrices are the central parameters for controlling the collapse and revival of the Rabi oscillations. It is shown that by choosing σ1 in the transformation generates the parabolic potential causing the total collapse of oscillations, while {σ2,σ3} yield the harmonic oscillator potentials ensuring the coherence of qubits.
Dirac's equation with static external fields is considered as an equation of relativistic quantum theory, which describes the quantum dynamics of an electron in four-dimensional space-time. It is shown that the Dirac electron has only one intrinsic degree of freedom -- spin. The quantum ensemble of an electron with a given energy consists of two inseparable from each other subensembles of heavy and light spin-1/2 quasiparticles to have opposite (relative) intrinsic parities. In the case of electrical scalar potential nonuniform in the direction of a particle motion, these subensembles behave as analogs of the electrical and magnetic components of the plane monochromatic electromagnetic wave propagating in a dispersive medium, nonuniform in the direction of a wave motion. The role of states with negative energies and the properties of the velocity operator are explained.
Quantum computation has traditionally been formulated by postulating abstract information carriers and subsequently identifying physical systems that realize them. We adopt the opposite viewpoint and ask what computational structure is supplied by a fundamental relativistic quantum system itself. Focusing on the simplest nontrivial massive spin carrier, spin-1/2, we show that its relativistic description supplies a native four-dimensional information carrier through the Dirac equation. The resulting Dirac information carrier possesses an intrinsic positive- and negative-energy decomposition that induces a physics-constrained computational structure comprising sector-preserving and sector-coupling quantum logic. Restricting the relativistic computational structure to the positive-energy sector and taking the nonrelativistic regime recovers the familiar Pauli qubit description. We distinguish mathematical unitary transformations from physically admissible gates and show how charge-conjugation structure, charge superselection, and associated reference-frame resources can constrain quantum logic beyond the sector-preserving structure. Finally, under appropriate physical assumptions, we establish the conditions under which this structure supports full single-carrier controllability. Our work establishes the Dirac information carrier as the massive spin-1/2 instance of a broader programme in which computational structures are derived from the relativistic representation carried by the underlying physical system.
Transistors play a vital role in classical computers, and their quantum mechanical counterparts could potentially be as important in quantum computers. Where a classical transistor is operated as a switch that either blocks or allows an electric current, the quantum transistor should operate on quantum information. In terms of a spin model the in-going quantum information is an arbitrary qubit state (spin-1/2 state). In this paper, we derive a model of four qubits with Heisenberg interactions that works as a quantum spin transistor, i.e. a system with perfect state transfer or perfect blockade depending on the state of two gate qubits. When the system is initialized the dynamics complete the gate operation, hence our protocol requires minimal external control. We propose a concrete implementation of the model using state-of-the-art superconducting circuits. Finally, we demonstrate that our proposal operates with high-fidelity under realistic decoherence.