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arXiv:0803.0766 · Physical Review A

Quantum computation in semiconductor quantum dots of electron-spin asymmetric anisotropic exchange

Xiang Hao, Shiqun Zhu

Published Oct 30, 2007• 11 citations
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Abstract

The universal quantum computation is obtained when there exists asymmetric anisotropic exchange between electron spins in coupled semiconductor quantum dots. The asymmetric Heisenberg model can be transformed into the isotropic model through the control of two local unitary rotations for the realization of essential quantum gates. The rotations on each qubit are symmetrical and depend on the strength and orientation of asymmetric exchange. The implementation of the axially symmetric local magnetic fields can assist the construction of quantum logic gates in anisotropic coupled quantum dots. This proposal can efficiently use each physical electron spin as a logical qubit in the universal quantum computation.

DOI
10.1103/physreva.76.044306

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The universal quantum computation is obtained when there exists asymmetric anisotropic exchange between electron spins in coupled semiconductor quantum dots. The asymmetric Heisenberg model can be transformed into the isotropic model through the control of two local unitary rotations for the realization of essential quantum gates. The rotations on each qubit are symmetrical and depend on the strength and orientation of asymmetric exchange. The implementation of the axially symmetric local magnetic fields can assist the construction of quantum logic gates in anisotropic coupled quantum dots. This proposal can efficiently use each physical electron spin as a logical qubit in the universal quantum computation.
We study the quantum computational power of a generic class of anisotropic solid-state Hamiltonians. A universal set of encoded logic operations are found, which do away with difficult-to-implement single-qubit gates in a number of quantum-computer proposals, e.g., quantum dots and donor atom spins with anisotropic exchange coupling, quantum Hall systems, and electrons floating on helium. We show how to make the corresponding Hamiltonians universal by encoding one qubit into two physical qubits, and by controlling nearest-neighbor interactions.
Some of the most promising proposals for scalable solid-state quantum computing, e.g., those using electron spins in quantum dots or donor electron or nuclear spins in Si, rely on a two-qubit quantum gate that is ideally generated by an isotropic exchange interaction. However, an anisotropic perturbation arising from spin-orbit coupling is inevitably present. Previous studies focused on removing the anisotropy. Here we introduce a new universal set of quantum logic gates that takes advantage of the anisotropic perturbation. The price is a constant but modest factor in additional pulses. The gain is a scheme that is compatible with the naturally available interactions in spin-based solid-state quantum computers.
We propose a method for quantum computation which uses control of spin-orbit coupling in a linear array of single electron quantum dots. Quantum gates are carried out by pulsing the exchange interaction between neighboring electron spins, including the anisotropic corrections due to spin-orbit coupling. Control over these corrections, even if limited, is sufficient for universal quantum computation over qubits encoded into pairs of electron spins. The number of voltage pulses required to carry out either single-qubit rotations or controlled-Not gates scales as the inverse of a dimensionless measure of the degree of control of spin-orbit coupling.
With the help of the spin-orbit interaction, we propose a scheme to perform holonomic single-qubit gates on the electron spin confined to a quantum dot. The manipulation is done in the absence (or presence) of an applied magnetic field. By adiabatic changing the position of the confinement potential, one can rotate the spin state of the electron around the Bloch sphere in semiconductor heterostructures. The dynamics of the system is equivalent to employing an effective non-Abelian gauge potential whose structure depends on the type of the spin-orbit interaction. As an example, we find an analytic expression for the electron spin dynamics when the dot is moved around a circular path (with radius RRR) on the two dimensional electron gas (2DEG) and show that all single-qubit gates can be realized by tuning the radius and orientation of the circular paths. Moreover, using the Heisenberg exchange interaction, we demonstrate how one can generate two-qubit gates by bringing two quantum dots near each other, yielding a scalable scheme to perform quantum computing on arbitrary NNN qubits. This proposal shows a way of realizing holonomic quantum computers in solid-state systems.
Universal quantum computation requires single-qubit control together with at least one entangling two-qubit gate. CNOT and CROT gates are two famous examples of such gates, wherein the state of one qubit (control) dictates the transformation applied to the other (target). By using a simple derivation motivated by symmetry, we show that current device architectures of semiconductor-based quantum dot devices prevent efficient implementation of a CNOT and other asymmetric control-target gates via Heisenberg exchange, Coulomb repulsion, or other interaction that is invariant under spin exchange. Guided by this general principle, we propose a heterogeneous blueprint of double quantum dot devices that enables efficient implementation of the CNOT by breaking the spin exchange symmetry. Crucially, our numerical simulations predict that this novel device blueprint can enable a single-pulse fault-tolerant 100-ns CNOT gate in isotopically purified silicon.
The exchange coupling between quantum dot spin qubits is isotropic, which restricts the types of quantum gates that can be formed. Here, we propose a method for controlling anisotropic interactions between spins arranged in a bus geometry. The symmetry is broken by an external magnetic field, resulting in XXZ-type interactions that can efficiently generate maximally entangled Greenberger-Horne-Zeilinger states or universal gate sets for exchange-only quantum computing. We exploit the XXZ couplings to propose a qubit scheme, based on double dots.