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arXiv:cond-mat/0009083 · Physical review. B, Condensed matter

Double-occupancy errors, adiabaticity, and entanglement of spin qubits in quantum dots

John Schliemann, Daniel Loss, Allan H. MacDonald

Published Feb 2, 2001• 239 citations
arXivHTMLPDF

Abstract

Quantum gates that temporarily increase singlet-triplet splitting in order to swap electronic spins in coupled quantum dots lead inevitably to a finite double-occupancy probability for both dots. By solving the time-dependent Schr\"odinger equation for a coupled dot model, we demonstrate that this does not necessarily lead to quantum computation errors. Instead, the coupled dot ground state evolves quasiadiabatically for typical system parameters so that the double-occupancy probability at the completion of swapping is negligibly small. We introduce a measure of entanglement that explicitly takes into account the possibilty of double occupancies and provides a necessary and sufficient criterion for entangled states.

DOI
10.1103/physrevb.63.085311

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Quantum gates that temporarily increase singlet-triplet splitting in order to swap electronic spins in coupled quantum dots lead inevitably to a finite double-occupancy probability for both dots. By solving the time-dependent Schr"odinger equation for a coupled dot model, we demonstrate that this does not necessarily lead to quantum computation errors. Instead, the coupled dot ground state evolves quasiadiabatically for typical system parameters so that the double-occupancy probability at the completion of swapping is negligibly small. We introduce a measure of entanglement that explicitly takes into account the possibilty of double occupancies and provides a necessary and sufficient criterion for entangled states.
We have calculated the exchange energy, double occupation probability of the lowest singlet state, and degree of entanglement of two holes in vertically coupled double Ge/Si quantum dots. We determined the conditions on which the exchange coupling is large enough for a fast swap operation in quantum computation and the double-occupancy probability is still low, thus maximizing the entanglement for a small computation error. We found that both the degree of entanglement and double-occupancy probability for quantum dots with different dot size collapse onto universal, size independent curves when plotted as a function of singlet-triplet splitting.
We study the corrections to adiabatic dynamics of two coupled quantum dot spin qubits, each dot singly occupied with an electron, in the context of a quantum computing operation. Tunneling causes double occupancy at the conclusion of an operation and constitutes a processing error. We model the gate operation with an effective two-level system, where nonadiabatic transitions correspond to double occupancy. The model is integrable and possesses three independent parameters. We confirm the accuracy of Dykhne's formula, a nonperturbative estimate of transitions, and discuss physically intuitive conditions for its validity. Our semiclassical results are in excellent agreement with numerical simulations of the exact time evolution. A similar approach applies to two-level systems in different contexts.
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We consider two quantum dots described by the Anderson-impurity model with one electron per dot. The goal of our work is to study the decay of a maximally entangled state between the two electrons localized in the dots. We prepare the system in a perfect singlet and then tunnel couple one of the dots to leads, which induces nonequilibrium dynamics. We identify two cases: If the leads are subject to a sufficiently large voltage and thus a finite current, then direct-tunneling processes cause decoherence and the entanglement as well as spin correlations decay exponentially fast. At zero voltage or small voltages and beyond the mixed-valence regime, virtual-tunneling processes dominate and lead to a slower loss of coherence. We analyze this problem by studying the real-time dynamics of the spin correlations and the concurrence using two techniques, namely, the time-dependent density matrix renormalization group method and a master-equation method. The results from these two approaches are in excellent agreement in the direct-tunneling regime for the case in which the dot is weakly tunnel coupled to the leads. We present a quantitative analysis of the decay rates of the spin correlations and the concurrence as a function of tunneling rate, interaction strength, and voltage.
Coherent spatial transport or shuttling of a single electron spin through semiconductor nanostructures is an important ingredient in many spintronic and quantum computing applications. In this work we analyze the possible errors in solid-state quantum computation due to leakage in transporting a single-spin qubit through a semiconductor double quantum dot. In particular, we consider three possible sources of leakage errors associated with such transport: finite ramping times, spin-dependent tunneling rates between quantum dots induced by finite spin-orbit couplings, and the presence of multiple valley states. In each case we present quantitative estimates of the leakage errors, and discuss how they can be minimized. The emphasis of this work is on how to deal with the errors intrinsic to the ideal semiconductor structure, such as leakage due to spin-orbit couplings, rather than on errors due to defects or noise sources. In particular, we show that in order to minimize leakage errors induced by spin-dependent tunnelings, it is necessary to apply pulses to perform certain carefully designed spin rotations. We further develop a formalism that allows one to systematically derive constraints on the pulse shapes and present a few examples to highlight the advantage of such an approach.
We study the two-qubit controlled-NOT gate operating on qubits encoded in the spin state of a pair of electrons in a double quantum dot. We assume that the electrons can tunnel between the two quantum dots encoding a single qubit, while tunneling between the quantum dots that belong to different qubits is forbidden. Therefore, the two qubits interact exclusively through the direct Coulomb repulsion of the electrons. We find that entangling two-qubit gates can be performed by the electrical biasing of quantum dots and/or tuning of the tunneling matrix elements between the quantum dots within the qubits. The entangling interaction can be controlled by tuning the bias through the resonance between the singly occupied and doubly occupied singlet ground states of a double quantum dot.