Spin quantum computation in silicon nanostructures

https://doi.org/10.1016/j.ssc.2004.12.037Get rights and content

Abstract

Proposed silicon-based quantum-computer architectures have attracted attention because of their promise for scalability and their potential for synergetically utilizing the available resources associated with the existing Si technology infrastructure. Electronic and nuclear spins of shallow donors (e.g. phosphorus) in Si are ideal candidates for qubits in such proposals because of their long spin coherence times due to their limited interactions with their environments. For these spin qubits, shallow donor exchange gates are frequently invoked to perform two-qubit operations. We discuss in this review a particularly important spin decoherence channel, and bandstructure effects on the exchange gate control. Specifically, we review our work on donor electron spin spectral diffusion due to background nuclear spin flip–flops, and how isotopic purification of silicon can significantly enhance the electron spin dephasing time. We then review our calculation of donor electron exchange coupling in the presence of degenerate silicon conduction band valleys. We show that valley interference leads to orders of magnitude variations in electron exchange coupling when donor configurations are changed on an atomic scale. These studies illustrate the substantial potential that donor electron/nuclear spins in silicon have as candidates for qubits and simultaneously the considerable challenges they pose. In particular, our work on spin decoherence through spectral diffusion points to the possible importance of isotopic purification in the fabrication of scalable solid state quantum computer architectures. We also provide a critical comparison between the two main proposed spin-based solid state quantum computer architectures, namely, shallow donor bound states in Si and localized quantum dot states in GaAs.

Introduction

During the past decade, the study of quantum computing and quantum information processing has generated widespread interest among physicists from areas ranging from atomic physics, optics, to various branches of condensed matter physics [1], [2]. The key thrust behind the rush toward a working quantum computer (QC) is the development of a quantum algorithm that can factorize large numbers exponentially faster than any available classical algorithm [3]. This exponential speedup is due to the intrinsic quantum parallelism in the superposition principle and the unitary evolution of quantum mechanics. It implies that a computer made up of entirely quantum mechanical parts (qubits), whose evolution is governed by quantum mechanics, would be able to carry out prime factorization of large numbers that is prohibitively time-consuming in classical computation, thus revolutionizing cryptography and information theory. Since the invention of Shor's factoring algorithm, it has also been shown that error correction can be done to a quantum system [4], so that a practical QC does not have to be forever perfect to be useful, as long as quantum error corrections can be carried out on the QC. These two key mathematical developments have led to the creation of the new interdisciplinary field of quantum computation and quantum information.

Many physical systems have been proposed as candidates for qubits in a QC. Among the more prominent examples are electron or nuclear spins in semiconductors [5], [6], including electron spin in semiconductor quantum dots [7], [8] and donor electron or nuclear spins in semiconductors [9], [10], [11]. The donor-based QC schemes are particularly interesting because all donor electron wavefunctions in a semiconductor are identical, and because doping makes a natural connection between quantum mechanical devices and the more traditional microelectronic devices: Doping in semiconductors has had significant technological impact for the past 50 years and is the basis of the existing microelectronics technology. As transistors and integrated circuits decrease in size, the physical properties of the devices are becoming sensitive to the actual configuration of impurities [12]. In this context, the important proposal of donor-based silicon quantum computer (QC) by Kane [9], in which the nuclear spins of the monovalent 31P impurities in Si are the qubits, has naturally created considerable interest in revisiting all aspects of the donor impurity problem in silicon, particularly in the Si:31P system.

In principle, both electron spin and orbital degrees of freedom can be used as qubits in semiconductor nanostructures. For example, electron orbital dynamics is quantized into discrete ‘atomic-like’ levels (on meV energy scale with a Bohr radius of the order of 10 nm) in semiconductor quantum dots, and two such quantized quantum dot levels could form the quantum two-level system needed for a qubit [13]. A great advantage of such orbital (or equivalently, charge) qubits is that qubit-specific measurements are relatively simple since one is essentially measuring single charge states, which is a well-developed experimental technique through single-electron transistors (SET) or equivalent devices [14]. A major disadvantage of solid-state charge qubits is that these orbital states are highly susceptible to interactions with the environment (which, in particular, contains all the stray or unintended charges inevitably present in the device), and the decoherence time is generally far too short (typically picoseconds to nanoseconds) for quantum error correction to be useful. A related problem is that inter-qubit coupling, which is necessary for the implementation of two-qubit gate operations essential for quantum computation, is often the long-range dipolar coupling for charge qubits. This makes it difficult to scale up the architecture, since decoherence grows with the scaling-up as more and more qubits couple to each other via the long-range dipolar coupling. Additional decoupling techniques have to be applied to ensure selective gate operations [15]. Since scalability is thought to be the main advantage of solid state QC architectures, little serious attention has so far been paid to orbital qubit based quantum computation in semiconductor nanostructures due to its fundamentally unscalable decoherence and entanglement properties.

Spin qubits in semiconductor nanostructures have complementary advantages (and disadvantages) compared with charge qubits based on quantized orbital states. A real disadvantage of spin qubits is that a single electron spin is difficult to measure, although there is no fundamental principle against the measurement of a Bohr magneton. The great advantage of spin qubits is the very long spin coherence times, which even for electron spins can be milliseconds (microseconds) in silicon (GaAs) at low temperatures. This six orders of magnitude coherence advantage in spin qubits over charge qubits has led to electron spin qubits in GaAs quantum dots and in P donor levels in silicon (as well as SiGe quantum structures) as the QC architectures of choice for the solid state community. In addition to the coherence advantage, spin qubits also have a considerable advantage that the exchange gate, which provides the inter-qubit coupling, is exponentially short-ranged and nearest-neighbor in nature, thus allowing precise control and manipulation of two-qubit gates. There is no fundamental problem arising from the scaling-up of the QC architecture since exchange interaction couples only two nearest-neighbor spin qubits independent of the number of qubits. In this review we provide a brief perspective on spin qubits in silicon with electron spins in shallow P donor levels in Si being used as qubits.

Although experimental progress in semiconductor-based solid state QC schemes has been slow to come during the past 5 years, they are still often considered promising in the long term because of their perceived scalability advantages. After all, the present computer technology is based on semiconductor integrated circuits with every smaller feature size. Therefore, semiconductor nanostructure-based QC architectures should in principle be scalable using the existing microelectronics technology. However, it still remains to be demonstrated whether and how the available (classical) semiconductor technology can help the scaling up of a quantum coherent QC architecture. For the spin qubits in silicon, for example, the key issues include clarifications of spin quantum coherence properties in the solid state environment, physical approaches to manipulate and entangle spins, fabrication of devices with atomic-scale precision, and measurement of single spins. In the following, we review our work on two of these important issues: donor electron spin coherence and spin interaction in silicon.

Section snippets

Spin coherence in silicon nanostructures

Before the seminal concept of quantum error correction was introduced in 1995 [4], it was widely believed that quantum computation, even as a matter of principle, is quite impossible since all quantum states decohere due to interaction with the environment, and such decoherence was thought to be fatal to QC operations. Although the quantum error correction principle has shown that a certain degree of decoherence can be corrected in QC algorithms, one still has severe limits on the amount of

Donor electron exchange in silicon

An important issue in the study of donor-based Si QC architecture is coherent manipulations of spin states as required for the quantum gate operations. In particular, two-qubit operations, which are required for a universal QC, involve precise control over electron–electron exchange [7], [9], [11], [31] and electron–nucleus hyperfine interactions (for nuclear spin qubits). Such control can presumably be achieved by fabrication of donor arrays with accurate positioning and surface gates whose

Summary

In summary, we have briefly reviewed two important issues related to donor-spin based quantum computing in silicon: quantum coherence of electron spins and exchange interaction among donor electrons. Our results show that the spin qubits based on donors in silicon have remarkable potential for very long quantum coherence times through isotopic purification. On the other hand, they also pose immense challenges in terms of precise nanostructure fabrications because of the degenerate nature of the

Acknowledgements

This work was supported by ARDA and LPS-NSA, as well as by CNPq, Instituto do Milênio de Nanociências and FAPERJ in Brazil, by ARO-ARDA at the University at Buffalo and the University of Maryland, and by DARPA SpinS program at the University of California at Berkeley.

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