Self-assembled nanoelectronic quantum computer based on the Rashba effect in quantum dots

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Quantum computers promise vastly enhanced computational power and an uncanny ability to solve classically intractable problems. However, few proposals exist for robust, solid-state implementation of such computers where the quantum gates are
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    a  r   X   i  v  :  q  u  a  n   t  -  p   h   /   9   9   1   0   0   3   2  v   6   7   N  o  v   2   0   0   0 A Self Assembled Nanoelectronic Quantum ComputerBased on the Rashba Effect in Quantum Dots S. Bandyopadhyay ∗ Department of Electrical EngineeringUniversity of NebraskaLincoln, Nebraska 68588-0511, USA Abstract Quantum computers promise vastly enhanced computational power and an un-canny ability to solve classically intractable problems. However, few proposals existfor robust, solid state implementation of such computers where the quantum gates aresufficiently miniaturized to have nanometer-scale dimensions. Here I present a newdesign for a nanoscale universal quantum gate. It consists of two adjacent quantumdots each containing a single electron. The spin of each electron encodes a qubit.Ferromagnetic contacts are used to coherently inject an electron into each dot witha definite spin orientation, thus defining the initial state of the qubit. To rotate thequbit, we exploit the fact that the ground state of the electron is spin-split becauseof the magnetic field caused by the ferromagnetic contacts, as well as the Rashba in-teraction arising from spin-orbit coupling. Arbitrary qubit rotations are effected bymodulating the Rashba splitting with an external electrostatic potential and bringingthe total spin splitting energy in a target quantum dot in resonance with a global acmagnetic field. . The controlled dynamics of the universal 2-qubit rotation operationcan be realized by controlling the exchange coupling between the two dots with yetanother gate potential which changes the overlap between the wavefunctions of thetwo electrons. The qubit (spin orientation) is read via the current induced betweenthe ferromagnetic layers under an applied potential. The ferromagnetic layers act as“polarizers” and “analyzers” for spin injection and detection. A complete prescriptionfor initialization of the computer and data input/output operations is presented. Fi-nally, we define a clear pathway towards self assembling this structure using chemicalsynthesis and some lithography. ∗ Corresponding author. E-mail: 1  1 Introduction There is significant current interest in quantum computers because they possess vastly en-hanced capabilities accruing from quantum parallelism [1, 2]. Some quantum computing algorithms [3, 4] have been shown to be able to solve classically intractable problems. i.e. perform tasks that no classical algorithm could perform efficiently or tractably. Quantumcomputers and quantum memories are also the backbone of quantum repeaters and telepor-tation networksExperimental effort in realizing quantum computers has been geared towards synthesizinguniversal quantum logic gates from which quantum computers can be built. A universal gateis a 2 qubit-gate [5, 6, 7] and has basically two attributes. First, it allows arbitrary unitary rotations on each qubit and second, it performs the quantum controlled rotation operationwhereby one of the qubits (the target qubit) is rotated through an arbitrary angle, if, and onlyif, the other qubit (the control qubit) is oriented in a specified direction. The orientation of the control qubit is left unchanged. It is this conditional dynamics of the controlled rotationoperation that is challenging to implement experimentally.Recently, it has been shown [8, 9, 10] that there exist universal falut-tolerant computers that can operate in a non-ideal noisy environment. They are usually a circuit composedof one- or two-qubit gates performing various unitary rotations on a qubit (e.g Hadamard,Pauli rotations through rational or irrational angles, etc.). They too can, in principle, berealized from the basic gate that we discuss here.The most vexing problem in experimentally realizing quantum computers is the issue of decoherence. Qubits are coherent superpositions of two-level states and, as such, are delicateentities. Any coupling to the environment will destroy the coherence of the superpositionstate and corrupt the qubit. Were it not for the recent discovery of quantum error correctingcodes [11] that can correct errors due to decoherence through the use of appropriate  software  ,quantum computing would have remained a theoretical curiosity.In the past, atomic systems were proposed as ideal testbeds for experimental quantumlogic gates because of the relatively long coherence times associated with the quantum statesof trapped atoms and ions [12]. Experimental demonstrations of quantum logic gates werecarried out in atomic systems [13, 14]. Recently, nuclear magnetic resonance (NMR) spec- troscopy has been shown to be an attractive alternative [15, 16] and there has been some reports of experimental demonstrations involving NMR [17]. However, there are also somedoubts regarding the efficacy of NMR based approaches when dealing with many qubits [18].The major drawback of both atomic and NMR systems is of course that they are unwieldy,expensive and inconvenient. Solid state (especially nanoelectronic) implementations wouldbe much more desirable because they are amenable to miniaturization. One would like aquantum gate that is one nanometer long and not one meter long. The technology base thatexists in the solid-state area with regards to miniaturization is unparalleled.While it is understood that solid state systems will be preferable vehicles for quantumcomputation, it is also well-known that the phase memory time of charge carriers in solidssaturate to only a few nanoseconds as the lattice or carrier temperature is lowered to a few2  millikelvins [19] (this is caused by coupling of carriers to the zero-point motion of phonons).Thus, solid state implementations of quantum gates where the qubits are coupled to chargedegrees of freedom will be always dogged by serious decoherence problems. Even thoughsuch systems have been proposed in the past [20, 21, 22], they will require clock speeds in the far infra-red frequency range to meet Preskill’s criterion for fault-tolerant computing[23].A possible solution of this problem is to use the spin degrees of freedom in solid statesystems to encode qubits since the spin is not coupled to electromagnetic noise and henceshould have much longer coherence times than charge. It has been shown that electronic andnuclear spins of phosphorus dopant atoms  31 P in silicon have very long spin-flip times (orthe co-called  T  1  times in the language of spectroscopy) of about an hour [24]. Consequently,nuclear spins of   31 P dopant atoms in silicon have been advocated as preferable vehicles forqubits [26, 25, 27]. However, the actual coherence time (or  T  2  time) of electron spin in P-doped silicon may be on the order of a millisecond. Compound semiconductors may exhibitsomewhat shorter spin coherence times, but spin coherence times as long as 100 ns havebeen experimentally demonstrated in n-type GaAs at the relatively balmy temperature of 5K [28]. Thus, it is practical to contemplate solid state quantum computers based on singleelectron spins.Not all semiconductors however are suitable hosts for qubits. Pyroelectric materials(uniaxial crystals without inversion symmetry) usually exhibit electric dipole spin reso-nance which can increase the spin flip rate significantly [29] by strongly coupling the spin tophonons. An advantage of quantum dots is that the spin-phonon interaction may be reducedbecause of a constriction of the phase space for scattering. Moreover, the phonon-bottleneckeffect [30] may block phonon-induced spin-flip transitions. Another obvious strategy to in-crease the coherence time is to decrease the phonon population by reducing the temperature.The temperature must be low in any case since the time to complete a quantum calculationshould not significantly exceed the thermal time scale ¯ h/kT   [31] irrespective of any otherconsideration.Quantum gates based on spin polarized single electrons housed in quantum dots havebeen proposed by us in the past [32] and more recently by Loss and DiVincenzo [33]. Here we adopt a different idea - which is still based on spin-polarized single electrons - to provide arealistic paradigm for the realization of a  self-assembled   solid-state, nanoelectronic universalquantum gate. 2 A Self Assembled Universal Quantum Gate Consider two adjacent  penta-layered   quantum dot structures shown in Fig. 1(a). In each dot,the two outer layers are ferromagnetic and the middle layer is a semiconductor. Insulatinglayers separate the ferromagnetic layers from the semiconductor layer in order to provide apotential barrier for an electron injected into the semiconductor layer. We will describe laterhow one can self-assemble this structure.3  T CAluminumAluminaSCFMFM I T CA´ 1  A´ 2 A 1  A 2 ↓↓ (a) (b) (c) zyxy (d) F  MI     S  C     y  x  Figure 1:  (a) Cross-section of a penta-layered quantum dot.  FM  refers to ferromagnet,  SC  refers tosemiconductor, and  I  refers to insulator. (b) The top view showing the ohmic contacts for measuringspin polarized current in the target ( T ) and control ( C ) dots, and the Schottky contacts  A 1 ,  A 2 , A ′ 1  and  A ′ 2 . These contacts are defined by fine-line lithography. (c) a “spin-wire” for transmittingquantum information is made out of exchange coupled single electron cells.  which induce the Rashabeffect and turn exchange coupling between  C  and  T  on and off. (d) The energy band diagram forthe structure along the x-direction (perpendicular to heterointerfaces) showing the spin-split statesand the corresponding spatial wavefunctions. 4  A dc potential pulse is applied between the two outer (ferromagnetic) layers to inject asingle spin-polarized electron coherently from one of the outer layers into the middle layer.We will discuss the ramifications of such spin injection later. In the middle layer, theelectron’s ground state is spin-split because of Rashba interaction [34, 35, 36]. The Rashba effect arises from spin-orbit coupling in the presence of a transverse electric field which isalways present at the interface of two dissimilar materials owing to the conduction banddiscontinuity. It is possible to electrostatically  modulate   this spin splitting [38] by applyinga transverse potential using lithographically delineated gate contacts. The applied potentialalters the interface field that causes the Rashba effect and hence changes the spin-splittingenergy. 2.1 Single qubit rotation A target qubit is selected for rotation by bringing its spin-splitting energy in resonancewith the external global ac magnetic field by applying a suitable potential pulse to the gatecontacts. Arbitrary qubit (spin) rotation is achieved by varying the pulse duration, i.e.the duration of resonance with the ac magnetic field. Such a procedure realizes the firstingredient of a universal quantum gate, namely arbitrary single qubit rotations. 2.2 Two qubit controlled dynamics In order to achieve the second and last ingredient of a universal quantum gate - namelythe conditional dynamics of a universal 2-qubit gate - we need to couple the rotation of onequbit (target qubit) with the orientation of another qubit (control qubit). This can be doneby exploiting the exchange coupling between two single electrons in two neighboring dots.The spin-splitting energy in any dot depends, among other things, on the spin orientationin the neighboring dot if the two dots are exchange coupled. It is obvious that the totalspin splitting ∆ target  in the target dot depends on the exchange interaction  J   with theneighboring (control) dot (and hence on the spin orientation of the control qubit) if the twodots are exchange coupled. After all, the exchange term will appear in the Hamitonian of thecoupled two-dot system. For instance, without the exchange interaction, the spin splittingenergy in the target dot is E  ↓  −  E  ↑  = ∆ (1)Now, let us turn the exchange interaction on which lowers the energy of the singlet statewith respect to that of the triplet state. Thus, if the control dot’s spin is pointing “up”,the target dot’s spin will prefer to be “down” if the exchange interaction is operative. Thisdecreases ∆. On the other hand, if the control dot’s spin is pointing “down”, then ∆ isincreased. The energy levels are shown in Fig. 2. Thus, the potential  V  target  that bringsthe  total   spin splitting energy ∆ target  in the target dot in resonance with the ac magneticfield  B ac  depends on the spin orientation in the control dot. Herein lies the possibility of conditional dynamics. We can find the  V  target  that will rotate the target qubit through anarbitrary angle only if the control qubit is in the specified orientation. Application of this5
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