Exchange in Silicon-Based Quantum Computer Architecture

Abstract

The silicon-based quantum computer proposal has been one of the actively pursued ideas during the past three years. Here we calculate the donor electron exchange in silicon and germanium, and demonstrate an atomic-scale challenge for quantum computing in Si (and Ge), as the six (four) conduction band minima in Si (Ge) lead to inter-valley electronic interference, generating strong oscillations in the exchange splitting of two-donor two-electron states. Donor positioning with atomic scale precision within the unit cell thus becomes a decisive factor in determining the strength of the exchange coupling-a fundamental ingredient for two-qubit operations in a silicon-based quantum computer.PACS numbers: 03.67. Lx, 71.55.Cn, Following the seminal proposal [1] by Kane there has been a great deal of activity [2][3][4] in efforts to develop a silicon-based quantum computer (QC) architecture. The basic ideas of the Kane proposal are simple and attractive: to use donor nuclear spins as quantum bits (qubits), and to utilize the vast infrastructure and technology associated with the Si industry to fabricate precisely controlled Si nanostructures, where exchange effects between electrons and nuclei in neighboring donor impurities (e.g. 31 P in Si) could serve as the two-qubit gates, similar to the electron-spin-based QC proposal by Loss and DiVincenzo [5]. The motivation for a Si quantum computer is obvious: Once the basic one-qubit and two-qubit operations have been demonstrated using donor impurities in Si nanostructures, computer chip fabrication technology associated with the existing and dominant Si industry will easily enable the scale-up of information processing involving large number of donor nuclear spin qubits. Indeed, one of the formidable stumbling blocks in developing working quantum computer hardwares has been the scale-up problem, as the demonstrated qubits in trapped ion and liquid state NMR techniques are not readily scalable in any significant manner [6].A great deal of experimental work is currently being aimed at developing suitable qubits in Si nanostructures with precisely introduced dopant impurities, using both a "top-down" approach with ion-implantation, and a "bottom-up" approach with MBE growth and scanning tunneling microscopy [4]. In the Si QC model [1,2], donor electrons act as shuttles between different nuclear spins. For two-qubit operations, which are required for a universal QC, both electron-electron exchange and electronnucleus hyperfine interaction need to be precisely controlled. These are unquestionably formidable experimental problems. In the original proposal, Kane used the Herring-Flicker exchange formula [7] for two hydrogenic centers to obtain an order of magnitude estimate of the electron exchange among donors in Si [1]. However, as he also pointed out, donor exchange in Si is not hydrogenic.In this Letter we show that exchange effects in proposed donor nuclei based Si QC architectures are actually very subtle due to quantum interference effects inherent in the complicated Si band...

Silicon-based spin and charge quantum computation

“…We show that strain may partially alleviate the exchange oscillatory behavior, but it cannot entirely overcome the problem. From the floating-phase HL approach results, our main conclusion is that, for all practical purposes, the previously adopted HL wavefunctions are robust, and the exchange sensitivity to donor positioning obtained by Koiller et al 2002a, b andWellard et al 2003 persists in the more sophisticated theory by .…”

“…where R = R A − R B is the interdonor position vector and J µν (R) are kernels determined by the envelopes and are slowly varying functions of R (Koiller et al 2002a, b). Note that Eq.…”

“…In previous studies by Koiller et al 2002a, b, as in the standard HL formalism presented in subsection 4.1, it is implicitly assumed that the phases e −ik µ ·R 0 in Eq. (3) remain pinned to the respective donor sites R 0 = R A and R B , as we adopt single donor wavefunctions to build the twoelectron wavefunction.…”

“…However, electron exchange in bulk silicon has spatial oscillations (Andres et al 1981, Koiller et al 2002 on the atomic scale due to valley interference arising from the particular six-fold degeneracy of the bulk Si conduction band. These exchange oscillations place heavy burdens on device fabrication and coherent control (Koiller et al 2002a), because of the very high accuracy and tolerance requirements for placing each donor inside the Si unit cell, and/or for controlling the external gate voltages. The potentially severe consequences of the exchange-oscillation problem for exchange-based Si QC architecture motivated us and other researchers to perform theoretical studies with increasingly sophisticated formalisms, incorporating perturbation effects due to applied strain (Koiller et al 2002b) or gate fields (Wellard et al 2003).…”

A computational workflow for designing silicon donor qubits

“…The donor electron wave function oscillates several times across the interaction region between qubits, due to interference effects arising from the degeneracy of the six valleys in the conduction band of silicon. The exchange coupling ( J) between donors is sensitive to these oscillations, which occur with a periodicity on the order of the silicon lattice constant [27]. Uncertainty in the position of the donor atoms therefore leads to unpredictable exchange couplings.…”