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For electron spin qubits in quantum dots, reducing charge noise sensitivity is a critical step in achieving fault tolerant two-qubit gates mediated by the exchange interaction. This work explores how the physical device geometry affects the sensitivity of exchange to fluctuations in applied gate voltage and interdot bias due to charge noise. We present a modified linear combination of harmonic orbitals configuration interaction (LCHO-CI) method for calculating exchange energies that is applicable to general quantum dot networks. In the modified LCHO-CI approach, an orthogonal set of harmonic orbitals formed at the center of the dot network is used to approximate the many-electron states. This choice of basis significantly reduces the computation time of the full CI calculation by enabling a pre-calculated library of matrix elements to be used in evaluating the Coulomb integrals. The resultant many-electron spectra are mapped onto a Heisenberg Hamiltonian to determine the individual pairwise electronic exchange interaction strengths, $J_{ij}$. The accuracy of the modified LCHO-CI method is further improved by optimizing the choice of harmonic orbitals without significantly lengthening the calculation time. The modified LCHO-CI method is used to calculate $J$ for a silicon MOSFET double quantum dot occupied by two electrons. Two-dimensional potential landscapes are calculated from a 3D device structure, including both the Si/SiO$_2$ heterostructure and metal gate electrodes. The computational efficiency of the modified LCHO-CI method enables systematic tuning of the device parameters to determine their impact on the sensitivity of $J$ to charge noise, including plunger gate size, tunnel gate width, SiO$_2$ thickness and dot eccentricity. Generally, we find that geometries with larger dot charging energies, smaller plunger gate lever arms, and symmetric dots are less sensitive to noise.