Abstract
We investigate a two-electron double quantum dot with both spin and valley degrees of freedom as they occur in graphene, carbon nanotubes or silicon and regard the 16-dimensional space with one electron per dot as a four-qubit logic space. In the spin-only case, it is well known that the exchange coupling between the dots combined with arbitrary single-qubit operations is sufficient for universal quantum computation. The presence of valley degeneracy in the electronic band structure alters the form of the exchange coupling and, in general, leads to spin–valley entanglement. Here, we show that universal quantum computation can still be performed by exchange interaction and single-qubit gates in the presence of an additional (valley) degree of freedom. We present an explicit pulse sequence for a spin-only controlled-NOT consisting of the generalized exchange coupling and single-electron spin and valley rotations. We also propose state preparations and projective measurements with the use of adiabatic transitions between states with (1,1) and (0,2) charge distributions similar to the spin-only case, but with the additional requirement of controlling the spin and valley Zeeman energies by an external magnetic field. Finally, we demonstrate a universal two-qubit gate between a spin and a valley qubit, allowing universal gate operations on the combined spin and valley quantum register.
Highlights
Since Loss and DiVincenzo [1] proposed quantum computing with electron spins in double quantum dots, there has been a substantial experimental progress in the field of coherent spin manipulation in semiconductors [2, 3, 4, 5, 6]
The majority of these experiments has been performed in gallium arsenide (GaAs) where the electron spin suffers from decoherence due to its coupling to a typically large number of nuclear spins, as well as spin relaxation due to spin-orbit coupling
Spin states in graphene quantum dots have been identified by transport measurements [9] but valley states have not been observed yet, whereas in carbon nanotubes (CNTs), a fourfold grouping of electronic states due to spin and valley degree of freedom have already been observed for a decade in transport measurements [10, 11, 12]
Summary
Since Loss and DiVincenzo [1] proposed quantum computing with electron spins in double quantum dots, there has been a substantial experimental progress in the field of coherent spin manipulation in semiconductors [2, 3, 4, 5, 6]. In recent experiments with siliconbased quantum dots, coherent spin manipulation with the exchange interaction has been performed successfully [19], and some control over the valley splitting has been demonstrated [20] Both in Si [21] and in graphene [22, 8, 23] there have been speculations that the valley degree of freedom might serve as an additional resource for classical or quantum information processing, i.e. as a classical bit for valleytronics [22, 8] or as a qubit [21, 23]. Including the valley degree of freedom, we end up with 16 (1, 1) states, six (0, 2) states, and six (2, 0) states
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