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Abstract:
As recently proved by Burgarth et al. [Phys. Rev. A 79, 060305(R) (2009)], a Heisenberg
spin chain can be controlled completely by acting locally on one of the spins at its ends. This
is the starting point for the present thesis. We consider mainly an isotropic spin- 1
2 Heisenberg
chain of length three and apply a control field to the first spin. The control is chosen to
be an element of the set of piecewise constant functions. Controlling the field in x- and
y-direction is sufficient for universal quantum computation. We perform operator control,
i.e. we find control functions such that the time evolution operator of the system reaches
a specified target unitary at some fixed final time, and analyse the time dependence of the
control fields found for some concrete gates. The implementation is performed by maximizing
the fidelity between the respective target gate and the dynamic time evolution of the system.
Throughout this work we use as standard gate a spin-flip gate which flips the last spin in the
chain and is achievable by control of the x-field only, but we implement as well entangling
gates like the controlled-NOT and square root of SWAP. We analyse the sensitivity of the
fidelity to random noise and search for smooth optimal control fields by applying filtering
techniques in the Fourier space. Concretely, we make use of two different filters having a
smoothing effect, namely of an ideal low-pass and a Gaussian filter. Furthermore, we extend
the three-spin system adding one more spin to the chain and estimate a minimal time for the
implementation of the spin-flip and CNOT gate depending on the number of spins whereas
we include results stemming from a chain of two, three, and four spins. Finally, we describe
how a quantum error correction circuit consisting of five qubits may be implemented by local
control of a Heisenberg spin chain of length five.
