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Figure 3.4: Bloch Sphere – a) Bloch sphere: Quantum states in two-level systems
can be depicted as vectors on a sphere. b) Rotations: Operations on the system are
visualized as rotations of the state vector around an axis defined by the operation.
c) Off-resonant rotations: If the qubit is driven off resonance, the rotation vector
points out of the X/Y-plane leading to rotations that can no longer cover great
circles.
the “Bloch Sphere” after Felix Bloch.
The states | n 〉 and | m 〉 are placed at
the poles of the sphere and any arbitrary superposition of the two is depicted by
a vector pointing to the surface of the sphere, called the “Bloch Vector”. The
spherical coordinates θ and ϕ that describe the direction of the Bloch Vector are
related to the described state via:
e iα | ψ 〉 = cos θ 2 | n 〉 + eiϕ sin θ 2 | m 〉 (3.37)
In this picture, the qubit interaction described above corresponds to a rotation
of the Bloch Vector around an axis pointing in the direction given by the prefactors
of σ x , σ y , and σ z . The sum of the squares of the prefactors is related to the rotation
angle. Since the I-part of the interaction only influences the overall phase factor
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