# Stern-Gerlach Simulations on a Quantum Computer

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In the Stern–Gerlach experiment, an unpolarized beam of neutral particles of spin 1/2 is directed through an inhomogeneous magnetic field (blue and red magnet), which produces separated beams of spin-up and spin-down particles. For simplicity, only the outgoing spin-up beam is shown in the graphic. This beam is then directed through a second magnet, for which the polarization can be rotated by an angle from the original. This further splits the beam (except when or ) into spin-up and spin-down beams with respect to the new polarization direction. Again, only the spin-up component is shown. The probability for a particle to emerge with spin-up (↑) or spin-down (↓) is given by and , respectively. The resulting probabilities of ↑ and ↓ are shown for five selected angles.

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Contributed by: S. M. Blinder (June 2017)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

The action of the single-qubit quantum gates can be represented by unitary matrices acting on the qubit :

identity (or IDLE): ,

Hadamard gate: ,

Pauli X (or NOT) gate: ,

phase (or gate: ,

gate: .

For example, the rotation is produced by the sequence

.

The probability of a result in the subsequent measurement is then given by

, or about 85% spin-up.

References

[1] Wikipedia. "Stern–Gerlach Experiment." (Jun 9, 2017) en.wikipedia.org/wiki/Stern-Gerlach_experiment.

[2] M. A. Nielsen and I. L. Chuang, *Quantum Computation and Quantum Information: 10th Anniversary Edition*, Cambridge: Cambridge University Press, 2010. doi:10.1017/CBO9780511976667.

[3] G. Fano and S. M. Blinder, *Twenty-First Century Quantum Mechanics: Hilbert Space to Quantum Computers*, New York: Springer Berlin Heidelberg, 2017.

## Permanent Citation