Single-Qubit Quantum Gates on a Bloch Sphere

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Using the Bloch sphere, a cubit can be represented as a unit vector (shown in red) from the origin to the point on the unit sphere with spherical coordinates . A single-qubit quantum gate operating on produces a rotated qubit , represented by the green vector. Check the box for "add gate 2?" to perform a second operation using gate . This produces another qubit , which is represented by the blue vector. You can choose from the gates H, X, Y, Z, S and T as defined in the Details.

Contributed by: S. M. Blinder (August 2017)
Open content licensed under CC BY-NC-SA



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


Hadamard gate: ,

Pauli X gate: ,

Pauli Y gate: ,

Pauli Z gate: ,

phase (or gate: ,

gate: .

In general, the operation gives a qubit of the form


with an overall phase factor of no physical significance. Multiplication by removes this factor and reduces the qubit to the canonical form



[1] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information: 10th Anniversary Edition, Cambridge: Cambridge University Press, 2010.

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

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