Most of the experimentalists can only (due to practical limitations) apply quantum gates in the lab that cause rotations around the Bloch sphere in the X-Y plane. However, many quantum algorithms rely on rotations around the Z-axis of the Bloch sphere. Are these Z-gates then a figment of the imagination of the theoretical quantum scientist?

In this video, Postdoctoral Researcher Ben Criger (QuTech) shows that this is not at all a problem. In fact, we can use a specific sequence of X and Y gates to effectively construct an arbitrary Z rotation around the Bloch sphere. We first mathematically show that the sequence synthesizes into a Z-gate, after which we show the same visually on the Bloch sphere. 

Prerequisite knowledge

• Bloch sphere
• Arbitrary rotations around the Bloch sphere (X, Y, Z rotations)
• Basic matrix multiplication algebra
• Hadamard gate
• Euler’s identity

Further thinking

Can you think of a similar sequence using X, Z rotations to synthesize a Y gate?

Solution

Further reading

For a recap on matrix notations of arbitrary Bloch sphere rotations, take a look at these lecture notes from Ian Glendinning: 

• http://www.vcpc.univie.ac.at/~ian/hotlist/qc/talks/bloch-sphere-rotations.pdf

A fun low-level read on the use of Bloch sphere rotations in quantum computing can be found at: 

• https://medium.com/@quantum_wa/quantum-computation-a-journey-on-the-bloch-sphere-50cc9d73530