# How Are Rotations on the Bloch Sphere Implemented in Practice?

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• kelly0303
kelly0303
Hello! I am curious about how different rotations on the Bloch sphere are done in practice. For example, assuming we start in the lower energy state of the z-axis (call it |0>), a resonant rotation on the Bloch sphere by ##\pi/2## around the x-axis will take you to ##\frac{|0>-i|1>}{\sqrt{2}}## (where ##|1>## is the excited state in the z direction). If we do the same thing around the y-axis we end up with ##\frac{|0>-|1>}{\sqrt{2}}##. This phase difference matters in practice in various scenarios (e.g. when doing a spin echo). But how do you change the rotation axis in practive? The field applied in the lab frame is ##E\cos{(\omega t + \phi)}##. You can make ##\omega## resonant and ##E## such that you get a ##\pi/2## pulse for the right time, but if you solve the Schrodinger equation in the rotating wave approximation, the ##\phi## term actually cancels in the final formula, so I am not sure what other degrees of freedom one has in order to achieve this. Thank you!

kelly0303 said:
Hello! I am curious about how different rotations on the Bloch sphere are done in practice.
It is different for photons (polarization), and for particles with a magnetic moment (spin). I recently fell into that trap:
gentzen said:
Well, I was thinking mostly in terms of optics and polarization. A more correct translation of that situation to an electron is that the spin states perpendicular to the direction of propagation are much easier to measure directly (by Stern-Gerlach type experiments) than the ones parallel to the direction of propagation.
[...]
My optics analogies were wrong, but the distinctions they suggested still remain somewhat true for electrons: Even so it seems easy to change the direction of propagation of a "particle" from y-direction to x-direction, it is only "theoretically easy" to do so without changing the spin in case the "particle" is not electrically neutral. But in that case, the Stern-Gerlach type experiment itself becomes difficult.

Let me be clear that my optical analogies had been more wrong than I was aware of. And because they were wrong, my post that you corrected was certainly confusing, both for experts and novices.
I am not sure how to exactly do it for particles with a magnetic moment. My guess is:
gentzen said:
But maybe one could use Lamor precession to rotate the spin of the "particle" instead of the direction of propagation. At least it seems possible "theoretically".

## What is the Bloch Sphere and why is it important in quantum computing?

The Bloch Sphere is a geometrical representation of the pure state space of a two-level quantum mechanical system (qubit). It is important because it provides an intuitive visualization of qubit states and their transformations, making it easier to understand quantum operations and algorithms.

## How are single-qubit rotations represented mathematically on the Bloch Sphere?

Single-qubit rotations on the Bloch Sphere are represented mathematically by unitary matrices, specifically by the Pauli matrices and the identity matrix. These rotations can be described using rotation operators, such as $$R_x(\theta)$$, $$R_y(\theta)$$, and $$R_z(\theta)$$, which rotate the qubit state by an angle $$\theta$$ around the respective axes (x, y, z) of the Bloch Sphere.

## What physical systems are used to implement rotations on the Bloch Sphere?

Rotations on the Bloch Sphere are implemented using various physical systems such as superconducting qubits, trapped ions, quantum dots, and nitrogen-vacancy centers in diamond. Each system employs different techniques to manipulate qubit states, such as microwave pulses, optical lasers, or magnetic fields.

## How are microwave pulses used to achieve rotations on the Bloch Sphere in superconducting qubits?

In superconducting qubits, microwave pulses are used to achieve rotations by resonantly driving transitions between the qubit's energy levels. The frequency, phase, and duration of these pulses are carefully controlled to perform precise rotations around desired axes on the Bloch Sphere, allowing for the implementation of quantum gates.

## What are the challenges in implementing precise rotations on the Bloch Sphere?

Challenges in implementing precise rotations include maintaining coherence and minimizing decoherence, accurately calibrating pulse parameters, and dealing with noise and imperfections in the physical system. Additionally, achieving high-fidelity operations requires precise control over experimental conditions and advanced error-correction techniques.

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