Developing the most basic app to understand Qubit state (3)

Sometimes we need to go back to basics!

In the third article in this series, I'll compare my app's results with that of IBM Quantum Composer to confirm that it's working correctly. In my app I can use quantum gates Z, X, Y, T, H.  The T is a quantum gate that adds the phase of e^iφ to the qubit |1>. By default, φ=π/4. Also, H is a Hadamard gate. Here, as an example, H, T, and H are successively applied in this order to the quantum bit initial state |0>.

Fig.1 shows (a) the execution results of my app and (b) the execution results of IBM Quantum. In (a), the probability (area of the colored disk) and phase (the angle of the straight line coming out from the center of the circle) are shown for |0> and |1>. On the other hand, in (b), the results are displayed only for |1>. Although the way the disks are displayed is slightly different, it can be seen that the probabilities and phases of both are the same. However, the values of probability amplitude (amplitude in (a) and Output state in (b)) seem to be different. This will be explained in Fig.2.

Fig.2 explains that although the expressions of the probability amplitudes of the two are different, they are the same quantum state. In my app, the phase with respect to |0> is zero, and the phase of |1> is the relative phase to it. This does not seem to be the case with IBM Quantum. In fact, when I input IBM's probability amplitude numbers into my app's function state2relphase, the results I got matched my app's representation of probability amplitudes. This result confirms that my app is working perfectly correctly, at least for this example.

Developing the most basic app to understand Qubit state (2)

 Sometimes we need to go back to basics!

This is a major update to the previous app. The aim was to help students become familiar with the most basic elements of quantum computing using only their smartphones. The example below shows the results of (1) starting from the initial state, (2) applying the Pauli X gate, (3) then applying the Hadamard gate, and (4) applying the Hadamard gate again. 

The probability amplitude, probability, and relative phase of each basis |0>, |1> are shown numerically and on a disk. Additionally, the values of θ and φ can be set using text boxes and sliders, making it easy to understand the position of the quantum bit on the Bloch sphere.

Developing the most basic app to understand Qubit state (1)

Sometimes we need to go back to basics!

As I have already written several times, I was able to almost completely understand the basic concepts and ideas of quantum computing by reading the book [1]. Although this book does not explain the Bloch sphere or phases in detail, I would not have been able to write this article without the knowledge I gained from the book.

I developed an app to understand the state of qubits on the Bloch sphere. As shown in the figure below, set the two angular parameters θ and φ using the sliders for a single qubit on the Bloch sphere. This app uses it to calculate the probability amplitude and probability for each of the two basis (|0> and |1>). Probability and phase are also illustrated in the two disks at the top right. Note that since the global phase can be ignored, the phase of |0> is set to 0, and the phase of |1> is expressed as a relative phase to that.

As you can see, the current app specifies θ and φ manually. In the next version, it will also be possible to apply basic quantum gates (X, Y, Z, H, etc.).

References
[1] Chris Bernhardt: Quantum Computing for Everyone, The MIT Press, 2020.
https://www.chrisbernhardt.info/