Create new fractal art and animations with Qiskit

Top row: Julia set mating fractal generated from a 2-qubit circuit. Below the same fractal pen plotted and framed. Bottom row: Julia set mating fractals generated from a 1-qubit circuit.

Superposition fractal animation from a 1-qubit circuit. In order from left: Bloch sphere, fractal animations with one complex numbers, and two variations of Julia set mating.

By Wiktor Mazin featuring @Russell Huffman

Superposition fractal animation from a 1-qubit circuit. In order from left: Bloch sphere, fractal animations with one complex numbers, and two variations of Julia set mating.

In this blog, two ideas are explored:

1. In order to get closer to mimicking the quantum information in a quantum circuit, all the complex amplitudes are leveraged in the fractal art generation. Instead of just utilizing one complex number as described in the previous blog, a rational function is leveraged, allowing both complex amplitudes to be included in the 1-qubit case.
2. Animations provide not only an artistic expression, but also a lens into how fractals change with rotations - as in the Bloch sphere. 

Julia set mating

Julia set mating enables tying two Julia sets together, thus, making it possible to explicitly include both complex amplitudes of a statevector from a 1-qubit quantum circuit. Pasting Julia sets together may happen via the rational function f(z) = (z2 + c1)/( z2 + c2), with z = (x + yi) at the point (x,y) in the complex plane, and c1 and c2 representing a state vector’s complex amplitudes. An alternate way to represent rational functions is f(z) = (c1*z2 +1-c1)/( c2*z2 + 1-c2).

The fractals at the bottom row in the top are generated with the latter of the mating functions, and in my view, are captivating images with their symmetries, repeating patterns, and astonishing details. The image at the very top is a very early, almost bewitching experimental work of a fractal generated with a 2-qubit quantum circuit. 

Fractal animations

Animations allow a closer connection to quantum circuit operations and reveal how fractals relate and change with incremental steps. The animation in the blog is an example of looking into quantum superposition via a 1-qubit quantum circuit. You can go from seeing a Hadamard gate and how 60 rotations around the Z axis (RZ gate) from the left relate to the Bloch sphere, to watching fractal animations generated with one complex number and examining the variations of Julia set mating described above. 

I think it's fascinating to see how the fractals almost morph into one another with each incremental step, clearly showing how closely related they are. It also demonstrates how small rotations on the Bloch sphere affect the visual expression of a fractal. 

We have only started to imagine the different animation paths on the Bloch sphere you can do. With the fractal art you can now generate, we are also speculating if we have the foundation to turn the quantum computing generated fractal images and animations into real artwork. Finally, we anticipate that the work may eventually be used as an artistic way of learning about quantum computing, gate operations, superposition, entanglement, and interference. More work is needed to dive deeper and further explore current and other avenues of meaningfully conveying quantum information.

Fractals as generative art - by Russell Huffman

Fractals are a unique example of how nature and mathematics can visually collide in a beautiful way. We as artists like to maintain creative control over our artwork, but to create a work of art with fractals means to give up a part of that creative freedom and let mathematics be the driver in the composition. However, there are still artistic decisions that can be made on top of the core concept in order to create an aesthetic object that serves as technical proof of concept and evidence that mathematics can be beautiful. It is possible to engineer the fractal output in a way that is still mathematically faithful, but open to creative manipulation. 

In this case, the output was traced and redrawn with a pen plotter and plotted with gel pens on black paper. The artistic decisions here were different in that the artist was more akin to a musical director. The quantum circuit and the fractal code determine the overall composition; then, the pen plotter (a drawing robot) affixes that composition to paper. Nevertheless, the artistry was still necessary at every step, and the final composition was only possible in collaboration between the artist and the science together.

In Summary

To reiterate from the previous blog, fractals have the potential to be an exciting way to visualize quantum information that bridges art, science, and nature. Both fractal images and animations generated by quantum computers reflect the complexity of quantum computing while providing a new lens through which to think about it. Head to the notebooks and start creating your own new, beautiful fractals and animations.