Making Music Using Two Quantum Algorithms

1 Making Music Using Two Quantum AlgorithmsEuan J. Allen1,2,*, Jacob F. F. Bulmer1, and Simon D. Small31 Quantum Engineering Technology Labs, H. H. Wills Physics Laboratory and Department ofElectrical & Electronic Engineering, University of Bristol, BS8 1FD, United Kingdom2 Centre for Photonics and Photonic Materials, Department of Physics, University of Bath, Bath,BA2 7AY, United Kingdom3 Tunnel of 6, 2022 AbstractThis document explores how to make Music Using Quantum computing text is an unedited pre-publication chapter which will appear in the book Quan-tum Computer Music , Miranda, E.

2 R. (Editor). This chapters provides the backgroundand specific details of a collaboration formed in 2021 between the Quantum EngineeringTechnology Labs - a Quantum computing and technology research group at the Univer-sity of Bristol - and Music artist, producer and audio engineer Simon Small. The goal ofthe collaboration was to explore how the data and concepts used in the research at theuniversity could be sonified to create sounds or even make audio outcomes of the collaboration can be heard at the following reference [1]or Reference [1] additionally includes thelab audio samples and sheet Music for other musicians to utilise the data and soundsgenerated from this 1: A pictorial representation of Grover s algorithm on a 3-qubit processor, plotted inthe style of the Joy Division album cover Unknown Pleasures.

3 [quant-ph] 5 Jan 20221 IntroductionComputers have and continue to shape the sound and Music landscape that we that be as tools to make Music with or inspiration for writing and composing Music ,the impact of computation in sound generation is not difficult to computers are a new type of device that completes computational tasks in a verydifferent way to the (classical) computers we experience in everyday life. Utilising aspects ofquantum physics, such as entanglement and superposition, Quantum computers are able tosolve problems which are very difficult for a classical computer to manner in which Quantum computers work is distinct from classical computers.

4 It istherefore of interest to explore what impact this technology could have on future sound andmusic generation, much like the impact conventional computers had in the early 20th , it is also of interest to explore how sound and Music might help in disseminatingconcepts of Quantum physics and computing to a wider audience - an audience that will likelyfeel the impact of this technology at some point in the computers work and process Quantum information, typically stored in quantumbits (qubits), whilst classical machines process classical information, typically stored in binarybits taking a value of 0 or 1.

5 Whilst there are examples of Quantum versions of sound oracoustic waves (phonons) [2], all Music and sound that is processed by the human ear in day-to-day life can be understood as being entirely classical. It is an interesting task to work outwhich Quantum concepts or computational Algorithms have a sufficiently interesting classicaloutput at the end of the computation to make them audially distinct or interesting from aclassical machine. This is part of the task of exploring Quantum computing chapters provides the background and specific details of a collaboration formed in2021 between the Quantum Engineering Technology Labs [3] - a Quantum computing andtechnology research group at the University of Bristol - and Music artist, producer and audioengineer Simon Small [4].

6 The goal of the collaboration was to explore how the data andconcepts used in the research at the university could be sonified to create sounds or evenmake project focused on two key concepts for sound generation: Quantum random walksand Grover s search algorithm. These two Quantum Algorithms formed the basis for sound andmusic generation, the culmination of which resulted in two full musical compositions. Thischapter is split in to three key sections. The first two provide a technical introduction to bothrandom Quantum walks and Grover s algorithm, covering how the Algorithms work and howto produce data from them that can be used to generate sound.

7 The final section covers howto take this data and use it for musical composition. Details of other techniques used in thecollaboration, such as audio samples from the laboratory, are also detailed in the final audio outcomes of the collaboration can be heard at the following references [1, 5].These links include the final pieces, audio samples, and a full musical pack for other musiciansto utilise the data and sounds generated from this Random Melodies from Quantum WalksA random walk is a great example of a place where there is a clear difference in the behaviourof Quantum and classical physics [6].

8 It is therefore frequently used as a thought experimentto teach the concepts of Quantum mechanics, but has also been used to inspired quantumalgorithms [7], and has been realised in experiments [8]. This section investigates how we canuse the difference between classical and Quantum random walks to create musically first describe how we define a Quantum random walk, and show how measurement canlead to classical dynamics. We then discuss how we simulate these systems. Although there area wide variety of excellent Quantum simulation libraries, we chose to implement our simulationusing only Python and NumPy.

9 This allows us to clearly see how all of our operations aredefined and Quantum Random WalksWe will define our random walk as follows, imagine that every time you want to take a step,you toss a coin. If it lands on heads, you step to the left. If it lands on tails, you step tothe right. We add a further rule of aperiodic boundary condition, which can be described bysaying that we have a fixed number of sites to stand on, if you reach the left edge, and needto step to the left, you move to the right edge and vice versa. This is equivalent to arrangingour sites in a circle where if you go all the way around the circle you end up back where wish for our sites to neatly correspond to musical notes in a scale, so we choose to use14 sites, labelled from 0 to 13, which can then be turned into notes from 2 octaves of an we assume that the state of our coin, and the site our walker is standing on are preparedindependently from each other, our Quantum system can be defined like:| =|C |S (1)where|C is the wavefunction describing the coin and|S is the wavefunction describingthe site.

10 The symbol is a tensor product, which here we can just think of as a symbolwhich shows that we are taking the two systems|C and|S and thinking of them as onesystem. We define coin basis states:|heads and|tails , and site basis states|j wherejcantake any value from 0 to 13 (j {0,1,2,..,13}).The operation we apply to simulate a coin toss is a Hadamard rotation. This maps thecoin states like:|head |heads +|tails 2(2)|tails |heads |tails 2.(3)These two states are equal superposition states between heads and tails, with a phasedifference between the two. We also need an operation which will move the site of the walker,depending on the outcome of the coin.