Abstract: We are interested in developing approaches to making music using gate-type quantum algorithms. We propose various principles and models to apply quantum gate circuits for music creation. In considering how we can connect between musical notes and qubits, and associate musical progression with quantum gate operations, we propose two distinct approaches: the wavefunction-based approach and the measurement-based approach. The former approach is intended to translate the quantum wavefunction directly to the musical expression and provide a scheme to perform a new improvisational live performance, referred to as quantum live coding. The latter is based on the quantum gate circuits’ measurement results, enabling various music creation schemes, including the quantum phase modulation model and the quantum game theoretical model.

In this article, we have proposed two different music creation approaches based on quantum gate circuits. 

(1) Wavefunction-based approach: An approach that applies the time evolution of the wave function of a qubit, which changes with the quantum gate operation, directly to musical expressions in an analog way. This approach relies on the use of full information on the quantum wavefunction. Therefore, we do not presume the use of an actual quantum computer unless the full information of the wavefunction is made available through quantum tomography, and instead offer our unique simulator-based music creation and performance system. This scheme is referred to as the wavefunction-based approach. As we explain later, this wavefunction-based approach can provide an interesting framework for a new musical performance called quantum live coding.　

In the article, the following sections are given to discuss the wavefunction-based approach.  

1. Basic principle of music generation based on the wavefunction-based approach

2. Example of music generation based on the wavefunction-based approach

(2) Measurement-based approach: An approach that applies the result of measurement (wave function collapse) for the quantum state associated with quantum gate operation to the musical expressions in digitally. This is a measurement-based approach. Therefore, we can assume the use of real quantum computers. This scheme is referred to as the measurement-based approach hereafter.

In the article, the following sections are given to discuss the measurement-based approach.  

1. Stochastic note generation model

2. Digital note-expression model

   1. Diatonic chord generation model

   2. Phase-modulated music generation model

   3. Two players model based on Quantum Game Theory