# Generative Music with Partitioned Quantum Cellular Automata

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## Abstract

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## 1. Introduction

## 2. Background

#### 2.1. Cellular Automata (CA)

#### 2.2. Quantum Computing

## 3. Partitioned Quantum Cellular Automata

#### 3.1. One-Dimensional PQCA

#### 3.2. Two-Dimensional PQCA

## 4. Music Mapping

#### 4.1. One-Dimensional PQCA Mapping

- Code A (bits 1 and 2) defines the source of the notes. There are four different sources to choose from (Figure 19).
- Code B (bits 3, 4 and 5) defines the duration of the cluster. There are eight different durations to choose from (Figure 20).
- Code C (bit 6) = rest switch.
- Code D (bits from 7 to 18) defines the notes of the cluster; these notes are picked from the source defined by code A

#### 4.2. Two-Dimensional PQCA Mapping

#### 4.2.1. Calculating the Pitch of a Cell

#### 4.2.2. Placing the Pitches in the Frame: Wait and Duration

## 5. Real Examples from Qubism

#### 5.1. Composing Rhythmic Clusters

`ibmq_toronto`, which is a 27-qubit superconducting Falcon processor made by IBM Quantum. We ran it for 50 cycles, with 30,000 shots per cycle. Figure 31 shows the resulting rhythmic sequence of clusters and Figure 32 depicts the cellular sequence that produced it.

#### 5.2. Composing Musical Forms

`ibm_washington`, which is a 127-qubit superconducting Eagle processor made by IBM Quantum. We ran it for 50 cycles of 80,000 shots per cycle. Figure 36 depicts the cellular sequence resulting from the first four cycles of the PQCA and Figure 37 shows the respective resulting musical forms: there are eight measures of music, two measures per cycle.

#### 5.3. Implementation Considerations

## 6. Concluding Discussion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 3.**A Bloch sphere representing a single qubit. Source: Smite-Meister, https://commons.wikimedia.org/w/index.php?curid=5829358 (accessed on 2 January 2023).

**Figure 4.**Circuit representation of the $\mathbf{CX}$ gate. In this case, ${q}_{1}$ will be flipped only if ${q}_{0}$ is $|1\rangle $.

**Figure 5.**An example of a quantum circuit showing a sequence of quantum gates, the first of which are two $\mathbf{H}$ gates applied to ${q}_{3}$ and ${q}_{4}$, followed by a $\mathbf{CX}$ gate with ${q}_{4}$ as a control to flip the state of ${q}_{0}$ and so on.

**Figure 7.**At the top is a bar of twelve cells partitioned into six supercells, two cells long each. At the bottom is a shifted version of the partition.

**Figure 10.**An example of five cycles of a one-dimensional PQCA. (Note: the colours of the squares stand for the measured values of the automaton. A black cell represents the number 1 and a white one the number 0.)

**Figure 17.**A pattern produced with nine cycles of the global update circuit shown in Figure 16. All cells started with $|0\rangle $.

**Figure 21.**An example of a cluster generated with A $=\left\{11\right\}$ and D $=\left\{011001001001\right\}$.

**Figure 27.**The musical rendering of the PQCA in Figure 22.

**Figure 30.**Global update circuit for the one-dimensional PQCA example. On the left side of the red dashed line are the gates to initialise the qubits.

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Miranda, E.R.; Shaji, H. Generative Music with Partitioned Quantum Cellular Automata. *Appl. Sci.* **2023**, *13*, 2401.
https://doi.org/10.3390/app13042401

**AMA Style**

Miranda ER, Shaji H. Generative Music with Partitioned Quantum Cellular Automata. *Applied Sciences*. 2023; 13(4):2401.
https://doi.org/10.3390/app13042401

**Chicago/Turabian Style**

Miranda, Eduardo Reck, and Hari Shaji. 2023. "Generative Music with Partitioned Quantum Cellular Automata" *Applied Sciences* 13, no. 4: 2401.
https://doi.org/10.3390/app13042401