APPLICATIONS OF QUANTUM ANNEALING TO MUSIC THEORY

Ashish Arya 
Ludmila Botelho
Fabiola Cañete
Dhruvi Kapadia
Özlem Salehi

"...might compose elaborate and scientific pieces of music of any degree of complexity or extent"

Ada Lovelace

Charles Babbage

Quantum annealing

Proposition: Find the minimum energy in a landscape

Ground state = optimal solution

$$ H(t) = \left(1-\frac{t}{T}\right)H_0+ \frac{t}{T}H_P $$

Different from gate based

$$ H_P = - \sum_{i > j} J_{ij}s_i s_j - \sum _i h_is_i, $$

$$ H(t) = \left(1-\frac{t}{T}\right)H_0+ \frac{t}{T}H_P $$

- Can be encoded in Binary Values

Ising model

Mathematical model to describe magnetization

$ s_i \in \{-1,+1\} $

NP-hard!

$ s_i \leftrightarrow x_i = \frac{1-s_i}{2} $

$ x_i \in\{0,1\} $

quadratic unconstrained binary optimization

Minimizing quadratic functions

$$ y=x^TQx $$

Penalty Method

$$ \text{min } y=f(x) $$

$$ \text{subject to: } x_1 +x_2 + x_3= 1 $$

$$\text{min }y=f(x)+P\left(\sum_{i=1}^3 x_i -1\right)^2 $$

Binary variables

Constants

And now for something completely different

Music Generation

rest

measure

Durations

Pitches

Music Generation

x_{i,j} = \begin{cases}% 1 & \text{note at position $i$ is $j$,}\\ 0 & \text{otherwise.} \end{cases}

$$\sum_{j \in P} x_{i,j} = 1$$

How to define the binary variables?

$$\left ( 1- \sum_{j \in P} x_{i,j} \right )^2$$

Pitches

Rules about consecutive notes

$$x_{i,C} + x_{i+1,D} \leq 1$$

$$x_{i,C} + x_{i+1,C} + x_{i+2,C} \leq 2 $$

$$P=\{p_1,p_2,\dots,p_k \}$$

$x_{i,j}$ for $i \in [n]$ and $j \in P$

$$P=\{\mathtt{C}, \mathtt{D}, \mathtt{E}, \mathtt{G}\}$$

Text

Melody Generation

Optimization

$$ -\sum_{ \substack{i\in [n-1] \\ j,j' \in P}} W_{j,j'} x_{i,j} x_{i+1,j'} $$

weights

Example: Ode to Joy excerpt

$$W_{F\#4,E4} = 2, $$

$$W_{F\#4,F\#4} = 2,$$

$$W_{F\#4,G4} = 1$$

melody Generation

$$\sum_{\substack{i \in [n] \\ j \in D}} d(j)y_{i,j} = L$$

Duration of notes

y_{i,j} = \begin{cases}% 1 & \text{note at position $i$ has duration $j$,}\\ 0 & \text{otherwise.} \end{cases}

Rhythm Generation

$$ \sum_{j \in D} y_{i,j} = 1$$

$$D = \{1,2,3,4\}$$

$$j \in D$$

And now for something Different

Markov random fields (MRF)

Set of random variables (0,1) that form an undirected graph and satisfy the Markov property.

Music with MRF

Antecedent: Algorythms, Generating music with D-Wave's Quantum Annealer (iQuHack 2021)

D-Wave's Markov Network

HARMONIZATION WITH MRF

Our work: harmonize a melody implementing music theory rules.

Results

Melody Harmonized melody

CONclusıon and future work

Quantum annealing is an alternative way for composing music.

Music completion, counterpoint generation, music reduction are some other candidate problems to investigate.

The work will appear as a chapter in the book "Quantum Computer Music: Foundations, Methods and Advanced Concepts", Miranda, E. R. (Editor).

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Harmonization

x_{i,j} = \begin{cases}% 1 & \text{chord at position $i$ contains note $j$,}\\ 0 & \text{otherwise.} \end{cases}

$$\sum_{j \in P} x_{i,j} = 3$$

$$(1-x_{i,p_{i_j}})$$