Mapping Quantum to Music

A quantum simulator is used to sample density matrices representing physical quantum states at various time steps. Density matrices can keep track of the uncertainty of the system's actual quantum state by containing all of the possible states that could have been produced by the noise model. The exact quantum state of a system is called a pure state. As incoherent noise is introduced to a density matrix representing a pure state, it can be thought of as changing from a single pure state to a statistical mixture of pure states due to the uncertainty introduced by the noise. When this happens, we say that the density matrix represents a mixed state. A density matrix can be written as its statistical distribution of pure states by performing an eigendecomposition where the eigenvectors are the pure states and the eigenvalues are the corresponding probabilities in the distribution.

Each pure state is a statevector representing a superposition of basis states, e.g. (statevector) = (a|00> + b|11>) / sqrt(2) = (a|0> + b|3>) / sqrt(2). The basis state numbers (e.g. 0 and 3) of the superposition of states are mapped (using a note map) to the MIDI note numbers to be played. The volume of each note is calculated as the probability of the basis state in the statevector (e.g. |a|^2 and |b|^2) multiplied by the probability of the pure state in the statistical distribution. So each pure state is a chord and because the statistical distribution can have multiple pure state terms, this means that multiple chords can be playing at the same time.

Each pure state of the statistical distribution is assigned a instrument collection. The instrument in the collection that will be used to play a note is determined by the corresponding basis state’s phase in the superposition. The angles are discretised to match the size of the collection, where an angle of zero corresponds to the first instrument. A list of up to 8 instrument collections can be specified when making the music video (see below example). The collections from the list will be assigned to pure states in the statistical distribution in order of decreasing probability. If there are less than 8 collections specified, the remaining pure states will use the last instrument collection in the list.